Die Grundlehren der
mathematisdien Wissenschaften in Einzeldarstellungen
Band 154
Ivan Singer
Bases in Banadi Spaces I

Ivan Singer
Bases in Banach Spaces I
Springer-Verlag Berlin • Heidelberg • New York 1970

Prof. Dr. Ivan Singer
Institute of Mathematics. Academy of the Socialist Republic of Romania, Bucharest
Geschaftsfuhrende Herausgeber:
Prof. Dr. B. Eckmann
Eidgenossische Technische Hochschule Zurich
Prof. Dr. B. L. van der Waerden
Mathematisches lnstitut der Universitat Zurich
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned,
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Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the
publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin ¦ Heidelberg 1970. Library of Congress Catalog Card Number 75-99014
Printed in Germany. Title No. 5137
Preface
This monograph attempts to present the results known today on
bases in Banach spaces and some unsolved problems concerning them.
Although this important part of the theory of Banach spaces has been
studied for more than forty years by numerous mathematicians, the
existing books on functional analysis (e. g. M. M. Day [43], A. Wilansky
[263], R. E. Edwards [54]) contain only a few results on bases.
A survey of the theory of bases in Banach spaces, up to 1963, has
been presented in the expository papers [241], [242] and [243], which
contain no proofs; although in the meantime the theory has rapidly
developed, much of the present monograph is based on those expository
papers. Independently, a useful bibliography of papers on bases, up to
1963, was compiled by B. L. Sanders [219].
Due to the vastness of the field, the monograph is divided into two
volumes, of which this is the first (see the table of contents). Some results
and problems related to those treated herein have been deliberately
planned to be included in Volume II, where they will appear in their
natural framework (see [242], [243]).
We hope that the present monograph will be useful both for special-
specialists in the field and for those who want to apply basis theory to other
problems. In order to make the book accessible to a larger circle of
readers, we have indicated, for the results of functional analysis which
we used, a reference to a treatise containing the proof of the respective
result; when we applied results which are not contained in such trea-
treatises, but only in journals, we have usually mentioned them as lemmas,
giving also their proofs.
The bibliography given at the end does not aim at being complete,
but wants merely to give useful orientation to the reader. Since some
of the results have deep roots in classical analysis and some have been
discovered and rediscovered, independently, by several authors, we did
not intend to trace down completely the history of all results. Some
of our unpublished results and remarks have also been included in the
present monograph, without any special mention. The references given
here concern the material of Volume I; the bibliography for Volume II
will be given separately in that volume.

VI
Preface
It is a great pleasure to acknowledge here the generous help of our
friend, Professor Aleksander Pelczynski, with whom we had numerous
stimulating conversations and correspondence during the preparation
of the present monograph. Also, we have profited from valuable remarks
in discussions and letters made by our colleagues and friends Professors
Czeslaw Bessaga, William J. Davis, David W. Dean, Ciprian Foias,
Gliceria Godini, Vladimir I. Guraril, Mihail I. Kadec, Bor-Luh Lin,
Joram Lindenstrauss, Charles W. McArthur, James R. Retherford and
William Ruckle (in alphabetical order). We are indebted to Dr. Clifford
Kottman for reading the entire manuscript and making valuable sug-
suggestions for its improvement.
Part of this monograph was completed in the Institute of Mathe-
Mathematics of the Academy of the Socialist Republic of Romania (over a
period of several years) and various parts of it were written while the
author was Visiting Professor at the University College of Swansea
D months), Florida State University C months), Pennsylvania State
University B months), University of Iowa F months) and Ohio State
University A month). We wish to express here our gratitude to Professor
Miron Nicolescu, President of the Academy of the Socialist Republic
of Romania and Director of the Institute of Mathematics and to the
Chairmen of the Departments of Mathematics of the above Universities,
Professors Jeffrey D. Weston (Swansea), Orville G. Harrold (F. S. U.),
Raymond G. Ayoub (P. S. U.), Robert H. Oehmke (U. I.) and Arnold
E. Ross (O. S. U.) for ensuring excellent working conditions. We extend
our thanks to all colleagues in these universities who attended our
seminars on selected topics of the theory of bases in Banach spaces,
for their stimulating interest and comments.
Finally, our thanks are due to Dr. Klaus Peters of Springer Verlag
for solving promptly and efficiently the various problems which appeared
during the preparation of this monograph.
June, 1969
Ivan Singer
After this book has been typeset, we felt it necessary to make the
following additional remarks:
1. The reader is warned to distinguish carefully between ф (small
phi = (p) and Ф (capital phi), which are used for different purposes. Also,
throughout the book 0 denotes the empty set and A\B the set-theoretic
difference {хеА\хфВ}.
2. We should note also the appearance, while the present book was
in press, of the introductory text book [276] on bases.
February, 1970
Ivan Singer
Contents
Chapter I. The Basis Problem. Some Properties of Bases in
Banach Spaces
§ 1. Definition of a basis in a Banach space. The basis problem. Relations
between bases in complex and real Banach spaces 1
§ 2. Some examples of bases in concrete Banach spaces. Some separable
Banach spaces in which no basis is known 10
§ 3. The coefficient functionals associated to a basis. Bounded bases. Nor-
Normalized bases 17
§4. Biorthogonal systems. The partial sum operators. Some characteriza-
characterizations of regular biorthogonal systems. Applications 23
§ 5. Some characterizations of regular ^-complete biorthogonal systems.
Multipliers 31
§ 6. Some types of linear independence of sequences 50
§ 7. Intrinsic characterizations of bases. The norm and the index of a se-
sequence. The index of a Banach space. Extension of block basic sequences 57
§ 8. Domination and equivalence of sequences. Equivalent, affinely equiv-
equivalent and permutatively equivalent bases 68
§ 9. Stability theorems of Paley-Wiener type 84
§ 10. Other stability theorems 93
§11. An application to the basis problem 109
§12. Properties of strong duality. Application: bases and sequence spaces . 112
§ 13. Bases in topological linear spaces. Weak bases and bounded weak bases
in Banach spaces. Weak* bases and bounded weak* bases in conjugate
Banach spaces 144
§ 14. Schauder bases in topological linear spaces. Properties of weak duality
for bases in Banach spaces 151
§ 15. (e)-Schauder bases and (ft)-Schauder bases in topological linear spaces 158
§16. Some remarks on bases in normed linear spaces 160
§17. Continuous linear operators in Banach spaces with bases 162
§18. Bases of tensor products 171
§19. Best approximation in Banach spaces with bases 174
§ 20. Polynomial bases. Strict polynomial bases. Г systems and Л systems . 184
Notes and remarks 200
Chapter II. Special Classes of Bases in Banach Spaces
I. Classes of Bases not Involving Unconditional Convergence
§1. Monotone and strictly monotone bases 214
§ 2. Normal bases 252
§ 3. Positive bases 261

VIII
Contents
§ 4. ^-shrinking bases 267
§ 5. Retro-bases in conjugate Banach spaces 279
§ 6. A>boundedly complete bases 284
§7. Bases of types wc0, (wc0)*, swc0 and (swc0)* 292
§ 8. Some properties of the set of all elements of a basis. Weakly closed and
(weakly closed)* bases 300
§9. Bases of types P, P*, a Panda P* 308
§ 10. Bases of types /+, (/+)*, at+ and (at+)*. The cone associated to a basis. 315
§11. Besselian and Hilbertian bases. Stability theorems 337
§12. Relations between various types of bases 359
§ 13. Universal bases. Complementably universal bases. Block-universal
bases 373
II. Unconditional Bases and Some Classes of Unconditional Bases
§ 14. Unconditional bases. Conditional bases 396
§ 15. Some separable Banach spaces having no unconditional basis .... 432
§ 16. Some characterizations of unconditional bases among ^-complete (or
total) biorthogonal systems and among bases. Some characterizations
by properties of the associated cone. Multipliers 458
§ 17. Intrinsic characterizations of unconditional bases. Some more separable
Banach spaces having no unconditional basis. Properties of strong
duality. Unconditional bases and sequence spaces 499
§ 18. Equivalence and permutative equivalence of unconditional bases. Uni-
Universal unconditional bases 529
§ 19. Best approximation in Banach spaces with unconditional bases .... 550
§ 20. Orthogonal bases. Strictly orthogonal bases. Hyperorthogonal and
strictly hyperorthogonal bases 555
§21. Subsymmetric bases 563
§22. Symmetric bases. Symmetric spaces 574
§ 23. Applications: Existence of non-equivalent normalized bases and condi-
conditional bases in infinite dimensional Banach spaces with bases 602
§ 24. Perfectly homogeneous bases. Application: Banach spaces with a
unique normalized unconditional basis 609
§25. Absolutely convergent bases. Uniform bases 621
Notes and remarks 622
Bibliography 646
Notation Index 659
Author Index 662
Subject Index 665
Volume II (in preparation):
Chapter III. Generalizations of the Notion of a Basis
Chapter IV. Applications to the Study of the Structure of Banach Spaces
Chapter V. Some Properties of Bases in Concrete Banach Spaces
Appendix I. Bases in General (not Necessarily Separable) Banach Spaces
Appendix II. Bases in Topological Linear Spaces
Chapter I
The Basis Problem. Some Properties
of Bases in Banach Spaces
§ 1. Definition of a basis in a Banach space. The basis problem.
Relations between bases in complex and real Banach spaces
The scalar field К for all (general or concrete) linear spaces con-
considered in the sequel will be either the field of complex numbers or the
field of real numbers.
Definition 1.1. A sequence {х„} in an infinite dimensional Banach
space E is called a basis of E if for every хеЁ there exists a unique
sequence of scalars {а„} с: К such that
x =
A.1)
(i.e. such that lim
х-
= 0).
A system of и elements {х^}"= j in a Banach space E of dimension
и < со is called a basis of E if it is a basis (in the usual algebraic sense)
of the underlying linear space, i.e., if for every xeE there exists a unique
n
system of и scalars {a.]}"=l^K such that x= ? a,x,.
For the sake of brevity, throughout the sequel we shall make the
following convention: we shall unify the infinite and finite dimensional
00 00
cases, by writing always ? a,x,-, instead of Xafxi when dim ? =00
" i = 1 i = 1 00
and ?а,Х; when dim?<oo. In other words, by ? a.x, we shall
actually mean ? а,х,.
i= 1
If a Banach space ? has a basis {х„}, then it is separable, since the
n
set of all finite linear combinations ? r,x,, where the r, are (complex
1 Singer, Bases in Banach Spaces 1

2 I. The Basis Problem. Some Properties of Bases in Banach Spaces
or real) rational numbers and и =1,2,..., is a countable dense set in ?.
It is not known whether the converse is true or not, i.e.:
Problem 1.1 (called "the basis problem"). Does every separable
Banach space possess a basis ?
By the Banach-Mazur theorem on the universality of the space
C([0,1]) for separable Banach spaces1, the basis problem is equivalent
to the following:
Problem 1.2. Does every subspace2 of C([0,1]) possess a basis?
In connection with this problem it is natural to ask
Problem 1.3. Does there exist an infinite dimensional separable
Banach space F, not isomorphic3 to L2([0,1]), such that every sub-
space of F has a basis?
Since by a theorem of Banach and Mazur every separable Banach
space is equivalent4 to a quotient space of I1, the basis problem is also
equivalent to the following:
Problem 1.4. Does every quotient space of I1 possess a basis?
In connection with this problem, it is natural to ask
Problem 1.5. Does there exist an infinite dimensional separable
Banach space F, not isomorphic to L2([0,1]), such that every quotient
space of F has a basis?
As we have already mentioned, the scalar field К for all (general
or concrete) linear spaces which we consider throughout this book,
can be either the field of complex numbers or the field of real numbers.
Since there are some well known relations between complex and real
Banach spaces, it is natural to ask how the bases of complex Banach
spaces are related to the bases of the corresponding real Banach
spaces and conversely. We shall now examine some aspects of this
question.
a) Restriction of the field of scalars. Let ? be a complex Banach
space. Then E is also a Banach space over the subfield R of real num-
numbers. This Banach space is denoted by ?(r) and it is called the real Ba-
Banach space obtained from E by the restriction of the field of scalars, or
1 See e. g. [10], p. 185, theorem 9. In [10] only real Banach spaces are considered.
Whenever we shall refer the reader to [10], we shall understand, without any special
mention, that the respective result of [10] can be extended to complex Banach
spaces with standard methods.
2 Unless otherwise stated, by "subspace" we shall mean: closed linear subspace.
3 We shall use the term "isomorphic" in the sense of Banach, i.e.: linearly
homeomorphic.
4 See e.g. [12], theorem e) or [133], p. 283, theorem A). We shall use the term
"equivalent" in the sense of Banach, i. e.: isometrically isomorphic.
1. Definition of a basis. Relations between bases in complex and real spaces 3
shortly, the real Banach space associated to E. A useful relation between
the bases of E and E(r) is the following:
Proposition 1.1. A sequence {х„}<=? is a basis of the complex Ba-
Banach space E if and only if the sequence {zn} defined by1
(n =
A.2)
is a basis of E{r) (but E(r) can have also other bases which are not of the
form A.2)).
Proof. Assume that {х„} is a basis of the complex Banach space E.
00 ОС
Then every xeE has an expansion x = ? (/?,• + i yj) x,- = Y,(Pjz2j-i +У]гц)
j=i j=i
with respect to {zn}. Furthermore, this expansion is unique, since the
GO OQ
relation ? 0?^-!+)>yz2j.) = 0 implies ? (/?,- + i}>,)Xj = 0, whence,
j=i j=i
since {х„} is a basis of E, Pj + iyj = O (/ = 1,2,...) and thus Pj = yj = O
(/=1,2,...). Consequently, {zn} is a basis of ?(r). .
Conversely, if the sequence {zn}<=? defined by A.2) is a basis of
?(r), then a similar argument shows that every xeE has a unique ex-
pansion x=
г. that {х„} is a basis of ?.
On the other hand, let ? be a one-dimensional complex Banach
space. Then every basis of ? is of the form {x1}, where Xj #0. However,
for any xe?, x^0, the couple z1 =x, z2 = (l +i)x is a basis of ?(r),
which is clearly not of the form {xt} и {iXj}. This completes the proof
of proposition 1.1.
In view of the relations between the coefficients fi} and yy above, or,
equivalently, between the coefficient functionals associated to the bases
{х„} and {zn} above (see §3, definition 3.1), let us mention the rela-
relationship between the elements of the conjugate spaces ?* and (?(r))*.
If /e?*, then for the functional g defined by2
g(x) = Ref(x) (хе?(г))
we have ge(E(r))* and
= Re[-i/(x)]= -Re/(ix)= -
1 Here ; = |/^1, but in general we shall use the letter i to denote positive
integers. Whenever we shall use ; as j/^-T, we shall make a special mention.
2 For any complex number ? = ri + i? {г}, С real) we use the notations R?

I. The Basis Problem. Some Properties of Bases in Banach Spaces
and thus f(x) = g(x) — ig(ix) is uniquely determined by g. Conversely,
if ge{E(r))*, then for the functional / defined by
f{x) = g{x)-ig{ix) (xeE)
we have feE*, since
f{ix) = g(ix)-ig{-x) = g(ix) + ig{x) = if{x) (xeE).
Furthermore, since ?(r) is a real Banach space and ge{E{r))*, g(x) is
real for all xeE, whence
Re/(x) = #(x), Im/(x) = -g(ix) (xeE).
Thus there exists a one to one real-linear1 mapping /-»# (where
g(x) = Ref(xj) of E* onto (?(r))*; in other words, this is a one to one
linear mapping of (?*)(r) onto (?(r))*. This mapping is also an isometry.
Indeed, if f{x) = rew (r>0 and 0 real), then, since |/(x)| is real, we have
гох|| = ||0||||х||,
whence, since this holds for all xe?, we obtain ||/1К||<7||. On the other
hand,
' " """ il (xeE),
whence ||g||<||/||. Consequently, ||/|| = ||0||, which proves our assertion.
In the above, to every complex Banach space ? there has been asso-
associated a real Banach space ?(r). Conversely, it is natural to ask, which
real Banach spaces F have the property that there exists a complex
Banach space ? with ?(r) = F. It is convenient to consider the following
more general question: for which real Banach spaces F does there exist
a complex Banach space E such that ?(r) is isomorphic to ?? In other
words: for which real Banach spaces F does there exist a real-linear iso-
isomorphism of F onto a complex Banach space ?? If a real Banach space
F has this property, then we say that F admits a complex structure2.
A characterization of such spaces is the following: A real Banach space
F admits a complex structure if and only if there exists an automor-
automorphism 3 и of F such that u2 (x) = — x (x e F). Indeed, if v is an isomorphism
of F onto ?(r), then u = v~1iv is an automorphism of F satisfying
ы2(х)= —x (xeF); conversely, if и is an automorphism of F satis-
satisfying м2(х)=-х (xeF), then, putting (a + ifi) x = ax + )8m(x),
|||x||| = sup ||ewx|| (xeF), and ? = ? endowed with this multiplication
OS9S2
1 I.e. additive and homogeneous with respect to real scalars.
2 For instance, we have seen in the above that for every complex Banach space
E the real Banach space Fl = (?(r))* admits a complex structure.
3 I. e. an isomorphism (linear homeomorphism) of F onto F.
1. Definition of a basis. Relations between bases in complex and real spaces
and norm, the mapping v: x—>x will be an isomorphism of ? onto ?(r)
(since ||x||<|||x|||<A + ||m||)||x|| for all xeF).
There are real Banach spaces ? which do not have this property,
e.g. if dim? = rc<oo, then the necessary and sufficient condition for ?
to admit a complex structure is that и be even; in this case dim? = —.
There are also infinite dimensional real Banach spaces which do not
admit a complex structure, e.g. the space J constructed in Ch. II, §4,
example 4.1'.
b) Extension of the field of scalars. Let G be a real Banach space.
Then G can be embedded (by a real-linear isometry) into a complex
Banach space ?, by the following procedure. Let ? be the cartesian square
GxG, endowed with the norm \\{y,z}\\ = (||y||2 + ||z||2)* (yeG,zeG).
Then the mapping
м:{у,г} -> {-z,y} (yeG,zeG)
is an automorphism of ? satisfying M2({y,z})= — {y,z}, whence, by a)
above, ? admits a complex structure2. Let ? be the complex Banach
space obtained from ? as in a), i.e. the space ? endowed with the multi-
multiplication by complex scalars
{y,z}||| = sup
OS9S2
and with the norm
Then the mapping y—>{y,0} is a real-linear isometry of G into ?,
since by
we have
= (oc2+p2)i\\y\\ =
(yeG; a,Д real),
,0}|||= sup ||(a + ij8){y,O}|| = ||y|| for all yeG.
|a+i0J=l
The space ? is called the complex Banach space obtained from G by
the extension of the field of scalars, or shortly, the complexification of E.
Since i{z,0} = {0,z} (zeG), we have
{y,z} = {y,0} + {0,z} = {y,0} + *{z,0} (yeG,zeG),
1 For a proof see [45].
2 In particular, it follows that a sufficient condition for a real Banach space F
to admit a complex structure, is that F be isomorphic to the cartesian square of a
real Banach space G.

6 I. The Basis Problem. Some Properties of Bases in Banach Spaces
and this decomposition is obviously unique. Thus, identifying the space
G with its isometrical image {G,0} in E, every xeE can be uniquely writ-
written in the form x = y + iz, where yeG,zeG. Furthermore, the mappings
x->y and x->z are projections of norm 1 onto G, since
+ 1|г||2)* = ||{з',г}||СхС<|||{>7,2}||| (yeG,zeG).
A useful relation between the bases of G and E is the following:
Proposition 1.2. Let G a real Banach space and let E be the complexi-
fication of G. A sequence {yn} с G is a basis of G if and only if it is a
basis of E (but E can have also other bases which ф G).
Proof. Assume that {у„} is a basis of G and let x = y + izeE be
arbitrary. Then there exist expansions with real scalars
whence y+iz= ? (Оу+г/^уу. Furthermore, let us show that this ex-
j=i =o
pansion is unique. Firstly, we observe that if ? (ау-И/^)уу converges,
00 QO J ~ *
say to yo + izo, then ]T a.}y} and Xft>j converge and we have
oo со j = 1 j = !
Уо = E aj>7> zo= E ft» Indeed, we have
whence, since the mapping y + iz—*y of ? onto G is of norm 1,
>0 as и ->оо,
and similarly zo=^j8j>'J-. Now, if ][] (a,-+ '/?;)У;= 0. then, by the
\yn)
preceding, both ^ y.jyi and ^ /^>^ converge to 0, whence, since \yn
is a basis of G, a_,- = ^ = 0 (/=1,2,...) and thus ay+i/?,- = 0 (/=1,2,...),
сю
which proves the uniqueness of the expansions ? (iXj + ip^yj. Conse-
Consequently, {у„} is a basis of E. J=1
1. Definition of a basis. Relations between bases in complex and real spaces 7
Conversely, if {yn} cG isa basis of E, then every yeG has an ex-
GO GO
pansion y= ? (ay + i/?,¦)y,-(a,-,/?y real), whence, by the above, y= X «уУ,
ос J=l J=l
+ 1^]/^Уу. Since every xoeE can be uniquely written in the form
j - 1 00 OO
xo = yo + izo, where y0, zoeG, it follows that ? /?yy~0, y= Хаууу.
Since {у„} is a basis of ?, this expansion is unique, and thus {у„} is a
basis of G.
On the other hand, let G be a one-dimensional real Banach space.
Then the complexification E of G is a one-dimensional1 complex Ba-
Banach space and any element xeE, x#0, is a basis of E. Hence there
exist bases of E which are not in G. This completes the proof of propo-
proposition 1.2.
In view of the relations between the coefficients a} and (Xj + iPj
above, or, equivalently, between the coefficient functionals associated to
the bases {у„} of G and {yn} of E (see § 3, definition 3.1), let us mention
the relationship between the elements of the conjugate spaces G* and
E*. If feE*, then for the functionals hl,h2 defined by
h2(y) = lmf(y) (yeG)
we have h1eG*,h2eG* and
= y + izeE),
and thus / is uniquely determined by the couple {/i,,/i2}eG*xG*.
Conversely, if {hl,h2}eG*xG*, then for the functional / defined by
f(x) = [h1{y)-h2{z)] + i[h2(y) + h1{z)] (x = y + izeE)
we have feE*. Furthermore, since for x = y + izeG we have y = x,
z = 0, we obtain
f{) Ki ih{y) (yeG),
whence Ref(y) = h1{y), Imf(y) = h2(y) for all yeG (because h^y),h2(y)
are real for all yeG).
Thus there exists a one to one real-linear mapping /-»{fr1,fr2}
(where h1(y) = Ref(y), h2{y) = lmf{y), yeG) of E* onto G*xG*; in
other words, this is a one to one linear mapping of (E*){r) onto G* x G*.
1 Whenever we write dimG, dim?, we understand the dimension with respect
to the field of scalars of G or E respectively.

I. The Basis Problem. Some Properties of Bases in Banach Spaces
This mapping is also an isomorphism1, when G* x G* is endowed with
the norm ||{M2}II = (IIM2 + IIM2)*- Indeed, we have
^ 2 + M2F
1|2\+
ill2 + IIM2)TIIMII (x = y + ize?),
and, similarly,
whence
= y + izeE),
IM2)"|
(xeE),
and thus У'Ы\/2\\{К^2}\\. On the other hand, since Му)А(У) are
real for all yeG, we have
whence
||. Similarly, p2K||/||, whence
Consequently,
1
/2
feE*
which proves our assertion.
In the above, to every real Banach space G there has been associated
a complex Banach space E. Conversely, it is natural to ask, which
complex Banach spaces E have the property that there exists a real
Banach space G whose complexification is E. It is convenient to con-
consider the following more general question: for which complex Banach
spaces E does there exist a real Banach space G such that the complex-
complexification of G is isomorphic2 to ?? A characterization of such spaces is
1 Hence, by part a), this mapping induces an isomorphism g-*{hi,h2} of
(E(r))* onto G* x G*, defined by
) = hl(y)-h1(z) (x = y + izeE(r)).
2 The answer cannot be obtained by the restriction of the field of scalars to
the reals. Indeed, as shown by the example of finite dimensional complex Banach
spaces E, the complexification of E(r) need not be isomorphic to E. On the other
hand, let us also mention that, as shown by the example of finite dimensional real
Banach spaces G, the real Banach space E(r) associated to the complexification E
of a real Banach space G need not be isomorphic to G.
1. Definition of a basis. Relations between bases in complex and real spaces
the following: A complex Banach space ? is isomorphic to the complex-
complexification of a real Banach space if and only if there exists an involution
on E, i.e. an antilinear1 automorphism w of E such that w2(x) = x (xeE).
Indeed, if v is an isomorphism of ? onto the complexification ?, of a
real Banach space G, then x: y + iz^-y — iz {yeG, zeG) is an involution
on El, whence w = v~xxv is an involution on ?; conversely, if w is an
involution on ?, then ? is isomorphic to the complexification of the
real Banach space G={xe?|w(x) = x} (since every xe? can be unique-
. . . x + w(x) x — w(x)
ly written in the form x = y+iz, where у = ё G, z = —: e G
and since ||x||<j/2(||y||2 + ]|z||2)i:
+ ||w||
2 2i
\\x\\ for all xeE).
If {х„} is a basis of a complex Banach space E admitting an invo-
involution w, then {w(xn)} is also a basis of E\ this remark is useful in the
study of bases of E xE.
If ? is a complex Banach space with dim? = n<oo, then there
exists a natural involution {?;}"->{?;}" on E, whence E is isomorphic
to the complexification of the и-dimensional real Banach space
{{?,-}" Е?|?!,...,?„ = геа1}. More generally, if a complex Banach space E
' GC
ajxj^ Y (^-c<xj)xj is con-
has a basis {х„} such that the mapping
ji ji
tinuous, then E is isomorphic to the complexification of the real Banach
space < Y, a.jXjeE\a.n =
U=l oo
(n = l,2,...)>, since the mapping
J
w:
Y, (<Xj + iPj)Xj-> Y, (aj~iPj)xj is an involution on E. Furthermore, if
j=i j=\
E is one of the usual complex Banach spaces, e.g. C([0, 1]), I/([0, 1])
(l<p<oo), c0 or/p(l ^p^<x), then the natural involution x(t) = y(t)
+ iz(t)^>xjf) = y{t)-iz{t) shows that C([0,1]) is isomorphic to the
complexification of the real Banach space C([0,1]), and similarly, each
of the other spaces is isomorphic to the complexification of the corre-
corresponding real space Z/([0,1]), c0 or I" respectively. This remark,
together with proposition 1.2 above, shows that every basis of the real
Banach space C([0,1]) is also a basis of the complex Banach space
C([0,1]), and similarly for I/([0,1]), c0 or I" respectively.
Let us also mention that some other specifically complex Banach
spaces are also isomorphic to complexifications of suitable real Banach
spaces. For instance, in the Banach space A of all complex functions
x(?) of a complex variable ?, which are analytic in {?| |?|<1} and
continuous in {?| |?|<1}, endowed with the norm ||x|| = max|x(?)|,
the mapping w: x(?)-»x(?) is an involution, whence A is isomorphic
I.e. such that w(Xx) = /,w{x) for all xeE and all (complex) scalars X.

10
I. The Basis Problem. Some Properties of Bases in Banach Spaces
to the complexification of the real Banach space G = {x6/4|x|[_1>1]
is real}. In the Banach space L(E,E) of all continuous linear mappings
of E into E, endowed with the norm ||м||= sup ||м(х)||, where E is
xeE
11*11= »
a complex Banach space admitting an involution w, the mapping
WjiM—>wmw is an involution1, whence L(E,E) is isomorphic to the
complexification of the real Banach space {ueL(E,E)\u(G) cG), where
G=jxe?|w(x) = x}.
§ 2. Some examples of bases in concrete Banach spaces.
Some separable Banach spaces in which no basis is known
The first idea which arises quite naturally in connection with the
basis problem is to examine whether or not the various concrete sepa-
separable Banach spaces occurring in practice possess a basis. In this section
we shall give some examples of such bases and some examples of sepa-
separable Banach spaces in which no basis is known. We shall use, for all
concrete Banach spaces occurring in the sequel, the standard notations.
Whenever we shall write F or If, we shall understand that p<oo.
Unless otherwise stated, the scalar field К for all concrete Banach spaces
considered in the sequel, can be either the field of complex numbers, or
the field of real numbers. Actually, there will be constructed only bases
consisting of real functions or real sequences respectively, but these
will be bases of the corresponding spaces considered either with complex
scalars, or with real scalars. This fact is quite natural, in the light of the
considerations of § 1.
Example 2.1. In the spaces c0 and lp(p> 1) the sequence2
(" =1.2,...)
B.1)
constitutes a basis.
Indeed, for х={?„}ес0 we have
x —
when и—>oo, while for х={?„} elp we have
i= 1
n
Х-У
= sup I
n+1 a< oo
=1 I KI.I-
1 Unfortunately, this is not an involution for the natural structure of Banach
algebra of L(E,E), since w1(uv) = wuvw = wuwwvw = wl(u)wl(v), which, in
general, ^w^i^w^u).
2 Throughout this book, du will be the Kronecker delta, i.e.
2. Some examples of bases in concrete Banach spaces 11
00 GC
»0 when n-»oo. On the other hand, ]Г ?;X,= ]Г п(х( implies, in c0,
lim max |^, — ^| = 0, while in lp it implies lim У \^ — t
n-»ool^i^n и^* °° I •= i
whence, in both cases, с( = ^, (i=l,2,...).
The basis B.1) is called the natural basis or the unit vector basis of the
spaces c0 and lp{p^\).
From the above it follows that in the space с the sequence
xo={l,l,...}, xn={6nj)f=l (n = 2,3,...) B.2)
constitutes a basis, namely, every x = {?n}ec has a unique expansion of
the form x=( lim ?„)хо+ У (с,- lim ?„)х;.
Vn-*a; / '-1^ n-»co /
Example 2.2. In the Banach space C([0,1]) the sequence
X0(f)=l, Xj(f) = t,
'2/-2 2/
0
1 for t =
2/-1
B.3)
Z-2 2/-Г
linear in | —r-^-, -r—r- I and
2
2k+l ' 2t+1
(/=1,2,...,2*; fc = 0,l,2,...)
constitutes a basis. More generally, let {«„}<= [0,1] be an arbitrary
sequence dense in [0,1] and for each и^2 let 9n denote that one of the и
subsegments of [0,1] determined by 0, a1,a2,...,an_ul (rearranged
in increasing order) which contains an. Then the sequence
*»(') =
0 for
1 for t = an_u
linear for the other f
B.4)
constitutes a basis of the space C([0,1]).
Indeed, for xeC([0,l]) satisfying x@) = x(l) = 0 let
<х„ = x(
s»W =
n -
fln-l)" I
n
i = 2
1
] a^,(a«-i) (
2
(
/1 = 3,4,...), B.5)
n = 2,3 ...). B.6)

12
I. The Basis Problem. Some Properties of Bases in Banach Spaces
Then, since х2,...,х„ are linear on each subsegment of [0,1] deter-
determined by {д;}?=1 (rearranged in increasing order), so is sn. Further-
Furthermore, by B.4) we have xJ+l(aj)=l and хг(а^) = 0 for i=j + 2,j + 3,...,n
(j=l,...,n—l), whence, taking also into account B.5), we obtain
j n
|>„М](Я/) = Z <xixi{aj) + <Xj+lxj + l(aj)+ Z aixi(aj) B-7)
i = 2 i=j + 2
j Г J 1
= z а;х,-(а,-)+ *к-)- Z «i^i(«j) =хК) 0'=i,...,n-i).
i = 2 L i=2 J
Consequently, sn(x) is the polygonal (piecewise linear) function inter-
interpolating the values of x in the points a_[ =0,a1,a2,...,an_1, 1 =ab i. e.,
if ah,ai2 are any two consecutive points of {aj^li (in the sense of
the natural order of [0,1]), then
B.8)
Let e>0 be arbitrary. Since x is uniformly continuous on [0,1], there
exists an r] = r](c)>0 such that |x(i') —x(i")|<c whenever r',r"e[O,l],
\t' — t"\<r\. Since the sequence {a}) is dense in [0,1], there exists a posi-
positive integer JV = JV[j/(e)] such that for n>N we have тах|аA — ai2\<r\,
where the max is taken over all couples of consecutive points of
{aJ?=-i- Now, let te[0,1] be arbitrary. Then there exists a X with
0<A<l, such that t = Xail + (\ — X)ah, where ah,ah are consecutive
points of {а^Ч!-). (n>N[ri(e)]) satisfying [ah,a^st. Then, by B.8),
= \a [x(t) - x(a,,)] + A - A) [x(t) - x(ai2)] |
< max |x(t')-x(t")|<e
and thus, since N\_y\{e)\ is independent of te[0,1],
||x-sj<e (n>N[jj(?)]).
Consequently, we have
QC
Now we drop the restriction x@) = x(l) = 0. Let xeC([0,1]) be
arbitrary. Then for the 3ceC([0,1]) defined by
x(t)=x(t)-x@)-[x(l)-x@)]t (re [0,1])
we have x@) = x(l) = 0, whence, by the above, x has an expansion of
2. Some examples of bases in concrete Banach spaces
13
the form x= ? a,x;, whence, putting ao = x@), a1=x(l) —x@), we
obtain ; = 2
x=
On the other hand, assume now that we have ? a;x, = 0. Then
i = 0
whence, putting t = 0, l,aba2,... and taking into account B.4), we
obtain, successively, а„ = 0 (и = 0,1,2,...), which proves that B.4) is a
basis of C([0,l]).
In particular, the basis B.3) is called the Schauder basis of C([0,1]).
The above construction can be generalized to the case of the space
C(Q), where Q is an arbitrary metric compact space (see Vol. II,
Chapter V).
Example 2.3. The sequence of equivalence classes1 {у„}, where yn
are the Haar functions, i. e. the functions defined on [0,1] by
for
-|/2fc for te
0 for the other t
21-2 21-1
t+i > 2fe+1
-1 2/
B.9)
constitutes a basis of the space Lp([0,1]) (p^ 1).
Indeed, let ax < ¦ ¦ ¦ < an_ 1 be the points of discontinuity of the func-
functions yu...,yn and let ао = 0, а„=1. Then from B.9) it follows, by in-
induction on n, that for ak<t<ak + 1 we have
0
for ak < t < ak +!,
for x<ak and afe+1
B10)
Indeed, for n= 1 formula B.10) is obvious. Assume now that B.10)
is true for some и ^ 1 and let a\ < ¦ ¦ ¦ < a'n be the points of discontinuity
1 We denote by x the equivalence class of the function x.

14
I. The Basis Problem. Some Properties of Bases in Banach Spaces
of j>i,...,j>n+i and a'0 = 0, a'n+i = l. Then there exists aj with
such that
21-2 21-
B.11)
for ге[а^_ьа})
2m for te[a'j,a'j+i)
0 for the other t,
лш+1' 2m+'
2/-1 2/
¦ym+ 1 ' лш
B.12)
where n+l=2ra + /, with 1
Now let 0</с<и and let а^<г<а^+1. If k?=j-l,j, then by B.12)
we have yn+i(t) = O, whence, by the induction hypothesis and by B.11)
we obtain
1
0
for a'k<t<a'k+u
for т < a'k and aj^. +1 < т,
i.e., B.10) with n+l instead of n. If k=j—\, then by B.12) we have
/г and
2ra for ajcjc+1,
— 2m for ait+] ^т<а^ + 2,
0 for t<a'k and aj< + 2 < т,
whence, taking into account that, by the induction hypothesis,
1
i= 1
we obtain
for т < a'k and a^ + 2 < т,
2m -i- ?m ?m + ^
2ra - 2m = 0
0
- for а'к<т<а'к+и
for a'k+i<T<a'k + 2,
for т < a^ and a^ + 2 < т,
i.e., B.10) with n+l instead of n. Finally, if к =j, then by B.12) we have
yn+i(t)= —1/2™ and similarly to the preceding case we obtain B.10)
for n+l instead of n, which completes the induction. This proves B.10).
2. Some examples of bases in concrete Banach spaces
15
Now, for xeLp([0,l]) let1
Then, by B.10) we have, for ak<t<ak+l,
[pH(x)](t)=
B.13)
B.14)
, B.15)
whence
However, by the Holder inequality for integrals, we have
j x(T)di
Ik
Consequently,
0 0
Since x has been an arbitrary element of Lp([0,1]), it follows that
for the linear operators sn defined on I/([0,1]) by
# = РЙ=ЁВД (xeI/([0,l])), B.16)
i= 1
we have
||sj|<l (и=1,2,...). B.17)
On the other hand, for every xeC([0,1]) it follows from B.15)
that [pn(x)](t) converges uniformly to x(t) on [0,1]\Л, where Л denotes
the countable set of all discontinuity points of the functions B.9). Hence
we have, in the norm of I/([0,1]),
lim||sn(x)-x||=0 (xeC([0,l])). B.18)
Since the set {x|xeC([0,l])} is dense in I/([0,l]), from B.17) and
B.18) we obtain
lim||sn(x)-x|| = 0 (x
i.e. every xeI/([0,1]) has an expansion of the form x= ? а;у(.
1 We denote by x an arbitrary function from the equivalence class x.

16 I. The Basis Problem. Some Properties of Bases in Banach Spaces
On the other hand, by B.9) we have the orthogonality relations
$yi(t)yj(t)dt = O (/#j;U=l,2,...)- B-19)
Indeed, it is obvious that jyi(t)y2k+l(t)dt = 0 A= l,2,...,2k;
= 0,l,2,...) and that for 1^, /2<2fe, ^ #/2, we have
since already y2lc + (l;y2lt + ,2 = 0. On the other hand' if ki>k2, then for
any lul2 with l</i<2'"((=l,2), the set {te[0,l]\y2k+h{t)*0} is
contained in an interval in which the function y2k,+h(t) has a constant
value A( = |/24 -]/2F2 or 0), whence
= 0,
which proves B.19). Now, from B.19) it follows that the coefficients a,
in the above expansions are uniquely determined, which proves that
{yn} is a basis of L"([0,1]). This basis is called the Haar basis of I/([0,1]).
Similarly to the extension B.4) of the Schauder basis B.3), one can
also define Haar functions {у„} with respect to a dense sequence {an\
с [0,1] and prove that the sequence {у„} constitutes a basis of Lp([0,1]).
The above arguments and constructions can be extended to prove
the existence of bases in separable Orlicz spaces and in the more general
separable Lx spaces, where X is an arbitrary levelling length function.
These spaces are defined as follows.
Let T be a set, IF a relatively complemented, countably additive
collection of subsets of T and v = v(e) a non-negative countably additive
measure defined for all ee3F (u-finiteness is not assumed). Let X(y)^ со
be a non-negative function defined for all non-negative measurable у
on T and let Lx consist of those measurable x for which x(|x|)<oo
(where |x|(t) = |x(t)| (teT)).
The function X is called a length function if the following conditions
are satisfied: a) X(y) = 0 whenever y(t) = O for almost all teT; b) ЛО)
<Я(у2) whenever yi(t)^y2{t) for almost all teT; c) X(y + y')
+ A(/); d) A(a;y) = aA(y)(a>0), and e) the relations y1(t)<y2(t)<...
(teT) imply A(supyJ = sup;.(>¦„). In this case L>' is a Banach space. A
length function X is called a levelling length function if it satisfies the
following conditions: f) A(y) = sup A(yxe), where %e denotes the
v(e) < so
3. The coefficient functional associated to a basis. Bounded and normalized bases 17
characteristic function of the set e, and g) X(y)^X(y) whenever у coin-
coincides with у except on some e with 0 <v(e)<oo, and y(t) = y{z)dv{z)
v(e)J
(tee). As we have mentioned above, every separable 1} space, where X
is a levelling length function, has a basis.
However, there are some concrete separable Banach spaces in which
no basis is known, e.g. the following:
Problem 2.1. Does a separable Lx space, where X is not a levelling
length function, posses a basis?
Problem 2.2. Does the Banach space Hl possess a basis?
We recall that Я1 is the subspace of the complex Banach space
L}(Q), where Q is the unit circumference (C||C| = 1], constituted by all
П
xel}(Q) such that j ein&x{»)d9- = 0 (n= 1,2,...).
— П
Let A be the Banach space of complex functions considered at the end
of§l.
Problem 2.3. Does the Banach space A possess a basis?
In order to be able to recognize whether or not a given sequence
{х„} in a Banach space E constitutes a basis of the space E, it is useful
to know the properties possessed by bases in Banach spaces. In the next
sections we shall present some properties of bases and various charac-
characterizations of bases in Banach spaces.
§ 3. The coefficient functionals associated to a basis.
Bounded bases. Normalized bases
Definition 3.1. Let {х„} be a basis of a Banach space E. The sequence
of linear functionals {/„} defined by
x= Xa,x,e?,; = l,2,...
C.1)
is called the sequence of coefficient functionals associated to the basis
{xn}, or, shortly, the associated sequence of coefficient functionals (we
shall write: a.s.c.f.).
Thus, if {х„} is a basis of the space E and {/„} the a.s.c.f., then
every xeE has a unique expansion of the form
x=Yfi(x)x,. C.2)
2 Singer, Bases in Banach Spaces I

18
I. The Basis Problem. Some Properties of Bases in Banach Spaces
Our next aim is to prove the fundamental fact that the coefficient
functionals associated to a basis of a Banach space E are continuous on
the space E. Let us first prove
Proposition 3.1. Let {х„} be a sequence in a Banach space E, such
that х„^0 (n=l,2,...) and let A l be the linear space of sequences of scalars
a;Xj converges >
endowed with the norm
{«„}!= sup
C.3)
C.4)
Then Ai is a Banach space-
Proof The number C.4) is finite, since the sequence
is convergent. Since all х„#0 (n= 1,2,...), C.4) is a norm on the linear
space C.3).
Let {а(„к)} (fc = l,2,...) be a Cauchy sequence in Ax. Then for every
e>0 there exists a positive integer N{s) such that
{«<*>}-{«<Г>}||= sup
Hence
<e (k,m>N(s)).
< 2e
whence, since all х„#0 (п= 1,2,...), it follows that
2e
|a<fc>-a<m>| < (
Consequently, for each n^l the sequence of scalars aj,k) (fc = l,2,...)
is convergent to a scalar ое„. Hence, from the inequalities
(Kf> -«И) X;
<e
we obtain, for m-> oo,
«i4-«,)x,-
e (k>N(s), n=l,2,...).
3. The coefficient functionals associated to a basis. Bounded and normalized bases 19
Then
n + i
v „ Y
! = П + 1
у a
(k>N(c);nJ=l,2,...),
whence, since each series ^ aj-k)x( is convergent and since E is complete,
it follows that ? а(х,- converges, i.e. {«„le/^. Moreover, by the
( = 1
above we have
||{а<к>Ыа„}||= sup
which completes the proof of proposition 3.1.
Proposition 3.2. Let E be a Banach space with a basis {х„} and let
{/„} be the a.s.c.f. Then
a) The Banach space Ax introduced in proposition 3.1 is isomorphic
to E, by the mapping
¦*»r
b) The numbers
N11= sup
I ft(x)x,
(xeE)
C.5)
C.6)
define a norm on the space E, equivalent to the initial norm of E.
Proof, a.) Since {х„} is a basis, we have х„#0 (n= 1,2,...) (by virtue
GO
of the uniqueness of the expansions ? а(х,), whence, by proposition
i= 1
3.1, Ax is a Banach space. The mapping C.5) of Ay into E is obviously
linear and of norm 1. Since {х„} is a basis of E, C.5) is one to one and
maps A j onto E (because of the uniqueness and existence of the expansions
GO
Yj <*jXj). Hence, by the inversion theorem of Banach1, C.5) is an iso-
i= 1
morphism of Ax onto E.
b) By part a), there exists a constant С ^ 1 such that
ас
I
i = l
CCiX;
^ sup
1 :? П < 00
n
Z xix<
i = l
I
a,-
xf
.1 = 1
E>
1 See e.g. [10], p. 41, theorem 5.

20 I. The Basis Problem. Some Properties of Bases in Banach Spaces
and it remains to observe that а„ = /„(х) (и =1,2,...) for all
x=
Theorem 3.1. Let {х„} be a basis of a Banach space E. Then the
coefficient functionals fn associated to the basis {х„} are continuous on E,
i.e. we have fneE* (n=l,2,...). Moreover, there exists a constant M
such that
UWII/»KM (n=l,2,...). C.7)
Proof. Let |||x||| be the norm on E defined by C.6). Then there exists,
by proposition 3.2 b), a constant C^ 1 such that
|||x|||<C|x| (xeE).
Since all х„#0 (п= 1,2,...), it follows that
I/»WI = -
и
z
1 = 1
ft
(x)xf
+
n-l
/ , J (\X) X^
Ikll
-^ 111*11
1С
2C
(xe?,n=l,2,...),
C.S
whence /„e?* and ||/J| <-—- (n=l,2,...). On the other hand, by the
IWI
definition of/„ we have 1 = /„(х„)<||/„|| ||х„|| (и=1,2,...), which com-
completes the proof.
Let us point out that in the above proof of proposition 3.2, and
hence also in that of theorem 3.1, the inversion theorem of Banach
(which amounts to the open mapping theorem or the closed graph
theorem) has played a decisive role.
From C.7) we infer
Corollary 3.1. Let E be a Banach space with a basis {xn) and let
{/„} с Е* be the a. s. c.f. Then
a) We have inf ||xJ>0 if and only if sup ||/J<oo.
l«n<co 1$п<сю
b) We have sup ||xj|<oo if and only if inf ||./J>0.
1 ^ Л < 00 1 ^ П < 00
In connection with this corollary, let us also mention some equi-
equivalent conditions for a sequence {х„} to be bounded from below or
from above:
Lemma 3.1. Let {х„} be a sequence in a Banach space E. Then
a) The following conditions are equivalent:
3. The coefficient functionals associated to a basis. Bounded and normalized bases 21
Г. ^infJxJX).
QO
2°. T/ie convergence of Y, atxi implies lim а„ = 0.
3°. The convergence of Z aix> <mP'<es SUP |а„| < oo.
b) The following conditions are equivalent:
1°. sup ||xj<oo.
X' ОС
2°. ?|а,-|<оо implies that ? а(х( converges.
i = 1 i = 1
Proof, a) If we have 1° and ? а,х;
; = i
converges, then |а„|
1
||а„х„||->0 as п-юэ. Thus, l°=>2°.
inf
1 « j < x ' J '
The implication 2° => 3° is obvious. j
Finally, if inf ||xJ=0, then, taking indices nk with ll*nJKx2i
(fc=l,2,...) and taking a =2k (fc=l,2,...) and а„ = 0 (и^иьи2,...),
n + p
i = n+ 1
we obtain that ? a,x; converges but sup |а„| = со. Thus, 3°=>Г.
; _ j 1 « И < 00
b) The implication 1 ° => 2° is obvious, since
< sup ||x_,|| X lafl-
1 « j < cc i = я + 1
Conversely, if sup ||xj = oo, then, taking indices nk with ||xnj|>2'c
1 ^ n < a; ]
(fc=l,2,...) and taking а„к = х^ (fc=l,2,...) and аи = 0(и#и1,и2,...), we
obtain that ?|а;|<ао, but ||а„кх„к||>1 (fc= 1,2,...), whence ? а,Х;
does not converge. Thus, 2° => 1°, which completes the proof.
In the terminology of § 8, condition a) 2° means that {х„} dominates
the unit vector basis of c0, while condition b) 2° means that the unit
vector basis of I1 dominates {х„}. Therefore, using the results of §8,
one obtains other equivalent conditions.
Definition 3.2. A basis {х„} of a Banach space E is called a bounded
basis if 0< inf ||xj< SUP ll*nll<0°- Tne basis {х„} is said to be
a normalized basis if ||xn|| = l (n=l,2,...).
From the definition 1.1 of a basis it follows that if {х„} is a basis
of a Banach space ? and {л„} an arbitrary sequence of scalars such that
л„#0 (n=l,2,...), then {я„х„} is also a basis of the space E; hence, in
Г x I
particular, s:—^У is a normalized basis of ?.

22
I. The Basis Problem. Some Properties of Bases in Banach Spaces
Theorem 3.2. Let {*„} be a bounded basis of a Banach space E. Then
there exists an equivalent norm \\x\\' on E, in which the basis {xn} is
normalized.
Proof. Let
(n = l,2,...)
C.9)
(the closed linear subspace of ? spanned by х1,...,хл_1,хл + 1,...) and
let {/„} с E* be the a. s. с f. to the basis {х„}. Then for any ? а(хге?(п)
we have, by C.7), l*n
<„- Z aixi
II/JI
/„K-Z«^
II/JI
inf
where M ^ 1 is a constant. Consequently, taking an a such that
0<a<— inf |xj|, we shall have
M 1
dist(xn,?(n))>a
Put
| = l,/c= l,2,...}uaS?] =
C.10)
C.11)
where SE={xeE\ \\x\\ < 1} is the unit ball of ? and where со A denotes
the closed convex hull of the set
We claim that the Minkowski functional 1|х1Г= inf к of the cir-
circled1 closed convex set S'E is a norm on E, having the required pro-
properties.
Indeed, ||x||' is a norm on ?, equivalent to the initial norm on ?,
since by C.11)and a<— inf ||xj| < mf II*J < SUP IIXJI wehave
sup \\xJ\SE
1 We recall that a set DaE is said to be circled, if /JDcD for each scalar /J
with |0| = 1.
4. Some characterizations of regular biorthogonal systems. Applications 23
and thus it remains only to prove that
||xj'=l (n=l,2,...). C.13)
Fix an arbitrary n and put
i2n = co[{j8xk||j8|= l,/c= 1 n-l,n + l,...}u«SE]. C.14)
Then obviously
whence, by C.10),
finc{xe?|dist(x,?(n)Ka},
*„<№„¦ C.15)
Furthermore, we have
А с {рхп\ \p\=l}vQn^c6A = S'E,
whence
C.16)
Now, by C.15) and since Qn is a circled closed convex set, there
exists1 an /e?* such that
/(xBJ*sup|/(x)|.
xsQn
Consequently, by C.16) and xneS?,
/(xn)=sup|/(x)| = ||/ir,
xeS'E
whence
and thus we have C.13), which completes the proof.
§ 4. Biorthogonal systems. The partial sum operators.
Some characterizations of regular biorthogonal systems.
Applications
Definition 4.1. Let ? be a Banach space. A pair of sequences (х„,/„),
where {х„} <=?, {/„} c?*, is called a biorthogonal system if
Мх} = 8ц (ij = 1,2,-¦¦)¦ D-1)
1 See e.g. [270], p. 109, theorem 4.

24
I. The Basis Problem. Some Properties of Bases in Banach Spaces
The biorthogonal system (х„,/„) is said to be E-complete if the
sequence {х„} is complete1 in ?.
From theorem 3.1 it follows that if {х„} is a basis of a Banach space
E and {/„} the a.s.c.f., then (*„,/„) is an E-complete biorthogonal system.
However, the converse of this statement is not true, as shown by
Example 4.1. Let Е = С2ж, the space of all continuous functions
on the real axis ( — 00,00) and having period 2% (i.e. x(t + 2n) = x(t)
for all te(— 00,00)), with the norm ||x|| = max \x(t)\ and let
o(t)si, x2n_l(t) = smnt, x2n(t) =
Г е( — ос , ас)
(te(-oo,со), и = 1,2,...),
if if (t2)
2„(х) = — x(i)cosnTdT, f2n+i(x) = ~ x(i)sin(n+ \)xdx
л J л I
(xe?, n = 0,l,2,...).
Then (х„,/„) is an E-complete biorthogonal system, since by the
theorem of Weierstrass the sequence {х„} is complete in E. However,
from the existence of continuous functions whose Fourier series are
not uniformly convergent it follows that {xn} is not a basis of E.
Remark 4.1. Actually, in every infinite dimensional separable Banach
space E there exist .E-complete biorthogonal systems (х„,/„) such that
{х„} is not a basis of E. Indeed, for infinite dimensional Banach spaces
with bases this follows immediately from proposition 4.3 below and
its proof. For separable Banach spaces which have no basis (if such
spaces exist at all), the assertion follows from the fact that in every
separable Banach space E there exist E-complete biorthogonal systems
(х„,/„) (embed E isometrically into C([0,1]) and orthogonalize the
image of a finitely linearly independent complete sequence {у„}с? in
C([0,1]), by the classical procedure of E. Schmidt). Let us also mention
that in Vol. II, Ch. Ill, we shall see many other examples of E-complete
biorthogonal systems (х„,/„) such that {х„} is not a basis of ? (T-bases
which are not bases, etc.).
From the remark preceding example 4.1 it follows that we have to
seek for the bases of a Banach space ? only among the sequences {х„}
1 We use this term in the following sense: the set of all finite linear combinations
n
? a,x, (a,eK, i=\,...,n; n=l,2,...) is dense in E. Banach [10] has used the term
i= 1
"fundamental" for such sequences and we also have done so in the expository
papers [241]—[243]. However, since Cauchy sequences are also called sometimes
fundamental sequences, we prefer to use here the term "complete sequence". This
term is frequently used, at least for sequences in Hilbert spaces.
4. Some characterizations of regular biorthogonal systems. Applications 25
belonging to biorthogonal systems (х„,/„). Therefore it is convenient
to give
Definition 4.2. Let ? be a Banach space. A biorthogonal system
(х„,/„) ({х„}с?, {/„}<=?*) is said to be regular, if the sequence {х„}
is a basis of the space E; otherwise (х„,/„) is said to be irregular.
Definition 4.3. Let E be a Banach space and (х„,/„)({х„}с?, {/„}с?*)
a biorthogonal system. For every xe?, the (convergent or divergent)
ОС
series ? fi(x)xt is called the formal expansion of x corresponding to
the biorthogonal system (х„,/„), and the following notation is used:
D.3)
Definition 4.4. Let ? be a Banach space and (х„,/„)({х„}с?, {/„}<=?*)
a biorthogonal system. The sequence of continuous linear operators
{sn}, where
s»(x)= E/iWXj (xe?, n=l,2,...), D.4)
i= 1
is called tfoe sequence of partial sum operators associated to the biorthogonal
system (х„,/„). If {х„} is a basis of ? and {/„} the a.s.c.f., then {sn} is
called the sequence of partial sum operators associated to the basis {xn}.
Theorem 4.1. Let E be a Banach space and (х„,/„) ({х„}<=Е, {/„}<=?*)
a biorthogonal system. Then the following statements are equivalent:
1°. (х„,/„) is a regular biorthogonal system.
2°. For every xeE the formal expansion D.3) is convergent and its
sum is x, i.e. we have lim sn(x) = x for all xeE.
П-* GO
3°. (х„,/„) is E-complete and sup ||sn(x)|| <oo for all xeE.
4°. (х„,/„) is E-complete and there exists a constant M^l such that
||sJ|^M (и=1,2,...). D.5)
Proof. If we have 1°, then {х„} is a basis of E and {/„} is the a.s.c.f.
(since ifwe denote by {gn} the a.s.c.f., then <?,-(x ,•)=/¦(*,¦) forall i,j=\,2,...,
whence gi(x) = fi(x) for all xe?, i=l,2,...). Hence we have 2°. The im-
implication 2° => 3° is obvious. The implication 3° => 4° is a consequence
of the principle of uniform boundedness1. Finally, if (х„,/„) is a bi-
m
orthogonal system, then for every finite linear combination p= ? XjXj
and all n^m we have J=1
1 See e.g. [10], p. 80, theorem 5.

26
I. The Basis Problem. Some Properties of Bases in Banach Spaces
Sn(P)= Z fi[ Z <*}Xj)Xi= Z Z aA'X'-= Z <xJxj = P^
i=l \j=i / i=i j=l j=l
whence it follows1 that 4° implies Г. This completes the proof.
The implication 1° => 4° is also a consequence of § 3, proposition
3.2. On the other hand, let us point out that in the proof of the implication
3° => 1° the principle of uniform boundedness plays a decisive role.
From the implication 4° => 1° of theorem 4.1 there follows
Corollary 4.1. Let E be a Banach space and (х„,/„)({х„} <= ?, {/„} <= ?*)
an irregular E-complete biorthogonal system. Then for every number a^l
m
there exists a finite linear combination p = Z ajxj such that ||p|| = l and
sup \\sn(p)\\ = a. J=1
m
Proof. There exists a finite linear combination q = Z Д,*,- such
j=i
that ||q|| = 1 and sup ||sn(g)|| > a, since otherwise the uniform bounded-
ness of sup ||sn(x)|| onadense subset of the unit sphere {xe?|||x|| = l}
1 ^ Л < 3D
would imply D.5), in contradiction with the irregularity of (х„, /„). By
biorthogonality we have A — X)x-[
ф(Х)= sup
1). Let
(\-Х)х1+лд \
(l-X)Xl+Xq\\)
Then ф(Х) is continuous on [0,1] (as the supremum of a finite family
of continuous functions) and we have, by biorthogonality,
ф@)= sup
1$
ф(\)= sup \\sn{q)\\=
11*1 II
= 1,
Hence there exists a Ao with 0<A0<1, such that ф(Х0) = а. Then for
p = — -— we have ||p|| = l and sup ||sn(p)|| = a.
Let us give now some applications of theorem 4.1.
Proposition 4.1. Let E be a Banach space with a basis {х„}. Let {(„}
be an increasing sequence of positive integers, {/„} the increasing sequence
1 We use the following simple fact: if {un} is a sequence of continuous linear
operators on ? such that ||«„||<М (n = l,2,...) and that limun(p) = p for all p in
П—* GO
a dense subset of E, then lim un(x) = x for all xeE.
4. Some characterizations of regular biorthogonal systems. Applications
27
of positive integers complementary to {(„} and со the canonical mapping
ofE onto1 ?/[х(п]. Then
a) {xin} is a basis of the space [x,J.
b) {a)(Xjn)} is a basis of the quotient space ?/[xin].
Proo/ Let {/„}<=?* be the sequence of coefficient functionals
associated to the basis {х„}.
a) Since/;m(xIn) = O(m,n= 1,2,...), we have /im(x) = 0 for all xe[x;j,
ОС
m=\,2,.... Hence x= Z fin(x)xtn f°r every xe[x,J and we can
n= 1
apply the implication 2C => 1° of theorem 4.1 to the biorthogonal
system2 (xinJin\[xJ.
b) Define
<№W]=/imM (xe?,m=l,2,...). D.6)
If (o(x) = w(x), then x — x'e[x,-J, whence/u(x — x') = 0(m= 1,2,...),
which proves that фт is well defined on ?/[x,J. Obviously, фт is linear
and фт[_(о(х1п)~]=8тя(т,п=1,2,...). Since for every xe? and e>0
there exists an x'eoj(x) with ||x'||< ||co(x)|| + ?, it follows from D.6) that
фт is continuous on ?/[xfJ (we have even ||$J| = ||/,m|| (m— 1,2,...)).
CO
Now, since co(xin) = 0 (n=l,2,...), we have a>(x)= Z Фп[_с)(х)\(о(х1п)
for every x=
> and thus we can apply the implication
i 1
2° => Г of theorem 4.1 to the biorthogonal system (ш(х,п), ф„).
Definition 4.5. A sequence {х„} in a Banach space E is said to be a
basic sequence it {х„} is a basis of the closed linear subspace [х„] of?.
According to proposition 4.1a), every subsequence of a basis is a
basic sequence. It is natural to ask whether a certain converse is also
true, namely:
Problem 4.1. Let {yn} be an infinite basic sequence in a separable
Banach space E. Does there exist a basis {х„} of ? with the property
that for each n there is an index in such that xin = ynl What if ? has a basis?
An affirmative answer to the first of these questions would imply
an affirmative answer to the basis problem, since every Banach space
contains an infinite basic sequence (see § 14, remark made after table
14.1). Moreover, the answer to the second question is also believed to
be negative, namely, it is conjectured that the basic sequence
1 We denote by [x,J the (closed linear) subspace of E spanned by se-
sequence {X(J.
2 We denote by /,n]Ulr]] the restriction of fin to [xin].

I. The Basis Problem. Some Properties of Bases in Banach Spaces
yjt) = sin 2" 7Tt
],n=l,2,...) D.7)
in C([0,1]) cannot be extended to a basis of C([0,1]).
We shall show now that if [у„] admits a complementary subspace
F with a basis, then the answer to problem 4.1 is affirmative. Indeed,
let us first prove
Proposition 4.2. Let G,F be two Banach spaces with bases {>¦„} and
{zn), respectively. Then the sequence {xn}<=GxF defined by
х2„-1 = {у„,0}, x2n={0,zn} (n=l,2,...) D.8)
is a basis of Gx F.
Proof. Let {y,z}eGxF be arbitrary. Then, since {yn), {zn} are bases
00 00
of G and F, respectively, we can write y= ? ч}у^, z= ? jSj-Zj-. Since
by the definition of the norm in G x F we have
" Z «j^.
- I °w
D.9)
it follows that ? аДу^О} converges to {>',0} and, similarly, ? fSj{O,Zj}
converges to {0,z}. Consequently,
= Z aj*2j-l+ Z 0j*2j-
To show the uniqueness of these expansions, observe first that if
j 7jXj converges, say to {>',z}, then, by
i
'~ Z Уг}-\
V- Ijlj-iyj:
n
'- У ¦" ¦ , V- Z-
2n
we have ? 1ц-\У; = У and, similarly, ^ y2jZj = z. Consequently, if
CO GO 00
Z Tjxj= {0,0}, then ZT2j-iyj = 0, ^ у2jzJ = 0, whence, since {у„},
4. Some characterizations of regular biorthogonal systems. Applications 29
{zn} are bases of G and F, respectively, yj = 0(j=l,2,...), which completes
the proof of proposition 4.2.
Definition 4.6. Let G,F be two Banach spaces with bases {yn} and
{zn}, respectively. The basis D.8) of G x F is called the cartesian product
of the bases {yn} and {zn}; sometimes, for brevity, we shall denote this
sequence by \{yn,0}}<j |{0,zn}} or by {у„} х {zn}.
The desired affirmative answer to problem 4.1 in the particular
case when [yj admits a complementary subspace F with a basis,
follows now from proposition 4.2, taking into account the following
well known
Lemma 4.1. Let G,F be two complementary subspaces of a Banach
space E, i.e., E=G@F (i.e., E=G + F, GnF = {0}). Then E is iso-
morphic to Gx F.
Proof. The mapping {y,z}—>-y + z of GxF into G@F is obviously
linear, one to one and onto. By virtue of
Ы
(yeG,zeF)
this mapping is also continuous, whence, by the inversion theorem of
Banach, an isomorphism, which completes the proof.
We shall see another important instance when the answer to problem
4.1 is affirmative in § 7, theorem 7.2.
Proposition 4.3. Let E be a Banach space with a bounded basis {х„}
and let {<xn} be a sequence of scalars such that а„#0 (n=l,2,...). The
sequence {yn} с Е defined by
is a basis of the space E if and only if the sequence
D.10)
is bounded.
Proof. Let {/„}<=?* be the sequence of coefficient functionals
associated to the basis {х„}. Then for the sequence of functionals
{#„}<=?* defined by
D.11)
we have gi(yj) = dij(i,j=l,2,...) and
: X /iW^- У„ (хеЕ, и=1,2,...). D.12)
an + 1

30 I. The Basis Problem. Some Properties of Bases in Banach Spaces
Assume now that
n+ll
is not bounded. Then there exists a
subsequence /?,n =
\\yj
(и =1,2,...) such that /?,„-> со and that
Y X; converges, say to zeE (we have used the fact that
sup ||xj<co). Then /;(z) = 0 for 1Фхп and j\n{z) = ——¦ (n=l,2,...).
Therefore
> whence, by D.12) and by Y fi(z)xi^z>
9i(z)yi fails to converge, and thus {у„} is not a basis of ?.
Conversely, assume now that
Ш
is bounded. Since {х„} is
a basis of ? with inf ||xJ>0, whe have /„(x)->0(xe?), whence, by
1 $n<co
n n
D.12) and Y /i(x)xi^xs we obtain ? д((х)>';—>x (xe?). Thus we can
i = l i=l
apply the implication 2° => 1° of theorem 4.1 to the biorthogonal
system (yn,gn).
In view of the next application of theorem 4.1 it is convenient to give
Definition 4.7. Let {х„} be a basis of a Banach space ?, with
inf ||х„||>0. We shall call block-perturbation of {х„} any sequence
1 ^n< со
{zk} с ? of the form
U for кфрп 2
,n + yn for k = pn
where
D.13)
D.14)
and where {т„}, {р„} are increasing sequences of positive integeres
such that mo = 0, mn_1 + l^pn^mn (n=l,2,...).
Proposition 4.4. Let {х„} be a basis of a Banach space E, with
inf ||х„||>0. Then every block perturbation {zk} of {х„} is a basis
1 ^n<co
ofE.
Proof. Let \zk) be of the form D.13) with {yn} satisfying D.14). Then
{zk} admits a biorthogonal sequence {hn} с ?* given by
-xkfPn for кфр„, mn_! + l<
for k = pn
5. Some characterizations of regular ?-complete biorthogonal systems. Multipliers 31
where {/„} is the a. s. с f. to the basis {х„}. Hence, for all xe?,
i i
Y fj{x)Xj-fpn(x) Y aiXi f°r mn-l +
= 1 i=mn_ i + 1
k=i
1
z.
J = l
for
Since {х„} is a basis of ?, there exists, by the implication 1°=j>4°
of theorem 4.1, a constant C^ 1 such that
(xe?,n=l,2,...),
whence
i
I
Z «ix;
= i+i
Since the basis {х„} satisfies inf ||xn|| >0, by § 3, lemma 3.1 a) we
1 ^n<co
also have lim/Pn(x) = 0 for all xe?. Consequently, for every e>0
and xe? there exists an integer N(e,x)>0 such that
i
z
- Y fj(x)xj
<?
Hence x= Z K(x)zk f°r au xe?, and thus, by virtue of the im-
fc = i
plication 2°^> Г of theorem 4.1, {zk} is a basis of ?, which completes
the proof.
§ 5. Some characterizations of regular ii-complete
biorthogonal systems. Multipliers
Let ? be a Banach spece and (х„,/„) ({х„} с ?, {/„} с ?*) an ?-com-
plete biorthogonal system. We shall use the notation
||x|||= sup
Y fi(x
(«?),
where |||x||| = oo is also possible. Let
{xe?| lim sn(x) exists}
E.1)
E.2)
E.3)

32
I. The Basis Problem. Some Properties of Bases in Banach Spaces
S2 = {xeE sup ||sn(x)|| < 00} =
{xe?| lim ||sn(x)|| exists and <oo}
lim |J sn(x) || does not exist},
и {xe?
E.4)
E.5)
where sn are the associated partial sum operators (see § 4, definition 4.4).
Then we have the inclusions
jp .— jp .— jp .— jp /c z:\
0 л i 0 . i 0 2 ' 0 3 . I »?.U I
If {х„} is complete in ?, the sets $Q, Su S2, ёъ are dense in ?, and
if {х„} is a basis of ?, we have S0 = S^=S2=Sz = E.
Proposition 5.1. Let E be a Banach space and (х„,/„) ({х„} с ?,
{/„} a E*) a biorthogonal system. We have S0 = S1 if and only if the
sequence {/„} cz E* is total1 on E.
Proof. Assume that S0 = Sl and let хе?,/„(х) = 0(и=1,2,...). Then
sn(x) = 0(« = l,2,...), whence, by ^0=^i. it follows that x = 0.
Conversely, assume that {/„} is total on ?, and let xe^. Then,
by biorthogonality and the continuity of the /„, we have
fix - lim ? f{x)xA = fj(x)-fj(x) = 0 0'= 1,2,...),
L n~"*> i = l J
n
whence, since {/„} is total on ?, x— lim ? /;(x)x, = 0, i.e. xeSo.
Proposition 5.2. Let E be a Banach space and (х„,/„) ({xn}czE,
{/„} cz E*) an irregular E-complete biorthogonal system. Then
a) The set S2 is of the first category.
b) The set E\&3 = {x e ? I lim || sn(x) || = 00} is of the first category.
c) If A is a subset of E, such that every xeA is the limit of a sequence
{yn} с Е satisfying sup |||yj| < 00, then A is of the first category.
Proof, a) Since (х„,/„) is irregular, for every positive integer m the
set \xeE sup j|sn(x)||^m> is nowhere dense (by the implication
4°=> 1° of theorem 4.1) and closed, whence the set
QO С
^2= U )XeE SUP llSn(X)H^W
is of the first category.
1 In the sense of Banach [10], i.e. {хеЕ |/„(х)=0 (и = 1,2,...)} = {0}.
5. Some characterizations of regular incomplete biorthogonal systems. Multipliers 33
b) Define a sequence of functions {/?„} on ^ou(?\^3) by
Ш = t4
Then each /?„ is continuous on 0*ou(?\^3), 0 ^ /?„(х) < 1
), and we have lim р„(х) = Р(х) (xeSou(E\S3)), where
ни for xsSo'
1 for хеЕ\,?ъ.
E.7)
Assume now that the set ?\<?3 is of the second category. Then, by
a well known extension of a theorem of R. Baire1, the function /? must
have at least one point of continuity in <?0 и (?\<?3), say x0.
Since (х„,/„) is ?-complete, So is dense in ?. Let {у„} be a sequence
in <?0 such that lim yn = x0. Then, taking into account E.7),
P(xo)=limP(yn)=\im
E.8)
On the other hand, ?\<?3 is dense in ?, since for хеЕ\,$ъ and
yeS0 we have x + ye?\<?3 and since So is dense in ?. Let {zn} be a
sequence in ?\<?3 such that limzn = x0. Then, taking into account E.7),
which contradicts E.8).
c) Define a sequence of functions {}>„} on ? by
..,., III*-WIII
(хе?,и = 1,2,...).
Then each yn is continuous on ?,
we have lim yn(x) = y(x), where
<
x
for xe<?2,
y(x) = i l+|||x||
1 for xbE\S2.
, and
E.9)
Since by a) above the set E\S2 is dense in ?, it follows that we
have y(x)=l for every point of continuity x of 7. Consequently, if x
1 We recall this result: Let {[!„} be a sequence of continuous real functions on
a metric space S, such that |/?„(х)|<М (xe<?, n= 1,2,...) and that lim/(n(x)
= p(x) (xeS); then the set of all points of discontinuity of (S is of the first category.
3 Singer, Bases in Banach Spaces I

34
I. The Basis Problem. Some Properties of Bases in Banach Spaces
is a point of continuity of у and {zn} a sequence in E such that lim zn = x,
then г / ч , ч i
lim y(zn) = y(x) = 1,
П-*ОО
whence, by E.9) and E.4), sup |||zj| = oo. Hence A is contained in
1 in< 00
the set of points of discontinuity of y, which is of the first category by
virtue of the extension of the theorem of Baire, used in the proof of b)
above. This completes the proof.
Proposition 5.2 a), b) says, in other words, that if (х„,/„) is an
irregular biorthogonal system, then almost everywhere on E (in the sense
of Baire category) sup ||sH(x)|| = oo and almost everywhere lim ||sn(x)||
does not exist. '*"<=c
Remark 5.1. In the proof of b) we did not use the hypothesis that
(хи>/и) is irregular. Actually, for regular (х„,/„) b) is trivially valid,
since then E\$3 is void. However, in a) and c) it is essential to assume
that (xn,fn) is irregular. Let us also mention the following short proof
of b); For every positive integer n the set
&n= {xeE\\\Sll+k(x)\\>\\x\\ + l (k = 0,1,2,...)}
is closed and nowhere dense (since So is dense in ?), whence, by
E\*3 c: Q J*,,
n = l
?\/3 is of the first category, which completes the proof.
Theorem 5.1. Let Ebea Banach space and (х„, /„) ({х„} с Е, {/„} с ?*)
an E-complete biorthogonal system. Then the following statements are
equivalent:
1°- (xn,fn) is regular.
до (р ев
For every xeE let {р„(х)} be a sequence of finite linear combinations
such that1 limpn(x) = x. Then the above statements are equivalent to
the following:
5°. sup \\\Pn(x)\\\ < oo (xeE).
1 ^П< 00
1 E.g. we can choose pn(x) to be an element of P(n) = [x,]"=1 for which
inf ||x — p\\ is attained. It is well known that there exists at least one such ele-
element (see e.g. [246], Ch. I, §2, corollary 2.2).
5. Some characierizations of regular ?-complete biorthogonal systems. Multipliers 35
Proof. The equivalence Г о 2° is nothing else but the equivalence
ГоЗ° of theorem 4.1. The implications Г=>3° and Г=>4° being
immediate (since Г implies S1=S2 = Si = E), it remains to prove that
3°=>1°, 4°=>1° and Го5°.
3°=>1°. Assume that (х„,/„) is irregular. Then, by the implication
4° => Г of § 4, theorem 4.1, we can successively construct an increasing
sequence of positive integers {т„} and a sequence {yn} cz E with the
following properties;
mn
Уп= I fiiyjx, (n=l,2,...;mo = O),
max
1
I MyJxt
i = mn~ i + 1
We claim that the element
*=I
E.П)
E-12)
E.13)
E.14)
satisfies xeS2\Sl, whence S^?=S2, which proves that 3° => Г. In
fact, by E.14), E.11) and biorthogonality we have, for mn_l +1
n-l 1 i к
)=? — y,+ — У
/W Л// ^^
j — 1 j и i — mn -l+l
whence, by E.13) and E.12),
sup
sup X — ВД
l
sup
p X
i.e. xeS2. On the other hand, for suitable integers kn with ти_!
</„ (и = 1,2,...; то = О) we have
kn
= М„
whence, by E.14), E.11) and biorthogonality,
l
= 1 (и=1,2,...),
which proves that x$Sx. Thus 3° => 1°.

36
I. The Basis Problem. Some Properties of Bases in Banach Spaces
4° => 1°. Assume that (*„,/„) is irregular. Then, by proposition
5.2 a) and b), the set S2 ls °f the first category, while ёъ is of the second
category. Hence ё2фёъ, which proves that 4° => 1°.
Г => 5°. If (*„,/„) is regular, then there exists, by § 3, proposition 3.2,
a constant C^ 1 such that
(xeE, n=
which, together with lim р„(х) = х (xeE), implies 5°.
5° => 1°. If (х„,/„) is irregular, then, by virtue of proposition 5.2 c),
we cannot have 5°. This completes the proof of theorem 5.1.
Remark 5.2. One can also give a direct proof of the implication
4° => Г of theorem 5.1, similar ot the above proof of the implication
3° => Г. In fact, if (*„,/„) is irregular, we can construct an increasing
sequence of positive integers {mn} and a sequence {у„} с Е satisfying
E.11), E.12) and
max
mn-i+ lik
i = mn - i + 1
ее
whence for the element x= ? У;е^ we shall have xe<?3\<?2 (because
<1 (n = l,2,...), whence xe<?3, but for kn such that
I
we have ||stn(x)|| =
^ и-1 (n= 1,2,...), whence jc?<?2).
Remark 53. From theorem 5.1 and the inclusions E.6) one can
derive new conditions equivalent to Г (e.g. <go = g2, etc.).
In the preceding we have considered Sl,S2 only as subsets of ?.
However, they are also linear subspaces of ?. Let us introduce on these
linear spaces the norm |||x||| defined by E.1), and denote by EUE2 the
normed linear spaces obtained in this way.
Proposition 5.3. Let E be a Banach space, (х„,/„) ({х„}<=?, {/„}<=?*)
a biorthogonal system, and E1,E2 the normed linear spaces defined above.
Then
a) ?j is a Banach space. Consequently, Et is a closed linear subspace
of E2 and if, Е^фЕ2, the sequence {х„} is not complete in E2.
b) {х„} is a basis of E^
c) // {/„} is complete in ?*, then E2 is a Banach space.
d) // {х„} is complete in E and {/„} is complete in ?*, and if the
sequence {ф„}<^Е* defined by
5. Some characterizations of regular incomplete biorthogonal systems. Multipliers 37
(хе?2,«=1,2,...)
E.15)
is complete in ?f, then {х„} is a basis of E.
e) // {х„} is complete in E and {/„} is total on E, and ifEi is reflexive,
then {х„} is a basis of E.
Proof, a) Since (х„,/„) is biorthogonal, all хпф0 (п= 1,2,...), whence,
by §3, proposition 3.1, the space A1 introduced in proposition 3.1 is a
Banach space. Since the mapping
{<Xn}-> X ZiXi
i= 1
is a linear isometry of At onto Eu it follows that ?t is complete,
b) Let
Since by E.1)
(хе?1,и=1,2,...).
-^ll/.WxJK^HMI (xeE,),
E.16)
we have ф„еЕ\ (и = 1,2,...). Since (х„,/„) is a biorthogonal system, it
follows that (хп,ф„) is a biorthogonal system. Taking into account
E.1), E.16) and E.3), we have
= sup
= sup
I fj(x)xj
¦0 (xe?,),
whence, by the implication 2° => Г of §4, theorem 4.1, {х„} is a basis
c) Let us first prove that if {/„} is complete in ?*, then
ilxKHMH (xe?2). E.17)
Let xe?2. Since sup
< 00 and since by biortho-
- - -- - hi — l и
gonality lim f, У ft(x)x, =/,(x) (/=1,2,...), from the completeness of
"^ L = i J

38
I. The Basis Problem. Some Properties of Bases in Banach Spaces
{/„} in ?* it follows that the sequence <! Z /{{х)хЛ is convergent
to x for the weak topology a(E,E*). Consequently, we have E.17).
Now let {yk} be a Cauchy sequence in E2. Then by E.17) and the
completeness of E, there exists an xeE such that lim>t = x. Since
YU
i= 1
= lim
k-* oo
sup
= 1,2,...)
(which is < oo because {yk} is a Cauchy sequence in E2), we have
xeE2. From the inequalities
<e
(k,m> N (в); n = 1,2,...)
we obtain, for wi—>oo,
л
z
<; я (k>N(B);n=l,2,...),
whence lim \\\yk — x||| = 0. Thus ?2is complete.
k-* oo
d) Since {/„} is complete in ?*, we have E.17), whence i,
n
(и=1,2,...). Let xe?2. Since sup Z/i(x)x" "^ °° an<^ smce by
Г" 1 Г" 
biorthogonality lim i/> Z /i W xi = lim // Z /iW xi =//(x) = "/Ых)
»-*<*> b=i J n"=° L=i J
(/=1,2,...), from the completeness of {(/>„} in ?| it follows that the
Г » "I
sequence < Z ii(x)xi fls convergent to x for the weak topology o-(?2,?f).
u=i 3
Thus x belongs1 to the (strongly) closed linear subspace of ?2 spanned by
the sequence <j Z /i(x)x;f> whence, since Z ii(x)x;e?i ("=1,2,...),
it follows that xeEl. Consequently, ?2 = ?1; whence, by the com-
completeness of {х„} in ? and by the implication 3°=>1° of theorem 5.1,
{х„} is a basis of ?.
e) Since E1 is reflexive and since by b) above {х„} is a basis of E1,
the sequence {^„}c?f defined by E.16) is complete in ?? (m fact,
00
otherwise there would exist2 an y= Z ^1(у)х(е?Д{0} such that
i=l
1 See e.g. [10], p. 134, theorem 2.
2 See e.g. [10], p. 58, theorem 7.
5. Some characterizations of regular incomplete biorthogonal systems. Multipliers 39
<^n(y) = 0 (и = 1,2,...), which is impossible). Let xe?2. Then, since
sup
1«П<00
< 00
and since by biorthogonality lim ф\ ? /;(х)х,
n^co |_j = 1 J
= lim/J Y.fi(x)xi =//(x) (/=1,2,...), from the completeness of {ф„}
""^ b=i J r- „ -j
in ?^ it follows that the limit lim ф У /;(х)х; exists for every феЕ*.
b-°° L=i J
Hence, since ?x is reflexive, there exists an element x'eEl such that
Итф\ У/1(х)хА = ф{х'){феЕ^). Thus, in particular, /(x)
/iMXi | = ^(x') (/=1,2,-•), whence, by E.16), /.(x) =//x')
i J
(/=1,2,...). Since {/„} is total on ?, it follows that x = x', whence
xeE1. Consequently, E2 = E1, whence, by the completeness of {xn}
in ? and by the implication 3°=>1° of theorem 5.1, {х„} is a basis of ?.
This completes the proof.
Remark 5.4. One has the following result, the proof of which is similar
to that of § 3, proposition 3.1:
Let {х„} be a sequence in a Banach space E, such that хпф0(п = 1,2,...),
and let A2 be the linear space of sequences ofscalars
sup
< oo
E.18)
endowed with the norm \\ {а„} || = sup
Z a,*.
. Then Ay is a Banach
space. (Moreover, we shall see in § 12, theorem 12.5 c), that A2 is
isomorphic to a certain conjugate Banach space if [л„| is a basis1 of ?).
However, one cannot use this result to give a proof of c) similar to the
above proof of a), since if (xn,fn) is a biorthogonal system such that
[/„] = ?*, then, in general, the mapping x—>{/„(х)} is an isometry of ?2
1 Actually, this result remains valid for any sequence {х„} с ? with
^0 (и=1,2,...). Indeed, by §8, proposition 8.1a), the unit vectors en = {<5nj}f=1
(n=l,2,...) are a basis of Ay = -J{an}c=X
s II
= sup
^ а,х( converges \ and by
i=i J
we have
sup
<ooj> and ||{а„}||=] sup
sup
i-l
< ОС > =
= sup
lS=n<00

40
I. The Basis Problem. Some Properties of Bases in Banach Spaces
onto a proper closed linear subspace of A2, i.e. for an {oin}eA2 there
does not exist, in general, an xe? such that /„(х) = а„ (и=1,2,...). In
fact, e.g. for ? = c0, {xn} = the natural basis of c0 and {/„} = the a.s.c.f.,
we have ?2 = ? = c0 and A2 = m.
We shall now characterize regular biorthogonal systems among
?-complete total biorthogonal systems in terms of properties of the
set of multipliers.
Definition 5.1. Let ? be a Banach space and (х„,/„)({х„}<=?, {/„}<=?*)
a biorthogonal system such that {/„} is total on ?. A sequence of scalars
{у„} is called a multiplier of an element xeE if there exists an element
х{Уп)еЕ such that x|7n)~ ? ykfk(x)xk, i.e., such that1
Jk(xlyJ = ykfk(x) (k=l,2,...); E.19)
since {/„} is total on ?, this х{Уп] is uniquely determined by x. The set
of all multipliers {yn} of x is denoted by M(x, (х„,/„)). A sequence of
scalars {у„} is called a multiplier of E if it is a multiplier of each xe?.
The set of all multipliers {у„} of ? is denoted by M(?, (х„,/„)). Thus,
M(?, (х„, /„)) =С\Щх, (х„, /„)). E.20)
Theorem 5.2. Let Ebea Banach space and (х„, /„) ({х„} с ?, {/„} с ?*)
аи E-complete biorthogonal system such that {/„} ;s total on E. The
following statements are equivalent:
1°- (*„,/„) is regular.
2°. M(?, (х„, /„)) гэ Ь и, t/ie set of all sequences of scalars {yn} such that
;= l
3°. M(?, (х„,/„)) contains every non-increasing sequence {у„} (i.e.,
such that yi^y2^ ¦¦¦) tending to zero.
Proof 1°=>2°. Assume that (х„,/„) is regular and let {yn}ebv be
arbitrary. Then, for any x e ? and и = 1,2,...
i- 1
fj(x)Xj.
Observe that if (х„,/„) is regular, then, obviously, хш=
5. Some characterizations of regular incomplete biorthogonal systems. Multipliers 41
Since we have, by § 4, theorem 4.1,
II i
I fj(x)x}
it follows that
i = 1
E.21)
it
Since lim ^ yif(xk)xj = ykxk (fe = 1,2,...) and since [х„] = ?, by
E.21) we infer that Y, У./
i= 1
for each xe? the element
converges for all xe?. Furthermore,
E.22)
obiously satisfies E.19). Thus, {у„}еМ(Е, (х„,/„)).
2° => Г. Assume that we have 2°. Then for each xe? we can define
a mapping ux:bv-*E, by
7kfk(x)xk ({yn}ebv).
E.23)
Since for xe?, {y\,l)}, {у(„2)}, {y,}eM(?,(x,,/J) and a = scalar,
we have
/*({*,> y (ах{Уп)) (к= 1,2,...)
and since {/„} is total on ?, it follows that
*,#,, + „,>„ = x.^+x,,,,,, (хеЕЛ^^.Ы^еМ^Дх,,,/,))), E.24)
ха(Уп| = ах(Уп) (хе?, {yn}eM(?,(xn,/n)),a = scalar), E.25)
whence each mapping ux is linear. We shall now prove that each ux
is closed (we recall that with the usual vector operations and with the
norm oo
-Нт|ув| ({yn}ebv), E.26)

42
I. The Basis Problem. Some Properties of Bases in Banach Spaces
feu is a Banach space). Let {y<,m)}^=1 (m=l,2,...) be a sequence in bv,
converging to an element {yn}ebv and such that lim их{{у(™у}?=1)
= zeE. Then, since fkeE*, we have m^co
as
On the other hand, since
E.27)
as
we have lim y(iT} = yk (k=l,2,...). Consequently,
Ш-* 00
уРЛ(*)-*У»Л(х) as m^o) (fc = 1,2,...),
whence, by E.27), we get
and therefore z = x{yn] = ux({yn}), which proves that wx is closed. Hence,
by the closed graph theorem, each ux is continuous on bv.
Now, if e, denotes the i-th unit vector {dni}™=1 = {0,...,0,1,0,...}
in bv, we have 'ТП~/
k\J*x\ei)i — Jk\xei) — °kiJk\x) —
whence, since {/„} is total on ?,
Consequently, since ||е1[|ь„ =
m
Z fiix)xt
rk(fi(x)xi) (xe?;i,fc=l,2,...),
(xe?,i = l,2,...). E.28)
= 2 (m = 2,3,...), we get
^2\\ux\\ (xe?,m=l,2,...),
whence, by §4, theorem 4.1 (implication 3°=>Г), (х„,/„) is regular.
The implication 2°=>3° is obvious, since every non-increasing se-
sequence tending to 0 belongs to bv.
3°=>2°. Every {yn}ebv can be written as the sum \yn- lim yk\
+ \ limyj, where the first sequence is in bv and tends to 0 and the
second sequence is constant. Furthermore, every real sequence {yjefer
tending to 0 can be written as the difference ^Z(^~7i+i
~ ) Z ("У* — Ti + i)~f> (where a,+ =a,-, a,r=0 if a,-^0 and а,^=0,
«Г = — ol{ if af^0) of two non-increasing sequences tending to 0. On
the other hand, by E.24) and E.25), М(?,(х„,/„)) is a linear space.
Hence, by 2° and since the constant sequences are in М(?,(х„,/„)), it
5. Some characterizations of regular incomplete biorthogonal systems. Multipliers 43
follows that every real {yn}ebv, whence also every {yn}ebv, is in
M(?, (*„,/„)), which completes the proof of theorem 5.2.
In the above proof we have seen that for any biorthogonal system
(*„,/„) such that {/„} is total on ?, the set of all multipliers M(?, (*„,/„)),
with the natural vector operations, is a linear space. Let us now give
some more properties of M(?, (*„,/„)).
Proposition 5.4. Let E be a Banach space and (*„,/„) ({*„} <= E,
{/„} a E*) a biorthogonal system such that {/„} is total on E. Then
a) With the natural vector operations, coordinatewise multiplication
and the norm
\{Уп}\\= sup
xeE
}.„} еМ(?,(х„,/„))),
E.29)
M(?,(*„,/„)) is a commutative Banach algebra, containing the identity
{1,1,1,...}.
b) The mapping {yn}->Vi-,n}, where
(xeE),
E.30)
is an isometrical algebraic isomorphism of М(?,(х„,/„)) into L(E,E)
(the Banach algebra of all continuous linear mappings of E into E, with
\\u\\ = sup ||m(x)||), satisfying
XeE
(xe?,n=l,2,...),
E.31)
where е„ denotes the n-th unit vector {Sni}j°=1 in M(?,(*„,/„)). Further-
v{M = yj*J 0 = 1,2,..), E.32)
i.e., each x} is a proper vector of v{yn], corresponding to the proper
value yj.
c) We have M(E,(xn,fj) cm( = /co) and the inclusion mapping is
continuous, namely
sup Ы<||Ы||«(?,(*„,/„» ({7„}бМ(?,(х„,/„))). E.33)
i =? j < °o
Proof a) Let {y(nl)}, {у<,2)}еМ(?,(х„,/„)). Then for every xeE there
exists an element x{ym\?E such that
and for this xiyB)( there exists an element (xl7,B)|)|yoIe? such that
(k= 1,2,...)
whence {y^)yi^)}eM(E,(xn,fn)). Thus, with coordinatewise multiplica-
multiplication, М(?,(х„,/„)) is a linear algebra, which obviously is commutative

44 I. The Basis Problem. Some Properties of Bases in Banach Spaces
and contains the identity {1,1,1,...}. Note that, since {/„} is total on ?,
we also have the relations
,(х„,/п)). E.34)
Now, if || {у„}||=0, then x(yni = 0 (xeE, \\x\\ <1), whence
Ук/к(х) = fk(xlyj = 0 (xeE, \\x\\ s: 1, k= 1,2,...)
and hence, putting x = —~(k=l,2,...), we get yt = 0(fc = 1,2,...). This,
llxjtll
together with E.24) and E.25), proves that E.29) is a norm on
M{E,(xnJn)).
Now let {y!,m)K°=i(m = l,2,...) be a Cauchy sequence in М(?,(х„,/„)),
i. e., such that
\\{7ТУ}-Ш\\= sup ||x(jr,.-x!yA))||<e (m,/>N(e)). E.35)
xeE
Then for every fixed к and xeE with /k(x)#0, ||x|| =1,
JtW Ук Ук\ — \Ук ]кУХ)~Ук Jk\X)\ — \
whence the limits lim y|ra) = yk (fc=l,2,...) exist and for every xeE we
have
Укт)Ш^укШ as m^<xj (/c=l,2,...). E.36)
On the other hand, taking any xe? and putting
we see from E.35) that {ym} is a Cauchy sequence in ?, whence
lim ym = yeE exists and
Therefore, taking into account E.36), we obtain
whence {у„}еМ(Е,(х„,/„)) and у = х(Уп). Furthermore, since
lim x(y(m,,= lim ym = y = x, ,, from E.35) for /->oo we get
m-*oo " m-*oo "
whence lim {у<,га)} = {у„}, which proves that М(?,(х„,/„)) is a Banach
m —¦ oo
space for the norm E.29).
Finally, by E.29) and formula E.38) below we have
п^г11*(Л= (]&) <||Ы|| (хб?,х#0,{7„}бМ(?,(х„,/„))),
5. Some characterizations of regular ?-complete biorthogonal systems. Multipliers 45
whence, by E.34), we get
'" ^||{уП1ШЙ||
and consequently, by E.29),
which proves that М(?,(х„,/„)) is a Banach algebra.
b) For each {у„}еМ(?,(х„,/„)) the mapping v{ln] defined by E.30)
is in L(?,?). Indeed, by
.) (x,ye?;/c=l,2,...)
and since {/„} is total on ?, we have
and similarly
(ссх){Уп) = а.х{Уп) (xeE, a = scalar),
E.38)
whence v{yn} is linear. Furthermore, each v{ynj is closed, whence con-
conv{yn}
{yn} {j
tinuous, since if lim Zj = zeE, lim v{y }(zj) = yeE, then, by fkeE*
as
as
whence fk{y) = ykfk(z) (k =1,2,...), i.e., y = z{yn) = vlyn)(z), which proves
our assertion.
Now, by E.24), E.25), E.34), E.29), E.28), and E.23), the mapping
{yn}^>v[yn} is an isometrical algebraic isomorphism of М(?,(х„,/„))
into L(?,?), satisfying E.31). Finally, from the relations
fklv{yjxj)] = fk((Xj)tyJ = ykfk(Xj) = ykSkJ = /к(уЛ) (k= 1,2,...)
we infer E.32) (since {/„} is total on ?).
c) By E.30) and E.32) we have
(/ = 1,2,...),
x}\\) °!|x,||
whence, by E.29), we get
rj 11^11
\u\
0 = 1,2,...),
i. e., E.33), which completes the proof of proposition 5.4.

46 I. The Basis Problem. Some Properties of Bases in Banach Spaces
Let us also observe that by the isometry {yn}^>v[yn} and by E.31)
we have
IIej =KII = sup |/„(х)| |jxj = ||/J ||х„|| ^ 1 (n=l,2,...); E.39)
xeE
if (*„,/„) is regular, then, by E.39) and § 3, theorem 3.1, we also have
sup ||ej<oo.
1 ^ П < 00
Remark 5.5. Similarly to E.23), one can define, for each xe?, a
mapping ux:M(E,{xn,fn))-+E, by
"«(Ы) = »ш(х) = х{Уп) ({у„}еМ(?,(х„,/„))). E.40)
Then, by E.24) and E.25), each ux is linear and by
К(Ы)\ < ||»,J| 11*11 = II Wll И ({}'„} еМ(?,(х„,/„)))
each ux is continuous.
This remark and the above proof of theorem 5.2 suggest the fol-
following corollary of proposition 5.4:
Corollary 5.1. Let E be a Banach space and (х„,/„) ({х„} с Е, {/„} с ?*)
an E-complete biorthogonal system such that {/„} is total on E. The
following statements are equivalent:
1°- (*„,/„) <s regular.
2°. There exists a constant C^l such that
E.41)
where е„ is the n-th unit vector {<5BJ-}JLi in М(?,(х„,/„)).
3°. There exists a constant C^l such that
Z 7,e,-
({у„}еМ(?,(х„,/„)),и = 1,2,...). E.42)
Proof. Г»2°. Since by proposition 5.4 the mapping {
a linear isometry we have, taking also into account E.31),
IS
n
Z et
i= 1
=
V.
= sup
xeE
II
x\\ « 1
=
n
i= 1
n
i= 1
/i(x)Xj
= sup
xeE
11*11 «1
(n = 1,
n
Z VeXX)
i = 1
2,...),
E.43)
whence, by § 4, theorem 4.1, the equivalence 1°<j>2° follows.
5. Some characterizations of regular incomplete biorthogonal systems. Multipliers 47
Г =>3°. If we have Г, then, by § 4, theorem 4.1, there exists a constant
such that for any {у„}еМ(?,(х„,/„)) and n= 1,2,... we have
v.
= sup
E
= sup
Z y,-ii-Wxi
= sup
xeE
Z /i(^(yn))X.-
< С sup |
xeE
ll*INi
,()
Finally, the implication 3°=>2° is obvious (taking yl=y2=--- = l).
This completes the proof.
Corollary 5.2. Let {х„} be a basis of a Banach space E, with the
a.s.c.f {fn}cE*. Then
a) The unit vectors em={<5mi},^1 (m=l,2,...) form a basic sequence
in M(?, (х„,/„)).
b) We have
М(?,(х„,/„))= {y
sup
Z у<е-
<ooV. E.44)
Proof, a) Let ht denote the coordinate functionals on M(?, (х„,/„)), i. e.,
ЧЬn}) = 7,- ({7„}еМ(?,(х„,/„)), ( = 1,2,...).
Then each ht is linear and by E.42) we have
Z Vjej
i-l
Z Vjej
2C
e,-
and thus each ht is continuous on M(E, (х„,/„)), whence also on [е„].
Since obviously /г,(е;) = E0 (;J=1,2,...), by E.42) and §4, theorem 4.1
it follows that {е„} is a basic sequence in M(?, (х„,/„)).
b) The inclusion с in E.44) is an immediate consequence of E.42).
Conversely, let {у„} с К be such that sup
Then lsSra<c0
= C'<oo.
sup
xeE
II jcII s? 1
= sup
Z Vi »«.(*)
= sup
V- (X)
Z i1^.-

48
I. The Basis Problem. Some Properties of Bases in Banach Spaces
Hence, since lim ? yJj(xt)Xj = ykxt (fc=l,2,...) and since [х„]=?,
it follows that lim ? yJA^x— ? 7,/i(x)x, = xjyii) exists for all xe?.
Since this x(yn) obviously satisfies E.19), we have {у„}еМ(?, (х„,/„)),
which completes the proof of corollary 5.2.
In general, {е„} need not be a basis of M(?, (х„,/„)), as shown by
Example 5.1. Let ? = c0 and let {х„} be the unit vector basis of ?.
Then M(?, (х„,/„)) = т with the norm ||{yn}||= sup sup |у„?„|
= sup |yj (i.e., with its usual norm), which is a non-separable space
1 ^ П < 00
and hence it has no basis.
Now we shall drop the hypothesis that {/„} is total on ? and give
another characterization of regular ?-complete biorthogonal systems,
suggested by the above results.
Theorem 5.3. Let E be a Banach space and (х„, /„)({*„} с ?, {/„} с ?*)
an E-complete biorthogonal system. Then (xn,fn) is regular if and only if
there exists a continuous linear mapping {?„}"* и{у„} of bv into L(?,?),
such that
«U*) =/»(*)*« (xeE,n=l,2,--), E-45)
where е„ is the n-th unit vector {Sni}°°=i in bv. Moreover, in this case for
every {yn}ebv such that Нту„ = 0 the mapping v{ln] is compact.
Proof. Assume that (х„,/„) is regular. Then {/„} is total on ? and
hence, by proposition 5.4, there exists a continuous linear mapping
{yK}-*v(yn) of М(?,(х„,/„)) into L(?,?) (with и(Уп) defined by E.30)),
satisfying E.45). Furthermore, by theorem 5.2, we have bv^M(E,(xn,fn))
and by E.21) the inclusion mapping feu->M(?, (х„,/„)) is continuous,
whence the restriction of the mapping {yn}~>vl7n] to feu is continuous.
Conversely, if the condition is satisfied, then
n n
V I VI — > 11 I V1 ^з > / I V1 V = г /v I -yiz. h y] — I / I
whence, since
(и = 1,2,...) and since the mapping {yn}-*v[yn)
of fe v into L(?, ?) is continuous, say of norm C, we get
SJ = Г.
and therefore, by §4, theorem 4.1, (х„,/„) is regular.
5. Some characterizations of regular incomplete biorthogonal systems. Multipliers 49
Finally, assume again that the condition is satisfied and let {yn}ebv
such that lim у„ = 0. Then, with С as above,
m
; = i
11 Г
/I Lei
=
v{yn}~
00
=
С У
=
hi-
'i + l
V
^0
% ¦'¦¦'¦
m
1 ) V^
i = l
as
whence, since by E.45) ? ytvei is of finite rank, we infer that vbn]
i= 1
is compact, which completes the proof.
From proposition 5.4 and theorem 5.3 we obtain
Corollary 5.3. A Banach space E has a basis if and only if there exists
a continuous multiplicative linear mapping {yn}-+v{yn) ofbv into L(?,?),
dimven(E)=l (и = 1,2,...),
such that
where е„ is the n-th unit vector {$„}?=! inbv.
E.46)
E.47)
Proof. The necessity part is an immediate consequence of proposition
5.4 and theorem 5.3.
Conversely, assume that the condition of corollary 5.2 is satisfied
and by E.46) let х„е?, х„#0 be such that
ven(E) = {axn | a = scalar} (n = 1,2,...).
E.48)
Then, by E.47), [х„] = ?. Furthermore, by veneL(E,E) and E.48)
there exist continuous linear functionals /„g?* such that
ven(x)=fn(x)xa (xe?, и = 1,2,...).
Since eiej = 5ijei(i,j=l,2,...) in fe и and since the mapping {}>„}
is multiplicative and linear, we have
Putting j = i, we get
(t,j=\,2,-¦¦)¦
E.49)
E-50)
(xe?, (=1,2,...),
whence 1>е.(х,.) = х; (г = 1,2,...), whence, by E.49), ^(xjx^x,- (;=1,2,...),
and hence _/i(x,-)= 1 (i= 1,2,...). Therefore we also have, for )Фи^{х})х{
= ve.(xj) = ve.(vej(xj)) = 6ijve.(xJ) = 0, whence /,(.4^ = 0. Thus (х„,/„) is an
?-complete biorthogonal system satisfying the condition of theorem 5.3,
and hence by this theorem, {xn} is a basis of ?. This completes the proof
of corollary 5.2.
4 Singer, Bases in Banach Spaces I

50
I. The Basis Problem. Some Properties of Bases in Banach Spaces
§ 6. Some types of linear independence of sequences
In the present section we shall consider three fundamental types of
linear independence of sequences, which will be frequently used
throughout the sequel. Later we shall also consider some other types of
linear independence (see Ch. II, § 11 and § 16 and Vol. II, Ch. III).
Definition 6.1. A sequence {*„) in a Banach space ? is said to be
a) finitely linearly independent, if every finite subsequence of {х„}
is linearly independent;
b) co-linearly independent, if
{а„}<=К,
,. = 0, imply а, = 0 (i=l,2,...);
F.1)
c) minimal, if
^^[^ь---,х„-,,х„+1,...] (и = 1,2,...;хо = 0). F.2)
Obviously, every minimal sequence is to-linearly independent and
every to-linearly independent sequence is finitely linearly independent.
For finite sequences the converse statements are also valid, but for
infinite sequences this is no longer true, as shown by the following
examples:
Example 6.1. a) Let ? = C([O,1]) and let
xn(t) = t
я-1
(ге[0,1];и = 1,2,...).
F.3)
Then {х„} is complete in E and си-linearly independent, but not2
minimal.
b) More generally, one can obtain examples of complete to-linearly
independent but not minimal sequences as follows. Let {х„} be an
to-linearly independent complete sequence in a Banach space ?, which
is not a basis of E (see e. g. § 4, example 4.1) and let x be an element
of E which does not admit an expansion of the form x =
the sequence {у„} с Е defined by i
a,x,
Then
F.4)
is complete in E and to-linearly independent, but not minimal.
1 We recall that by [X[,..., х„ _ j, х„ +1,...] we denote the (closed linear) sub-
space of E spanned by the elements x1,...,xn-1,xn+l,...
2 E.g. by virtue of the theorem of Miintz (see [174], Ch. Ill, §3, theorem 2).
6. Some types of linear independence of sequences
51
Example 6.2. Let {х„} be a basis of a Banach space E, with the
a. s. с f. {/„} and let x be an element of ? such that /„(x)#0 (n= 1,2,...),
e g. one can take oo i
Z F5)
Then the sequence {yn} <= E defined by F.4) is complete in E and
finitely linearly independent, but not cu-linearly independent.
Obviously, every basis of a Banach space ? is a minimal sequence.
The converse is not true, as shown by § 4, example 4.1. Furthermore,
a sequence {х„} с ? is a basis of ? if and only if {х„} is to-linearly
independent but no supersequence of {х„} is to-linearly independent.
Let us recall now a lemma from the general theory of normed linear
spaces, which will be useful in the sequel. For an arbitrary Banach
space В we shall use the notations
x||<l}, F.6)
= {x'eB
= 1}.
F.7)
Lemma 6.1. Let F,G be two closed linear subspaces of a normed
linear space E. Then
a) We have
dist (<rF, aG) ^ dist {aF, G)^\ dist (aF, aG). F.8)
b) // Gi is a closed linear subspace of G and x#0 an element of
G such that1 G = [x] + Gu then
1 / x \
dist (aF, (TGl) ^ dist {<rF, oG) ^ - dist (aF, aGi) dist I —, F + G^ I. F.9)
Proof, a) Since aG^G, the first inequality of F.8) is obvious. In
order to prove the second one, we shall first prove that
dist (SF,ffG)^3f dist {aF,aG). F.10)
For an arbitrary e>0 choose yeSF and z'eaG such that
||3' + z'|| <dist(SF,rrG) + fi. F.11)
If 1-Tdist(ov,oy.)^ Ilyll, then we have
If l--2-dist((TF,(TG)<||y||, then from the inequalities
у
-
у +Z'
Ilyll
1 For A,BczE we use the notation
={a + b\aeA,beB}.

52
I. The Basis Problem. Some Properties of Bases in Banach Spaces
У
у-
У
+ z
= 1-IMI <-dist(fff,GG),
dist (ay, aG),
it follows that
|| у + z || > dist [aF, aG) - \ dist {aF, <xG) = \ dist (aF, aG).
Thus in all cases we have, taking into account F.11),
dist (SF, aG) > \ dist (aF, aG) - в,
which, since e>0 is arbitrary, proves F.10). By changing the roles of F
and G in F.10), we also have
dist (oF, SG) ^ \ dist{aF, aG). F.12)
Now, take arbitrary y'eaF and zeG. If ||z||^l, we have, taking
into account F.10),
llzll llzll
dist (SF, aG) ^ - dist [aF, aG).
If ||z||<l, we have, by F.12),
|| y' + z || Js dist (o>, SG) ^ i dist (o>, crG).
Thus, in all cases, we have
which proves the second inequality of F.8).
b) Since (jGl <=. <jg, the first inequality of F.9) is obvious. In order
to prove the second one, take arbitrary y'eo>, z'e<rG. Since G = [x] + G1
x
and x^O, we have z' = / \-zt for a suitable scalar л and a suit-
suitable z1eG1.
If |1| > jdist(o>,(TGl),we have
x
+ -
Ixll Ul
1 I X
- dist(<tf,<tGi) dist f
If |Я|<jdist(t7f,t7G ), we have
y' + ^H- A-
Now, by the inequalities
lly' + ZiH-
F.13)
zi ~]
6. Some types of linear independence of sequences
53
we have
||y' + z,|| ^ dist(Gf,GGi) — jdist(o>,(TGl) = f dist((TF,crGl).
x
Hence, taking into account F.13) and 1 =
we obtain
2 ,.
3
¦¦ - dist(fff,GGl) - -dist(o-F,ffGl)
1 1 / X
= -dist((jf,GGl) ^-dist((jf,GGi)dist ——,
Thus, in all cases we have
1 / x
dist((jf,GG) ^-dist(o-f,GGl) dist f
which completes the proof of lemma 6.1.
Let us give now some characterizations of minimal sequences. We
shall use, for a sequence {х„} <= E, the notations
Р(п)=[хь...,х„] (и=1,2,...), F.14)
Р("»=[хп + 1,х„ + 2,...] (и=1,2,...), F.15)
i=l,2,...), F.16)
ч=1,2,...), F.17)
«7(И)={Х6Р(И)|||Х|| = \} = <Трм
аи.. ,апеК;п=\,2,...\= (J Рм,
F.18)
О for /c = 0
\ а;х; for
а,х; for
а,.х,.еР , F.19)
= 0,1,2,...). F.20)
Theorem 6.1. For a sequence {х„} in a Banach space E the following
statements are equivalent:

54
I. The Basis Problem. Some Properties of Bases in Banach Spaces
1°. \xn\ is minimal.
2°. There exists a sequence {/„} <= E* such that (х„,/„) is а Ы-
orthogonal system.
3°. There exists a sequence of constants dn>0 (n= 1,2,...) such that
we have
E l«
,x,-
F.21)
for all finite sequences of scalars a1,a2,...,an.
4°. The relation lim ?а<п)Х; = 0, where a^eK (i= 1,2,...,т„;
и=1,2,...), implies lima!n) = 0 (i= 1,2,...).
n-*oc
5°. // lim ? а!п)х( exists, t/ien so do the limits lim а<п) (г = 1,2,...).
n->oc j=1 n->co
//а// х„^0 (и = 1,2,...I, t/ierc t/ie abore statements are equivalent
to the following statements:
6°. VFe have2
Р=[х1,х2,...]=Р(„)©Р("»
F.22)
7°. There exists a sequence of endomorphisms3 {un}czL(E,E) such
that
ы„(х) = х (хбР(п);и=1,2,...),
ы„(х) = 0 (хбР(п);п=1,2,...).
F.23)
F.24)
8°. For each positive integer n there exists a constant Cn, 1 ^ С„< со,
such that we have
I««x,
E «,*«
F.25)
for all positive integers m and all a1,oi2,.-.,ocn + meK (w=l,2,...).
9°. We have
dist(<j(n),P(n))>0 (и=1,2,...). F.26)
10°. We have
п)Уп))>0 (и = 1,2,...), F-27)
1 Of course, this condition is satisfied whenever we have 1°.
2 I.e. P = Pln) + Pln) and P^nP1"^^} (n=l,2,...).
3 I. e. continuous linear mappings of ? into E. For Banach spaces E, F we
denote by L(?, F) the Banach space of all continuous linear mappings of E into
F endowed with the usual norm ||«||= sup
xeE
6. Some types of linear independence of sequences
55
11 °. For each positive integer n there exists a constant С'„, 1 ^ С'„ < со,
with the following property: for every poeP(n) there exists an feE*
such that
f(Po) = \\Poh F.28)
/(y) = 0 (yeP(n)), F.29)
KII/KQ. F.30)
12°. We have
13°. We have
sup ||Sn(p)|| < со (и=1,2,...).
sup 1|К„(р)|| < со (и=1,2,...).
F.31)
F.32)
Proof. The equivalence V'o2" is an immediate consequence of a
well known corollary of the Hahn-Banach theorem1.
2°=>3°. Let
*.=г4л! (и=1>2'-)- F33)
Then, for all finite sequences of scalars <x1,a2,-¦¦,<*„ we have
1
whence
E«,
E aixi
(/•=1,2,. ...и),
E atxi
The implication 3° => 4° follows from the fact that for any fixed i0
we have, by 3°,
E «i-
4°=>5°. Assume that we have 4° and that lim Y, «-"'х,- exists. Then
/т„+к т„ \ "^°° " = 1
lim XI а!"+(с)х,- Е а!"Ч =° whence,by4°, lim |а<.и + A)-а|п)| = 0
n,fc->cc у ; = 1 i=1 у п,к->ж
(i=l,2,...), whence the limits lima!"' (i=l,2,...) exist.
5° ^2°. Let
Ф.Ц 0LiX\ = aj (t a,xteP;j=l,2,...,n).
1 See e.g. [10], p. 58, theorem 6.

56
I. The Basis Problem. Some Properties of Bases in Banach Spaces
If xeP, x= Нтр„, рп= ? а}и)х,-(и=1,2,...), then, by 5°, we can put
! — 1
Then ф] is a continuous linear functional on P, satisfying (^ (x,-) = <5y
(i,j=\,2,...). Extending each ф-} to an f}eE* (/=1,2,...), we obtain 2°.
Thus \°o2°oToA°o5°.
2°=>6°. If we have 2°, let {sn\ be the sequence of partial sum oper-
operators associated to the biorthogonal system (х„,/„). Then wn = sn\p is
a continuous linear projection of P onto P(n), along P(n) (и = 1,2,...),
whence we have F.22).
6° => 7°. If we have 6°, let wn be a continuous linear projection of
P onto P(n), along P(n) (n=l,2,...). Then, extending each wn to an
uneL(E,E) (this is possible, since wn is of finite-dimensional range), we
have F.23) and F.24).
7° =>8°. By F.23) and F.24) we have
П
i.e. F.25) with С„=||иЛ (и=1,2,...).
8°=>9°. By F.25) we have, for every ? а,-х,-б«7(п) and а„ + 1, а„+2,.-.,
1 Д 1
eK,
whence F.26).
The equivalence 9°<=>10° is an immediate consequence of lemma
6.1 a) applied to F = P(n), G = P(n).
9°=>11°. Let poeP{n). Ifpo = O, there exists an feE* satisfying F.28),
Po
F.29) and 11/11 = arbitrary. If po^O, we have -—-e<r(n), whence,
by 9°, dist
IIpoII
, P(n)) ^ dist((j(n), P(n)) = dn>0. Then there exists1
1
an/e?* satisfying F.28), F.29) and ||/|| =—.
1Г=>12°. By F.28), F.29) and F.30), for every p= ? a;x,eP and
we have, putting po= E a;x; = Sn(P)>
1 See e.g. [10], p. 57, lemma.
7. Intrinsic characterizations of bases. Norm and index. Extension of blocks 57
The equivalence 12°<=>13° is obvious by F.20).
12°=>3°. Assume now that all х„^0(и = 1,2,...) and that we have
12°, and put
where 1„= sup ||Sn(p)|| (и=1,2,...). Then <5„>0 and for every finite
peP
sequence of scalars al5a2,...,а„ we have
S-l У ах
->j+i|
whence
1
< —¦
which completes the proof of theorem 6.1.
Remark 6.1. If {х„} is complete in E, the sequence {«„jcL(?,?) in
7° is uniquely determined and coincides with the sequence {sn} of
partial sum operators associated to the biorthogonal system (х„,/„).
Indeed, this follows e.g. from 6°.
Remark 6.2. Condition 4° is of similar type as those occurring in the
definition of finitely linear independence and co-linear independence,
but more restrictive. Therefore the sequences {х„} satisfying 4° have
been called, by A. I. Markushevich [156], strongly linearly independent
sequences. Note that if all х„/0 (и = 1,2,...), condition 4° is equivalent
to the continuity of the linear mappings Sn (n= 1,2,...) on P, while 12°
is nothing else but the boundedness of the Sn (и = 1,2,...) on P.
§ 7. Intrinsic characterizations of bases. The norm and the
index of a sequence. The index of a Banach space.
Extension of block basic sequences
In § 4 we have seen some characterizations of those minimal se-
sequences {х„} in a Banach space E, which are bases of E, while in § 5
we have seen some characterizations of those complete minimal se-
sequences {х„}сЕ which are bases of E. All these characterizations

58
I. The Basis Problem. Some Properties of Bases in Banach Spaces
have explicitly used the sequence {/„}<=?* of the biorthogonal system
(xn,fn)- We shall now give necessary and sufficient conditions for a
complete sequence {xn}cE, х„#0 (и=1,2,...) to be a basis of ?, in
terms only of the properties of the sequence {xn}. Thus these will be
intrinsic characterizations of bases.
We shall use, for a sequence {х„}<=?, the notations PM,Pin\ <j(n),
a("\ P, Sk and Rk of the preceding section (see F.14)-F.20)) and
the notations
(«=1,2,...), G.1)
= sup ||Sk(p)||= sup
1 ^fc<00 1 rSfcrgfl
E a;x;
p=ljaixieP). G.2)
; = i
Theorem 7.1. Let E be a Banach space and \xn} a complete sequence
in E such that х„фО {n=\,2,...)} Then the following statements are
equivalent:
Г. {х„} is a basis of the space E.
2°. For every sequence {pn\ a P converging to 0 we have
sup
|<оо.
G.3)
3°. There exists a sequence of endomorphisms {м„} <= L(?, ?) satisfying
F.23), F.24) and
1<C1= sup ||mJ<oo. G.4)
1=
n
E ««*,
i= 1
«SC2
п + m
E a;xi
In this case, the sequence \un} is uniquely determined and coincides
with the sequence {sn} of partial sum operators associated to the basis {х„}.
4°. There exists a constant C2 with 1 ^ C2 < oo, such that we have
G.5)
G.6)
G.7)
for all positive integers n,m and all a1,a2,...,an+me.K.
5°. We have
C3= inf
1 ^П < oo
6°. We have
inf dist(«7(n),t7(B))>0.
1 Of course, these conditions are satisfied whenever {*„} is a basis of E.
7. Intrinsic characterizations of bases. Norm and index. Extension of blocks 59
7°. We have
/ X \
0, G.8)
х„
inf dist —^,
««<» \\\xn\
inf dist(ff(n),(T("+'I))>0.
G.9)
8°. We have G.8) and there exists a positive integer k0 such that
inf dist(o-(n),cr(n + fco))>0. G.10)
9°. There exists a constant C4,1 s$ C4 < со, with the following property:
for every n and every poeP{n) there exists an feE* satisfying F.28),
F.29) and
G.П)
10°. We have
C5= sup sup ||5„(р)||= sup |||p|||<co.
l:Sn<oo peP peP
11°. We have
sup sup ||i?B(p)||<GO.
G.12)
G.13)
Furthermore, for the above constants we have
G.14)
w/iere < со holds if and only if {xn} is a basis of the space E.
Proof. 1° => 3°. If {х„} is a basis of E, then, by the implication 1° => 4°
of § 4, theorem 4.1, the sequence {sn} of partial sum operators associated
to the basis {х„} satisfies F.23), F.24) and G.4) and, by virtue of § 6,
remark 6.1, it is the only sequence having these properties.
3° => 4°. By F.23), F.24) and G.4) we have
E-
i= 1
4°=>5°. By G.5) we have, for every E aix;eG(n) and an+i, a,J
E aix.-~ E a*x;
E a;x<-
C,

60
I. The Basis Problem. Some Properties of Bases in Banach Spaces
The equivalence 5°<=>6° is an immediate consequence of § 6, lemma
6.1 a) applied to F = P(n),G = Pin).
6°=>7°. Assume that we have 6° and take an arbitrary xeEin) = P(n_l)
+ P(n). Then x = y(n_1) + z(n) for suitable y{n-X)eP{n_y),z(n)eP(n\ whence
— x
By §6, lemma 6.1a), applied to F = P(" 1( and G = P(n_1), we have
1 С
2 '
where C= inf dist^V^O by 6°. On the other hand, by §6,
I ^ n < oo
lemma 6.1 a), applied to F = P(n), G = P(n\ we have
х„
.»
У (я - i
Consequently, we have
— x
C2
У{п- I
(хе?(п), и=
-.
which proves G.8).
Since ff(" + '°cG(")(n,/c = l,2,...), we also have
dist(ff(B), <x<"+*>) ^ dist(ffw, ff(B>) ^ С (и, /с = 1,2,...),
whence G.9). Thus 6° => 7°.
The implication 7°=>8° is obvious.
8°=>6°. For a fixed positive integer n, applying §6, lemma 6.1b)
successively to F = P(n), G = P<" + (c-1>, G^P^^, x = xn + k (k= 1,2,...,/c0)
/ x
and taking into account that dist "+k ,P(n) + P(n +
/ \ \llxn+tll
> dist —!i±f-, ?<"+ k> , we obtain
Vllxil
7. Intrinsic characterizations of bases. Norm and index. Extension of blocks 61
whence, by multiplication,
distK^^^^dist^,^^"») П dist
> l
(fc=l,2,...,/c0),
G.15)
k=l
Assume now that we have 8° and put С = inf dist (тт^тг, E(n)),
l«n<» VIIх»
which is >0 by G.8). Then, by G.15), we have
inf
I 5n<00
c'Y
— inf dist(ff(n),(T(n + ''o)),
3 / 1«„<оо
whence, by G.10), we obtain G.7). Thus 8°=>6°.
5°=>9°. Let роеР(и). If po = o> ^еге exists an feE* satisfying
F.28), F.29) and ||/|| = arbitrary. If po#O, we have --^-etr^, whence,
, ч ¦ llPoll
by 5°, dist(-^-,P(n) )^C3. Then there exists1 an feE* satisfying
VIlPll /
F.28), F.29) and
dist
VllPoll
9°=>10°. By F.28), F.29) and G.11), for every p= ^ а;х,еР and
n i — 1
we have, putting p0= X a>xf = ^n(P)>
The equivalence 10°<=>ll° is obvious by F.20).
10° => 2°. If \pn} <= P is a sequence converging to 0, we have
SUP IIpJ<00> whence, by 10°,
1 ^ n < oo
sup HIpJI ^ C5 sup ||pj| < go.
2°^» 10°. If 10° is not satisfied, there exists a sequence \pn} <= P
with ||р„||<1 (и = 1,2,...) and lim |||ри||| = оо. Then for qn = -
П-*ОО
have ||^„||->0 and |||^„|||->эо, which contradicts 2°.
Р"
lipji
we
1 See e.g. [10], p. 57, lemma.

62
I. The Basis Problem. Some Properties of Bases in Banach Spaces
10°=>l°. If we have 10°, then, since х„^0 (n=l,2,...), from the im-
implication 12°=> 2° of §6, theorem 6.1 it follows that there exists a
sequence {/„} <= ?* such that (х„,/„) is a biorthogonal system. Then
for the partial sum operators sn associated to (х„,/„) we have, by bi-
orthogonality,
s,(p) = Sn{p) (peP,n=l,2,...), G.16)
whence, by 10° and since {х„} is complete in E, sup ||sj <oo. Con-
1 ^П< 00
sequently, by the implication 4°=>1° of §4, theorem 4.1, {%„} is a
basis of E.
Furthermore, from the above proofs of the implications 3° => 4° => 5°
=^9°=> 10° it follows that
inf C2
inf C4 ^ C5
1.
Let us prove that С5 = Сг. If {х„} is not a basis of E, then, by the
above, we have C5 = C1 = oo. On the other hand, if {х„} is a basis of E,
then, by the proof of Г =>3° given above, we have Ct = sup ||sj|<oo,
1 ^П<00
where {sn} is the sequence of partial sum operators associated to the
basis {х„}. Hence, by G.12) and G.16), C5 = C1<co, which completes
the proof of theorem 1.5.
Remark 7.1. Let us also mention the following alternative proof of
the implication 10°=>l°. If we have 10°, then, since х„#0 (и = 1,2,...),
from the implication 12°=>2° of § 6, theorem 6.1 it follows that there
exists a sequence {/„} <= E* such that (х„,/„) is a biorthogonal system.
Then, since {х„} is complete in E, from 10° and the implication 5°=> 1°
of § 5, theorem 5.1 it follows that {х„} is a basis of E.
Remark 7.2 A comparing of the conditions 7° -13 ° of §6, theorem 6.1
with the conditions 3°-6° and 9°-11° of theorem 7.1 shows that the
bases of a Banach space E could be called "uniformely minimal" se-
sequences. Furthermore, with the condition 4° of § 6, theorem 6.1 (con-
(condition of "strong linear independence", see remark 6.2), it is natural to
compare condition 2° of theorem 7.1. However, note that in 2° of
theorem 7.1 it is also assumed separately that х„#0 (и=1,2,...), while
in 4° of § 6, theorem 6.1 this is not assumed, but follows as a consequence.
Finally, let us mention that condition G.8) is equivalent to the existence
of a sequence {/„} с Е* such that (х„,/„) is a biorthogonal system
satisfying sup ||xj ||/J| <oo (this follows from Ch. II, §2, corol-
corollary 2.1, applied to
). Of course, every basis of E satisfies this
condition (see § 3, theorem 3.1, formula C.7)).
7. Intrinsic characterizations of bases. Norm and index. Extension of blocks 63
Definition 7.1. Let {х„} be a complete sequence in a Banach space ?,
such that х„=?0 (и = 1,2,...). Then the number G.14) is called the norm
of the sequence {х„}; we shall denote this number by v = vtXnj. The
number y = y{Xn] = — is called the index of the sequence {х„}.
These numbers admit a geometrical interpretation. Indeed, we recall
that if F, G are subspaces of a Banach space ? and o> = {xeF| ||x|| = 1},
the number ^-^
(F;G) = dist(<rf,G) G.17)
is called the inclination of F to G. With the aid of this notion and of
the notations F.14), F.15), we can write
?„„,= inf (Р^Ч G.18)
1 ^ П < 00
By theorem 7.1, we have
1 «S v|Xn,< со, 0<7{Xn(<l; G.19)
the sequence {х„} is a basis of ? if and only if v(Xn,< со, or, equivalently,
G.20)
Definition 7.2. Let ? be a Banach space. The number
Г(Е) = supylXni,
where the sup is taken over all complete sequences {х„}с:? with
xn ф 0 (n = 1,2,...), is called the index of the space E.
By G.19) we have, for all Banach spaces ?,
1, G-21)
and the space ? has a basis if and only if Г(?)>0. It is natural to ask
whether or not there exist Banach spaces with bases such that Г(Е)< 1,
i. e. Banach spaces ? such that 0<Г(?)< 1. We shall see in Chapter II,
§ 1, that the answer is affirmative for finite dimensional Banach spaces,
but unknown for infinite dimensional Banach spaces.
Let us give now some corollaries of theorem 7.1.
Corollary 7.1. Let E be a Banach space and {х„} a complete sequence
in E such that
([хь...,х„];[хи
G.22)
where [xH + 1] is the one-dimensional subspace of E spanned by xn + l.
Then {х„} is a basis of E, of index у {Хп)^ p.

64
I. The Basis Problem. Some Properties of Bases in Banach Spaces
Proof. Put pa = ([Xl,...,Xa];[xn + 1])(n=l,2,:-), P=UPn- Then
l and for any finite sequence of scalars аь...,а„+ш we have,
by G.22),
n + m
У a x
i= 1
=
n + m — 1
i= 1
^ Pn+m-IPn+m-2
^ + m-lA + m-2
+ mxn
n + m —
z
i= 1
¦ •A,
+m
2
л
z
i=l
a,-
A.
+m-i
n + m- 1
z «
1=1
>¦¦¦
П
z **,
i= 1
whence, by the implication 4°=>1° of theorem 7.1, {xnj is a basis of E,
of index y{Xn]~^p, which completes the proof.
The following is a' useful tool for proving that certain normed linear
spaces are complete:
Corollary 7.2. Let E be a normed linear space and {xn} a complete
sequence in E, with xn=/=0 (n = l,2,...), satisfying any one of the equivalent
conditions of theorem 7.1. Then E is complete if (and, obviously, only if)
30
every Cauchy series of the form ? a,x; is convergent to an element ofE.
i= 1
Proof. By virtue of theorem 7.1, {х„} is a basis of the completion ?л
of E. Let yeEh be arbitrary. Then у has an expansion of the form
00 00
y= Y, aixi and therefore ? а,х, is a Cauchy series. Hence, by our
1 = 1 qo 1 = 1
hypothesis, ? a;x,- is convergent to an element of E, and thus yeE.
i- 1
Since yeE" has been arbitrary, it follows that E is complete, which
concludes the proof.
The next corollary gives a method of "piecewise" construction of
new bases in Banach spaces with bases.
Corollary 7.3. Let {х„} be a basis of a Banach space E, {mn} an
increasing sequence of positive integers, mo = O, and {zn} a sequence
in E such that {z,}^mn_1 + 1 is a basis of [xmn_1 + 1,...,xmj (и= 1,2,...),
of norm
V;,..». , sSM<oo (n=l,2,...). G.23)
Then \zn} is a basis of E.
Proof. It is obvious that [zn] = ?. Let I,k be arbitrary positive
integers and let n, q be positive integers with n^q, such that
7. Intrinsic characterizations of bases. Norm and index. Extension of blocks 65
„, т,_, + 1 ^l + k^m,,. Furthermore, let <хи...,х1+к be
arbitrary scalars, let
/J,x, be the representation of
with respect to the basis {х„}™=1 (/= \,...,q — 1) and let
ptxt
l + k
be the representation of
a,z;. Then
i —mq- l + 1
т„- l
"In — 1
mn - i
У Bx-
i = l
+
+
i — n
M
M
z
nZt
z
<2v|Xn)(v|Xn)+l)M
Z
Z
= 2v|Xn)(vUii)+l)M
Z «,-z,-
whence, by theorem 7.1 (implication 4°=> 1°), {zn} is a basis of E, which
completes the proof.
Let us observe that if we omit in theorem 7.1 the hypothesis [х„] = E,
then we obtain various characterizations of basic sequences'. We shall
use now this remark to prove
Corollary 7.4. Let {х„} be a basis of a Banach space E, let {т„} be
an increasing sequence of positive integers, mo=Q, and let
yn= X «i*f> У«*° (« = 1,2,...).
i=mn- i + 1
G.24)
Then {yn} is a basic sequence, with vlyn]^v{Xn].
Proof. By the implication Г => 4° of theorem 7.1 and by G.24) we have
Z yjyj
Z;,- Z
V У " a x
j=l i = mj- , + 1
n + g
Z Z ?/«,¦*!
j= 1 ,=mj_ t+l
= V,
х„)
V - v
/ , ' j -У j
j=i
1 See §4, definition 4.5.
5 Singer, Bases in Banach Spaces 1

66
I. The Basis Problem. Some Properties of Bases in Banach Spaces
for all positive integers n, q and all y1,...,yn+qeK. Consequently, by
the implication 4°=>1J of theorem 7.1, {у„} is a basic sequence, with
v(yn)<v(;Cri), which completes the proof.
Definition 7.3. Let {х„} be a basis of a Banach space E. A sequence
{у„} <= E is said to be a block basic sequence {with respect to the basis
{*„}), if it is of the form G.24), where {т„} is an increasing sequence of
positive integers and mo = O.
By corollary 7.4, every block basic sequence is necessarily a basic
sequence. The block basic sequences will have many applications in
the sequel. In the present section we shall show only that for this
important class of basic sequences the problem of their extension to
a basis of the whole space E (§ 4, problem 4.1) has an affirmative answer.
For this purpose, we need
Lemma 7.1. Let F and G be two (и—1)-dimensional subspaces of an
n-dimensional Banach space E, where n<co. Then there exists an iso-
isomorphism T from F onto G such that
(xeF).
G.25)
Proof. There existsl a projection и of F onto its (n — 2)-dimensional
subspace Fr^G, such that ||/f-u||=l (whence |[u||<||/f-u|| +||/f|| =2)
and similarly, a projection v of G onto FnG, such that ||/G — v\\ = l
(whence ||t>||<2). Take yoeF, \\yo\\ = 1, such that u(yo) = O, and zoeG,
||zo|| = l, such that v(zo) = O. Then the mapping T:F->G defined by
G.26)
(xeFnG,aeK)
is obviously linear and we have, for any x + ayoeF,
||T(x + ayo)\\ = \\x + azo\\ «c \\x\\ + \a\=\\x\\ + \\ayj
Similarly, for any x + ayoeF we have
|x + ayo|| sS ||x|| + |a|= ||x|| + ||azo|| = ||u(x + azo)
azo
whence G.25), which completes the proof of lemma 7.1.
1 Indeed, in any reflexive Banach space В there exists a projection и onto any
hyperplane H={xeB \f(x) = 0}, such that ||/g-u|| = l, where IB denotes the
identical mapping of В onto itself (take xleB with ||:>Cil| = l,/(xi) = H/ll and put
u(x) = x - y~*i for all xeB).
7. Intrinsic characterizations of bases. Norm and index. Extension of blocks 67
Now we can prove
Theorem 7.2. Let \xn} be a basis of a Banach space E and let
у„=
(n=l,2,...;mo =
G.27)
be a block basic sequence with respect to \xn\. Then there exists a basis
\zn] ofE such that
гтп = У„ (и=1,2,...), G.28)
В"„., + 1 = М="„-,и (и = 1,2,...). G.29)
Consequently, for the sequences of coefficient functionals {/„}, {hn\ <=?*
associated to the bases \xn} and {zn} respectively, we have
[«„] = [/„]• G-30)
Proof. For each n there exists1 an (т„ — т„_1 — \(-dimensional
subspace Gn of ?n = [x,.]j"=mn_] + 1 which admits a projection ?/„: ?n->Gn
such that Un(yn) = 0, ||l/J<2. Furthermore, by lemma 7.1 above, for
each и there exists an isomorphism Tn from ^"[xj'/'i'"^^ onto Gn
such that
il|x||<||^(x)||<3||x|| (xeFJ. G.31)
Put
^(xO for m,.1 + Ui<m,-l („=1,2,...). G.32)
for г = ш„
Then for any scalars fimn_1 + l,...,fimn we have Un
mn — 1
^ ^j zf (since Un(zmJ = Un(yn) = 0, and zi = Tn(x{
for
mn
i = mn - l + 1
1
;„-1), whence taking into account G.31), it follows that
.— 1 we have
for any such scalars and any / with mn_
I PiTn(xd
«S3
I P,x,
= 9v.
Z Pi*i
ил Z
Л =тп - i
Xj)
I
<18v.
xj\
Z
i=mn- i + 1
1 Indeed, in any Banach space B, for any xoeB there exists a hyperplane
G<=B which admits a projection U.B^G such that U(xo) = 0, ||l/||<2 (take
feB* with H/ll =-i-,/(xo) = l, and put G={xeB \f(x) = 0}, U(x) = x-f(x)xo
for all xeB). l|xoll

68 I. The Basis Problem. Some Properties of Bases in Banach Spaces
and therefore {z,-}'>mn_1 + 1 is a basis of [x;]"'=mn-, + i> of norm
Consequently, by corollary 7.3, {zn} is a basis of E, which, by G.32),
satisfies G.28) and the inclusion <= in G.29), whence also the equality in
G.29).
Now, for {/in}<=?* with ^(г^) = ёи we have, by G.29),
Mxj) = 0 (i = mn_l + l,...,mll; j=f=mn_1 + l,...,mn; и =1,2,...),
that is,
On the other hand, for {/„}<=?* with /((х7) = ^- we obtain, as in
Ch. II, § 4, formula D.7),
т„_1 + 1 И[*;Ьт„-1 +!,..., J1 («=1,2,...). G.34)
From G.33) and G.34) we infer
№„„., +^[/fe.-. + i ("=1,2,...), G.35)
whence G.30), which completes the proof of theorem 7.2.
Remark 7.3. Actually, proposition 4.4 of § 4, on block perturbations
of bases, may be also regarded as a theorem on extension of block basic
sequences of a particular form.
§ 8. Domination and equivalence of sequences. Equivalent,
affinely equivalent and permutatively equivalent bases
Definition 8.1. Let E,F be two Banach spaces. A sequence {xn}cE
is said to dominate a sequence {>'„}<=? provided that for all sequences
{а„} of scalars
GO GO
? ? (8.1)
a,x; converges
a,y; converges.
In this case we shall use the notation {х„}>{у„}.
We shall say that {xn}cE dominates strictly the sequence {у„}<=?
and we shall write {х„)>^-{у„}, provided that there exists a continuous
linear mapping ueL([xn], [у„]) such that
The sequences {х„}с? and {у„}с:? are said to be equivalent if
we have simultaneously {х„}>-{у„}>-{х„}, and strictly equivalent if
8. Domination and equivalence. Affinely and permutatively equivalent bases 69
we have {xn}>?~{yn}>P~ {х„}. In these cases we shall use the notations
{*„} ~ {У„} and {х„} к {у„} respectively.
Finally, we shall say that the sequences {х„}<=?, {yn}cF are
fully equivalent, and we shall write {xn}v {}'„}, provided that there
exists an isomorphism1 и of E onto F satisfying (8.2).
It is obvious that strict domination => domination and that full
equivalence => strict equivalence => equivalence. As we shall see below,
the converse implications are not valid. It is also immediate that if
{х„}~{у„}, then the mappings ueL([xn], [у„]) and !?eL([yn], [х„])
in the definition of strict equivalence are such that и is an isomorphism
of [х„] onto [у„] and v is the inverse isomorphism, carrying [у„] onto
[х„]. Hence, if both {х„} and \у„} are complete in E and F respectively,
we have {х„}«{у„} if and only if {х„} 2 {у„}.
Theorem 8.1. Let E,F be two Banach spaces and let {х„\ c?, {у„} <=F.
Then
a) The following statements are equivalent:
i°. W>W-
2°. T/iere exist a positive integer n0 and a constant C>0 smc/i
that we have
(8.3)
/or all finite sequences of scalars ano, аио+1,...,а„о+т (w=l,2,...).
b) The following statements are equivalent:
no + m
i = n0
^C sup
ПО^:к ^ПО + III
к
E atxi
i = nn
2°. T/iere exists a constant C">0 suc/i t/iat we
n
i- 1
n
i= 1
(8.4)
/o/* all finite sequences of scalars аьа2,--.,а„.
3°. For every i//e[yn]* the system of equations
/las a (unique2) solution 0е[х„]*.
// {х„} /s minimal, these statements are equivalent to the following:
4°. There exists a positive integer n0 such that {хп}™и'>?'{уп}щ1.
If (х„,ф„) ({фп} <= [*„]*) and (у„,ф„) {{фп} с[Уп]*) аге biorthogonal
systems such that {ф„} is total on [у„] , these statements are equivalent
to the following:
' We recall that by "isomorphism" we mean: linear homeomorphism.
2 Since {%„} is complete in [х„], the solution of (8.5) is unique.

70 I. The Basis Problem. Some Properties of Bases in Banach Spaces
5°. For every хе[х„] the system of equations
Фп(х) = фп(У) (и =1,2,...)
(8.6)
has a (uniqueI solution уе[у„~].
V {xn} (S a basis of [х„] and {yn)^F is arbitrary, these statements
are equivalent to the following:
6°. {xn}>{yn\.
c) The following statements are equivalent:
2°. There exist a positive integer n0 and a constant C>0 such
that we have (8.3) and
na + m
E «;*;
sup
no ^ к ^ no + m
E «ij;
(8.7)
n
E
i= 1
X;
n
E
i= 1
a,-
У;
for all finite sequences of scalars аяо,яио+1,...,аио+т (m=l,2,...).
d) The following statements are equivalent:
2°. There exists a constant C">0 suc/i t/iat we /iat'e (8.4) and
S.8)
/or all finite sequences of scalars aba2,...,an.
3°. 77ie system of equations (8.5) /las a (unique) solution фе [х„]*
for each фе[у„\* and a (unique) solution фе[у^\* for each 0е[х„]*.
7/{х„}, {у„} are minimal, these statements are equivalent to the following:
4°. T/iere exists a positive integer n0 such that {х„}™«{у„}™.
(/" (Х„Ж) ({Фп} с [хя]*) and (у„, tfg ({^„} с [у„]*) are biorthogonal
systems such that {ф„}, {ф„} are total on [х„] and [у„] respectively,
these statements are equivalent to the following:
5°. The system of equations (8.6) has a (unique) solution уе\Уп]
for each хе[х„] and a (unique) solution хе[х„] for each ye[y^\.
If {*n}' {yn} are bases of [х„] and [у„] respectively, these statements
are equivalent to the following:
6°. {Х„}~{У„}-
Proof a) The implication 2° => 1° is obvious.
1°=>2°. Assume that 2° is not satisfied. Then we can successively
find an increasing sequence of positive integers {mn} and a sequence
of scalars {а„} such that
1 Since {ф„} is total on {у„}, the solution of (8.6) is unique.
8. Domination and equivalence. Affinely and permutatively equivalent bases 71
1 =
E
max
E
а;х,-
(n=l,2,...;mo =
Then ? aixi is convergent and ? a,y; divergent, i.e. Г is not
satisfied. i=1 i=1
b) 1°=>2°. If «6L([x],[y ]) satisfies (8.2), then for any а1,а2,...,а„бК
we have
C=\\u\\.
n
E а;У;
i=i
=
( " \
u[ ? а;х;
\,-=i /
? С1{Х{
i=l
, i.e. (8.4) with
n
2°=>1°. Assume that we have 2° and for any p= ? а;х,еР put
" i = i
uo(p)= 2] «(У,-- Then м0 is a linear mapping of the dense subspace P of
i= 1 n
[х„] into [у„] (ц0 is well defined on P, since ? а;х, = 0 implies, by (8.4),
n ; = i
that ? а;у, = 0), satisfying ио(хп) = у„ (и=1,2,...). Since by (8.4) this
i= 1
mapping u0 is also continuous on P, it can be extended to an ueL([xn],
[у„]) satisfying (8.2).
Г=>3°. If ueL([xn], [у„]) satisfies (8.2), then for every фе[уп~]*
the system of equations (8.5) has the (unique) solution ф = и*(ф), where
u*eL([yn]*, [xn]*)is the adjoint of и. п
3°=>Г. Assume now that we have 3°and for any p= ? а,х;бР
" i=l
put uo(p)= Z a;y;. Then uo(p) is well defined. Indeed, assume the
contrary, i.e. that there exists a finite sequence of scalars а1)а2,...,а„
n n
such that Y, а,х, = 0, ? а;у,#0. Then there exists а фе[у„}* such that
'A I E а;У; р=°- ВУ 3° to this i//e[yj* corresponds a 0e[xj* such that
we have (8.5), whence ф(О) = ф
= Ф
i = 1
а,х; = ? а;0(х,) = ? а^(у;)
i=l / 1=1 1=1
а,-у,-1 # 0, which is impossible. Thus u0 is a well defined linear
mapping of P into [у„] satisfying мо(х„) = у„(п=1,2,...). Now, for every
фе\_уп~]* let us denote by Т(ф) the functional 0е[х„]* corresponding
to ф by 3°. Then T: ф^>Т{ф) is a linear mapping of [у„]* into [х„]*.
Indeed, for arbitrary ф,ф'e\_y^\* and scalars a,a'eK we have
[ T(a ^ + «>')] (x;) = (а ^ + а>') (у;) = а ^(

72 I. The Basis Problem. Some Properties of Bases in Banach Spaces
whence, since {xn} is complete in [xn],
Т{аф + а'ф') = а Т{ф) + а'Т(ф').
We claim that T is also continuous on [у„]* for the norm topology.
Indeed, let {iAn}<=[.yn]*, »Ae[vn]* and 0e[xn]* be such that
lim Т(ф„) = ф.
Then
,)= lim
(/=1,2,...),
whence ф=Т(ф). Consequently, by the closed graph theorem1, T is
continuous on [у„]*.
Now, since we have
imi 1Ж1 llpll (peP.^eW*),
where || T\\ < oo by the above, it follows that u0 is also continuous on P.
Hence u0 can be extended to a mapping ueL([xn], [у„]) satisfying (8.2).
Thus we have 1°.
The implication 1°=>4° is obvious.
4°=>1°. Assume that {xn} is minimal and that we have 4°. For any
p=
put
HQ— 1
i = l
u2{p) =
af у,- if
if
if
if
where и0 is as in 4°. Then щ, u2 are linear mappings of P into F (they
are well defined on P, since {xn} is minimal) and we have M1=i'1w1,
w2 = r2iv2, where w, denotes the linear projection of P onto ^„0-d
1
along
«o 1
», ()l the mapping ? «,¦*,.
i=1
a,-y,- of ^„„„d into [>-„],
1 See e.g. [10], p. 41, theorem 7.
8. Domination and equivalence. Affinely and permutatively equivalent bases 73
vv2 the linear projection of P onto f'"" along f(no_1) and t?2 the mapping
ajj. Here W!,w2 are continuous by the implication
? a,x;-> ?
i - но i = no
1°=>6° of §6, theorem 6.1, vl is continuous by the linear independence
of x1?x2,..., xno_, and v2 is continuous by 4°. Consequently, ut
and u2, whence also uo = ul+u2, are continuous on P. Extending this
u0 to an ueL([xn], [у„]), we obtain a continuous linear mapping of
[xn] into [>¦„] satisfying (8.2).
1°^> 5°. Assume that (х„,ф„)({фп} с: [х„]*) and (у„,^„)({^п} ^ [>>„]*)
are biorthogonal systems such that {(//„} is total on [у„] and that we
have Г. Then, if ueL([xn], [>'„]) satisfies (8.2), we have
] j № M (U= 1.2,...),
whence, since [х„] is complete in [х„],
0,.(х)=>А,.[ы(х)] (хе[х„],г = 1,2,...).
Consequently, for every xe[xj, the element ы(х)е[>'„] is a solu-
solution of (8.6).
5° =>1°. Assume that (хп,фп)({ф„} cz [х„]*) and (yn,i)({i}c[yj*)
are biorthogonal systems such that {ф„} is total on [>-„] and that we
have 5°. For every хе[х„], let us denote by u(x) the element уе[>'„]
corresponding to x by 5°. Then м:.х->м(х) is a linear mapping of [х„]
into [>'„]. Indeed, for arbitrary zr,z2e\_x^\ and scalars aux2€K we
have, by the linearity of ф1 and i//; (i= 1,2,...),
Ф1 [u{ai zi + a2 z2j] = ф((сс1 zy + a2 z2) = at ^^z^ + a2 0((z2)
(r= 1,2,...),
whence, since {ф„} is total on [у„],
We claim that и is continuous on [х„], i.e. that ueL([xn], [)>„]).
Indeed, let {zn} с [х„], ze[xn] and г'е[у„] be such that
lim zn = z, limu(zn) = z'.
Н-*0О П-+00
Then, by the continuity of ф{ and ф{ (г' = 1,2,...),
0,.(z)= Hm0,.(zn)= Нт^[и(г„)] = ^(г') (г = 1,2,...),
П-*СО Н-+0О
whence z' = u(z). Consequently, by the closed graph theorem,

74
I. The Basis Problem. Some Properties of Bases in Banach Spaces
Finally, from the relations
(U= 1,2,
it follows, since {фп} is total on [у„], that м satisfies (8.2). Thus we have 1°.
The implication Г=>6° is obvious.
6° =>4°. Assume now that {х„} is a basis of [х„] and that we have 6°.
Then, by the implication 1°=>2° of part a), proved above, there exist
a positive integer n0 and a constant C>0 such that we have (8.3).
Hence, by the implication 1°=>4° of § 7, theorem 7.1, applied to the
basis {xn} of [х„] and to а1 = --- = а„о^1 =0, there exists a constant С
such that we have
no + m
E
i — но
У1
no + m
У W Y
i = но
for all finite sequences of scalars ano, ano + i,---,ano + m- Consequently, by
the implication 2° => Г of part b), proved above, we have {х„}*0У-*-[у„}*0.
Finally, c) and d) are immediate consequences of a) and b) respec-
respectively. This completes the proof of theorem 8.1.
Remark 8.1. Let us also mention the following alternative proof of
the implication 6° => 1° of part b). If {х„} is a basis of [xn], condition 6°
ensures the convergence of the series
GO
u(x)= X «,Уг
for all x= ? a,x,e[xn]. The mapping и: [х„]-»[)>„] defined in this way
is linear and satisfies (8.2). Furthermore, since we have u(x)= limun(x)
«-¦ОС
" ( °° \ •
(xe?), where un(x) = ? а;у, I x= ? а(х,-е[х„], «= 1,2,... I, and since
by § 3, theorem 3.1, each и„ is continuous, from the Banach-Steinhaus
theorem it follows that и is continuous. This completes the proof.
In the implication 6°=>1° of b) it is essential to assume that {х„}
is a basis of [х„], and in the implication 6°=> 1° of d) it is essential to
assume that both {х„} and {yn} are bases of [х„] and \_yn~] respectively.
Indeed, this follows e. g. from the following general result (see also
theorem 8.2 below):
Proposition 8.1. Let {х„} be a sequence in a Banach space E, such
that х„фО (и=1,2,...), and let Av be the Banach space of sequences of
scalars introduced in § 3, proposition 3.1. Then
a) The unit vectors en={5nj}f=l (и=1,2,...) constitute a basis of Av
b) We have {х„}
- е„
and {е„
8. Domination and equivalence. Affinely and permutatively equivalent bases 75
00
Proof, a) If {а„}еАу, i. e. if ? a,x,- converges, then
{a»}- E *iei
= sup
m + 1 ^ fc < oo
E
a,x,-
>0 for m->oo,
whence ? «i^ converges to {а„}. On the other hand, if ?«;*; = <
then, by the above,
IK} 11= sup
E a*x;
= lim
E a^i
lim sup
E а;е.-
E a.x;
whence а„ = 0 (n=l,2,...). Thus, every {an}€Al has a unique expansion
00
Y otje,-, i. e. {en} is a basis of X^
b) We have seen in the above proof of a) that the convergence of
00 30
У а,х, implies that of У а;е,-. The converse implication follows from
; = i
the inequality
; = n +1
sup
E a'xi
E а'е.-
and
from the completeness of E. Thus {х„}~{е„}. Finally, by
sup
У a-x-
У at?,-
and the implication b) 2°=>b) 1° of theo-
theorem 8.1, we also have {еп\^~{х„}. This completes the proof.
Proposition 8.1 shows that if a sequence {х„} с Е, with х„^0
(«=1,2,...), is not a basis of [х„], then {х„}~{<?„} (and hence {х„}> {(?„})
but {xn}#{en}1 (actually, {х„})^-{е„}). Thus domination and equi-
equivalence do not imply strict domination and strict equivalence respec-
respectively.
We shall now show that the situation when domination implies
strict domination, i.e., the implication bN°=>b)l° of theorem 8.1,
characterizes, in a certain sense, bases.
Theorem 8.2. Let E be a Banach space and {х„} a complete sequence
in E, such that хпф0 (и =1,2,...). The sequence {xn} is a basis of E if
and only if
{?„} с К, {х„} > {A} => {х„}>ИЛ,}, (8.9)
1 Indeed, if {х„}«{е„}, then there exists an isomorphism u of Л, onto [х„],
such that и(е„) = х„ (п= 1,2,...), whence {х„} is a basis of [х„].

76 I. The Basis Problem. Some Properties of Bases in Banach Spaces
i. e., if and only if for every sequence of scalars {/?„} with the property
00 DO
that the series Y a(A converges for all {а„} such that Y y.ixi con-
converges, the system of equations
/00 = А, (и = 1,2,...) (8.10)
has a solution feE*.
Proof. The necessity is nothing else but the particular case F = K
of the implication b) 6°=>b) Г of theorem 8.1.
Conversely, assume that condition (8.9) is satisfied and consider the
space Al of proposition 8.1. Since {е„} is a basis of Au for any heAf
we have
whence {xn}>{h(en)}. Consequently, by condition (8.9), {xn}y^{h(en)},
i. e., there exists an feE* such that
/(*„) = МО (и=1,2,...). (8.11)
Thus, for any heA* = [e^\* the system of equations (8.10) has a
solution /е?* = [х„]*, whence, by the implication b) 3°=>b) 1° of theo-
theorem 8.1, we infer {х„}>>^{<?„}. Since by proposition 8.1 {е„}>?-{х„}, it
follows that {xn} x {е„}, whence {х„} is a basis of E, which completes
the proof.
Proposition 8.1 above also shows that the usual equivalence of
sequences does not conserve any one of the following properties: 1. com-
completeness, 2. linear independence of any type, 3. being a basis. This is
also true for sequences of the same space E, as shown by
Example 8.1. Let {х„} be a basis of a Banach space E and let {>•„} с Е
be the sequence defined by
71 — X-i, y2 — ZXb у„ — Х„ [П — 5,Ч,...). (ИЛ/.)
Then {у„} is not complete in E (and hence not a basis of E), not
finitely linearly independent (and hence not «-linearly independent and
not minimal) and we have {хп}~{у„}, {х„}#{у„}, {у„}$>~{х„} (since
*>eL([>n],?), v(yl) = xl, v(y2) = x2 would imply x2 = v{y2) = v{2xl)
= 2t'(x1) = 2L'(y1)=2x1, which contradicts the assumption that {xn} is
a basis of E).
Similarly, even a stronger condition than the strict equivalence of
sequences, in which we require the existence of an ueL(E,F) satisfying
(8.2) and of a veL(F,E) satisfying v(yn) = xn(n=\,2,...), does not imply
their full equivalence and does not conserve completeness (and hence
also the property of being a basis), as shown by
8. Domination and equivalence. Affinely and permutatively equivalent bases 77
Example 8.2. Let ? = c0, {xn} = the natural basis of c0, and let
{>¦„
? be the sequence defined by
(8.13)
{Уп
Then {>>„} is not complete in E and we have \xn} «{>'„} in the above
stronger sense, but {х„}%{уп}. Indeed, the correspondence xn->x2n_!
generates a continuous linear mapping и of c0 into c0 satisfying (8.2),
while the correspondence x2n_t->xn, x2n->0 generates a continuous
linear mapping v of c0 onto c0 satisfying v(yn) = xn (и= 1,2,...), but the
(unique) continuous linear mapping ueL(co,co) satisfying (8.2) maps c0
onto the proper subspace [x2n_!] of c0.
However, the strict equivalence conserves all types of linear indepen-
independence of sequences considered in § 6 (and also those in Ch. II, § 11 and
§ 16) since they are, in a certain sense, "properties of {х„} with respect to
[х„]" (and not "with respect to ?"). More generally, the strict domination
also conserves these properties, as shown by
Proposition 8.2. Let E,F be two Banach spaces and let {xn}<=?,
\^F, {yn}»\xn}. Then
a) If all х„^0(и=1,2,...), then all у„фО (n = \,2,...).
b) // {xn\ is finitely linearly independent, so is {>¦„}.
c) If {xn} is oj-linearly independent, so is {>'„}.
d) // {xn} is minimal, so is {yn\.
Proof. Let veL{[yn], [х„]) be such that
v(yn) = xn (и=1,2,...). (8.14)
a) If у„ = 0, then by (8.14), xn = v{0) = 0.
n
b) If aua2,...,aneK are such that ? а;)>; = 0, then, by (8.14), we
have Y, <*iXi = v I Y а'У> ) = г;@) = 0' whence, if {х„} is finitely linearly
;=i \i=i /
independent, а1 = а2= ••• =а„ = 0.
c) The proof is similar to that of part b).
d) If {х„} is minimal, there exists, by the implication Г=>2° of
§ 6, theorem 6.1, a sequence {/„}<=?* such that /(Xj) = <5iy {i,j= 1,2,...).
Then for the sequence {gn}^F* defined by gn = v*{fn) (и=1,2,...) we
have, by (8.14),
whence, by the implication 2°=>1° of § 6, theorem 6.1, {у„} is minimal,
which completes the proof. Let us mention that d) can be also easily
derived e.g. from the equivalence 1°<=>3° of § 6, theorem 6.1 and the
implication b) l°=>bJ° of theorem 8.1.

78
I. The Basis Problem. Some Properties of Bases in Banach Spaces
Finally, it is obvious that the full equivalence conserves all properties
of sequences considered above, since they are invariant under isomorphic
mappings of ? onto another space F. One can also add to these properties
the more restrictive types of completeness of sequences, known under
the names of {an}-completeness and completeness of order p. We re-
recall that a sequence {xn} in a Banach space E is said to be {an)-com-
plete in E, where {а„} is a given sequence of non negative numbers,
provided that for every xeE and every ?>0 there exists a finite sequence
ofscalars а1)а2,...,ая such that
• - E
< ?, E \ai\ai<?
(8.15)
We also recall that the sequence {х„} <=? is said to be complete of order
p in E, where l^p^oo, provided that for every xeE and every ?>0
there exists a finite sequence of scalars al,a2,...,an such that
-•-!
e,
<?
1 1
- + -= 1
p q
(8.16)
where in the case p=oo the second inequality is to be replaced by
max |a,|<?. It is obvious from these definitions that both {an}-com-
1 sS i sS n
pleteness and completeness of order p imply completeness in the usual
sense. It is also immediate that if {xn} <= E and {у„} с F are fully equivalent
and if {х„} is complete in any one of the above senses, then so is {yn}.
Indeed, e.g. if {х„} is \an)-complete, и an isomorphism of E onto F
satisfying (8.2), у an arbitrary element of F and ?>0, then for the
a1;a2,..
lave
y-
eK
n
-E
i= 1
satisfyin
g
=
[
)- E
< \\u\\
E
we
which shows that {у„} is {an}-complete in F.
A similar remark is also valid for most of the properties of sequences
which we shall consider in the sequel, since they are invariant under
isomorphic mappings of ? onto another space F. However, an exception
is e. g. the property of being a normalized sequence (see § 3, definition 3.2).
In finite dimensional Banach spaces all bounded bases {хя} (see
§ 3, definition 3.2) are equivalent. It is natural to ask whether the converse
is also true1, i.e. whether this property characterizes finite dimensional
1 It is obvious that a boundedness condition is necessary, since in infinite
dimensional Banach spaces a bounded basis {xn} is not equivalent to the basis
Ш defined by yn = nxn(n=l,2,-.).
8. Domination and equivalence. Affinely and permutatively equivalent bases 79
Banach spaces among Banach spaces with a basis. We shall see in Ch. II,
§ 23, that the answer is affirmative and, moreover, that in every infinite
dimensional Banach space with a basis there exist a continuum of
mutually non-equivalent normalized bases.
Even for bases the condition of being equivalent (in the sense of
definition 8.1) is sometimes too strong, since a transformation of the form
уя = Кхп(п=\,2,...), where
0< inf \1„\
sup |Я„|< oo,
1 sSn< oo
(8.17)
leads to a basis {yn}, which in general is not equivalent to the basis {х„}.
Therefore the following less restrictive condition of affine equivalence
seems to be useful:
Definition 8.2. A basis {х„} of a Banach space E is said to affinely
dominate (respectively, to be affinely equivalent to) a basis {>•„} of a Banach
space F, if there exists a sequence of scalars {).„} satisfying (8.17), such
that the basis {л„х„} of E dominates (respectively, is equivalent to)
the basis {>¦„} of F in the usual sense, i.e. such that
И-;а,х; converges
а.-у, converges
(8.18)
(respectively, Y "Aiaixi converges <=> Y лгУг converges).
i=l i= 1
Problem 8.1. Do there exist in every infinite-dimensional Banach
space with a basis two normalized bases which are not affinely equivalent?
Of course, one can also consider, similarly to definition 8.1, affine
domination and affine equivalence of arbitrary sequences (instead of
bases) as well as the corresponding "strict" notions1.
The following notion of equivalence is also useful (see Ch. II, § 13
and §18):
Definition 8.3. We shall say that a basis {х„} of a Banach space E is
permutatively equivalent2 to a basis {>'„} of a Banach space F, and we
shall write {х„} ~ {у„}, if there exists a permutation a of the set
^={1,2,3,...} such that the sequence {.ve(B)} is a basis of ? and that
the bases {xe(B)} and {>>„} are equivalent.
1 It is easy to verify that {х„} is affinely equivalent to \yn\ in the sense of de-
definition 8.2 only if {х„} and {>>„} affinely dominate each other.
2 Naturally, one can also define permutative domination, but we shall not
use it in the sequel.

80 I. The Basis Problem. Some Properties of Bases in Banach Spaces
Remark 8.2. The relation of permutative equivalence of bases is
reflexive, symmetric and transitive. Indeed, reflexivity is obvious. Assume
now that {х„} ~ {yn} and let a be a permutation as in definition 8.3.
Then, by theorem 8.1, implication d) 6°=>d)l°, {xa(n)} x {yn\, i.e.,
there exists an isomorphism и of E onto F such that м(хст(м)) = yn (n = 1,2,...).
Consequently, м(хя) = ув-1(я)(и=1,2,...), whence {уя ,(n)} isabasisofF
equivalent to the basis {х„} of E, i.e., {у„} ~ {х„}. Finally, if {х„}
~ Ы ~ {*„}, say {хаая)}~{уя}, {y«,2(n)}~{zn}, then и{хаия)) = уа,
r(>v2(n)) = zn (и=1,2,...) for suitable isomorphisms u,v, whence
vu{xt,iaAn)) = v(ya2{n)) = zn (и=1,2,...) and thus {х„} ~ {zj. We shall
use script letters Ж,1®,... to denote the equivalence classes with respect
to the relation of permutative equivalence.
We shall need to generalize the notion of cartesian product of two
bases (§ 4, definition 4.6) to an infinity of factors. We recall that for a
sequence {?„} of Banach spaces we denote by (Ev xE2 x ...),2 the
Banach space of all sequences у={у„} such that у„еЕ„ (и =1,2,...)
and
= E Ы12 <co-
Proposition 8.3. Let {?„} be a sequence of Banach spaces and for each
n let En have a normalized basis {xkn)}, such that the norms of these bases
satisfy sup vu,n)(<oo. Then the sequence {ek} defined by
for j=\,...,
(8.19)
l;
N = 0,1,2,...
is a normalized basis of the space E = {ElxE2x •••),2, of norm
(8.20)
Proof. Let у={у„}е{Е1 хЕ2х ...),2 be arbitrary. Then, since у„еЕ„
and since {xi.n)} is a basis of En (и =1,2,...), there exist scalars 4"»
00
(k,n = 1,2,...) such that >-„= ? ^х[п) (и = 1,2,...). Put
k= 1
ft = 4n), (8-21)
where k,n are defined by the natural one to one correspondence e,-«->xiB)
induced by (8.19) (i.e., by е;={0,...,0,х1п),0,...}) and let e>0 be arbi-
2
trary. Then there exist an N = N{s) such that
g
||>>;||2 < — and an
Domination and equivalence. Affinely and permutatively equivalent bases 81
= m\_N{sj] such that
Уп
%k Xk
< — {n=l,...,N). Since
by (8.21) i E zilLl\--, E oi[N)xkN\O,O,-- > is of the form E }'¦<?,•,
l к = 1 к = 1 J i = 1
where M < go and where some yt = /3, and the other y, = 0, we obtain
M
у- Е 7iei
*= 1
{yi,...yN,0A...}-] Y.a^4l\..., E
' « \| / JV
= ( E IWlM+(E
<?,
which proves that [ek]=(El xE2x ...),2.
Now let pt be arbitrary scalars and define 4"' by (8.21). Then for
any integers N and j with Ky < 2 JV +1 we have
E"
j
E
k= 1
, 2^ ak
if
' .V + 1 N + l
k= 1
N+l
vfJVt^N) V „(JV+1) (iV+1) г, г,
к к ' ' ? • ¦ ¦
ak xk 7
if
(where ? af +
is the (JV+l)-th coordinate, i.e., for 7 = 2JV+1
the term E o^~N)x\/~N)= E o4iV+1)xliiV+1) does not occur), whence
k=l t=l
E
i= 1
E <
V-jv-i
E
k=l
E-
k= 1
+ E
n=j-N
N+l
if
E <4n)
/ (Xk Xk
if
6 Singer, Bases in Banach Spaces I

82
I. The Basis Problem. Some Properties of Bases in Banach Spaces
whence it follows that for any integers JV, and^ with
such that Nj+ji^N2+j, we have
N2+j
E fte,- < sup vtxir
Consequently, by §7, theorem 7.1, {ek} is a basis of (?t x?2x •••),2,
of norm satisfying (8.20), which (since obviously ||ej = l, fc = 1,2,...)
completes the proof of proposition 8.3.
Definition 8.4. Let {?„} be a sequence of Banach spaces and for
each и let En have a normalized basis {x(tn)}, such that the norms of
these bases satisfy sup v!x,n),<oo. The basis (8.19) of (?, x?2x •••),2
lSn<oo k
is called the cartesian product of the bases {x^"'} (и=1,2,...) and we
shall denote it by
x[l)}
x {х
[2)}
Let us define now the operations of cartesian product and infinite
power for equivalence classes of bases (with respect to the relation of
permutative equivalence).
Definition 8.5. Let E,F be two Banach spaces with bases {х„} and
{у„}, respectively, and let Ж and & denote the equivalence class of all
bases which are permutatively equivalent to the bases {xn} and {у„},
respectively. We shall call cartesian product of the classes Ж and <3/ and
we shall denote by Жх'З/, the equivalence class (with respect to the
relation of permutative equivalence) of the basis {х„} x {yn} of E x F
(see §4, definition 4.6). We shall call infinite power of the class Ж and we
shall denote by Ж*1 the equivalence class of the basis {ek} of (?t x E2 x ¦ ¦ -),2
defined by (8.19), where En = E and xkn) = xk (и,к = 1,2,...).
Remark 8.3. The definition of the classes Ж х <2/ and Жт does not
depend on the particular choice of the representatives from the classes
Ж and c&. Indeed, let {х'„} ~ {х„} and {y'n} ~ {>•„} be bases of E, E and
F, F, respectively. Then there exist permutations a l, a2 of Jf= {1,2,3,...}
such that {x^i(n)} and {y'a2in)} are bases of E and F, respectively, and
{х:.,„Л~(х„!, {^.,„Л~(у„К whence also {x;i(n)} »{х„}, {у^,„и~\уп}-
Let
by
t/kj ~~ l-V
{i)} ^
x {.Уп}, and define a permutation a of
(8.22)
Then {z;(k)} = {x;i(,1)}x{_4(,1)}, whence {z;(fc)} is a basis of E x F.
Furthermore, if u and i> are isomorphisms of E and F onto E and F,
respectively, with u(x'aiW) = xn, v(y'a2(n)) = yn (и=1,2,...), then the
mapping
{uxv)({x,y})={u(x),v(y)} (xeE,yeF)
(8.23)
8. Domination and equivalence. Affinely and permutatively equivalent bases 83
is an isomorphism of E'xF' onto E x F, satisfying (uxv)(z'a(k)) = zk
(fc = l,2,...), whence {z'a(k)}x{zk}, whence {z'k} ~ Jzk}. A similar argu-
argument shows that {x^} - {х„} implies {xj,} x [x'n\ x ¦¦¦ ~ {х„} х {х„} х •••.
Proposition 8.4. Let E,F,G be three Banach spaces with bases {х„},
{yn} an(l {г„}, respectively, and let Ж,<& and % be the equivalence class
of all bases which are permutatively equivalent to the bases {xn\, {у„\ and
{zn}, respectively. Then
a) fxf = f xf;
Proof, a) Let {zk} = {xn\ x {yn}, i.e., z2n_1 = {xn,0}, z2n={0,yn\
(и=1,2,...), and define a permutation a of ,Ж= {1,2,3,...} by
(тBи-1) = 2и,
(и = 1,2,...).
(8.24)
Then г„12н_1)={0,уя}, zaan)={xn,0} (n=l,2,...), whence {zaW}
{}'„} x {*„} ^ F xE, bythemapping {x,y}->{y,.x} {xeE, yeF), whence
(х„).
b) Since {х„} х ({у„} х {zn}) is the basis
{хь{0,0}}, {0,{>ч,0}}, {x2,{0,0}}, {0, {0,Zl}},...
and ({х„} x {yn}) x {zn} is the basis
we have {х„} х({у„} x {zn})^({xn} x {yn})x {zn}, via a suitable per-
permutation a of Ж and the mapping {x, {y, z}}->{{x,y}, z] (xeE, yeF,zeG).
c) Since {х„) x ({х„} х {х„} х •••) is the basis
{МОД...}}, {0Дхь0,0,...}}, {х2,{0,0,...}}, {0,{0,хь0,...}},...
and ({х„} х {х„} х • • •) is the basis
{х„0,0,...}, {0,хь0,...}, {0,х2,0,...}, {х2,0,0,...},...
we have {х„} х ({х„} х {х„} х •••)~({х„} х {х„} х •••), via a suitable per-
permutation a oft/T and the mapping {x,{y,z,...}}^>{x,y,z,...}(x,y,z,...eE)\
this mapping is an isomorphism by Ch. II, § 18, lemma 18.5.
d) Since ({х„}х{у„})х({х„}х{у„})х ••• is the basis
{{*i,0}, {0,Л}, {x2,0}, {0,j2},...} x {{хь0}, {0,У1}, {x2,0}, {0,y2},...} x •••
= {{xb0}, {0,0},...}, {{0,0}, {xb0}, {0,0},...},

84 I. The Basis Problem. Some Properties of Bases in Banach Spaces
and ({х„}х{х„}х •••)х({у„}х{у„}х •••) is the basis
{{.х„0Д...},{0Д...}}, {{0,0,...}, {уь0,0,...}},
{{0,х1Д0,...},{0,0,...}}, {{0,0,...}, {0,>-ь0,0,...}},...
we have ({х„} х {уп}) х ({х,,} х {>-„})ix ••• ? {{х„} х {х„} х •••) х ({>¦„}
х {>'«}х ¦")> у'а а suitable permutation a of Jf and the mapping
{{h,si}, if2^s2}» ¦•¦}-*{{fi>^2» •••}> {si>s2> •••}} (ti,f2» •¦ - e?, s1,s2,... eF);
by Ch. II, §18, lemma 18.5, this mapping is an isomorphism, which
completes the proof of proposition 8.4.
§ 9. Stability theorems of Paley-Wiener type
In this section as well as in § 10 we shall see that if a sequence {yn}
in a Banach space E is "sufficiently near" to a given sequence {х„} in E,
then {xn}x{yn} and, if [х„]=?, then {х„} %{>'„}. Hence it will follow
that various properties of sequences {x,,} in a Banach space E are
"stable" in the sense that they are conserved by every sequence \yn)
"sufficiently near" to the sequence {х„}.
In the literature, the emphasis in the formulations of stability theo-
theorems is put on the conservation of certain properties of sequences "suf-
"sufficiently near" to a given sequence {х„}. However, in our opinion, the
main assertions of the stability theorems are those of the strict equi-
equivalence, or full equivalence if [х„] = E, of "sufficiently near" sequences
{х„} and {>'„}, while all assertions of "stability" are consequences of
these assertions. In the sequel we shall carry through this point of view,
in the formulations of stability theorems.
The implications b) y) and b) 5) of the following theorem are called
the Paley-Wiener theorem (see the Notes and remarks for more details):
Theorem 9.1. Let {х„} and {y,,} be sequences in a Banach space E.
Assume that there exists a constant /., 0^л< 1, such that we have
(9.1)
for all finite sequences of scalars otl, a2,..., a,,. Then
a) We have {х„} ~{у„}- Consequently, we have the following equi-
equivalences :
а) х„^0(и=1,2,...) if and only if у„=?0(п=\,2,...).
P) {х„\ is finitely linearly independent if and only if {yn} is finitely
linearly independent.
}') {xn} !S co-linearly independent if and only if {yn\ is co-linearly
independent.
n
E«;
— i
[xi-Уд
n
i= 1
9. Stability theorems of Paley-Wiener type
85
<5) {х„} is minimal if and only if {у„} is minimal.
e) {х„} is a basic sequence if and only if {yn\ is a basic sequence.
b) // {xn} is complete in E, we have {xn} »{>•„}. Consequently, we
have the following implications1:
я) // {х„} is {an}-complete in E (а„^0, п = 1,2,...), so is {yn}.
P) V lxn} !S complete of order p in E A <p< oc), .so is {yn\.
}') // {х„} is complete in E, so is {yn}.
<5) //' {xn} is a basis of E, so is {у„}.
Proof, a) From (9.1) it follows that we have
whence
Ya-x-
L—i ' * 1
i= 1
-
(=1
i= 1
A-я)
Ea;xf
Еа!Л
= A+A)
Ea.x.-
(9.2)
for all finite sequences of scalars a1,a2, ...,а„. Hence, by the implica-
implication dJ°=>d)l° of §8, theorem 8.1, we have {х„}х{у„}. The equi-
equivalences a)- c) are now a consequence of § 8, definition 8.1.
b) Assume that [х„] = Е. Then, by part a) proved above, there
exists an (unique) isomorphism и of E into E (onto [>'„]), satisfying (8.2).
For this isomorphism we have, by (9.1), ||/? — u\\^?. (where IE denotes
the identical mapping of E onto E). Since л<1, it follows that the in-
verse mapping u~l=
of ? onto E. Hence {
— и)к exists and thus и is an isomorphism
{х„}%{у„}. The implications a) — S) are now con-
consequences of the properties of full equivalence (see § 8). This completes
the proof of theorem 9.1.
Remark 9.1. If we have (9.1) with
(9.1) we have
^, then, since by (9.2) and
E ai
for all finite sequences of scalars al5 a2, ¦
E Wi
1-Я
, я„ and since 0
1-Я
< 1,
from part b) above it follows that we have {xn}S{yn} whenever {yn}
1 Actually, from theorem 9.2 below it follows (see remark 9.4) that if we have
(9.1) and if {у„} is complete in E, then still {xn}x{yn}. Consequently, the converse
implications to a), j8), y), 5) of b) are also true.

86
I. The Basis Problem. Some Properties of Bases in Banach Spaces
is complete in E. We shall see below that here we may drop the restric-
restriction ).<\, i.e. the same conclusion holds if we have (9.1) with O^A<1.
Remark 9.2. Actually, b) is equivalent to y) of b). Indeed, we have
seen above that b) implies y) of b). Conversely, if {х„} is complete in E,
then, since by a) we have {х„}к{уп}, from y) of b) it follows that {%„}
% {>•„}. In view of this remark let us also give the following alternative
proof of the implication y) of b):
Assume that {xn} is complete in E but {yn} is not complete in E.
Then G=[yn~]=?E, whence, by a well known lemma of F. Riesz1, there
exists an xeE\G such that ||x|| = 1, dist(x, G)>A, whence
<ydist(.x,G).
(9.3)
For this x, by a corollary of the Hahn-Banach theorem2, there
exists an/e?* such that
/(y,) = 0 0=1,2,...),
1
dist(x,G)
Since the sequence {х„} is complete in E, there exists a sequence
such that
lim
Then, taking into account (9.1), we obtain
/
= lim
/
lim H/ll
lim
dist(x,G)«-*
dist(x,G)'
which contradicts (9.3) and completes the proof.
A comparing of the two proofs above of b) y) shows the power of
the operator technique used in the first proof. This operator technique
is useful in stability problems, as we shall see in the sequel.
1 See e.g. [10], p. 83, lemma.
2 See e.g. [10], p. 57, lemma.
9. Stability theorems of Paley-Wiener type
87
In the limit case A=l theorem 9.1 is no longer valid, as shown by
Example 9.1. Let {х„} be a basis of a Banach space E and let yn = 0
(и=1,2,...). Then {х„}, {yn} satisfy (9.1) with A=l, but {yn} is not
even complete in E.
However, one can prove e.g. the following positive result:
Proposition 9.1. Let {х„} and {yn} be sequences in a reflexive Banach
space E, such that {xn} >- {yn} and that
(9.4)
for all sequences of scalars {an} such that ? а,х,е?. Then, if {х„} is a
basis of E, {у„} is complete in E. '=1
Proof. Assume that {yn} is not complete in E. Then G=[yn]^?,
whence, since E is reflexive, there exists an xeE\G such that
||x|| = l=dist(x,G).
Then, by a corollary of the Hahn-Banach theorem, there exists an
feE* satisfying f{yt) = 0 (i=l,2,...), /(x)=l and ||/|| = l. On the
00
other hand, since {х„} is a basis of E, we have an expansion x= ? я,х,,
whence ' = i
Л
= 11x11 = 1
which is impossible. This completes the proof.
The next theorem and its corollary show that condition (9.1) of
theorem 9.1 can be replaced by a weaker condition which is symmetric
with respect to the sequences {х„}, {у„} and which implies that {xn}
«{у„}, respectively, that {х„}х{у„} if either [х„]=? or [>„] = ?.
Theorem 9.2. Let {xn} and {yn} be sequences in a Banach space E.
Assume that there exists a constant A, 0^A< 1, such that we have
(9.5)
for all finite sequences of scalars ab a2,..., а„. Then
a) We have {хп}«{у„}. Consequently, we have the equivalences
a) a) — e) of theorem 9.1.
n
V a(x — v-)
= 1
/
^A
\
n
i = l
+
i= 1

I. The Basis Problem. Some Properties of Bases in Banach Spaces
b) // either {х„\ or {yn} is complete in E, then {xn}S{yn}. Conse-
Consequently, we have the implications b) a) — <5) of theorem 9.1 as well as the
converse implications.
Proof, a) From (9.5) it follows that we have
1-Я
Г+я
!= 1
I ад
1+Я
(9.6)
for all finite sequences of scalars ab a2,..., an. Hence, by the impli-
implication dJ°=>d)l° of §8, theorem 8.1, we have {х„}«{у„}.
b) Assume that [х„] = ?. Then, by part a) proved above, there
exists an (unique) isomorphism и of E into E (onto [у„]), satisfying
(8.2). For this isomorphism we have, by (9.5),
(хеЕ).
(9.7)
We shall now show that и maps E onto E, which will complete the
proof. To this end, it will be sufficient to show that u* is one to one;
indeed, then u(E) = E (since otherwise there would exist an feE*,
/VO, with [m*(/)](x) = /[u(x)] = 0 for all xeE, whence u* would not
be one to one), whence, u(E) being complete (because E is complete
and и is an isomorphism), u(E) = u(E) = E.
Put
We claim that
1-Я
(xe?,-oo<a<e).
(9.9)
Indeed, assume, a contrario, that there exist an xeE and an ae( — go,г)
such that
Их)-ах||<е||х||. (9.10)
Put p = e — a and y = u(x) — ax. Then
= y + (e-/3)x, whence, by (9.7), (9.8), and (9.10),
and u(x) = y + ocx
and, on the other hand, again by (9.10),
a contradiction which proves the claim (9.9).
Now let
for all feE*,a<p\. (9.11)
9. Stability theorems of Paley-Wiener type
Then D=?0, since for Ро=~(Н1 H—) and апУ %<Ро we nave
|а|>||м|| Н—, whence
11«*(/)-«/11>(|я|-||«
and thus poeD.
Furthermore, let
= supp.
peD
(9.12)
We shall prove that e^E, whence, by (9.11) with a = 0 (and with
any peD such that 0<p<e),
> | ll/ll (feE*), (9.13)
which will complete the proof.
Assume, a contrario, that e<s and let a<e be arbitrary. Then, by
(9.9), u — aIE is an isomorphism of E into E. On the other hand, since
there exists a peD with a<p<e, we have
(9.14)
whence (u — aIE)* = u* — aIE, is one to one and consequently the iso-
isomorphism u — ocIE maps E onto E (by the same argument as that made
above for м). Furthermore, by (9.9),
1
whence
?
(9.15)
a crucial sharpening of (9.14). In particular, for a = e < e we obtain
«*(/)-«/--/
>\\u*(f)-af\\--\
(feE*),
and hence for any ft with e^/3<e + — we have
f)-Pf\\ = \\u*(f)-ef-(P-e)f\\>\\u*(f)-ef\\-(P-e)\\f\\

90
I. The Basis Problem. Some Properties of Bases in Banach Spaces
which, together with (9.14), shows that e H—e?>. However, this con-
contradicts the definition (9.12) of e and thus the proof of theorem 9.2 is
complete.
Remark 9.3. If we have (9.5) with 0^Я<^, one can give the follow-
following simple proof of b): For the isomorphism и of E onto [_у„], satisfying
(8.2), we have (9.7) and, by virtue of (9.6),
1+л
Nil
whence, taking into account
Consequently, there exists the inverse mapping u~l= ? {IE — u)k,
k = 0
and thus и is an isomorphism of E onto E and {х„} %{>'„}• This com-
completes the proof.
Remark 9.4. Condition (9.1) implies (9.5), and thus theorem 9.1 is
also a consequence of theorem 9.2; moreover, it follows that the impli-
implications <x) — S) in theorem 9.1 b) can be replaced by equivalences.
Corollary 9.1. Let {х„} and {yn} be sequences in a Banach space E,
satisfying one of the following two conditions:
1°. There exist three constants /с,ЯьЯ2 with k>0,0^ЯьЯ2
<min(l,21" <<), such that we have
(9.16)
Я',/i, v<l, such that
(9.17)
ф
\
п
Е
i= 1
а;х,-
+я2
п
Е
1= 1
for all finite sequences of scalars al, a2,..., «„.
2°. There exist three constants ?.',n,v with
we hate
п
ЕаД*;-)';)
— 1
2
<Я'
i= 1
2
-l-2/i
n
Еа.х<
i — l
п
У а-v-
[ = J
+ V
п
!= 1
/or all finite sequences of scalars rxl, a2, •••, «„¦
we /шг;е а) and b) of theorem 9.2.
( A
Proof. If we have (9.16), then for l = max(A1,/l2)max\4l,2'' ) we
have (9.5). If we have (9.17), then for A=[max(A',Ju,v)]i we have (9.5).
Hence in both cases we can apply theorem 9.2.
9. Stability theorems of Paley-Wiener type
91
Remark 9.5. Actually, the conditions occurring in theorem 9.2 and
corollary 9.1 are equivalent. In fact, we have seen that Г=>(9.5) and
2°^-(9.5). On the other hand, (9.5)=>1° with fc= 1,/1=Я2 = /., and
(9.5)=>2° withl' = fi = v = /.2. Consequently, theorem 9.locorollary 9.1,1°
ocorollary 9.1, 2°.
In the limit case Я=1 theorem 9.2 is no longer valid, as shown
again by example 9.1. Corollary 9.1, Г with /c^l is no longer valid in
the limit cases ?.l = l or Я2 = 1 and corollary 9.1, 2° is not valid if
a' = \ or v=l (takej.'n = 0(n= 1,2,...), respectively Е = 12,х„ = еп+1—еп,
yn=en+1(n=l,2,...), where {en} is the unit vector basis of I2). Corollary
9.1, Г with 0</c^l is no longer valid if Я1 = 22 = 21~? (take [х„] = ?
and у„ = и(х„) (и =1,2,...), where u:?->? is a linear isometry such
that и{Е)фЕ).
In the above conditions (9.1), (9.5), (9.16) and (9.17) the coefficients
of X; and >', have been the same. However, one can replace (9.1) (and
similarly, (9.5), (9.16) and (9.17) respectively) by a condition in which
this is no longer required. In fact, we have
Theorem 9.3. Let {х„} and \yn) be sequences'in a Banach space E.
Assume that there exists a constant Я, 0^Я<1, such that for every
finite sequence of scalars а,,х2,
satisfying
х„ there exist scalars ft,/?2, ...,/?„
E («,¦>'.-
Eft;
Еа;>';
(h=\,2,...,n). (9.18)
Then, if {xn} is a basis of [х„], the sequence {у„} is a basis of [у„].
Proof. By (9.18) we have
whence
and similarly
ft
i= 1
ft
1
ft
i= 1
1 + Я)
+ Я
-Я
ft
;Eft*.-
ft
i
= i
i= 1
ft
i= 1
1+Я
1-Я
E aiyt

92
I. The Basis Problem. Some Properties of Bases in Banach Spaces
On the other hand, since {xn} is a basis of [х„], there exists, by §7,
theorem 7.1, a constant C^ 1 such that we have
I,
I
i= 1
PiX,
П
I
i= 1
PiXi
Consequently, we have
i= 1
1-Я
i= 1
IP,X,
n
С
L/U-
i= 1
for every finite sequence of scalars ac1, a2,..., а„. Hence, by § 7, theorem
7.1, {>>„} is a basis of [>„], which completes the proof.
Condition (9.18) is a weakening of (9.1) (and even of (9.5)), but we
lose some of the conclusions of theorem 9.1, namely, (9.18) does not
imply that [х„] and [>„] are isomorphic (hence (9.18) does not imply
that {х„}к{уп}), as shown by
Example 9.2. Let ? = cox/\ endowed with the norm ||{x, >>}||
= max(|)x||Co, ||}>||/i)- ^ W} (' = 0,1) denote the unit vector bases of c0
and I1, respectively, let
xn={e<°),0},
= 1,2,...
(9.19)
Then {х„} and {yn} are basic sequences in E, satisfying (9.18), but
[х„] is not isomorphic to [yn~\ (and hence {х„}#{у„}).
Indeed, for any finite sequence of scalars a1;..., а„ let
i=l n).
(9.20)
Then we have
= maxl sup
\lsSisSft
_ 1
= i
i.e., (9.18) with 1 = ^. However, [xn]sc0 and [yj^/1 and thus [х„]
is not isomorphic to [_yn].
h
i= 1
+
i= 1
10. Other stability theorems 93
§ 10. Other stability theorems
The following theorem is a consequence of §9, theorem 9.1, but we
state it separately because of its importance for applications.
Theorem 10.1. Let E be a Banach space, (х„,/„) ({х„} с Е, {/„} с ?*)
a biorthogonal system, and {yn} a sequence in E. If
Illx.-^IMI/iH^i, (lo.i)
;= l
t/геи we have (9.1). Consequently, {х„}&{у„} (and hence {yn} is minimal)
and we also have b) of theorem 9.2.
Proof. If we have A0.1), then, for any finite sequence of scalars
n
acua2,...,an we have, putting x= ^а;Х;,
and thus we can apply § 9, theorem 9.2.
Remark 10.1. In particular, condition A0.1) is satisfied if
sup ||/„||<со and
1«я<ш ¦*- 1
У llx,-v,ll < — (Ю.2)
sup
(since
sup ||/„|| Х||х,--}>;||) and conversely, from
i=l , 1
this particular case of theorem 10.1 it follows already theorem 10.1 in
the general case (considering the sequences {х„||/„||}, {}>„||/и||})-
Remark 10.2. The conclusion of theorem 10.1 also holds under
weaker assumptions, namely, condition A0.1) can be replaced by any
one of the weaker conditions
sup sup
^ л < со xeE
SUp SUp Y,\\Xi-yi\\\fi(x)\^A.<l,
Un<co xeE ¦_ i
llllSl
or if sup ||/J <oc, by
1 S
sup II
Н/1И1
1
«
A0.3)
A0.4)
A0.5)

94
I. The Basis Problem. Some Properties of Bases in Banach Spaces
Indeed, it is obvious that A0.4)=>A0.3) and the above proof of
theorem 10.1 shows that A0.3)=>(9.1). Finally, if sup ||/„||<оо, let
1 ^n < oo
ab..., an be arbitrary scalars and take a gneE* with \\gn\\ = 1 such that
/ "
9n\ 1«,.(Х;-^
V;=i
and by A0.5),
n
?«.¦(*,¦-}',¦)
i= 1
\
n
^a.fX;-});)
1=1
П
= Ха,#„(х;-}>,)«:
i=l
sup 11/,-H
L ^ j < cc
n
X XiXi
. Then,
; max \<Xj
n
Z \вп(Х
i=l
txixi)(j=l,...,n)
sup ||
which, since /i sup ||/-|| <1 (by A0.5)), completes the proof.
If we weaken A0.1) by replacing X<\ with A<oo, we obtain
Corollary 10.1. Let E be a Banach space, (х„,/„) ({х„} c E, {/„}<=?*)
a biorthogonal system and {yn} a sequence in E. If
A0.6)
t/геи t/геге exists a positive integer n0 such that
a) {xn}Zx {У»}по (and hence {yn}?0 is minimal).
b) 7/ (хЛто 's a foas's °/ Мад ("'" particular, if {х„} is a basis o/ ?;,
then {yn}Z0 is a basis of [yn~]^0-
c) // {х„} is a basis of E, then the sequence {xn}"°~lu {yK}Z is a
basis of E, fully equivalent to {х„}. Hence, in this case1
Proof. If we have A0\6), then there exists a positive integer n0 such
X
that ? ||x, —y;|| ||/j|| <1. Hence, by theorem 10.1 applied to the se-
sequences {х„}вд, {}>„}вд, we have a) and b). Furthermore, from theo-
theorem 10.1 applied to the sequences {х„}, {х„}"°~1и {у„}°0 we have c),
which completes the proof.
In the situation of c) above, we also have the following results:
Theorem 10.2. Let E be a Banach space with a basis {х„} and let {yn}
be a sequence in E, satisfying A0.6), where {/„} is the sequence of coeffi-
coefficient functionals associated to the basis {xn}. Then
1 We recall that if ? is a Banach space and G a closed linear subspace of E,
then, by definition, codimE G = dim E/G.
10. Other stability theorems
95
a) The following statements are equivalent:
1°. {yn} is complete in E.
2°. {yn} is oi-linearly independent.
3°. {yn} is minimal.
4°. {yn} is a basis of E.
5°. Ы*{уя}.
b) The sequence {yn} can be transformed to become a basis of E, by
changing suitable к elements of it, where /c = codim?[_yn] < со.
c) Among the possible relations of co-linear dependence
there exist к relations
A0.8)
A0.9)
where к = codim? [у„], such that any other relation A0.8) is a linear
combination of the relations A0.9).
Proof, a) 2°=>5°. By A0.6) the mapping
s(x)=f1fi(x)(xi-yd
A0.10)
is well defined on E and it is an endomorphism of norm ^ a. Conse-
Consequently, the mapping u = IE-s (where IE is the identical mapping of ?
onto E) is continuous. Since we have
) = x-s(x)=
(xeE),
A0.11)
and since {>>„} is co-linearly independent, и is one to one. On the other
hand, the mapping s is compact, since by A0.6) we have lim||s — t'J =0,
n и—ев
where vn(x) = Х/,(х)(х,-}>;) (xeE, и = 1,2,...). Consequently1, u = IE-s
i= 1
maps E onto E. Hence, by the inversion theorem of Banach, и is an iso-
isomorphism of E onto E, which satisfies, by A0.11), и(х„) = у„ (и =1,2,...).
Thus {х„}~{Л}-
1 We apply here the following well known result (see e.g. [10], p. 154, theo-
theorem 14): If for a compact s the equation x — s(x)=0 admits the unique solution
x=0, then for every yeE the equation y = x — s(x) has a solution x.

96
I. The Basis Problem. Some Properties of Bases in Banach Spaces
The implications 5G=>4O=>3O=>2° and 5°=>1 are obvious.
Г=>2°. Assume that 2° is not satisfied, i.e. that we have A0.8) with
sup |ая|#0. Then there exists a positive integer; with
such that aj#O, where n0 is as in corollary 10.1 (since otherwise
sup |а„| = 0, because {yn}^0 is minimal). Hence y~ ]
and consequently [_у„] = [>>„]„Фу Since by corollary 10.1 c) we have
codim?[_yn]^ = n0 — 1 and since l^j^n0 — 1, it follows that
codim?[_yn]^l, i.e. that Г is not satisfied.
b) If /c = codim?[_yn]=0, then, by the implication l°=>4° of part a),
proved above, {yn} is a basis of E. If 1^/с^и0 — 1, where n0 is as in
corollary 10.1, then among the elements yi,y2,.-.,yno-l there exist
no-k-l elements, say y'1,y'2,...,y'no-k-l, such that \уп~\ = {{у'п}"Гк~1
и {>>„}?] and that {y
j \no~k- 1
¦n/1
u
>'„}вд is a basis of [_у„]. Thus, replacing
the к elements, say y'no-k,y'no-k+1,-~,y'«o-i of the set {^}
\{y'i>y'2> ¦¦¦> Упо-к-1}> by a basis {zn}\ of an arbitrary subspace G of E
such that G® |>Я] = Е, we obtain a basis {zj^u Ш^"^ {у„}„°°0
of E.
c) If /c = codim?[_yn]=0, then, as we have seen above, there exists
no relation A0.8) with sup |а„|#0. If 1 ^k^n0 — 1, where n0 is as in
1 ^ Л < 00
corollary 10.1, rearrange the finite sequence {yn}"ioi into {у'„}Т~к~1
u {УпУпГ-l as in part b) above. Then, since y'no_k, ...,y'no-
иЫЯ and since ШТ^^
there exist к relations of the form
o [{}
{,,„}« is a basis of [{/в}Т>~*~М)'..}"о]>
which, obviously, can be rewritten in the form A0.9). Consider now an
arbitrary relation of the form A0.8). Then
к по —к - 1 go
-HfX'no-jy'no-j= T, ^i+Z
j — 1 i=l i = no
A0.13)
where {a^}"° i is the rearrangement of {а„}"° i corresponding to
ШТ~\ and since {y'n}1°~k~lv {yn}% isabasisof [{у»}"*"^^}^],
the coefficients a-, a,- in the second member of A0.13) are uniquely de-
determined. Hence, taking into account A0.12), we obtain
10. Other stability theorems
97
Consequently,
oo It no-k-l к со к
о=Т*х1У'=Т<<хпо~]Упо-]- Z IXo-.Ao-j..}';- Z IXo-Ao-;,;>'¦¦
i= 1 j=l i=l j= 1 i = no J = l
It Г no-k-l =o "j
= ZK»o-J Упо-j- Z Pno-j.ty'i- Z Pno-j,iyi I
j= 1 L i = 1 i = "o J
which completes the proof.
Remark 10.3. In particular, condition A0.6) of corollary 10.1 and
theorem 10.2 is satisfied if sup ||/„|| < со and
A0.14)
i= 1
(and in the particular case when 0< inf ||/J ^ sup \\fn\\ < со, con-
1^П<0О 1^П<СО
ditions A0.6) and A0.14) are equivalent, since in this case we also have
ZllXi-ttll < ¦ , ., r ,, Z IIxi~У>II II/.-ID and conversely, this parti-
i=i , mf II;B|| ,-=i
1 Sn< oo
cular case of corollary 10.1 or theorem 10.2 implies already corollary
10.1 or theorem 10.2 in the general case (considering the sequences
{хя||/Л},Ы1/.И})-
Remark 10.4. The conclusions of corollary 10.1 and theorem 10.2
also hold under weaker assumptions, namely, condition A0.6) can be
replaced by any one of the weaker conditions
lim sup
n-* oo xeE
z /¦
i = n+ 1
=o,
A0.15)
00
Z \\xi~yi\\fi is unconditionally convergent, A0.16)
i= 1
A0.17)
or, if sup ||/J<co, by
1 irl< 00
QO
Т,(Х1~У>) is unconditionally convergent.
i= 1
Indeed, A0.16) implies (by Ch. II, § 16, lemma 16.1)
lim sup J ||х;-^|И/;(х)|=0,
which obviously implies A0.15). Furthermore, if we have A0.15), then
there exists a positive integer n0 such that
7 Singer, Bases in Banach Spaces I

98
I. The Basis Problem. Some Properties of Bases in Banach Spaces
sup sup
по ^ n < x1 xeE
Z Я
< 1,
i.e., we have A0.3) for the pairs of sequences {х„}^, {у„}™0 and {х„},
(xn}io1 u {>'n}n0' whence, by remark 10.2, the desired conclusion
follows. Finally, A0.17) implies (by Ch. II, § 16, lemma 16.1)
lim sup
л-x. feE*
\f(Xi-yi)\ =
whence, if sup ||/J < со, there exists a positive integer n0 such that
sup
sup
i.e., we have A0.5) for the pairs {xn}Z, Ш%, and {х„}, {х,}Г'и {}/„}?.
Let us also observe that in the case when {х„} is a basis of E, con-
condition sup ||/„|| < oo in remarks 10.1-10.4 can be replaced by the
1 S»< 00
equivalent condition inf ||х„|| >0 (by § 3, corollary 3.1a)).
1 ^ П < GO
The following theorem and its corollary 10.2 below will be called
the Krein-Milmann-Rutman theorem (see the Notes and remarks for mo-
motivation):
Theorem 10.3. Let E be a Banach space and let {xn) be a minimal
sequence in E. Then there exists a sequence of constants yn > 0 (n = 1,2,...)
with the following properties; if a sequence {>'„} с Е satisfies
(и=1,2,...),
A0.18)
then we have (9Л) with k<\. Consequently, {хи}~ {yn} (and hence {у„}
is minimal) and we also have b) of §9. theorem 9.2.
Proof. Since {х„} is minimal, there exists, by the implication 1°=>3°
of §6, theorem 6.1, a sequence of constants Sn>0 (и=1,2,...) such
that we have
|а(|<5,-
I
for all finite sequences of scalars
«„«j, ...,«„. Then, putting 'Уп = л5п(п= 1,2,...), where A<1, we have,
by A0.18),
E «Л*;-)',)
E
E aixi
and thus we can apply § 9, theorem 9.2.
10. Other stability theorems
99
Remark 10.5. Let us also mention the following alternative proof of
theorem 10.3. Since {х„} is minimal, there exists, by the implication
1°=>2° of §6, theorem 6.1, a sequence {/„}<=?* such that (х„,/„) is a
1
biorthogonal system. Putting in A0.18) yn = -^—— (n=l,2,...), we
¦^ II 7л II
have then A0.1) with k^j, and thus we can apply theorem 10.1.
An important immediate consequence of theorem 10.3 is the fol-
following:
Corollary 10.2. // E is a Banach space with a basis [x,,J, then in
every dense subset D of E there exists a sequence {yn} с D which is a
basis of the space E. In particular, there exists a basis {у„} of E of the
form
Уп= ? а(.")х. (и=1,2,...). A0.19)
;= l
We shall consider now the problem, which sequences yn > 0 (и = 1,2,...)
have, for a given minimal sequence {х„} <= E, the property occurring in
theorem 10.3.
Theorem 10.4. Let Ebea Banach space and (х„,/„) ({х„} <= Е, {/„} <=. E*)
a biorthogonal system. In order that a sequence }'„>0 (и =1,2,...) have
the property that every sequence {yn} <= E satisfying A0.18) also satisfies
(9.1) with X<\ (and hence is strictly equivalent to {х„}) it is sufficient
that
b = sup sup
I Ej I = 1 1 S Л < 00
Z v/;/-
< 1.
A0.20)
In order that every sequence {у„} <= E satisfying A0.18) be finitely
linearly independent (in particular, in order that every {у„} <= E satis-
satisfying A0.18) be minimal, or strictly equivalent to {xn}, or satisfy (9.1)),
it is necessary that
bs=l. A0.21)
Proof. Sufficiency. Assume that A0.20) holds. Let {у„) с Е satisfy
л
A0.18) and let <xl, ...,«„ be arbitrary scalars. Then, putting x = ^ а;х;
i= 1
and e, = sign/(x) for /(x)/0, e, = l for /(x) = 0, we have |e,-| = l and
\f(x)\yi
n
E *;(*.—Л)
;= l
=
„
i= 1
П
= E ef/iWVi <
= 1
П
< E i/iWHi^-^ii < E \J
i=l i=l
л
E ^/-
,¦ =
1
[•tjl
||x||sSb
n
E а;х;
i= 1
i.e., (9.1) with k = b<\.

100
I. The Basis Problem. Some Properties of Bases in Banach Spaces
Necessity. Assume that b>\. Then there exists a functional
n
fo=T.$>)yifi with N0)l = l (»=1 и) such that \\fo\[Xj]\\
i= 1
m
= sup |/0(p)|>l, whence also a polynomial p = Y a,x,eP such that
P
Then /0(z0)=l and ||zo|| =-—
1
ting
we obtain
and
х._й(°)л..2 fOr j=] n
X; for ( = и+1, и + 2,...
A0.22)
1. Hence, put-
A0.23)
\\xt-yt\\
('¦= 1Д •••)
i = я + 1 i = l
i = 1
= zo-zo/o(zo) = O if
Thus, {}'„} satisfies A0.18), but {у„} is not finitely linearly inde-
independent. This completes the proof of theorem 10.4.
Corollary 10.3. Let E be a Banach space, (х„,/„) ({х„} с Е, {/„} с ?*)
an E-complete biorthogonal system such that [/„] contains no subspace
isomorphic to c0 and let у„>0 (п= 1,2,...). Тйе following statements are
equivalent .-
1 °. There exists a positive integer n0 such that every sequence {yn} <= E
satisfying A0.18) also satisfies (9.1) "from n0 on", i.e.,
A0.24)
Z a'(X
= no
n
Z a;x.-
i = «o
for all scalars ocno, ocno+1, ...,«„ and some constant X<\.
10. Other stability theorems
101
2°. There exists a positive integer n0 such that for every sequence
{yn} a E satisfying A0.18) we have {х„}?« {у„}„°°0-
3°. There exists a positive integer n0 such that for every sequence
{yn} с Е satisfying A0.18) the sequence {у„}^0 is minimal.
GO
4°. Yj У if is unconditionally convergent.
i= 1
Proof. The implication 1°=>2° is a consequence of § 9, theorem 9.1.
Since {xn}^0 is minimal, the implication 2° => 3° is obvious.
3°=>4°. If we have 3°, then by [xj = ? and by the necessity part of
theorem 10.4 we have
sup sup
and therefore
sup sup
sup sup
|?f I — 1 По^ П < 00
I Mfi
I Wfi
no— 1
i= 1
(Ю.25)
Hence, by Ch. II, § 15, lemma 15.1 and lemma 15.8, ? у,/; is un-
unconditionally convergent. i='
4°=>1°. If we have 4°, then, by Ch. II, §16, lemma 16.1, there exists a
positive integer n0 such that
sup
ФЕ"
Let |e,-| = l (i=l,2,...) and и be arbitrary. Take a fe?** with
|| У || = 1 such that V
e,yifl} =
. Then
T, ?iVi/i
= I E.
sup
11*11=5 1
whence bno = sup sup
: — < 1. Consequently, by the
sufficiency part of theorem 10.4 and by [х„] = ?, we have A0.24),
which completes the proof of corollary 10.3.
In the necessity part of theorem 10.4 the condition b^ 1 cannot be
replaced by b<\, as shown by
Example 10.1. Let E = c0, {xn}=the unit vector basis of c0, {/„} = the
a.s.c.f. to {xn} and

102
I. The Basis Problem. Some Properties of Bases in Banach Spaces
Then b= sup sup
| =
Y -f-
1
A0.26)
= Y, — — 1' but for every sequence
1
{yn}aE satisfying A0.18) we still have {х„}~ {>>„}.
Indeed, assume that {yn}aE satisfies A0.18) with у„ = —. Ob-
serve first that for every ? a,-x,eco\{0} there exists an index i0 such
;= l
that |a,-J< max |a,| and hence
I «;(*,--Л
max
i= 1
. A0.27)
Furthermore, for any scalars a2,...,an we have
i = 2
( = 2
д l i.A
" la I " 1
I ~< I ^ 2m?<Jaj
i=2
}1,,}^, i-e-> there exists an iso-
isoand hence, by §9, theorem 9.1,
morphism и of [xB]f onto [yn~]':
Фп) = Уп (и = 2,3,...). A0.28)
Therefore, since {х„} is a basis of E, in order to prove that {х„}«{_yn}
it is sufficient to prove that .Vi^lJvl^- Assume, a contrario, that
У1е1Уп]2- Then, since by A0.28) {yn}% is a basis of [у„]^, we have an
00 ^
expansion _yx= ^^Уь whence, again by A0.28),
i=2
i=2
Consequently, taking into account A0.27), we obtain
i = 2
a contradiction. This proves that {х„}»{_у„}.
Finally, assume that [yn]#? = c0. Then there exists a continuous
linear functional w
f(x)=?tidi (x = {Qec0) A0.29)
i= 1
such that ||/|| = 1, f(yn) = O (и=1,2,...). Hence, by A0.18) for yn = -,
10. Other stability theorems
103
which, together with ^|?ув| = ||/|| = 1, implies
n= 1
\Чп\ = \\Хп-у„\\ =y» (и=1'
00
However, if yl= Y^)xb tnen
i= 1
A0.30)
whence l/^'l^j and hence, by A0.30), l^i/?^1'! ^ i- Furthermore, since
lim/n(x1-j;1) = 0 and ||x, -yj =i#0, we have
1
i = 2
i=2
1 " 1 1
Consequently,
о=|/Ы1 =
R(D
i=2
a contradiction, which completes the proof of the assertions of example
10.1.
In theorem 10.5 below we shall see that such an example is no longer
possible if E is reflexive and that in the sufficiency part of theorem 10.4
the condition b<\ cannot be replaced by b^l.
Let us first prove
OO
Lemma 10.1. Let ? x; be an unconditionally convergent series in a
i= 1
Banach space E. Then the set
м= ;>>*¦¦
e,- =-
(i=l и), и=1,2,..
A0.31)
is conditionally compact.
Proof. Let e>0 be arbitrary. We shall find a finite e-net for M,
which will complete the proof.
By Ch. II, §16, lemma 16.1 (implication 1°=>6°) there exists a posi-
positive integer N = N(e) such that
sup
l/(x.-)l < ^ •
1
A0.32)

104 I. The Basis Problem. Some Properties of Bases in Banach Spaces
Let {e;} with |e;| = l (i'=l,2,...) be arbitrary and let p be an arbi
trary positive integer. Take a geE* with ||#|| = 1 such that g\ 2, E;x;
N + p
. Then, by A0.32),
i = N + 1
N + p
N + p
N+p
On the other hand, the subspace ?JV=[x1,...,xJV] of ? is finite-
dimensional and the set
и=1,...,ЛГ> A0.34)
in this subspace is bounded (e.g. by the number У,!|х;||) and hence
;= l
conditionally compact. Let yu...,yt be a finite —net for M, and let
n n
be arbitrary. If «<iV, then ^е;х;еМ, and hence there
exists a v; such that
i= 1
j and hence there exists a ^ such that
i= 1
< — < e. If и > N, n = N + p, then
E_
~2'
which, together with A0.33), implies
N + p
T, ?ixi~yj
JV
l?ixi-yj
i- 1
+
N+p
У EX
i = N + 1
Thus, y1,..., уi is a finite e-net for M, which completes the proof of
lemma 10.1.
Theorem 10.5. Let E be a reflexive Banach space, (*„,/„) ({х„} <= E,
{fn}<=^E*) a biorthogonal system and yn>0 (и = 1,2,...). The following
statements are equivalent:
1°. Every sequence {yn}aE satisfying A0.18) also satisfies (9.1) with
A<1.
2° Every sequence {_у„}<=? satisfying A0.18) is strictly equivalent
to {*„}.
3°. Every sequence {yn} <=? satisfying A0.18) is minimal.
4°. For the number b defined by A0.20) we have
b<\.
A0.35)
10. Other stability theorems
105
Proof. The implication 1°=>2° is a consequence of §9, theorem 9.1.
Since {х„} is minimal, the implication 2°=>3° is obvious.
3°=>4°. Assume now that we have 3°, but not 4°, i.e., b^l. Then,
by 3° and theorem 10.4, b^l and hence b = l. Therefore the set
E Sibfi
i = 1,..., n),
A0.36)
is bounded, whence, by Ch. II, § 15, lemma 15.1 (implication 4°=>6°)
andCh. II, §15, lemma 15.8, the series ? yj{
is unconditionally con-
vergent. Consequently, by lemma 10.1, the set A0.36) is conditionally
compact and hence there exists a functional foeE* with fo\[Xj]
, where |б(;п)| = 1, such that
\= SUP SUP
|?i| = l l«k<o
= b=l.
A0.37)
Since ejn)y,= У ?<iI)yifi(xJ)-+fo(Xj) as и->оо, the limits li
1
= -fo(xj) (/=1,2,...) exist and |бH)| = 1 (/=1,2,...)- Hence, by A0.37)
and since by our assumption [xj is reflexive, there exists an element
zo=lim Yj af)xjeLxj] such that
||zo|| = l=/o(zo)=lim
П>СО
Put
Then, by A0.39) and A0.38),
A0.38)
A0.39)
\\xi-yi\\ = \\e\(»yiz0\\=yt
lim
а}">х(-г0 ton
and, since ||zo|| = 1, there exists at least one index i such that
Thus, {yn} satisfies A0.18) but {у„} is not minimal (by §6, theorem
6.1, implication Г=>4°), a contradiction with the assumption that we
have 3°.

106
I. The Basis Problem. Some Properties of Bases in Banach Spaces
Finally, by theorem 10.4, the implication 4°=>1° is valid for any
Banach space E. This completes the proof of theorem 10.5.
For stability properties another terminology is also used. Namely,
it is convenient to give the following definitions.
Definition 10.1. Let {х„} and {yn} be sequences in a Banach space E
and let
pn = \\xn-yn\\ (и=1,2,...). A0.40)
The sequence {yn} is said to be
a) PW-near to {х„}, if there exists a constant A, O^A<1, such that
we have (9.1) for all finite sequences of scalars ab a2,..., an;
b) PH-near to {xn}, if there exist three constants /с, АЬА2 with
k>0, 0^A,,A2<min(l,21^^) such that we have (9.16) (as we have
observed in § 9, this is equivalent to the existence of a constant A, 0 ^ A < 1,
such that we have (9.5)) for all finite sequences of scalars a1;a2,...,an;
c) N-near to {xn}, if there exist three constants k',fi,v with
0 ^ А', ц, v < 1 such that we have (9.17) for all finite sequences of scalars
d) in the {y^-neighbourhood of {х„}, where {yn} is a sequence of
positive numbers, if we have
pn<cTn (И=1,2,...); (Ю.41)
e) near to {х„}, if we have
limpB = 0;
П-* 00
f) strictly near to {xn}, if we have
A0.42)
A0.43)
g) weakly near to {*„}, if we have
lim sup
A0.44)
or, equivalently (by Ch. II, §16, lemma 16.1), if Х(х,-}>;) is uncon-
unconditionally convergent. i = 1
If (xn,fn) ({xn} <= E, {/„} с Е*) is a biorthogonal system, a sequence
{yn} <=. E is said to be
h) KL-near to {xn}, if
|>Ш1<«>; (Ю-45)
10. Other stability theorems
i) strongly KL-near to {xn}, if
107
A0.46)
Definition 10.2. A property SP of a sequence {х„} in a Banach space
E is said to be
a) PW-stable, if every sequence {yn} cE, which is PW-near to {х„},
has property SP;
b) PH-stable, if every sequence {у„} <=. E, which is РЯ-near to {х„},
has property вР;
c) N-stable, if every sequences {_у„} с ?, which is iV-near to {х„},
has property SP:
d) stable, if there exists a {yn}-neighbourhood V of {xn} such that
every sequence {jJeK has property SP;
e) strongly stable, if every sequence {у„} <=. E, which is near to {xn},
has property SP;
f) strictly stable, if every sequence {>>„} <= E, which is strictly near
to {xn}, has property 0>\
g) weakly stable, if every sequence {_у„} c ?, which is weakly near
to {xn}, has property SP.
If (х„,/„) ({х„} с ?, {/„} <=. E*) is a biorthogonal system, a property
^ of {х„} is said to be
h) strictly KL-stable, if every sequence {yn} <= ? which is KL-near
to {xn}, has property 3P\
i) KL-stable, if every sequence {_у„} <= ?, which is strongly KL-near
to {xn}, has property SP.
Then § 9, theorem 9.1 shows that the properties of a sequence {xn} <= E
of being a) constituted of non-zero elements, ft) finitely linearly inde-
independent, y) co-linearly independent, 6) minimal, e) a basic sequence,
?) complete (of any type) in E, and n) a basis of E are PW-stable. Theo-
Theorem 9.2 and corollary 9.1 show that these properties of {х„} are also
РЯ-stable and iV-stable. Theorem 10.1 shows that if (х„,/„) is a bi-
biorthogonal system, the properties 6) — r\) of {xn} are KL-stable. Theo-
Theorem 10.2 and remark 10.4 show that if {х„} is a basis of E with the
a.s.c.f. {/„}, then the property of {xn} of being a basis is strictly KL-stable
and, if {xn} is a bounded basis, weakly stable, in various classes of se-
sequences. Consequently, the property of being a bounded basis is strictly
stable in the same classes of sequences. Finally, theorem 10.3 shows
that the properties ё) — п) are stable.
Every sequence {yn} a E is in the {yn}-neighbourhood of any
sequence {х„} <= E, for у„=||х„ — у„\\ (и=1,2,...). We also have, obvi-
obviously, the implications: PW-near => iV-near; strictly near => weakly
near => near and, if {xn} is a minimal sequence, strongly KL-near

108
I. The Basis Problem. Some Properties of Bases in Banach Spaces
=> XL-near. By §9, remark 9.5 we have iV-near <=> РЯ-near. From
theorem 10.1 it follows that strongly KL-near => PW-near, while from
the proof of theorem 10.3, given in remark 10.5, it follows that if V
1
denotes the
ЛП+ 1
-neighbourhood of {х„}, then {у„} e V => {yn}
is strongly KL-near to {*„}, whence also PW-near; on the other hand,
obviously, PW-near => {yn} belongs to the {||xj}-neighbourhood of
Concerning any property & of a sequence {xn}, we have, obviously,
the implications: strongly stable => weakly stable => strictly stable
=> stable (the last of them follows taking {yn} to be a sequence of positive
numbers satisfying ^ y; <co) and strictly KL-stable => KL-stable.
i= 1
From the remarks made above it also follows that РЯ-stable <=> iV-stable
=> PW^stable => КL-stable => stable and that, if {xn} is a bounded
basis, strictly .KL-stable <=> strictly stable.
Finally, let us mention that certain stability properties characterize
bases among complete minimal sequences {xn} with inf ||xJ>0.
For this purpose, let us give Kh<c°
Definition 10.3. A complete minimal sequence {xn} in a Banach
space E is called
a) strictly stable, if every co-linearly independent sequence {yn} <= E,
which is strictly near to {х„}, is complete in E;
b) weakly stable, if every co-linearly independent sequence {у„} <=. E,
which is weakly near to {xn}, is complete in E.
Theorem 10.6. Let {xn} be a complete minimal sequence in a Banach
space E, with inf ||х„||>0. The following statements are equivalent:
1°. {х„} is a basis of E.
2°. {xn} is strictly stable.
3°. {xn} is weakly stable.
Moreover, if one of these statements holds, then every co-linearly
independent sequence {yn} <= E, which is weakly near (or, in particular,
strictly near) to {xn}, is a basis of E, equivalent to {xn}.
Proof. The implication 1°=>3° and also the last statement of the
theorem, are contained in theorem 10.2a) (implication 2°=>5°), taking
also into account remark 10.4.
The implication 3°=>2° is obvious.
2°=>1°. Assume that {х„} is weakly stable, but not a basis of E.
Then there exists an xoeE such that ? У|(хо)х; does not converge,
i= i
11. An application to the basis problem
109
where {/„} с Е*, /;(Xj) = <5,-j (i,j= 1,2,...). Hence there exists an index и
such that /„(xo)#O. Put
Xj for i = l, ...,n— 1, n+l,n + 2,...,
A0.47)
1
х„ -
x0 for i = n.
Then
Xf-V, =
is strictly near to {xn}. Furthermore, we show that {yn} is co-linearly
CO
independent. Indeed, assume that ]T a,}>; = 0. Then, applying f to the
relations i=1
I *jXj= I Zj(Xj-yj) =
777 I ZjfniXj) =
Jn\xO) j=l Jn\xO
«„,
we obtain
fi(x0)
a,- = ——- а„
fn(x0)
A0.48)
Since Y, aixi converges and ? Mxo)xi does not converge, the
i = 1 i = 1
equalities A0.48) are possible only if а„ = 0, whence a, = 0 (i=l,2,...),
which proves that {yn} is co-linearly independent.
Since by our hypothesis {х„} is strictly stable, it follows that {yn}
must be complete in E. However, this contradicts the relations
»(}>;) =/»(*.-) - 77~/»W = <
Thus, 2°=>1°, which completes the proof of theorem 10.6.
Some other types of nearness of sequences and some other stability
theorems will be given in Ch. II, § 11 and § 16.
§ 11. An application to the basis problem
The basis problem (see § 1, problem 1.1) seems to have a negative
answer, i.e. it is probable that there exists a separable Banach space
which has no basis. In this section we shall give a sufficient condition
for the existence of a separable Banach space having no basis, which
suggests a possible method of constructing such a space.

по
I. The Basis Problem. Some Properties of Bases in Banach Spaces
For a Banach space E and a closed linear subspace G of E we shall
denote by ^(E,G) the set of all continuous linear projections of E
onto G (i.e. the set {иеЦЕ,Е) | u2 = u, u(E) = G}). We shall denote the
statement "G is a closed linear subspace of the Banach space E" by
G<E. A1.1)
For an arbitrary Banach space E let us consider the functions
inf ||u|| (и=1,2,...). (П.2)
ф„(Е)= inf
G<E
dim G = n
We have then
A1.3)
If ? is a Hilbert space, we have
ф„(Е)=1 (и=1,2,...). A1.4)
Furthermore, from §4, theorem 4.1 it follows that for every Banach
space E with a basis {х„}, we have
sup фп{Е)< со, A1.5)
1 ^n < oo
since the partial sum operator sn is a projection of E onto the «-dimen-
«-dimensional subspace P[n) = [x1,...,xn] of ? (и=1,2, ..-).1 Consequently, if
there exists a separable Banach space E with sup фп(Е)= oo, then
this space E has no basis, and thus it yields a negative solution of the
basis problem. However, this sufficient condition for the existence of a
separable Banach space having no basis does not yield a method for
constructing such a space. Making use of §10, corollary 10.2 one can
give the following more constructive result:
Theorem 11.1. Let {akj be an increasing sequence of positive num-
numbers such that 1ш%=со. // there exists an increasing sequence of
Banach spaces ~>°°
?3-<?4-<?5-< ••¦, (П-6)
dim?n = « (и = 3,4,...), A1.7)
such that for every pair of positive integers к, п with n = n
there exists a positive integer N = N(k,n) with the properties
1 Thus, in particular, for every Banach space E having a basis {xn} of norm
v= sup ||sj = l, we have A1.4). Furthermore, since ||s*|| = ||sj, for the conju-
gate space E* of a Banach space E having a basis we have sup ф„(Е*)< oo and,
if{xn} is of norm v = l, then </>„(?*)=! (n = l,2,...).
11. An application to the basis problem
111
A1.8)
фк+1(Ек)>ак, (П.9)
then there exists a separable Banach space having no basis.
Proof. By A1.6) there exists a natural norm on the linear space
oo / °o V
\J Е„. We claim that the completion E = I \J Е„ I of the normed
linear space \J En is a separable Banach space having no basis.
Indeed, E is separable by A1.7). Assume now that E has a basis.
00
Then, by § 10, corollary 10.2, \J En must contain a basis {*„} of E.
Since 0<a1<a2< "' limak=oo, for the sequence of partial sum ope-
lt->00
rators {sn} associated to the basis {xn} there exists, by § 4, theorem 4.1,
an <xk such that
=1,2,...). A1.10)
Furthermore, since {х„} <= \J En, there exists a positive integer
n=3
n = n(k)^k+l such that Р{к+1) = [хи...,хк+1]<Е„. Let N = N{k,n)
be a positive integer with the properties A1.8) and A1.9). Then, by A1.8)
and A1.6) we have En<EN, whence P(k + 1)<EN. Since dimP(k + 1) = k+l,
from A1.9) it follows that we have ||u|| >ak for all ue0>(EN,P{k+l)).
However, for u = sk+1\EN this contradicts A1.10), completing the proof
of theorem 11.1.
Remark 11.1. For the numbers k, N and ak satisfying A1.9) we have
— >
N
N k+1
Indeed, according to a result of F. Bohnenblust [28] we have
и+1
whence, by induction,
n
t
N
A1.11)
A1.12)
A1.13)
which for n = k+l gives, taking into account A1.9), the inequality A1.11).
Remark 11.2. From Ch. II, § 2, theorem 2.2 and the Hahn-Banach
theorem it follows that for every Banach space E we have

112 I. The Basis Problem. Some Properties of Bases in Banach Spaces
ф„(Е)^п (n=l,2,...). A1.14)
Consequently, condition A1.9) in theorem 11.1 is non-void only if
we assume that
ixk<k+l (fc=l,2,...). A1.15)
Concerning the possibility of practical application of this sufficient
(but not necessary) condition for the existence of a separable Banach
space having no basis we remark, however, that the effective calculation
and even a suitable lower evalution of 0t+i(?iv) seems to be difficult.
We shall see in Chapter II, § 1, that there exists an E3 with ф2(Е3)> 1,
but for the calculation of ф2(ЕА) for an ?4>?3 one has to consider
new two-dimensional subspaces of ?4.
§ 12. Properties of strong duality. Application: bases
and sequence spaces
Let ? be a Banach space with a basis {х„} and let {/„} с Е* be
the a.s.c.f. Then it is natural to ask, what can we say about the sequence
{/«}¦ We shall call properties of duality any relations between bases
{х„} of Banach spaces ? and their a.s.c.f. {/„}<=?*. When both ?
and ?* are endowed with their norm-topologies, we shall call these
relations properties of strong duality; when ? and ?* are endowed with
the weak topology <r(?,?*) and the weak* topology <r(?*,?) respec-
respectively, we shall use the term: properties of weak duality. In the present
section we shall turn our attention to properties of strong duality.
If {х„} is a basis of a Banach space ?, then, in general, the a.s.c.f.
{/„} need not be a basis of ?*, since it may even happen that ?* is non-
separable (and hence ?* has no basis at all), as shown by
Example 12.1. Let ? = /4 Then E has a basis, but the conjugate
space1 E* = m is non-separable.
However, one can also give the following positive result:
Theorem 12.1. Let {х„} be a basis of a Banach space E and let {fn} <= E*
be the a.s.c.f. Then {/„} is a basic sequence (in E*) and we have
f=
A2.1)
1 For two Banach spaces F and G we denote the assertion "F is isometrically
isomorphic to G" by F = G.
12. Properties of strong duality. Application: bases and sequence spaces 113
Proof For the adjoint s* of the n-th partial sum operator sn asso-
associated to the basis {х„} we have
= g| iyi(x)xj = | X^xj/i
whence
ii (geE*,n=\,2,...).
A2.2)
Hence, for every finite linear combination g = ? fjjfj we have, by
biorthogonality, J=1
i=l i = 1
Now, let /е[/„] and й>0. Then there exists a finite linear СОШН-
СОШНИК
nation g = ? ft-ffj such that
v + l
where v= sup ||sj<oo. Hence, taking into account ||s*|| = ||sj
1 =Sn<oo
(и =1,2,...) and A2.3), we obtain
l|s*(/)-/li «S \\s*(f)-s*(g)\\ + \\s*{g)-g\\ + \\g-f\\
s e
<v 1 = e (n = m?, mE+l,...),
v + l v + l
cc
which, by A2.2), proves A2.1). On the other hand, if ? *ifi = ®, then'
by biorthogonality, we have i=1
(/=1.2,...)-
;= l
Thus, {/„} is a basis of [/„], which completes the proof of theorem 12.1.
In the sequel we shall denote by n the canonical mapping of ? into
?**, i.e.
(хе?,/е?*). A2.4)
Corollary 12.1. Let E be a Banach space and let (xn,fn) ({xn} с ?,
{/„} с ?*) be a biorthogonal system. If {/„} is a basis of the space ?*,
then {х„} is a basis of the space E.
Proof. By biorthogonality, the a.s.c.f. to the basis [/„] of ?* is
nothing else but {я(х„)}. Hence, by theorem 12.1 applied to the basis
8 Singer, Bases in Banach Spaces I

114
I. The Basis Problem. Some Properties of Bases in Banach Spaces
{/„} of ?*, [n(xn)\ is a basis of n(E). Since л is a linear isometry, it
follows that {х„} is a basis of E.
Corollary 12.2. Let [xn] be a basis of a reflexive Banach space E
and let {/„} с E* be the a.s.c.f. Then {/„} is a basis of the space E*.
Proof. Since E is reflexive, we have к(Е) = Е**, whence, since ж
is a linear isometry, \к(хп)} is a basis of ?**. Hence from corollary 12.1
applied to the biorthogonal system (/„, n(xn)) it follows that {/„} is a
basis of ?*.
In Ch. II, §4, we shall give necessary and sufficient conditions for
a basis [xn] to have the property that the a.s.c.f. {/„} is a basis of E*.
We have seen in example 12.1 that if a Banach space E has a basis,
its conjugate space E* need not have a basis. However the converse
problem is unsolved:
Problem 12.1. If the conjugate space E* (of a Banach space E) has a
basis, does the space E possess a basis?
A partial result is given in corollary 12.1 above. The following related
questions are also unsolved:
Problem 12.2. a) If the conjugate space E* (of a Banach space E) is
separable, does E* possess a basis? b) If E* is separable, does E possess
a basis?
Naturally, problems 12.1 and 12.2 can be also considered as par-
particular cases of § 1, problem 1.1. Problem 12.2 b) is intermediate be-
between problems 12.1 and 1.1 (since E* has a basis => E* is separable
=> E is separable).
For a linear subspace V of the conjugate space E* (of a Banach
space E) we shall denote by ф the canonical mapping of E into V*, i.e.
the continuous linear mapping of E into V* defined by
[<?(*)] (/) = /(*) [xeEJeV). A2.5)
In other words, we have
ф(х) = п(х)\у (xeE), A2.6)
where, as before, к denotes the canonical mapping A2.4) of ? into ?**.
We shall also use the notation
К1={Ф6?**|Ф(/) = 0 for all feV}.
Then for any total subspace V of E* we have, obviously,
A2.7)
n(E)nV1 = {0}. A2.8)
If ? is a Banach space with a basis {xn\, the a.s.c.f. {fn}<=E* is total
on ?, whence we have A2.8), ф is one to one, and the norm-closed linear
12. Properties of strong duality. Application: bases and sequence spaces 115
subspace (/=[yJ of ?* is dense in ?* for the weak* topology <r(?*,?).
However, in the next theorem we shall give considerably stronger
results.
Let us recall that the characteristic of a linear subspace V of a con-
conjugate Banach space ?* is the greatest number r = r(V) such that the
unit cell {/eK|||/||<l} of V is a(E*, ?)-dense in the r-cell
{/e?* | Il/H^r} of ?*. Obviously, 0^r(F)^l, but it may also happen
that r(V) = 0 for a <r(?*,?)-dense linear subspace V of ?*. However,
in the next theorem we shall see that if V= [/„], where {/„} is the a.s.c.f.
to a basis {x,,}, then this pathological case cannot occur.
Theorem 12.2. Let E be a Banach space with a basis {х„}, let V= [/„]
be the norm-closed linear subspace of ?* spanned by the a. s. c.f. {/„} с ?*
to {xn} and let v = v{Xnj be the norm of the basis {х„}, i.e., v= sup ||jn||,
where {sn} is the associated sequence of partial sum operators. Then
a) We have
i.e., the unit cell {feV
— cell {feE* | H/ll <-
r(V) ^ - > 0,
A2.9)
|<1} of V is a(E*,E)-dense at least in the
b) The closure IE of the unit cell SE = {xeE | ||x||<l} of E for the
weak topology a(E, V) is bounded, namely
c) We have
inf sup
xeE fsV
хФО ||/||=S1
- > 0.
v
A2.10)
A2.11)
d) For the canonical mapping к of E into ?** we have
mf
A2.12)
e) The canonical mapping ф of E into V* is an isomorphism, satisfying
1 .. .
where we use the notation
r(Vy
(xe?),
A2.13)
= \\ф(х)\\= sup \f(x)\ (xeE).
feV
imia
A2.14)

116
I. The Basis Problem. Some Properties of Bases in Banach Spaces
U {xn} is a monotone basis1, the above inequalities and inclusions
become equalities. If r(V)= 1 fin particular, if {х„} is a monotone basis),
then ф is an isometry.
Proof. By well known results2 on the characteristic r(V), we have
1
Г7Г = inf SUP
SUD X xeE feV
ФО ||/||«П
= inf
хеЕ.хФО
A2.15)
and thus the statements a), b), c), d) are equivalent, while e) follows
x
from c), via
(xeE). Therefore it will be sufficient to prove
any one of the assertions a), b), c), d). Let us prove b).
Let {yd}dEA be a net in SE which is a{E, F)-convergent to an xeIE.
Then lim/•()',,) = /-(x) (/= 1,2,...), whence, in the norm-topology,
\imsn(yd) = sn(x) (n=l,2,...).
deA
Let f,>0 be arbitrary. Then there exists a do = do(e,n)eA such that
whence, by \\yd
(deA) and sup ||sj=v<oo,
Consequently, since ||x||^ sup ||sB(x)|| (xeE) and since c>0 and
xeIE were arbitrary, we get ' ^"<°°
HxKv (xeIE),
i.e., A2.10), which completes the proof of theorem 12.2.
Remark 12.1. The weaker assertion r(V)>0 can be also derived
from § 14, theorem 14.1, formula A4.8) and a theorem of S. Banach3.
Remark 12.2. In general, r(F)#—. Indeed, if E is reflexive, then
v
for every basis {х„} of E we have, by corollary 12.2, V=E* (where
^=[/n]> fi(xj) = uij)> whence r(F)=l, while it is easy to construct
bases of E for which — is arbitrarily small.
v 1
Let us also mention that in a certain sense the constant — in theo-
v
rem 12.2 is the best possible lower bound for r(V). Indeed, if {x,,} is a
monotone basis (i.e. with the norm v= 1), then r(V)=\ = —.
1 I.e., v= 1. Such bases will be studied in Ch. II, § 1.
2 See [47] or [33], p. 121, exercise 14.
See [10], p. 213, theorem 2.
12. Properties of strong duality. Application: bases and sequence spaces 117
Now we shall show that r(V) can have any value a such that 0<Л< 1.
Example 12.2. Let 0<яг%1 and let {х„} be the following basis of
the space E = ll:
1 1
=-eu х„=--е,+е„
A A
A2.16)
where {е„} denotes the natural basis of the space E = l1. Then r(V) = X.
Indeed, let {/„} and {hn} be the a.s.c.f. to the bases {х„} and {е„}
respectively. Then
A2.17)
We claim that for every /= ?a,-/,eF (see theorem 12.1) with
IKl we have i=1
In fact, by A2.17),
[CO I*
Ahl(x)+ ?>*(*) + Xе
i = 2 J i=2
A2.18)
whence
(xel1),
= 2,3,...).
A2.19)
On the other hand, by /= ? a,/; and ||/J = 1 (n= 1,2,...), we have
i= 1
liman = 0, which, together with A2.19), implies A2.18).
"-»*• ОС DO
Consequently, for every x= ^^е,-е1' and /= ^aJ.eF with
H/il^l, we have , i=1 i=1
Considering, for any fixed x= X ^г^6'1» *пе sequence of functionals1
/=i
1 We recall that
)^y for a#0,
signa= < |a|
U for a = 0.

И 8 I. The Basis Problem. Some Properties of Bases in Banach Spaces
gn(z) = (sign ^)fx (z)+ X (- sign ^ + sign ?jf(z)
i=2
it follows that
sup
(ze/\n=l,2,...),
i = 2
x=
Hence, for every x=??ieie/1 with ||x|| = Y, 1^1 = 1
sup |/(x)| = 1— A— a)|^i 1^1— A— A) = A,
and the value X is attained for x = e1e/1. Since by the theorem of
Dixmier mentioned above, we have r(V)= inf sup
fV
f\
' \ X
o Ц/1И
follows that we have r(V) = X, which proves our assertion.
Let us give an application of theorem 12.2e).
Proposition 12.1. Let E,F be Banach spaces with bases {х„} and {}'„},
respectively, and let {/„} с ?*, \hn} с F* be the a.s.c.f. to {х„} and {у„},
respectively. Then
a) {xn}>{yn} if and only if {hn}>{fn}.
b) {х„}~Ы if and only if {/„MU-
Proof, a) If {х„} >{>¦„}, then, by §8, theorem 8.1b) (implication
6°=>1°), there exists a continuous linear mapping и: Е—>F such that
u(xn) = yn (n=l,2,...).
Hence, for the adjoint mapping u*: F*->?* we have
[m*(A,-)] (^) = h,[u(xj)] = hAyj) = 5y = /,(х7) (ij = 1,2,...),
whence, since [х„] = E,
u*(ht) = f, (i = l,2,...).
Thus {hn\y?-{fn} and hence {hn}>{fn}.
Conversely, assume now that {hn}>{fn}. Since the a.s.c.f. to the
basis {йи} of [/!„] and to the basis {/„} of [/„] are nothing else then
{фР(у„)} and {фЕ(х„)}, respectively, where фР,фЕ denote the canonical
mappings of F into [/!„]* and E into [/„]*, respectively, we have, by
12. Properties of strong duality. Application: bases and sequence spaces 119
the "only if part of a) proved above, {фг(х„)} >{фЕ(у„)}- Since by
theorem 12.2e) both фЕ and фр are isomorphisms, it follows that
Ы>{уя}.
Finally, b) is an immediate consequence of a), which completes the
proof of proposition 12.1.
Theorem 12.1 suggests naturally the question, what are the relations
between the norms of the bases {х„} of E and {/„} of [/„]. The follow-
following theorem gives upper and lower evaluations of v{fn) by means of vUni.
Theorem 12.3. Let {х„} be a basis of a Banach space E, let {/„}<=?*
be the a.s.c.f. to {xn} and let F =[/„]. Then
1 ^r{V)v,Xn]^v{fn]^v{Xn). A2.20)
Hence, in particular, if r(V)=\,1 then
v{/n! = v!Xnl. A2.21)
Proof. From formula A2.2) for the adjoint s* of the partial sum
operator sn, it follows that
,)/f= Z
A2.22)
where ф denotes, as before, the canonical mapping of E into V*. Hence,
since {ф(х„}} с V* is the a.s.c.f. to the basis {/„} of K=[/n],
%„)= sup ||s*|r||< sup ||sJ=v(Xn!,
and taking also into account A2.13),
vj/ni= sup
A2.23)
v\\= sup sup sup I[>*(/)](x)
l=Sn<x feV xs?
Н/1И1 llxll si
= sup sup sup |/'[sn(-v)]|= sup sup ||sn(x)||F
l=Sn<or, feV xeE l=Sn<x1 ||x||Sl
" НЛИ1 ||x||=Sl
>r{V) sup sup \\sn(x)\\=r(V)v{Xn). A2.24)
lS»<a l|x||Sl
Consequently, by A2.9), A2.24) and A2.23) we have A2.20). which
completes the proof of theorem 12.3.
Let us also mention separately the following part of formula A2.24),
which may be useful for other applications:
vf/n,= sup sup ||s,,(x)||K.
l«x l|||=Sl
A2.25)
1 This happens e.g. when the basis {х„} is monotone (by theorem 12.2) or
when E is reflexive.

120
I. The Basis Problem. Some Properties of Bases in Banach Spaces
The inequalities A2.10) are, in a certain sense, the best possible,
since for monotone bases they all become equalities. Let us show that
they are, in general, strict inequalities. For this purpose we shall give
an example in which simultaneously all of them are strict.
Example 12.3. Let {%„} be the following basis of the space E=l1:
x1=-e1, x2 = e2-e3, xn=--e1
/ A
(n = 3,4,...), A2.26)
where {е„} denotes the natural basis of the space I1 and A is an arbitrary
number such that 0<A^l (this basis differs from A2.16) only in the
term x2), let {/„} <=?* be the a.s.c.f. to {х„} and let F=[/n]. Then
1
v =2 + —,
and thus, for 0 < Я < 1,
, = 3,
-= v{
A2.27)
i.e., all inequalities in A2.20) are strict.
Indeed, if {hn} is the a.s.c.f. to the natural basis {е„} of /\ we have
(xe/1),
(" = 4,5,...).
A2.28)
Thus (/=[/„] is the same as for the basis A2.16), whence, by ex-
example 12.2, r(V) = l
Furthermore, by A2.26), for any positive integer n and any scalars
«!,...,«„ we have
i = 3 A
, = 3 '-
a3-a2| +
Hence, since for ^ |а;|#0 (n>3), m= 1,2,... we have
i=3
a,
,k a
1 N
A l={
1 L—i "\
+ |а3-а2|
12. Properties of strong duality. Application: bases and sequence spaces 121
we obtain
i=3
A ?3 A
a2| + |a3-a2|+ ? |а,|
i = 4
«i \-; «,-
т ~ 2. т
i=3
A2.29)
Z aix.-
whenever Z |a,-| =#0 (n>3), m=\,2,.... Since obviously
i = 3
_ l«il l«il
l|aiXl" =T^T + 2|c
А л
у
A ff3 A
2
Za;x;
i= 1
i+iiaiV1
i = 4
/ Я
A2.30)
A2.31)
Tax
i=i
(A; > 3), it remains to consider
«sc
. Taking in the inequalities
A2.32)
n-\
the scalars o^ =<x3 = l, <x2 = , a4= ••• =at = 0, we get
n
— +
а,х,-
Z aix.-
= с
1
whence, for n-+oc, we obtain — + 2^C, and thus, taking into account
A2.29), A2.30) and A2.31), k
1
A
However, we have here the equality sign, since from
a, a,
a, -a,
a3 —
A i = 4 /.
к
+ |a3—a2| + Z I

122
I. The Basis Problem. Some Properties of Bases in Banach Spaces
we obtain
2
Z a,
5>«
)(
/V
к
Z
«1
Я
aiXi
к
z
1 = 3
ОС;
A
(fc>2). Therefore v|Xnl = - + 2.
Finally, let us compute v|/n|. By A2.28), for any positive integer n
and any scalars a1,...,an we have
i= 1
= sup
хеЕ = 1г
IWI=S1
+ a,
i =
a^./i^x
00
n+ 1
= max (
з =s ; =s n
A2.33)
Hence for ? |а;|#0 (и^З), m= 1,2,... we have
i=3
Z u-Ji
= max
3==;==»
max
Since obviously
(fc>2), it remains to consider
I «
a;./I
A2.34)
A2.35)
. Taking in the inequalities
2) the scalars at = 1, a2 = 2, a3= -2, a4= ¦•• = at = 0, we get
max A,3) =
2
z«,/,
;= l
к
У а- /'•
1 = 1
whence 3^C and thus, taking also into account A2.34) and A2.35),
v^j-^min
12. Properties of strong duality. Application: bases and sequence spaces
However, we have here the equality sign, since
Z
= max(|ai|
|, \ax+ixi\) =
z«.-/«
123
(fc>2); indeed, if
,, .. l«2|
while if
<|ai|, then |ai|+ |a2|<|a1| + 2|a1| =
\ax |, then |at | + |a2
3|a2|
and, on the
other hand,
,+a3)-(ai+a3)|
2 2
!«! + a3|). Therefore v{J-n} = 3, which completes the proof of the asser-
assertions of example 12.3.
This example shows that there exist bases {х„} with v)/n) = 3 (where
{/„}<=?* is the a.s.c.f. to {х„}) and with v(Xiii taking any arbitrarily
large preassigned value. It is natural to ask whether the limit case
v|Xnl= со is also possible, i.e., whether there exists a biorthogonal system
(х„,/„) with [*„] = ? and {/„} total on E, such that {/„} is a basic
sequence, but {х„} is not a basis of E. We shall show that the answer
is affirmative (this also shows that a certain converse of the first statement
of theorem 12.1 is not valid).
Example 12.4. Let1 ?=(?1x?2x ¦¦¦){1=l1, where E~ll (/= 1,2,...)
and for each; let {xj/*} be the basis A2.26) of E~l\ with Я = —.
XX ^
Since the set [j [j [0 0, xj/', 0,...] in ? is countable, let {х„} be
j=i -1 = 1 "T^T""
an arbitrary numbering of it, and let {/„} be the corresponding num-
numbering of the functionals {0 0, fj\ 0,...} e(?? x ?? x • • •),„ = E* = m,
where {/J/'} is the basic sequence A2.28) in E* = m, with Я = —.
1 We recall that (?, x ?2 x ••¦),, (respectively, ?, x E2 x •••)„) is the space of
all sequences {х„} with х„е?„ (n=l,2,...), for which {||хл||}е/' (respectively,
||х„||}еш), endowed with the norm
{х„}|| =
||xn|| (respectively,
=sup ||х„||). There exists a natural linear isometry (/' x Z1 x ¦¦¦),, = I' (see Ch. II,
1 ^ n< OG
§18, lemma 18.5c)).

124
I. The Basis Problem. Some Properties of Bases in Banach Spaces
Then (х„,/„) is a biorthogonal system, [х„] = ?, {/„} is total on ? and
it is a basic sequence in ?*, but {х„} is not a basis of ?.
Indeed, it is obvious that [х„] = ? and that {/„} is total on ?.
Since by A2.27) we have v{xu)| = 2+; (/=1,2,...), it follows, by §7,
theorem 7.1, that {х„} is not a basis of ?. On the other hand, since
{x(nJ)} is an unconditional basis1 of ?j, {/!/'} is an unconditional basic
sequence in EJ (by Ch. II, §17, theorem 17.7) and by A2.33) its un-
unconditional norm v\u}U)j does not depend on;'. Since for {gi,g2, ¦¦¦}
e(??x?*x ••¦)„, we have \\{gl,g2,...}\\ = sup \\gj\\, it follows that
1 =5 j < »
{/„} is a basic sequence in ?*, which completes the proof of the assertions
of example 12.4. Let us also mention that r(V) = 0 (where V= [/„])•
We don't know whether the converse of the last statement of theo-
theorem 12.3 is true, i.e., whether for a basis {х„} of a Banach space ?,
with the a.s.c.f. {/„}, the equality v{/, = v{JC ( implies r(V)=\ (where
V= [/„])•
If {х„} is a basis of a Banach space ?, with the a.s.c.f. {/„}, then
{/„} isabasisof F=[/n] (bytheorem 12.1), with the a. s.c.f. {ф(хп)} с V*
where ф denotes the canonical mapping of ? into V*, whence, by virtue
of theorem 12.3, we have
1 ^r(W)v{fni^v{lHxn)]^v{fnj, A2.36)
where W =[ф(хп)] = ф(Е) с V*. It is natural to ask whether one can
say more about the constant v^(Xn)i. We shall now show that this is
indeed the case, namely, that we always have r(W)=\, whence the
last two inequalities in A2.36) are actually equalities.
Theorem 12.4. Let V be a linear subspace2 of a conjugate Banach
space E*, let ф be the canonical mapping of E into V* and let \?=ф(Е)<^ V*
Then
r(W)=\. A2.37)
Hence, in particular, if {х„} is a basis of a Banach space E, {/„}
{}
the a.s.c.f to {х„}, V={Q с ?* and \?=[ф(х„)] =
r(W)=\,
V*, then
A2.38)
Proof. By formula A2.15) applied to \?=ф(Е)^ V* and the re-
relations ||0(x)|| = i[x||F^ ||.x|| (.xeE), we have
1 For the notions of unconditional basis, unconditional basic sequence, un-
unconditional norm of a basic sequence, see Ch. II, § 14, § 17.
2 We need not assume that r(V)>0. Let us mention that the semi-norm ||x||F
is a norm on E if and only if V is total on E (or, equivalently, ф is one-to-one).
12. Properties of strong duality. Application: bases and sequence spaces 125
r(W) = inf sup
feV 4>eW
f*0 11^11=51
> inf sup
fsV xsE
f*0 \\x\\il
= inf sup
/eV xeE
f*0 \\x\\ $
-(x)
(X)
= inf
f
whence, since the characteristic of any subspace is always < 1, we infer
A2.37).
The second assertion of theorem 12.4 follows from the first one,
taking into account that [ф(х„)]=ф{Е) whenever {х„} is a basis of ?
and ф the canonical mapping of ? into V* = [/„]*, and applying the
last statement of theorem 12.3 to the pair ({/„}, {ф(х„)\). This completes
the proof of theorem 12.4.
Remark 12.3. Formula A2.37) is equivalent to the statement that
the canonical mapping и of V into W* = ф(Е)*, defined by
«(/)[0(x)] = O(x)](/)=/(x)
is an isometry, i.e.,
V, ф(х)е W)
sup |/(x)|=||u(/)|| = |
xeE
= sup |/(x)| (feV), A2.39)
xsE
11*11=51
and this latter formula follows also directly from the relations ||.x||v< ||x||
(xe?) and ||u|| <1.
Combining theorems 12.3 and 12.4, we obtain the following rela-
relations between the norms of {xn} and {ф(х„)):
Corollary 12.3. Let {xn} be a basis of a Banach space E, {/„] the
a. s. c.f. to {xj, V= [/„] с ?*, and ф the canonical mapping of E into V*.
Then
*^v!<Mxn)l<v|x, A2.40)
Remark 12.4. By A2.25) and A2.22) we have the following formula
for the computation of х,ф,х r:
}= sup sup \\s*(f)\\w= sup
l=Sn<cc /eV l=Sn<o
Н/1И1
sup ||sn(x)||F. A2.41)
xsE
With the aid of this, it is easy to obtain again, directly, the second
equality in formula A2.38). Indeed, by A2.41), ||x||^^||.x|| (xe?) and
A2.25) we have
SUp \\sn(x)\\v = V{fn},
1 =Sn< cc xeE
11*11=5 1

126 I. The Basis Problem. Some Properties of Bases in Banach Spaces
whence, since by theorem 12.3 we have v!4,(Xri)l«Sv(/nl, we infer v,MXn)}
Proposition 12.2. Let E be a Banach space with a basis {xn}, V= [/„]
the norm-closed linear subspace of E* spanned by the a.s.c.f. {/„} to
{х„}, and v the norm of the basis {xn\, i.e., v= sup ||sj<oo, where
1 ^ И < OO
{sn} is the associated sequence of partial sum operators. Then
a) We have
H/ll < sup Yf(xi)fi <v||/|| (feE*). A2.42)
1=1
Conversely, if for a sequence of scalars {«„} <= К we have
sup
z «¦/«
<oo, then there exists an feE* such that
b) We have
; sup
1 «n<co
Z
/(*„) = «„ (n=l,2,...).
n
z
A2.43)
s; sup
v 1 =Sn< =o
1 = 1
A2.44)
Conversely, if for a sequence of scalars {аи} <= К we have
sup
Z a;x<
<co, then t/iere exists a 4*eV* such that
с) Же /шие
sup
^ П < 00
(Фе?**).
A2.45)
A2.46)
< oo,
A2.47)
Proof, a) Let feE* and c>0 be arbitrary ami let x = xCJeE be
such that ||x|| = 1, |/(x)|^||/|| —c. Then
Conversely, if for a sequence {«„}<= К we have sup
then there exists а ФеЕ** such that i=S"<«
: SUp
1 ^ П < 00
+ e = lim
1 = 1
Z ж-)/.-
12. Properties of strong duality. Application: bases and sequence spaces
whence, since s>0 and feE* were arbitrary,
127
: sup
1 $П<00
Z
(feE*).
On the other hand, by A2.2),
sup
1 ^И<СС
!= 1
= SUp ||5„*(/Ж
= sup
1 S<
and thus we have A2.42).
Conversely, assume now that sup
1 ^ П < 00
any finite sequence of scalars /?,,...,/?„ we have
1 = 1
= M<ao. Then for
Z«,/, Z ft*.
n
i= 1
Z ftx,
j= 1
у о
IP,*,
whence, by a well known theorem of E. Helly1, there exists an feE*
satisfying A2.43).
b) and c). The first inequality in A2.44) follows from part a) applied
to the basis {/„} of F=[/n], since it has the a.s.c.f. {ф{х„)) (whence
z
!= 1
is obvious.
Z nf,)x(
) and the second inequality in A2.44)
Furthermore, by A2.2) we have
\i= 1
whence, by A2.4),
^eE**,feE*,n=l,2,...),
(ФеЕ**,п=\,2,...).
A2.48)
where n is the canonical mapping of E into ?**. Since к is an isometry,
it follows that we have
Z *Ui)xt
(ФеЕ**, п=\,2,...),
i.e., A2.46). In particular, if WeV*, then, taking an arbitrary extension
Фе?** of V, with norm ||Ф|| = Ц^Ц, we obtain the last inequality in A2.44).
1 See e.g. [10], p. 55, theorem 4.

128 I. The Basis Problem. Some Properties of Bases in Banach Spaces
n
Conversely, assume now that sup Z aixi <c0- Then
sup
1 5=П< CO
: sup
1 S Л <OO
i=\
< со, whence, by part a) applied
to the basis {/„} of F=[/n], there exists a WeV* satisfying A2.45).
Taking an arbitrary extension ФеЕ** of W, we shall also have A2.47),
which completes the proof of proposition 12.2.
Let us also mention the following alternative proof of the converse
part in a): If sup
Z «;/;
< со, then, since E is separable, there
exist1 an feE* and a sequence {и,-} of positive integers such that
lim ?a,./,.(x) = /(x) (xeE),
j
whence, taking х = х„ (n = 1,2,...), we obtain A2.43), which completes
the proof.
Theorem 12.5. Let E be a Banach space with a basis {%„}, V=\_fn]
the norm-closed linear subspace of E* spanned by the a.s.c.f. {/„} to
{х„}, ф the canonical mapping of E into V* and n the canonical mapping
of E into E**. Then
a) The space E* is isomorphic, by the mapping
ri:f^{f(xn)} (feE*),
to the Banach space of sequences of scalars
sup
< со
in which the norm is defined by
Wlr, 111 =
Z«,.
A2.49)
A2.50)
A2.51)
b) The restriction v\v of the isomorphism A2.49) to V maps V onto
the Banach space
Z OLi.fi converges (strongly)>, A2.52)
;=i J
in which the norm is defined by A2.51).
1 See e.g. [10], p. 123, theorem 3.
12. Properties of strong duality. Application: bases and sequence spaces 129
c) The space V*( = E**/V1) is isomorphic, by the mapping
т :«"->{«"(/«)} (VeV*), A2.53)
to the Banach space of sequences of scalars
sup
1 ^ n < gc
n
I
i —
j aiXi
1
<co
A2.54)
defined in § 5, remark 5.4.
d) The restriction х\ф(Е) of the isomorphism A2.53) to ф(Е) maps ф(Е)
onto the Banach space
A2.55)
a,- x,- converges >,
introduced in §3, proposition 3.1.
V {*«} l's a monotone basis, the isomorphisms in a) —d) above are
isometries.
e) We have
<?(?)=
1=1
converges^,
3
A2.56)
converges},
J
A2.57)
and ф(Е) is norm-closed in V*.
f) We have
п{Е)@У1=\феЕ**
and k(E)@ Vl is norm-closed in E**.
Proof, a) is an immediate consequence of proposition 12.2 a).
b) For every feV we have {/(х„)}еЛ3 (by A2.1)). Conversely, if
X> GO
{«„}eA3, i.e., Z a;/i converges to an /e V, then «„= Z a;/i(xJ = f(xJ
i= 1
(n=l,2,...), whence {«„}=?/(/).
c) is an immediate consequence of proposition 12.2b). It also fol-
follows from observing that by proposition 12.2c), Ф->{Ф(./„)] (ФеЕ**)
is a continuous linear mapping of E** onto A2, having the kernel V1,
whence it induces, by the open mapping theorem1, an isomorphism of
E**/VL onto A2, and taking into account that V* is canonically iso-
isomorphic to E**/VL, by the mapping2 f -+{ФеЕ**\Ф\у= f} (feF*).
1 See e.g. [10], p. 40, theorem 4.
2 See e.g. [43], Ch. II, § 1, lemma 2.
9 Singer, Bases in Banach Spaces 1

130
I. The Basis Problem. Some Properties of Bases in Banach Spaces
Moreover, c) also follows from a) above applied to the basis
{/«} °f ^=[/n]. taking into account theorem 12.2e). Conversely,
let us observe that a) can be derived from c) above, since by c) ap-
applied to the basis }/„} of [/„], the space [ф(х„)]* = ф{Е)* is iso-
morphic, by the mapping h^ {к(ф(х„))} = {(ф*(к))(х„)} (кеф(Е)*) to
и/
¦к
sup
I*,./,
<со> = /14 and since (by theorem 12.2e))
E* is isomorphic to ф(Е)*, by the mapping {ф*) 1: ф*Aг)-+1г феф(Е)*).
e) Since {х„} is a basis with the a.s.c.f. {/„}, for every {Р = ф(х)еф(Е)
x; x
the series Z f (/,)x;= Z Мх)х( converges. Conversely, if for a fe V*
i=l i=l
00
the series ? Ч*{/г)х{ converges, say to xeE, then, since {xn} is a basis
with the a.s.сf. {/„},
whence
(n=l,2,...) and consequently tF(f) = f(x) for all feV, i.e., ? = ф{х)
еф{Е), which proves A2.56).
Furthermore, ф(Е) is complete, whence norm-closed in V*, since by
theorem 12.2e) it is isomorphic to E. However, one can also derive the
norm-closedness of ф(Е) directly from A2.56). Indeed, if {?„}<= ф(Е),
4*eV*, ||fn-f II->0 as n->co, and c>0, then we have, by proposition
12.2 b),
Z
Z
z ^
h X:
+ 2v\\4>-tFk
which is <c for n>N = N(c), if we take А;0 = А;0(к) such that ||f-
< — and N = N(e) such that
4v
Z '
i = n + 1
<-for
,/=1,2,... .
Hence, by A2.56), Ч'еф(Е), which proves that 0(?) is norm-closed in V*.
d) For every Уеф(Е) we have, by e), {^(fjjeA^ Conversely, for
{xJeA^At let T = Tl({oLn}). Then {fP(/B)} = T(fP)= (aB} and thus
the series Z lP(fi)xi= Z а>х' converges, whence, by e), Ч'еф(Е).
i = 1 i = 1
Let us also observe that from d) it follows again that ф(Е) is norm-
closed in E*, since by d) ф(Е) is complete.
f) For any Фея(?HК± we have Ф\у={Реф(Е), whence, by e),
X' ОС ОС
the series Z ®(fi)xi= Z ^U\)xi converges. Conversely, if
12. Properties of strong duality. Application: bases and sequence spaces 131
converges, then
converges, where f = <?|KeF*, whence,
by e), Ф
Ф\у = п(х)
=? = ф(х)еф(Е) for a suitable xe?. Then, by A2.6),
, whence Ф-ж(х)еУ1 and thus Феп(Е)® VL. This proves
A2.57).
Finally, the norm-closedness of л(Е)@ V1 is well known1 to be a
consequence of r(V)>0 (which holds by theorem 12.2a)), but it can be
derived also directly from A2.57), with an argument similar to that
used in the above proof of e), making use of proposition 12.2c). This
completes the proof of theorem 12.5.
As another application of proposition 12.2, we shall give now a
relation between bases and sequence spaces.
A sequence space (or a coordinate space) is a set S of sequences of
scalars which is closed under coordinatewise addition and scalar multi-
multiplication, i. e., a linear subspace of the space of all sequences.
Definition 12.1. We shall say that a sequence space S is associated
to a basis of a Banach space, if there exists a Banach space E with a
К
Z
basis {х„} such that S coincides with the set At = {{%„}¦
converges}.
We shall consider the problem of characterization of sequence
spaces associated to a basis. For this purpose we recall some notions of
the theory of sequence spaces.
The y-dual of a sequence space S is the sequence space Sy defined by
к
sup
Z
< со for all {ос„}еБУ A2.58)
For every sequence space S we have, obviously, S a Sri. A sequence
space S is said to be y-perfect, if Syy = S.
A sequence space S is called a BK-space, if it is a Banach space and
the coordinate functionals are continuous on S, i. e., the relations
х„ = {а)"»),х = Ц}б5, 1ш1х„ = х imply lima;."> = aj.(/=l,2,...).
In the sequel we shall need
Lemma 12.1. //' T is a BK-space containing all unit vectors e,,, then
its y-dual Ty can be normed so as to become a BK-space, by
= sup
sup
A2.59)
See e.g. [47].

132
I. The Basis Problem. Some Properties of Bases in Banach Spaces
Proof Let us first show that ||{^}||<oo for all {|3,}eT7. For any
fixed {j3j}e Ty consider on T the linear functionals
where f is the i-th coordinate functional on T, i. e.
№;}) = «< (<=U,.-.).
A2.60)
A2.61)
Since T is a BX-space, the functionals f, whence also Fn, are con-
continuous on T and by the definition of Ty we have
sup |Fn({<Xj})| = sup
<in<oo
E Pi*i
< со
whence, by the principle of uniform boundedness,
\\{pj}\\= sup sup №j})l= sup ||FJ|<oo. A2.62)
Obviously, \\{Pj}\\ is a norm on T7. To show that T7 with this
norm is a Banach space, let {/fy0} be a Cauchy sequence in T7, i.e.,
sup sup
E (Йч-
< ? (fc>JV(e),p=l,2,...). A2.63)
Taking in these inequalities {a,-} = —— we obtain
II * II
whence the limits lim /Sj,fc) = /?„ (и =1,2,...) exist. Taking p->oo in
A2.63), we obtain к"°°
sup sup
lajlsT 1 ^ л < зо
E i/r-frK-
A2.64)
whence we infer that {{if }-{(jj}eTy (k>N{e)). Consequently, {/Зу}еТ7>
and lim || {fif} - {/?,.} \\ = 0, which proves that Ty is complete.
Finally, to show that Ty is a BX-space, let {fif}, {/З^еТ7 satisfy
A2.64). Then for {a,} = -^- it follows that
Ik II
whence lim /^ = /?„ (и = 1,2,...), which completes the proof.
к-* со
This being said, we can give
12. Properties of strong duality. Application: bases and sequence spaces 133
Theorem 12.6. A sequence space S is associated to a basis of a Banach
space if and only if S contains all unit vectors е„ and there exists a y-perfect
BK-spaceT such that [en]T = S. In this case, T=Syy.
Proof. Assume that S is associated to a basis {xn} of a Banach space E,
i.e., S = A^ =
a.n]
К
Е aixi converges> . We shall show that the
Banach sequence space A2 = < {а„} а К
{а„}||= sup
1 $
E ал
i= 1
J
sup
E а>х*
< oo > , with
, introduced in § 5, remark 5.4, is a y-perfect
BX-space; since S = A1 = \_en]A2, this will prove the necessity of the
condition of theorem 12.6.
Let !A{oi..}6/J be such that lim {af} = {«:}, i.e.,
lim sup
E ("r-«j)xj
J=l
= 0.
Then for each i = 1,2,... we have
. Ык)-а,)х,\\ 1
i-l
sup
E №-*
0 as
and thus Л2 is a BX-space.
To show that Л2 is y-perfect, observe first that
A2.65)
Indeed, if {fin}eA\, then, since {fn(x)}eA2 for all xeE (because
{х„} is a basis of E), we have, by the definition of A\,
n
for all xeE, whence, by the principle of uni-
sup
Hn< so
E ft/
< ОС
form boundedness, sup
E Pifi
< oc and hence, by proposition
12.2a), there exists an feE* such that f(xn) = jin (n=l,2,...); since
[х„] = ?, this / is obviously unique. Conversely, if feE*, then
sup
E
11/11 sup
1 S
E aixt
< oo for all {а„}Е-42, i.e.,
I7,, which proves A2.65).
Now, if {an}eAy2\ then, by A2.65) and the definition of Ay, we
have sup
1 $П<00
= sup
E a./(^,)
<oc for all feE*,

134
I. The Basis Problem. Some Properties of Bases in Banach Spaces
whence, by the principle of uniform boundedness,
sup
s: n < x
E <*.¦*.•
< 00,
i.e.,
Thus
Ay2y'
A2, whence A2 is y-perfect. This proves
that the condition is necessary.
Let us prove now that we also have S' = A"i,, whence Syy = A'2=A2.
By virtue of S = At с A2 we have Sy:
•A\.
Assume, conversely, that
{Pn}eSy. Then, since {fn(x)}eAl=S for all xeE (because
basis of E), we have, by the definition of Sy, sup
E
is a
< ОС
for all xeE, whence, as in the above proof of A2.65), we infer that
there exists an feE* with /(*„) = /?„ (n=l,2,...) and hence, by A2.65),
{|8„}е/4|, which proves S"' = A\ and Srt = A2.
To prove the sufficiency of the condition of theorem 12.6, assume
that it is satisfied, i. e., that S contains all unit vectors en and there exists
a y-perfect BK-space T such that [en]r = S. We shall show that {en}
is a basic sequence in T, whence S is associated to the basis {х„}= {е„}
of the Banach space E = S.
By lemma 12.1 applied to the y-dual space Ty endowed with the
norm A2.59), the space T= Tn is a BK-space also for the norm
= sup
T
sup
S n < oo
A2.66)
We claim that this norm is equivalent to the initial norm on T.
Indeed, let us denote by 7\ the space T endowed with the norm A2.66)
and let u:x-*x be the identical mapping of T onto T,. In order to
show that и is continuous, let xn={aff)}, х={ау}еГ be such that
Итх„ = х, Ити(хя) = у={бЛеТ1. Then, since both T and T, are
/1-* ОС П^ /
BK-spaces, we must have lim j.{") = aj, lim y.f} = Pj (/= 1,2,...), whence
fij—y-j (j= 1,2,...), i.e., у = м(х) and thus, by the closed graph theorem,
и is continuous. A similar argument shows that u~l is also continuous,
which proves that A2.66) is equivalent to the initial norm on T.
Now, in the norm A2.66) we have, obviously,
n
E W,
' l
= sup
||'{/>У)н«1
=
n + m
sup
к
E fta<-
^ ' i — 1
sup
sup
E
for all finite sequences of scalars au...,an + m. Consequently, by §7,
theorem 7.1, [en] is a basic sequence in Tendowed with the norm A2.66),
12. Properties of strong duality. Application: bases and sequence spaces 135
whence also in T with its initial norm. This proves the sufficiency part
of theorem 12.6.
Finally, let us show that if the above condition is satisfied, we have
necessarily T=Syy. By S a T we have Sy => Ty, whence Syy с туу=Т.
Conversely, let {xj}eT. Then, since by the above the norm A2.66) is
equivalent to the initial norm on T, there exists a constant C>0 such
that
E
= С sup sup
E
С HI {a.}
.} H
whence sup
E
< oo, i.e., {а„}еЛ2 (for the basis {е„} of S).
Since we have proved in the above that Syy = A2, it follows that Та А2
= Syy, whence T=Syy, which completes the proof of theorem 12.6.
As an application of sequence spaces associated to bases, we shall
now consider the problem, which are the infinite matrices а = (а{])
which "preserve bases". Let us first recall one more notion of the theory
of sequence spaces.
The [1-dual of a sequence space S is the sequence space Sp defined by
Ш
к
E PiiXi converges for all {an}eS\. A2.67)
;= l
For every sequence space S we have, obviously, Sp c: Sy. Hence,
by lemma 12.1 and the argument of the second part of its proof, we ob-
obtain
Lemma 12.2. // T is a BK-space containing all unit vectors en, then
its fi-dual T11 can be normed so as to become a BK-space, by A2.59).
In the sequel we shall assume that T1' is endowed with this norm.
Proposition 12.3. a) Let {>'„} be a sequence in a Banach space E,
such that у„#0 (и=1,2,...) and let Al = Al({yn}) be the Banach space
of sequences of scalars introduced in § 3, proposition 3.1. Then Ai({yn}f
= Ai({y«})*> by the mapping {/?„}-+&, where
The inverse mapping is given by
A2.68)
A2.69)
where en are the unit vectors in Al({yn})

136 I. The Basis Problem. Some Properties of Bases in Banach Spaces
^x^f^E*, by the mapping
b) If {х„) is a basis of E, then
„}—>/, where
The inverse mapping is given by
./-{/(*„)}•
A2.70)
A2.71)
Proof, a) The mapping {ft,}-»fc is obviously linear. Furthermore, if
for h defined by A2.68) we have h = 0, then /?и = й(е„) = О (и =1,2,...).
Thus, the mapping {/3„}-»й is one to one.
Now let heA^y^f be arbitrary. Then for pn = h{en) (n= 1,2,...)
we have, taking into account that {en} is a basis of Лх ({>>„}) (by §8,
proposition 8.1),
02.72)
onto Ax({>>„})* and the inverse map-
mapThus, {/3„}->й maps
ping is given by A2.69).
Finally, we have
= , sup
'""{'«„ill/i"'
sup
sup
H!"n!ll«l
theorem of Banach, the mapping (/?„}
i \\P nntn A.iiv \\*
fc is
whence, by the inversion theorem of Bana
an isomorphism of A^y^f onto >!,({>'„})*.
b) If {х} is a basis of E, then, by §3, proposition 3.2, А^{х„)) = Е,
rphism of A^y^f
b) If {х„} is a basis of E
by the mapping
and hence E*^A1({xn})*, by the mapping ufXnr.f-*h, where
fc({aB})=[«rXn)(/)]({a«}) = /["(x»!({«-})] = E я'/^)
A2.74)
Consequently, taking into account part a) above, it follows that
E* = Al(\xn}f, by the mapping /->{й(е„)} = {/(х„)}, i.e., by A2.71),
which, since the inverse of A2.71) is obviously the mapping (/?„}-»/
defined by A2.70), concludes the proof of proposition 12.3.
12. Properties of strong duality. Application: bases and sequence spaces 137
Remark 12.5. For the sake of completeness we observe that the iso-
isomorphism (ufXn})~i: A^lx^)*^E* is nothing else than the mapping
h->f, where
E *i*i = E «i/M = E «!>
E W
A2.75)
which, together with part a) above, implies again that the mapping
{&}->/ defined by A2.70) is an isomorphism of Ax({xn}f onto E*,
inverse to A2.71).
Let us now formulate the problem of characterizing the infinite
matrices which "preserve bases". Let {х„} be a basis of a Banach space
E and let {у„} с Е, упф0. Then there exists an (unique) infinite matrix
а = (а^) such that
CO
*= HbjXi (/=1,2,-),
i= 1
or, with the matrix notations Ж = (х„), ^ = (>'„), such that
A2.76)
A2.77)
We shall consider the problem, what conditions on a = (au) are
necessary and sufficient in order that {у„} be a basis of E. The following
proposition will be useful:
Proposition 12.4. Let {xn} be a basis of a Banach space E and let
{>>„}<=?, у„#0(п =1,2,...) be defined by A2.76). Then
a) The matrix а = (аи) induces a continuous linear mapping va: Ax(\yn})
{} d
A2.78)
or, with the matrix notation a = l a2 I, by1
ra(a) = aa. A2.79)
b) Tfte transposed matrix a' = (a,-,-) induces a linear mapping
¦ Ai({xn)f ^>Ax({yn}f defined by
A2.80)
In A2.79) and A2.81) we consider va(a) and co.(jS) as one-column matrices.

138 I. The Basis Problem. Some Properties of Bases in Banach Spaces
or with the matrix notation fl = \ [i2 I, by
and for the adjoint v*\ АЛ{х„})*-> АЛ{у„})* of va we have
{[v*(hj](eyj)}^=1 = vat{{h{e^)}) (НеАЛ{х„})*), A2.82)
where in the left side e\ denote the unit vectors in АЛ{)!„}) and in the
right side e* denote the unit vectors in АЛ{х„}).
с) Consequently, we have
vat = w{ylniv*w{Xn}, A2.83)
where w{Xn},w{yni denote respectively the isomorphisms Л1((х„})'!
^АЛ{х„})* and \({yn}f = AX{{yn})* of proposition 12.3a).
Proof, a) If {<хп}еАЛ{уп}), then, by A2.76),
where {/„}c?* is the a.s.c.f. to {х„}. Hence each series Y,aijy-j
30 J X)
converges and i Y,aijaj\ e^i(i-x-i;)' namely,
J=l
go / gc
1A^ф=1^Уг О2-84»
i=l V=l / J=l
Furthermore, the mapping va defined by A2.78) is obviously linear
and for every [а„}еЛ1({у„}) we have
i'a((a,,})l!= sup
E
;=i V = l
Un)
E ^>v
V!xn) SUP
jp
< X
n
E
'X.
E
1= 1
Xjyj
( x
\j =
= \
whence ||r0
b) Let "heAi({xn})* be arbitrary. Then, since by A2.76) {fl,v}/=i
еАЛ{х„}) (/=1,2,...) and since {<?*) is a basis of АЛ{х„}), we have,
taking also into account A2.78),
12. Properties of strong duality. Application: bases and sequence spaces 139
Hence the series ? ао-й(е?)(/=1,2,...) converge for all
i= 1
and, since v*(h)eAЛ{yп})*, from proposition 12.3a) we infer that
e^i({}'n}f for all НеАЛ{х„})*. Consequently, again
by proposition 12.3a), the series Y,aijPt (/=1,2,...) converge and
^ЛШ)" for all {Pn)eA^xxn}f. Thus, var. АЛ{х„})р
,({>'„}? is well defined by A2.80). Furthermore, by A2.80) we have
j=l
which, together with A2.85), gives A2.82).
Finally, the linearity and continuity of va, can be seen directly, but
they also follow from formula A2.83), which will be proved below.
c) By the definition A2.69) of the isomorphisms w^(, w,"^ and by
A2.82) we have
tVwj-n11№ = iv({^)})={[t'a*W]H)}f=1=wjynV*W (he АЛЫ*),
whence, putting h = w,xJ{^}) ({р„}еАЛ{хп})р), we infer A2.83), which
completes the proof of proposition 12.4.
Now we can prove
Theorem 12.7. Let {xn} be a basis of a Banach space E, let [yn] a E,
упф0 (и =1,2,...) and let а = (аи) be the (unique) infinite matrix satis-
satisfying A2.76). Furthermore, let va: АЛ{уя})^> АЛ{хп}У and i>: A?xxn}f
-»Ax ({>'„}/ be the mappings defined, in proposition 12.4. The following
statements are equivalent:
1°- !>'n] is a basis of the space E.
2°- [>'n] = ?- and there exists a constant C^l such that we have, for
any finite sequence of scalars at,..., an + nl
x /n + m
E
3°. va maps АЛ{уп}) one to one onto АЛ{х„}).
4°. а = (ац) has an inverse b = (bh) such that
Ы'^еАЛЫ? (/=1,2,...),
A2.86)
A2.87)
A2.88)
Ji=i

140
I. The Basis Problem. Some Properties of Bases in Banach Spaces
5°. vat maps A1({xn}f one to one onto Ax({yn}f.
6°. va(A1({yn})) = A1({xn}) and vat(A1({xn}f) = A1({yn}f.
T. (i)va is one to one on A^y,,}); (ii) v^A^y^j) is closed in /11({х„});
(iii) vai is one to one on A^x^f.
8°. (i) va is one to one on AY({>>„}); (ii) vat(Ax{{xn}f) is closed in
Ai({yn}f', (iii) vat is one to one on A^x^f.
9°. (i) Al({yn}f = {{f(y^}\feE*}\ (ii) ty is one to one on A^x^f.
Proof. By A2.84) we have
CO
Z
CO / И
and hence the equivalence I°o2° is a consequence of §7, theorem 7.1.
I°o3°. Let us first observe that in any case we have
"(»,} = Mf*»itJ«> A2.89)
where u,Xnj denotes the isomorphism A2.73) of ^({x,,}) onto E and
GO
u{yn) denotes the continuous linear mapping {а„}-> ? Оуу,- of Л,({у„})
into E. Indeed, by A2.78) and A2.84), j = i
Х«Л )= E I>t/«j Ц
i Ji=i/ \ /
Consequently1, r0 is onto if and only if for each xeE there exist
GO
scalars a, such that x= ? Оуу,- and гд is one to one if and only if {yn}
j=i
is co-linearly independent.
°. If {}»„} is a basis of ?, there exists a (unique) matrix b = (bi])
t
such that
(/=1,2,...).
A2-90)
;= l
Hence, by proposition 12.4 a) applied for [xn] interchanged with
GC
{у„}, for every {}>„}е/^({x,,}) each series ? b^yj converges and
1 Let us also mention the following alternative argument: by A2.89) we have
3°, or equivalently (by the inversion theorem of Banach), va is an isomorphism
of /),({у„}) onto /Ii({.*„}), if and only if ulyn} is an isomorphism of /4i({yn}) onto
E, which holds (since by §8, proposition 8.1, {е„} is a basis of ^({}/„}) and since
u{yn>(ej) = >'j f°r i= 1>2,...) if and only if {у„} i's a basis of E.
12. Properties of strong duality. Application: bases and sequence spaces 141
00 ) GO
Mly*}). i-e-' we have A2-87) and A2.88). Furthermore,
by A2.84) for {х„} interchanged with {yn},
<Xj /CO
E E btiy})yi= E УМ
;=i \j=i /
A2.91)
Since by A2.76) we have {yJ} = {aJI[}JL1eA1{{xt,}l applying A2.91)
to this particular sequence we obtain
CO /CO
E E bijajk b,- = E aJkXj=yk (k=1,2,...),
1 = 1 \J=1 / J-1
whence, since {}'„} is a basis of ?,
J=l
i.e., ba = the identity matrix. Similarly, from A2.84) for {Xj} = {bJk}f=i
GAi({yn}) (by A2.90)), taking also into account that {х„} is a basis of E,
we obtain that a b = the identity matrix, and thus b is the inverse of a.
4°=>3°. Assume that we have 4°. If for a sequence {а„}еЛ1({>>„})
we have и„({аи}) = 0, i.e., аа = I Еаоа./) = °' t'len' ЬУ 4°'
\j=l /f=l
а = Ьаа = 0, i.e., а„ = 0 (п=1,2,...). Thus, va: A^y,,})-»-Ai({xn}) is one
to one.
Now let у = {у„}е/11({х„}) be arbitrary. Then, by 4°, Ьуе^Цу,,})
and y = aby = va(by). Thus, ca maps ^id^}) onto A^x^).
3°«.5°. We have 3° if and only if v* maps ^({x,,})* one to one
onto A^ly,,})*, which, by proposition 12.4, happens if and only if we
have 5°.
3°^6° by 3°=>5°.
The proof of the implication 6° => 3° is similar to the above proof
of the implication 5° => 3°.
3°«>7°. va(A]({>¦„})) is dense in A^jxJ) if and only if1 r* is one
to one on Л1({х„))*, which, by proposition 12.4, happens if and only
if we have (iii) of 7°. Consequently, va maps ^4i({>'n}) onto ^({x,,})
if and only if we have (ii) and (iii) of 7°.
3° => 8°. If we have 3°, then we obviously have (i) if 8° and, by the
implication 3°=>5° proved above, we also have (ii) and (iii) of 8°.
8°=>7°. If we have 8°, then by (ii) and by proposition 12.4,
vUAi({xn}f) is closed in /^({y,,})*, whence there follows2 (ii) of 7°.
1 See e.g. [50], p. 479, lemma 8.
2 See e.g. [50], p. 488, theorem 4.

142
I. The Basis Problem. Some Properties of Bases in Banach Spaces
I°o9°. By §8, theorem 8.2, {х„} is a basis of E if and only if we
have (i) of 9° and [}>„] = ?. Now, if va, is not one to one on Л1((х„})'!,
then there exists a sequence {/3„}е/41({х„})", {/?„}#0, such that
va'({Pn\) = ) E aijPn = 0- Hence, for the functional feE* defined
Li=i Jj = i
by A2.70) we have /У О and
/(}',)=/(l>,r^ = f>n& = 0 G=1,2,...)
and therefore [>„]#?. On the other hand, if t> is one to one on
and /еР,/(}ь) = 0 0=1,2,...), then {/(xI1)}e,41({x11})'1 and
«-'..({Я*.)}) = { E ay/(^.-)} = {/( E «yXi
Li=1 J j=l I \i=l
whence /=0 and therefore [>¦„] = ?. This completes the proof of
theorem 12.7.
Remark 12.6. In the above proof we have also shown that the fol-
following statements are equivalent:
1°. [yn]=E.
2°. va(Ai({y»})) is dense in Л,({х„}).
3°. vat is one to one on ^([x,,)/.
This observation, together with theorem 12.7, yields necessary and
sufficient conditions in order that {>'„} be a basic sequence in E.
Definition 12.2. We shall say that a sequence space S is a multiplier
algebra for a basis of a Banach space, if there exists a Banach space E
with a basis {х„} such that S = M(E, (х„,/„)), where {/„) с E* is the
a. s. с f. to {x,,}.
We shall consider the problem of characterization of such sequence
spaces.
We recall that a sequence space S is called a BK-algebra if S is a
BK-space and S is closed under coordinatewise multiplication, i.e., S is
a subalgebra of the algebra of all sequences. In a BK-algebra S the
functions x—>>x and x->x>' are continuous in x for fixed у by the
continuity of the coordinate functional and the closed graph theorem.
Consequently1, every BK-algebra is a Banach algebra (in general,
without identity).
Theorem 12.8. A sequence space T is the multiplier algebra for a
basis of a Banach space if and only if T is a y-perfect BK-algebra con-
containing all unit vectors en and the identity e= {1,1,1,...}.
See [51], p. 860—861.
12. Properties of strong duality. Application: bases and sequence spaces 143
Proof. Assume that T=M(?, (х„,/„)), where {х„} is a basis of a
Banach space E, with the a.s.c.f. {/„}. Then е„еТ (n=\,2,...), eeT
and by § 5, proposition 5.4 a), T is a Banach algebra for the norm E.29).
Furthermore, by § 5, proof of corollary 5.2 a), the coordinate functional
are continuous on T. Finally, by § 5, corollary 5.2, T is the space
A2 = A2({en}) for the basis {е„} of S=[en]T, whence, by the above
proof of the necessity part of theorem 12.6, T is y-perfect (moreover,
T=Sn).
Conversely, assume now that T is a y-perfect BK-algebra con-
containing all unit vectors е„ and the identity e= {1,1,1,...}. Then, by
the above proof of the sufficiency part of theorem 12.6, {e,,\
is a basic sequence in T (moreover, we also have T= [ejp'
A2.92)
sup
1 < n < go
n
1= 1
< oo >). We shall prove that
T=M(S,(en,hn)),
where S=[en]T and where {hn} is the a.s.c.f. to.the basis {е„} of S,
which will complete the proof.
Let {у„}е Т be arbitrary. Then, since Tis a BK-algebra, the mapping
v[ynj defined by
»{у„}(Ы)= {>¦„«„} = ЫЫ ({я.} 6
A2.93)
is a continuous linear mapping of T into T, as we have observed above.
m
Since for any finite linear combination E «j^eS we have iv.n, ( E aiei
= E yi'xieh it follows that c|Ii?,(S)cS = [cJT and hence, since {е„} is
' = 1 X ОС
a basis of S, ^ У;^,- converges for every ^ a^eS. Consequently,
{yn}eM(S,(en,ftj) and thus Tc M{S,(en,hn)\
Now let {yn}eM(S,(е„,й„)) and {Р„}еТу be arbitrary. Then, by
the above proof of theorem 12.6, formula A2.65), there exists a functional
heT* such that
h{en) = pn (и =1,2,...). A2.94)
Furthermore, by § 5, proposition 5.4 b), for the mapping r,7n, defined
by A2.93) we have
:eL(S,S). Since eeT, we have for а [ер1'
cTyy=T (because for any {<xn}ebv and {ft}e[e]y we have
2_, |a, —aj+1|<oc, sup E ft < ^^ whence, by E aifi'i

144
I. The Basis Problem. Some Properties of Bases in Banach Spaces
= ? (a,-a,+ i) E #• + «„+! ? P'r we8et SUP
i=i j=i j=i 1«"<1
? *<ff
< oo, i.e.,
{а„}е[е]тг). Hence, since the sequence I ? еЛ is bounded in bv and
u = i 3
since the inclusion map of BK-spaces is continuous (by the same argu-
argument as that used in the above proof of formula A2.66)), it follows that
"] e,;} a S a T is bounded in S. Consequently,
E y>Pi
E У Me,)
h[ ? У^
,i= 1
sup
h\ "(y»>( Ee,-
whence {у„}еТуу= Т. Thus M(S,(en,hn))<= T, which, together with the
opposite inclusion proved above, gives A2.92). This completes the
proof of theorem 12.8.
§ 13. Bases in topological linear spaces. Weak bases and bounded
weak bases in Banach spaces. Weak* bases and bounded
weak* bases in conjugate Banach spaces
The notion of a basis has the following natural extension to general
topological linear spaces:
Definition 13.1. A sequence {х„} is a topological linear space U is
called a basis of U if for every xeU there exists a unique sequence of
scalars {а„} с К such that я,
i= 1
convergence of the series being that of the topology on U.
A topological linear space U which possesses a basis {.хя} is clearly
separable1, since the set of all finite linear combinations ?r,x,-, where
the r, are rational scalars, is countable and dense in U. It is natural to
ask whether or not the converse is also true, i.e. whether or not every
separable topological linear space U possesses a basis. The answer is
negative, as shown by
1 I.e. (see e.g. [50]. p. 21, definition 11) there exists a countable subset of U,
which is dense in U; see also the Notes and remarks to this §.
13. Bases in topological linear spaces. Weak and weak* bases
145
Example 13.1. Let U be a separable complete metric linear space,
such that the only continuous linear functional on U is f=0 (e.g. one
can take C/ = S([O,1]) = the space of all equivalence classes of meas-
measurable functions on [0,1], endowed with the metric p(x,y)
It, or [/ = Lp([0,l]) with 0<p<l, where a
fundamental system of neighbourhoods of 0 is given by the sets
i )
xeU $\x(t)\pdt<t:\, ?>0). Then U has no basis. In fact, let us
о J
sketch the proof of this statement. With a method similar to that used
in the proof of §3, theorem 3.1, one can prove1 that the coefficient
functionals /„ associated to any basis \xn} of U (see § 14, definition 14.1)
are continuous on U. Hence, by our hypothesis, /„ = 0 (и =1,2,...).
However, this is impossible.
We shall not present here the theory of bases in topological linear
spaces, since it would take us beyond the scope of this book. We shall
consider only the particular cases when U is a Banach space endowed
with one of the usual associated locally convex topologies; it will turn
out that the bases of these spaces are closely related to the bases of
Banach spaces endowed with their norm topologies.
Definition 13.2. A sequence {х„} is a Banach space E is called a
weak basis (bounded weak basis) of E, if {х„} is a basis of the topological
linear space U obtained by endowing E with the weak topology a(E,E*)
(respectively, with the bounded weak topology2). A sequence \hn} in the
conjugate space E* (of a Banach space E) is called a weak* basis (bounded
weak* basis) of E*, if {hn} is a basis of the topological linear space U
obtained by endowing E* with the weak* topology a(E*,E) (respec-
(respectively, with the bounded weak* topology). For the sake of brevity, we
shall use for such bases the terms w-basis, fow-basis, w*-basis and bw*-
basis respectively.
Lemma 13.1. a) Let {у„) be a sequence in a Banach space E and let
yeE. Then, w-\imyn = y if and only if bw-\im yn = y.
Л-* CO П-* GO
b) Let {gn} be a sequence in the conjugate space E* (of a Banach
space E) and let geE*. Then, w*-lim д„ = д if and only if bw*-\im gn = g.
1 See the Notes and remarks to § 3.
2 We recall that the bounded weak topology (b vv-topology) for E is the strong-
strongest topology which coincides with the weak topology a(E, E*) (vv-topology) on
each t-cell {xeE | ||x||<t} of E. The bounded weak* topology (frw*-topology) for
E* is the strongest topology which coincides with the weak* topology a(E*,E)
(vv*-topology) on each r-cell of E*.
10 Singer, Bases in Banach Spaces I

146 I. The Basis Problem. Some Properties of Bases in Banach Spaces
Proof, a) If bw-lim yn = y, then, since the fow-topology is stronger
than the w-topology, we have w-lim yn = y.
П—» GC
Conversely, if w-lim у„ = у, then there exists a!>0 such that \,yn <(
n-* x
(n=l,2,...) and ||.y||<t. Let >U be an arbitrary open bw-neighbour-
hood of v and put fSt = [xe? | ||x|| ^t}. Then <%ntSE is relatively
w-open in tSE. Hence there is a w-neighbourhood f of у such that
f ntSE<=J/ntSE. Since w-lim у„ = у, there exists a positive integer
/V such that yne i nrSE for all n^N, whence упеШ for all n^N and
so b w-lim yn — y.
П-* GC
The proof of part b) is similar.
Theorem 13.1. Let be a Banach space, {xn\ a sequence in E, and {hn}
a sequence in E*. Then
a) The following statements are equivalent:
1°. {xn} is a w-basis of E.
2°. {х„} is a b w-basis of E.
3°. {х„} is a basis of E.
b) The following statements are equivalent:
1°. {hn} is a w*-basis of E*.
2°. {hn} is a bw*-basis of E*.
Each of these latter statements implies the following .-
3°. {hn} is a basic sequence.
4°. r([hn])>0 (where г([й„]) = the characteristic of the subspace
[>„] of E*).
Proof. The equivalences 1 Oo2° of a) and b) are immediate conse-
consequences of lemma 13.1.
a) 3°=>1°. If {х„} is a basis of E, every xeE has an expansion
? / " \
x= 2_, aixn whence also a w-expansion /I ? a,X; )-» f(x) (feE*). This
w-expansion is unique, since from the relations /I ? a,x,-)->0 (feE*)
\i=l J
we obtain, taking /=/} (where {/„} is the sequence of coefficient func-
tionals associated to the basis {xn}), that rx.J = 0 (/=1,2,...). Thus (xA}
is a w-basis of ?.
If we have а) Г, then the w-closed linear span of {х„} coincides
with E, whence1 also [xn] = ?. Therefore, in order to prove a) 1°=>3°,
it is sufficient to prove that a) 1 =>[х„} is a basic sequence. However,
this latter implication, as well as the implication b) 1°=>3°, are conse-
consequences of the following more general result:
1 See e.g. [10], p. 134, theorem 2.
13. Bases in topological linear spaces. Weak and weak* bases
147
Proposition 13.1. Let E be a Banach space, and let M be a subset of
the conjugate space E*, having the following two properties:
@>x) If {zn}<=E and /(zJ->0 for all feM, then sup ||zj<oo.
For every ?>0 there exists a <5>0 such that the relations
||х„К<5(п=1,2,...), /(*„)-Дхо)(/еМ) imply \\хо\\^в.
Then every "basis with respect to M" of the space E, i.e. every se-
sequence {xn} c? such that for each xeE there exists a unique sequence
of scalars {а„} с К satisfying /( ? а,-х,-J-»/(x) (feM), is a basic
sequence. ^' = 1 '
Proof. For the sake of brevity, we shall say that a sequence {zn} a E
is (M)-convergent to xeE, and we shall write zn^lx, or (M)-limzn = x,
if we have f(zn)->f(x) (feM). From @>2) it follows that the (M)-limit
is unique, i.e. that the relations zn^lx, zn{-^Xx' imply x = x' (since
2n-zn{^Xx~x' and \\zn-zn\\^S for all <5>0). If f yt""
x i = 1
shall write (M) У yi = x.
, then we
i 1
Assume now that {xH} is a basis with respect to M of the space E,
and let A\M) be the linear space of sequences of scalars
, a,-*,- (M)-converges to an element of E>, A3.2)
endowed with the norm
IIWII= sup
A3.3)
By @{) the number A3.3) is finite for every {а„}еА\М). Since {xn{
is a basis with respect to M of the space E, we have хи#0 (n= 1,2,...),
whence A3.3) is indeed a norm on A\M).
We claim that A\M) is a Banach space. In fact, let {^} (fc= 1,2,...)
be a Cauchy sequence in A[M}. Then for every e>0 there exists a positive
integer N(e) such that for k.m>N(r.) we have
(и =1,2,...),
A3.4)
where <5>0 is chosen as in BP2). Hence there exists, by the arguments
used in the proof of §3, proposition 3.1, a sequence of scalars [а„] such
that
= а„ (n = l,2,...). A3.5)

148 I. The Basis Problem. Some Properties of Bases in Banach Spaces
Let us prove that {xn}eA\M). Put
Then, by A3.4) and CP2) we have
\\Ук-Ут\\^с (k,m>N{e)),
whence, by the completeness of the space E, there exists an xeE such
that
lim>'k = x. A3.6)
ft-* go
We shall show that x = (M) Z a,x,., which will prove that {х„\еА\М).
We have the inequalities i=1
fix- Y,aix,
.1 \Ук — L, ai Xi
A3.7)
Let t]>0 and feM be arbitrary, and put
П
e =
4Ц/11
A3.8)
Then by A3.6) there exists a k>N{z) (where /V(e) is as in the above),
such that ||x-yk|| <e, whence, by A3.8),
ц
A3.9)
On the other hand, since k>N(?X from A3.4) for m->cc we obtain,
by A3.5),
n n
у а(*)х._ у a.X; ^min(f;<)) (n=l,2,...), A3.10)
i=l i=l
whence, by A3.8),
= i i = i
A3.11)
Furthermore, since yk = {M) Z^'x,-, there exists, by the definition
i = 1
of (M)-convergence, a positive integer n0 such that we have
13. Bases in topological linear spaces. Weak and weak* bases
149
- (п>п0).
Thus, by A3.7), A3.9), A3.11) and A3.12) we have
/ x -
(n>n0),
A3.12)
which proves that x = (M) Z «,-x,- and that {^j
i= 1
Now, by A3.10) (which is actually true for all k>N(e)) and A3.3)
we have
whence lim {а*,'1)} = {а„} in A[M). This proves that Л(,М) is a Banach
space.
Hence, by arguments similar to those used in the proof of § 3, pro-
proposition 3.2a), the mapping
{«.}¦
A3.13)
is an isomorphism of A^ onto E. Consequently, as in the proof of § 3,
theorem 3.1, the coefficient functionals /„(х) = а„ (xeE, n=\,2,...) are
continuous on E, i.e. fneE* (и = 1,2,...). Thus, by the implication
3°=>1° of §4, theorem 4.1, {х„} is a basic sequence, which completes
the proof of proposition 13.1.
Now, applying proposition 13.1 for M = E*, we obtain the implica-
implication a) 1°=>3° of theorem 13.1 (taking into account that for a w-basis
{х„} of E we have [х„]=?, as remarked above). Finally, applying
proposition 13.1 for E* instead of E, {hn} instead {х„}, and for M = the
canonical image of E in ?**, we obtain the implication b) Г => 3°.
b) Г=>4°. Let us denote by V the subspace [й„] of ?*, by V1 the
set of all w*-limits of all w*-convergent subsequences of V, and by
the union of the w*-closures of the t-cells {feV
t<oo. If we have Iе, then every feE*
n
/(x)=lim У а,/2,(х) (xe?), whence Кх = ?
\ } of V, for all
has a w*-expansion
Since1 Vl^V(i\ it
follows that F<1» = ?*, whence, by a theorem of J. Dixmier2, we have
r(V)>0, i.e. 4е. This completes the proof of theorem 13.1.
1 See e.g. [47], theorem 1 (a).
2 See e.g. [47], theorem 6.

150
I. The Basis Problem. Some Properties of Bases in Banach Spaces
One cannot replace b) 3° by the assertion: {hn} is a basis of ?*.
In fact, it may happen that ?* has a w*-basis, but it is not even separable,
as shown by
Example 13.2. Let ? = /'. Then the unit vectors in E* = m consti-
constitute a w*-basis of ?* (see § 14, theorem 14.1), but E* is not separable
(and hence it has no basis at all).
Conversely, it is natural to ask whether or not a basis {hn} of ?*
is also a w*-basis of ?*. The answer is again negative, as shown by
Example 13.3. Let ? = c0 and define {hn}<=E* = ll by
fti=/i, K = L-i~fn (n = 2,3,...), A3.14)
where {/„} is the natural basis of I1. Then {«„} is a basis of ?*, but not
a w*-basis of ?*.
Indeed, observe first that by § 4, proposition 4.3, the sequence
^„={1,...,1,0,0,...} (п=1Д...) A3.15)
isabasisof ? = c0. IdentifyingcanonicallyI1 with ?* = cg, thesequence
{и„+1} is obviously the sequence of coefficient functional associated
to the basis {>'„} of ? = c0, whence, by §12, theorem 12.1, {«„+1j is
30
a basic sequence. Now, hl$\_hn+1~\, since for the functional Ф0(f) = ? b
i= 1
(f={QeE* = ll) we have Ф0(й1)=1, Ф0(йи+1) = 0(п=1,2,...) and {«„}
is complete in ?*, since the relations Фе?**, Ф(и„) = 0 (и=1,2,...)
imply Ф = 0. Consequently, («„j is a basis of ?*. Furthermore, although
every /e?* has a w*-expansion /(x) = ? а,й;(х) (хе?) (since {й„}
isabasisof ?*), this w*-expansion is not unique, since by й,(х) — 5] и,(х)
=./„(х) (xe?, « = 2,3,...) we have /i,(x)- E «,(x) = 0 (xe?). Thus,
{й„} is not a w*-basis of ?*.
Since there exists no locally convex space U satisfying the hypotheses
of example 13.1, it is natural to ask whether or not every separable
locally convex space (in particular1, every vv*-separable conjugate
Banach space) possesses a basis (respectively, a w*-basis). The answer
is negative, as shown by
Example 13.4. The conjugate space E* = m* of the Banach space
E = m is w*-separable, since the sequence {/„}<=?* defined by
1 Note that every conjugate Banach space E* endowed with the vv*-topology
a(E*, E) is a sequentially complete locally convex space.
14. Schauder bases in topological linear spaces. Properties of weak duality 151
, n=\,2,...)
A3.16)
is total on E (whence its w*-closed linear hull is ?*). However, the space
E* = m* has no w*-basis. In fact, assume the contrary, i.e. that m* has
a w*-basis {/„}. Then, by definition, for every fern* there exists a
unique sequence of scalars {а„} с К such that
lim /(x) - ? a,/-(x) = 0 (xem).
However, by a result of Grothendieck1, from these relations it
follows that
lim# [f- У xji = 0 (Фет**).
п-а> \- .t\ )
Consequently, m* is w-separable, whence2 also separable for the
norm-topology. However, this is impossible (since m itself is not separa-
separable). Thus m* has no w*-basis.
Problem 13.1. a) Does the conjugate space E* of a separable Banach
space ? possess a w*-basis? b) What about a separable conjugate Banach
space ?*? c) What about a conjugate Banach space E* having a basis?
Concerning a), we shall see in § 14 that if ? has a basis, then E*
has a w*-basis. However, the converse problem is unsolved:
Problem 13.2. a) If the conjugate space ?* (of a Banach space ?)
has a w*-basis, does the space ? possess a basis? b) Under the same
hypothesis, is the space ? separable?
§ 14. Schauder bases in topological linear spaces. Properties
of weak duality for bases in Banach spaces
Definition 14.1. Let {х„} be a basis of a topological linear space U.
The sequence of linear functional {/„} defined by
x=
i= 1
A4.1)
is called the sequence of coefficient functionals associated to the basis
{х„} (we shall write: a.s.c.f.).
1 Let us recall this result of Grothendieck (see e. g. [43], Ch. VI, § 4, proposition
D) (a), or [88], p. 168): If {gn}cm* and limgn(x) = 0 for all xem, then ИтФ(/) = О
for all Фет**.
2 See e.g. [10], p. 134, theorem 2.

152
I. The Basis Problem. Some Properties of Bases in Banach Spaces
Thus, if {х„} is a basis of U and {/„} the a.s.c.f, then every xeU
has a unique expansion of the form
A4.2)
Definition 14.2. A basis {xn} of a topological linear space U is said
to be a Schauder basis of U, if all coefficient functionals /„ (n= 1,2,...)
are continuous on U (i.e. if fneU* for и = 1,2,...).
By § 3, theorem 3.1, every basis of a Banach space ? is a Schauder
basis of this space. However, we shall see that in other locally convex
spaces1 this is, in general, no longer true. Among the properties of
bases which are not Schauder bases, let us mention the following:
Proposition 14.1. Let {xn} be a basis of a topological linear space U,
which is not a Schauder basis of U, and let {/„} be the a.s.c.f. If fx ф U*,
then
a) xl6[xn]? = l/.
b) {xn}i is not a basis of [х„] J" = U.
Proof, a) If х1ф[хп~]%, then there exists a linear2 functional f on U
such that
/(x1)=l, f(y) = 0 (ye [*„]?).
Since {х„} is a basis of U, every xeU has a unique expansion x = a.1x1
GC
+ Yj aixi and hence the null-subspace of/ is [xn]J. Since this sub-
i — 2
space is closed, / is continuous on U, i.e. fell*. On the other hand,
00
for every x= ?/;(х)х;еС/ we have
/W=/ Z fi(x)x, =/
and hence f1=feU*, which contradicts the assumption f\фU*. Thus
¦VieWi, whence C/ = [xn] = [x1; [xn]^] = [xH]J'.
1 When considering more general topological linear spaces, it is obvious that
those spaces on which the only continuous linear functional if/ = 0, do not admit
any Schauder basis.
2 A priori we don't know that / is continuous, since U is an arbitrary topo-
topological linear space.
14. Schauder bases in topological linear spaces. Properties of weak duality 153
b) We claim that xx has no expansion with respect to {xn} J', whence
00
{xn\i *s n°t a basis of [xn]^ = t/. Indeed, if x: = ? ^x,, then
ОС IT
Xj — Z Ax. = 0' and thus 0 has two distinct expansions ^ 0.x, and
i = 2 i = 1
30
xt — Z Pixt> which contradicts the assumption that {х„} is a basis
i = 2
of C/. This completes the proof of proposition 14.1.
In the sequel we shall be concerned again with the particular cases
when U is a Banach space E endowed with one of the locally convex
topologies considered in §13. Since a linear functional on E is con-
continuous if and only if it is w-continuous, which happens if and only if
it is b w-continuous, it follows that in a Banach space E the notions of
Schauder basis, w-Schauder basis and bw-Schauder basis are equivalent.
Moreover, by theorems 3.1 and 13.1 these notions are also equivalent
to the notions of basis, w-basis and bw-basis. In conjugate Banach
spaces endowed with norm-, w*- and bw*-topologies, the situation is
different. Since a linear functional Ф on E* is w*-continuous if and only
if it is bw*-continuous on ?*, it follows that in E* the notions of w*-
Schauder basis and bw*-Schauder basis are equivalent. However, a
w*-Schauder basis of E* need not be a (Schauder) basis of ?*, as shown
e.g. by the sequence considered in § 13, example 13.2. Conversely, a
(Schauder) basis of E* need not be a w*-Schauder basis of ?*, as shown
by §13, example 13.3. Furthermore, a w*-basis of E* need not be a
w*-Schauder basis of ?*, as shown by
Example 14.1. Let E = c0 and let {/„} be the sequence in ?*s/!
defined by . n+1
where {е„} denotes the natural basis of I1. Then {/„} is a basis of E* in
the norm-topology and a vv*-basis of ?*, but not a w*-Schauder basis
of this space.
Indeed, let {Ф„} с ?** be the a.s.c.f. to the basis {en} of E* and put
j=2
Then (/„, yn) is a biorthogonal system and we have
n Г oo —1 n
¦=1 L j=2 ' ' J
s=2
Z (-
(/е?*,и=1,2,...).

154
I. The Basis Problem. Some Properties of Bases in Banach Spaces
whence, since ? |Ф//)|<°° (f^E* = lx), for every e>0 and feE*
there exists an integer N(e,f)>0 such that
Z tif)fj - Z *
<e
(n>N(e,f)).
Hence /= Z 10)fj
f$E*, and thus, by virtue of the
implication 2°=> 1° of § 4, theorem 4.1, {/„} is a basis of E* in the norm-
topology.
Furthermore, from the above it follows that every feE* also has a
w*-expansion x
and we shall now show that this expansion is unique. Assume that for
a sequence of scalars {«„} we have
(xeE).
A4.5)
Since for x = bn (n = 1,2,...), where {bn) is the natural basis of E = c0,
we have, by A4.3),
}=(-iy+1, /ДЬ„) = <5> (« = 2,3,...; j=l,2,...),
from A4.5) for х = Ь„(и=1,2,...) it follows that
X (-l)J'+4- = 0, «2 = a3= ••• =«„= ••• =0,
j=i
whence an = 0(n=l,2,...). Thus {/„} is a w*-basis of ?*.
On the other hand, by A4.4) we have х1фж{Е), whence ул is not
w*-continuous, and thus (/„} is not a w*-Schauder basis of E*. This
completes the proof of our assertion.
We can now summarize the preceding results concerning a sequence
{hn} с Е* in the following diagram of implications and non-impli-
non-implications:
Г Schauder basis ~|
[_ (<=> basis) J
w*-Schauder basis
> bw*-Schauder basis)
A4.6)
Г Basic sequence"!
[with г([/г„]) >OJ
[w*-basis ~|
(<=> bw*-basis)J
14. Schauder bases in topological linear spaces. Properties of weak duality 155
We have the following theorem of weak duality:
Theorem 14.1. A sequence {/„} in a conjugate Banach space E* is a
w*-Schauder basis of E* if and only if E has a basis {х„} whose a.s.c.f
is {./„}•
Proof. Assume that {/„} is a w*-Schauder basis of E*. Then the
associated coefficient functionals
«„ = (/)„(/) (n=l,2,...)
are all w*-continuous on E*, whence there exists a sequence {х„} <= E
such that
</>„(/) = /(*„) (feE*, «=1,2,...).
Then, from </>;(./}) = <3iy we infer that
i.e. that (х„,/„) is a biorthogonal system. Since {/„} is a w*-basis of E*,
with the a.s.c.f. {ф„}, we have
lim
^(/)/.W=/W (xeE, feE*). A4.7)
Hence, for the sequence of partial sum operators sn(x) = Y, fi(x)xi
i= 1
(xeE, n= 1,2,...) we have sup ||sn(x)|| <co (xeE). Thus, by the impli-
cation 3°=>1° of §4, theorem 4.1, {х„} is a basic sequence. We claim
that [х„] =-Е. In fact, otherwise there exists an xoe?, хо^[х„], whence
also an feE* with /0(x0)=l, /0(xn) = 0 (n=\,2,...). Thus, by A4.7),
0= ^ /oW/iW^/o^oK'' which is impossible. This proves that
i= 1
[х„] = ?, and thus {*„} is a basis of E, with the a.s.c.f. {/„}.
Conversely, assume now that [x,,J is a basis of ? with the a.s.c.f.
{/„}. Then
f(x) = X /(x,)/,M (*e?, /e?*), A4.8)
i.e. every feE* has a w*-representation by {/„}. In order to prove the
uniqueness of this vv*-representation, assume that for a sequence of
scalars {я„} с К we have
lim
(xeE).
Then for х = х7 G=1,2,...), taking into account the biorthogonal-
ity relations /i(xj) = <5u, we obtain

156
I. The Basis Problem. Some Properties of Bases in Banach Spaces
<X; = 0 0=1,2,...),
and thus {/„} is a w*-basis of E*. Finally, the coefficient functionals
*, и =1,2,...)
of the w*-representation A4.8) are obviously w*-continuous on E*, and
thus {/„} is a w*-Schauder basis of E*, which completes the proof of
theorem 14.1.
An immediate consequence of theorem 14.1 is the following:
Corollary 14.1. The conjugate space E* of a Banach space E has a
w*-Schauder basis if and only if the space E has a basis. Consequently,
the problem whether the conjugate space E* of an arbitrary separable
Banach space E possesses a weak* Schauder basis is equivalent to the
basis problem (problem 1.1).
However, the general w*-Schauder basis problem (i.e.: does every
w*-separable conjugate Banach space E* possess a w*-Schauder basis?)
and hence also the Schauder basis problem in locally convex spaces
(i. e.: does every separable locally convex space U possess a Schauder
basis?) has a negative answer, as shown by § 13, example 13.4.
Consider now the following assertions concerning a Banach space E:
1°. E is separable.
2°. E has a basis (<=> E* has a w*-Schauder basis).
3°. ?* is w*-separable.
4°. E* is separable.
5°. E* has a basis.
6°. E* has a w*-basis.
Then the preceding results concerning these assertions can be
summarized in the following table of implications:
1°
2"
3r
4°
5°
6°
1°
+
+
— m
+
+
7
2°
9
+
— m
9
9
7
Table 14.
3°
+
+
+
+
+
+
1.
4°
-I1
-P
-l\m
+
+
5°
-I1
-I1
-l\m
9
+
6"
9
+
— m
7
7
+
14. Schauder bases in topological linear spaces. Properties of weak duality 157
Here the symbol + (respectively —) in the и-th row and m-th column
means that the implication /j°=>m° is (is not) valid, while the symbol ?
means that the problem of the validity of the implication is unsolved.
The only implications which have not been proved in the preceding, are
Г=>3° and 4°=>1°, but they are well known1. On the right side of each
symbol —we have also written the concrete spaces E which yield the
desired counter-examples (naturally, there exist also other counter-
counterexamples). The symbols? are nothing else but the problems 1.1, 12.1,
12.2, 13.1 and 13.2; we have not formulated separately problems con-
concerning w*-Schauder bases in E*, since they are equivalent to the cor-
corresponding problems concerning bases in E (by corollary 14.1). We have
not considered the property of existence of a basic sequence in E*, since
such sequences exist in every Banach space (indeed, this follows from
Ch. II, § 15, proposition 15.1, since every separable Banach space E can
be embedded isomorphically in a space with a basis, namely, in C([0,1])
and since obviously there exists a sequence {yn} с E such that ||yj = 1,
/,(>'„) = 0 for i=l,...,n; и =1,2,...). One can also combine some of the
above hypotheses, e.g., the question whether 2° and 4° together imply
5° is nothing else than the first question in Ch. II, § 5, problem 5.1 a).
It is natural to raise the problem of the extension of the known
results for bases in Banach spaces to bases in locally convex spaces
and, in particular, to w*-Schauder bases in conjugate Banach spaces.
We shall not treat this problem in detail, since this would take us
beyond the scope of the present book, but we shall give here an ex-
example which shows that this extension is not always possible. For this
purpose, let us consider the Krein-Milman-Rutman stability theorem
(see corollary 10.2). It is natural to give
Definition 14.3. We shall say that a topological linear space U
having a basis (Schauder basis) has the KMR property for bases (Schau-
(Schauder bases) if every dense linear subspace of U contains a basis (Schauder
basis) of the space U.
By § 10, corollary 10.2 and §3, theorem 3.1, every Banach space with
a basis has the KMR property for (Schauder) bases. However, for
locally convex spaces this is, in general, no longer true, as shown by
Example 14.2. Let E = c0 and let G be a cr(?*,?)-dense norm-closed
linear subspace of E* = l\ of characteristic r(G) = 02. Then E* has a
w*-Schauder basis (by corollary 14.1), but G contains no w*-basis of
the space E*, and thus E* endowed with the w*-topology does not have
the КМЯ-property for w*-bases or w*-Schauder bases. In fact, assum-
1 See e.g. [10], p. 124, theorem 4 and p. 189, theorem 12.
2 Such a subspace exists, see e.g. [159], [47]; see also [33], p. 121, exercise 14f).

158
I. The Basis Problem. Some Properties of Bases in Banach Spaces
ing the contrary, let \gn} <= G be a w*-basis of the space E*. Then, by
the implication 1°=>4 of § 13, theorem 13.1 b), г{[дп~])ф0. On the other
hand, from [gJ с G and r(G) = 0 it follows that r([^n])^r(G) = 0, i.e.
г([G„]) = 0. This contradiction proves our assertion.
Remark 14.1. It is natural to ask whether or not with the method
of example 14.2 one can also obtain an example of a separable Banach
space having no basis, i.e. a negative solution of the basis problem.
However, the answer is negative. In fact, all known subspaces Gc?*
with r(G) = 0 possess bases and thus the statement 4° of §13, theorem
13.1 b) does not remain valid if we replace in it the w*-basis {h,,} of E*
by a basic sequence [hn] in E*.
§ 15. (e)-Schauder bases and (b)-Schauder bases in
topological linear spaces
Definition 15.1. Let \xn} be a Schauder basis of a topological linear
space U, with the a.s.c.f. {/„}. The sequence of continuous linear oper-
operators {sn}, where
t (xeU,n=l,2,...), A5.1)
{х)
is called the sequence of partial sum operators associated to the basis
In the case when U is a Banach space, we have, by the implication
1°=>4° of §4, theorem 4.1, sup ||sj<oo. A corresponding property
1 ^ tt < OO
can be defined for Schauder bases in general topological linear spaces
in several ways, of which the following two seem to be useful:
Definition 15.2. Let {xn} be a Schauder basis of a topological linear
space U, and let [sn] be the associated sequence of partial sum opera-
operators. {xn} is said to be
a) an (e)-Schauder basis of [/, if 0 is a point of equicontinuity of the
sequence {<;„} (or, equivalently1, if (sn} is an equicontinuous subset of
the linear space L{U,U) of all continuous linear mappings of U into
itself).
b) a (b)-Schauder basis of I/, if {sn} is a bounded subset of the space
L(U, U) endowed with the topology of uniform convergence on bounded
subsets of U.
Since every equicontinuous subset of L(U,U) is bounded2, every
(e)-Schauder basis of U is a (b)-Schauder basis of U. In the case when U
is a Banach space, every basis of U is an (e)-Schauder basis and a
1 See e.g. [32], p. 9, proposition 7.
2 See e.g. [33], p. 26, proposition 7.
15. (e)-Schauder bases and (ft)-Schauder bases in topological linear spaces 159
(b)-Schauder basis, as we have observed above. However, for general
topological linear spaces this is no longer valid. To illustrate this, we
shall consider here the case of Banach spaces endowed with w-topologies
and conjugate Banach spaces with w*-topologies.
Proposition 15.1. Let E be an infinite-dimensional Banach space. Then
a) E has no w-(e)-Schauder basis.
b) E* has no w*-(e)-Schauder basis.
c) // E has a basis, then every basis of E is a w-(b)-Schauder basis
of E.
d) // E* has a basis which is also a w*-Schauder basis of E*, then
every such basis of E* is a w*-(b)-Schauder basis of E*.
Proof, b) Assume that {/„} is a w*-Schauder basis of E*. Then, by
§14, theorem 14.1, E has a basis {xn} whose a.s.c.f. is {/„}. Let
Then
1
A5.2)
A5.3)
— 2"||xJ * ' '
Consider now the w*-neighbourhood
W=WX0.A(Q)={feE*\\f(x0)\<\},
and let V=Vyitym;l.@) be an arbitrary w*-neighbourhood of 0. Since
LVi > • • • > >'m] ^ E, there exists an feE*, /#0, such that /(x) = 0 for all
xe[yb ...,ym], whence ctfeV for all scalars cteK. On the other hand,
n
since ./V0 and since /=w*-lim ^ /(x;)/,-, there exists a positive in-
integer и such that /(х„) # 0. Let и0 be the least positive integer with this
property. Then, denoting by Уп the sequence of partial sum operators
n
associated to the w*-Schauder basis {/„} of E* (i.e. Sfn{f)= X /(Xf)/,-,
feE*, n= 1,2,...), and putting ;=1
"VIO+ 1 I
a =
we have, taking into account A5.3),
!= 1
2+1\\xJ\\f(xJ\
|/(х„0)|2"°||х0||
= NI/(xno)||/no(xo)|
= 2,

160
I. The Basis Problem. Some Properties of Bases in Banach Spaces
and thus 9"no{a.f)$W, which proves that {/„} is not a w-(e)-Schauder
basis of E*.
a) Since the intersection of the null spaces of a finite number of
non-zero continuous linear functionals on E contains a non-zero ele-
element, an argument similar to that used in b) above proves a).
d) Let {/„} be a basis of E* which is also a w*-Schauder basis of E*
and let {У„} be the associated sequence of partial sum operators. Then,
by §4, theorem 4.1, we have sup ||^|| <oo. Let A be an arbitrary
w*-bounded subset of E* and V=Vyu...,ymiAty an arbitrary w*-neigh-
bourhood of 0 in E*. We have to prove that there exists a l>0 such
00
that (J Уп{А) с X V, i. e. such that
n=l
|[W)](y/)|<AE (feA,n=l,2,...;i=\,...,m). A5.4)
Since A is a w*-bounded subset of ?*, it is also strongly bounded, i.e.
sup|| /I! < oo. Consequently, taking X >- sup \\?fj sup||/|| max ]|>';|[,
feA R l^n<oo fsA 1Щп
we have A5.4), which proves that {/„} is a w*-(b)-Schauder basis of E*.
c) Since every w-bounded subset of a Banach space E is strongly
bounded, an argument similar to that used in d) above proves c). This
completes the proof of proposition 15.1.
§ 16. Some remarks on bases in normed linear spaces
In the preceding three sections we have defined bases, Schauder
bases, (e)-Schauder bases and (b)-Schauder bases in general topological
linear spaces U, and we have considered the particular cases when U
is a Banach space endowed with one of the usual locally convex topo-
topologies: w,bw,w*,bw* (and of course, the norm topology). Now we
shall consider another important particular case, when U is a normed
linear space N (with its norm topology). We shall see that the results
for bases in Banach spaces are no longer valid for bases in general
normed linear spaces.
Firstly, let us observe that in normed linear spaces there exist bases
which are not Schauder bases, as shown by
Example 16.1. Let {xn} be an co-linearly independent but non-minimal
sequence in a Banach space E, such that [*„] = ? (see §6, example 6.1)
and let N be the linear space
«; xi converges
A6.1)
16. Some remarks on bases in normed linear spaces
161
endowed with the norm induced by E. Then {xn} is a basis of N, but
not a Schauder basis of N. Indeed, if there exists a sequence {(/>„} <=N*
such that ф{(Х]) = ёи {i,j= 1,2,...), then, since N = E, one can extend
each (/>„, by continuity, to an fneE* and we have, obviously, f;(Xj) = <5,v
(ij= 1,2,...), in contradiction with the assumption that {*„} is not
minimal.
The same example also shows that a basis {xn} of a normed linear
space N need not be a basis of the completion E = NA of N.
Furthermore, in normed linear spaces there exist Schauder bases
which are neither (e)-Schauder bases, nor (b)-Schauder bases, as shown by
Example 16.2. Let ? be a separable Banach space, let (*„,/„)
({х„} czE, {/„} c?*) be an ^-complete biorthogonal system such that
{х„} is not a basis of E (see e.g. §4, example 4.1; we have seen in §4,
remark 4.1 that every separable Banach space E admits such a biortho-
biorthogonal system) and let N be the linear space
N = {xeE | limsn(x) = x} A6.2)
n—* oo
,-(x)x,., xeE, n= 1,2,...), endowed with the norm in-
xn) is a Schauder basis of N, but not a (b)-Schauder
|| < со, then, since N = E, we also have
(where sn{x)= ?.
; = i
duced by E. Then
basis of N. In fact, if
whence, by
sup ||
1 =Sn<oo
sup I!
^ П < OO
<0O.
J4, theorem 4.1,
{*.}
is a basis of E, in
contradiction with our hypothesis. Since for every normed linear space
N the notions of (b)-Schauder basis and (e)-Schauder basis are ob-
obviously equivalent, it follows that {xn} is not an (e)-Schauder basis of N.
The same example also shows that a Schauder basis {*„} of a normed
linear space N need not be a basis of the completion E = NA of N.
Furthermore, the arguments used above also show that a basis {xn} of
a normed linear space N is a basis of the completion E — NA of N if
and only if {xn} is a (b)-Schauder basis (or, equivalently, an (e)-Schauder
basis) of N. From this latter remark it follows that the following ex-
existence problems are equivalent:
1°. The basis problem (problem 1.1).
2°. Does every separable normed linear space N posses a basis?
3°. Does every separable normed linear space N possess a Schauder
basis?
4°. Does every separable normed linear space N possess a (b)-Schau-
der basis (an (e)-Schauder basis)?
In fact, if the answer to Г is affirmative, then so is the answer to 4
(by the preceding remark), whence so is the answer to 3 , whence so is
the answer to 2°. On the other hand, it is obvious that if the answer to
2° is affirmative, then so is the answer to Г.
11 Singer, Bases in Banach Spaces I

162
I. The Basis Problem. Some Properties of Bases in Banach Spaces
Finally, let us observe that problems 2° and 3° above can be de-
decomposed into the sum of problem 1° and respectively one of the fol-
following unsolved problems:
Problem 16.1. a) Does every incomplete separable normed linear
space N possess a basis? b) What about a Schauder basis?
§ 17. Continuous linear operators in Banach spaces with bases
Theorem 17.1. Let E be a Banach space with a basis {xn}, let {/„}
be the a.s.c.f. to {*„}, and let F be a Banach space. Then the Banach
space L(E,F) of all continuous linear mappings of E into F is isomorphic,
by the mapping u-*{u(xn)}, to the space of sequences of elements
endowed with the norm
V \\\ =
sup sup
^ n < oo xeE
sup sup
^n<oo xeE
Z
<OD)
Namely, we have
H\^\\{u(xn)}\\^v{xju\\ (ueL(E,F)),
where v{Xn} is the norm of the basis {xn}.
The inverse mapping {у„}->м is given by the formula
«M=
A7.1)
A7.2)
A7.3)
A7.4)
// {xn} is a monotone basis1 of E, this correspondence is an isometry
and we have
- lim sup
«-* oc xeE
(ueL(E,F)). A7.5)
Proof. Let us first observe that A7.2) is indeed a norm on B, since
ll{yn}ll=O, then from sup||/i(x)>',|| =0 we infer y\=Q, whence by
sup
Z Шу(
= sup Ц/2МУ2И =0, we get y2=®, whence, continu-
ing in this way, we obtain yn = 0 (n = 1,2,...).
1 I.e. v(Xjij= 1. Such bases will be studied in more detail in Ch. II, § 1.
17. Continuous linear operators in Banach spaces with bases
Now let ueL(E,F) be arbitrary. We have then, putting
n
(n=l,2,...) and и„(х)= Z/,(x)y,-(xe?, и=1,2,...),
u(x) = u
whence, by the principle of uniform boundedness1,
163
yn = u(xn)
sup sup
^ n < 00 xeE
= sup ||mJ<co,
1 =Sn< 00
and thus {у„} = {и(х„)}еВ and
||m||< sup ||м„||= sup sup
1 <in< 00 xeE
On the other hand, for every xeE and n = 1,2,... we have
n
Z ЛМл
XX:
whence
\H{xn\)\\^v{xJu\\,
and thus we also have A7.3). This proves that the mapping м-»{м(х„)}
is an isomorphism of L(E,F) into B.
In the particular case when the basis {xn} is monotone, we have
whence
Hull = lim su
n-* go xe
lP
SUP Wl =\
= sup sup
1 =Sn< oc xeE
/,-(*)tt
and thus the mapping к-»(м(х„)} is now an isometry of L(E,F) into B,
and we have A7.5).
Now let {yn}eB be arbitrary. We shall show that formula A7.4)
defines a mapping ueL{E,F) (satisfying, obviously, и{х„) = у„ for all
n= 1,2,...), which will complete the proof. Put
(хб?,и=1,2,...).
A7.6)
1 See e.g. [10], p. 80, theorem 5.

164 I. The Basis Problem. Some Properties of Bases in Banach Spaces
Then,since {у„}еВ, wehave sup ||Mj|| = ||{у„}|| <co. Furthermore,
/ » <a / " \
by fix •) = <5 • • we have un + m\ У a,- x • I = и А У a,- x,- for all n, m = 1,2,...
\i=i / \,= i /
/ " \
and all scalars aY,...,an, whence lim мД ^ а,х; I exists. Since the set
of all Y oiixi ^ dense in E and since F is complete, it follows that the
i— 1
operator u(x) = lim м„(х) (хе?), i.e. A7.4), is well defined and in L(E,F),
x-* x
which completes the proof of theorem 17.1.
Remark 17.1. In the particular case when F = K (the field of scalars),
from theorem 17.1 above we obtain again theorem 12.5 a) of § 12; indeed,
in this case for any {а„) <= К wehave || {я„} || = sup sup
l«n<oo xeE
= sup
z *Ji
1 = 1
Theorem 17.2. Let E be a Banach space with a basis {х„}, let {./„} be
the a.s.c.f to {х„}, and let F be a Banach space. Then the Banach space
L(F, E) of all continuous linear mappings of F into E is isomorphic, by the
mapping r-> {г*(./„)}, to the space of sequences of functionals
9i(y)xi converges in E for every yeF>, A7.7)
endowed with the norm
\{вп)\\ = sup sup
1 ^ л < x >'eF
Namely, we have
Z di(y)xi
(veL(F,E)),
A7.8)
A7.9)
where v,Xn, is the norm of the basis {х„}.
The inverse mapping {g,,} -> v is given by the formula
A7.10)
// {xn\ is a monotone basis of E, this correspondence is an isometry
and we have
Ml = 11 {Я»} II = lim sup
h-»oo yeF
Z 0.-(>')*(
(veL(F,E)). A7.11)
17. Continuous linear operators in Banach spaces with bases
165
Proof. Let us first observe that A7.8) is indeed a norm on D, since by
the principle of uniform boundedness it is finite for each {gn}eD and
n
since if \\{gn}\\ =0 then from ? gt{y)Xi = 0 and the linear independence
i = l
of *,,...,xB we infer g{{y) = 0 (yeF, ||y||<l, i=\,...,n; n=l,2,...),
whence gn = 0(n=\,2,--¦)•
Now let veL(F,E) be arbitrary. We have then, putting gn = v*(fn)
i = 1 i = 1
and thus {gn}={v*(fn)}eD and
(yeF),
II г [I = sup
yeF
Z et(y)xt
sup sup
i= 1
— \\tr*( fUll
— II I1 \Jn))\
On the other hand, since
we have
II {"*(/„)} II = sup sup
UK33 ye F
z
/ . J i L V
(xeE,n=l,2,...),
( = 1
sup
and thus we also have A7.9). This proves that the mapping u—>{u*(_/n)}
is an isomorphism of L{F,E) into D.
In the particular case when the basis {xn} is monotone, we have
sup
F
Z ffi
sup
yeF
1ЫИ
Z 3iiy)Xi
sup
л=1,2,...) and vUrii=\, whence
n
1Ф
and thus the mapping t'-»{!>*(/„)} is an isometry of L{F,E) into D
and we have A7.11).
Now let {gn}eD be arbitrary. Then, by the Banach-Steinhaus
n
theorem applied to the sequence of operators vn(y)= ? ^(у)*, (yeF,
n=l,2,...), the operator v(y)=\imvn(y)(yeF), i. e. A7.10), is in L(F,E),
which completes the proof of theorem 17.2.
Remark 17.2. One may also consider the spaces В and D above
endowed with the norms ||{jn}||= sup
Z ffcbi
1 = 1
and \\{gn}\\

166
I. The Basis Problem. Some Properties of Bases in Banach Spaces
= sup
X 9ii
respectively, and in this case, obviously, the map-
mappings м->{м(х„)} and v-*{v*(/„)} become isometries. However, the
norms A7.2) and A7.8) are more suitable for applications.
Remark 17.3. One cannot replace in theorem 17.2 the space D by
the space ) {gn}
F*
sup sup
X 9i
< oo (, since for the ele-
elements {gn} of this latter space some series X 9i{y)xi may be divergent,
as shown by the example F = E = c0, {xn} = the unit vector basis of E,
9i(y) = rii+rii{y={rin}eco,i=U2,...) and yo = {n°}eF with <#0; in-
n n
deed, in this example, for the operators vn(y) = X 9ib>)Xi = X (^i +tlt)xi
i = 1 i = 1
we have ||i;J =2 (n= 1,2,...) but limvn{y0)$c0.
n-»oo
Let us give now some applications of theorems 17.1 and 17.2 to
concrete spaces.
Corollary 17.1. Let F be an arbitrary Banach space. Then L(P,F)
is equivalent1, by the mapping м-> {u(xn)}, where {xn} is the unit vector
basis of I1, to the space of sequences of elements
sup
endowed with the norm
IIWII= sup bn\\-
1 Sn< oo
The inverse mapping {jn}—>м is given by the formula
A7.12)
A7.13)
A7.14)
Proof. Take in theorem 17.1 E-l1 and {х„} = the unit vector basis
of E. This basis in monotone (whence L(E,F) = B) and its a. s. с f. {/„}
is nothing else but the sequence of coordinate functionals
fi(x)=t;i (x={Qel1). A7.15)
Formula A7.2) becomes
= sup ,
| с j I s= 1
= sup ||yj,
i. e. A7.13), which completes the proof.
1 I. e. linearly isometric.
17. Continuous linear operators in Banach spaces with bases
167
Corollary 17.2. Let F be an arbitrary Banach space. Then L(F,P)
is equivalent, by the mapping v^>{v*(/„)}, where /„ are the coordinate
functionals on I1, to the space of sequences of functionals
? 1
X 9t is weakly* unconditionally Cauchy> A7.16)
¦¦ = i J
endowed with the norm
||{#n}ll = sup sup
1 ^ л < oo yeF
1Ы1«1
\= sup
1 =S n < oo
sup
The inverse mapping {gn}^>v is given by the formula
v(y)={gM iyeF).
A7.17)
A7.18)
Proof. Take in theorem 17.2 ? = /' with its (monotone) unit vector
basis {xn}. Then by A7.7) we have
i.e. A7.16), and by A7.8) we have
П
\\{gn}\\ = sup sup X \g?y)\,
eF > — i
|«1
i. e. the first equality of A7.17). Now,
sup
Il
= sup
sup
F
and, on the other hand
whence
sup
\9iiy)\,
sup
l^|
sup X \9ib>)\ < sup
n
\ рилу)
n
X №1
1
A7.19)
A7.20)
From A7.19) and A7.20) we get the second equality of A7.17), which
completes the proof of corollary 17.2.
Corollary 17.3. Let l<p<co and let F be an arbitrary Banach space.
Then L(P,F) is equivalent, by the mapping u->{u(xn)}, where {х„} is
the unit vector basis of lp, to the space of sequences of elements

168
I. The Basis Problem. Some Properties of Bases in Banach Spaces
lim , sup
i- 1
n
Z б*
i= 1
= lim sup
n-> oo ge F*
( "
, Z \o^y
\I
.)l« Iе
/
endowed with the norm
||{jn}||= lim _ sup
1 1
where —I— = 1.
p q
The inverse mapping {yn}-m is given by the formula
i= 1
exists and <<x>( A7.21)
A7.22)
A7.23)
Proo/. Taking in theorem 17.1 E = l" and {xn} = the unit vector basis
of E, we obtain A7.21) and the first equality of A7.22). Furthermore, from
, SUP
i
Z С;Л
i= 1
= n sup sup
у ii iP<] jeP
Z Ь
= sup
IffO'.-)!
i
we obtain the second equality of A7.22), which completes the proof.
Corollary 17.4. Let 1 <p<co and let F be an arbitrary Banach space.
Then L{F,lp) is equivalent, by the mapping r—>{;•*(/„)}, where/„ are the
coordinate functionals on I", to the space of sequences of functionals
A7.24)
A7.25)
endowed
III,.!
with the norm
|| = lim sup 1
n -* x yeF >
1 1
where —1— = 1.
p q
The inverse mapping
*
(
' n
\l= 1
{gn
00
z
Ы
}-
\gM*f
v is given
< со for all
П-* Cf
by
yeF J"
Z i8-^
the formula
A7.26)
Proq/! Taking in theorem 17.2 E = V and {xn} = the unit vector
basis of E, we obtain A7.24) and the first equality of A7.25). The second
equality of A7.25) follows as in the above proof of corollary 17.3.
Corollary 17.5. Let F be an arbitrary Banach space. Then L(co,F) is
equivalent, by the mapping м-»{м(х„)}, where {xn} is the unit vector
basis of c0, to the space of sequences of elements
17. Continuous linear operators in Banach spaces with bases
169
J; is weakly unconditionally Cauchy
endowed with the norm
1= SUP
SUP
|l
= sup
sup 2.
gsF* i=i
The inverse mapping {у„}->и is given by the formula
00
«w= z ^ (^={Ueco)-
A7.27)
A7-28)
A7.29)
i= 1
Proof. Take in theorem 17.1 ? = c0 with its (monotone) unit vector
basis {*„}. Then by A7.1) we have
sup
sup
Z ^У1
i== 1
<CC
A7.30)
and by A7.2) we have the first equality of A7.28). The second equality of
A7.28) follows as in the above proof of corollary 17.2. Finally, from
A7.30) and the second equality of A7.28) we infer A7.27) ,which completes
the proof of corollary 17.5.
Remark 17.4. Since in conjugate Banach spaces the weakly* and
weakly unconditionally Cauchy series coincide1, from corollaries
17.2-17.5 it follows that L{F,lv) = L{c0,F*) and L(F,lp)=L{l\F*), where
l<o<co, - + -=1. However, this is well known and obvious via
p q
L(F,E*) = {F®yE)* =(E®yF)* = L{E,F*) (for any Banach spaces F,E).
Corollary 17.6. Let F be an arbitrary Banach space. Then L{F,c0) is
equivalent, by the mapping i;->{u*(/„)}, where /„ are the coordinate
functionals on c0, to the space of sequences of functionals
= {{gn}
endowed with the norm
F* | {gn} weakly* converges toO} A7.31)
\\{gn}\\= sup \\gj. A7.32)
The inverse mapping {gn}—>v is given by the formula
v(y)={gM iyeF). A7.33)
Proof. Take in theorem 17.2 E = c0 and {xn} = the unit vector basis of E.
1 See e.g. Ch. II, § 15, lemma 15.1, equivalence 1"о6°.

170
I. The Basis Problem. Some Properties of Bases in Banach Spaces
Theorem 17.3. Let E be a Banach space with a basis {xn}. Then E has
the approximation property in the sense of Grothendieck, i. e. the identity
operator /?:?—>? can be approximated, uniformly on every compact
subset of E, by linear operators of finite rank1. Namely, the sequence of
partial sum operators {sn} associated to the basis {х„} approximates IE
uniformly on each compact set M с ?.
Proof Let ?>0 be arbitrary, let {>>(},!=1с? be a finite ? net
for the compact set Mc?, where v[x ,= sup \\sn\\, and let xeM be
2A
and a
<- for all n>N(s)
arbitrary. Then there exist an y} such that ||x—j,-||
positive integer N = N(s) such that \\y(-sn'
and i'=l,...,/, whence
||Х-5„(Х)|| < ||X-^.|| + Il^-S.^)!! + \\Sn{yj)-Sn(x)\\
< \\х~У]\\(^+^{х„})+ \\yj-sn(yj)\\ <- + - = ? {n>N{s)),
and thus, since xeM and e>0 have been arbitrary,
lim sup]|x-.sn(x)||=0,
n—» oo xeM
which completes the proof of theorem 17.3.
Remark 17.5. Since the basis problem is unsolved, no example is
known of a separable Banach space which does not have the approxi-
approximation property, but neither has it been proved that every separable
space has this property. This is one of the famous classical problems of
functional analysis, of great importance for applications, as shown also
by corollary 17.7 below.
Remark 17.6. It is also not known whether the converse of theorem
17.3 above is valid, i.e. whether a separable Banach space having the
approximation property must have a basis or not. However, for sequen-
sequentially complete separable locally convex spaces the answer to this latter
question is negative, as shown by example 13.4 of § 13. Indeed, as we have
seen there, the space m* endowed with the weak* topology a(m*,m)
is locally convex, separable and sequentially complete, but has no basis.
On the other hand, let us show that every conjugate Banach space ?*
endowed with the weak* topology a(E*,E) has the approximation
I. e. of finite-dimensional range.
18. Bases of tensor products
171
property1. Let Mc?* be <r(?*,?)-compact and let V=Vx1,...,xn-,A0)
be a <t(?*, ?)-neighborhood of 0 in ?*. Then there exists a continuous
linear mapping v: E^E of finite rank, satisfying
e.g. one can take a norm continuous linear projection of ? onto [x;]"= 1.
Hence
\ (/eM,i=l n),
and thus v*(f)-feV (feM). Since i;*:?*^?* is continuous for
<r(?*,?), <r(?*,?) and of finite rank, this proves our assertion.
From theorem 17.3 we infer
Corollary 17.7. Let E be a Banach space with a basis {х„} and let F
be an arbitrary Banach space. Then every compact linear operator
ueL(F, E) can be uniformly approximated by linear operators of finite rank.
Proof. Let ueL(F,E) be an arbitrary compact linear mapping of F
into ?. Then the set M = u(SF) с ? is compact2, whence, by theorem 17.3,
||m-sbm||= sup ||mQ')-sb[m(>')]|| = sup||x-s,i(:>c)||->0 as n->oo,
yeF XeM
which, since snueL(F,E) is of finite rank (by &imsn\u(F)~\^n<a)),
completes the proof of corollary 17.7.
Remark 17.7. The importance of corollary 17.7 (and hence also of
the spaces with bases) for applications is obvious. The problem, whether
every separable Banach space E has the property ocurring in corollary
17.7 is known3 to be equivalent to the problem whether every subspace
of c0 has this property and also to4 the problem mentioned in remark 17.5.
§18. Bases of tensor products
In this section we shall consider the problem, whether the tensor
product E®XF (i.e. the completion of the algebraic tensor product
with respect to the cross norm a) of two Banach spaces E,F
1 We recall that a locally convex space U is said to have the approximation
property (in the sense of A. Grothendieck [89], p. 165), if for every compact sub-
subset M of U and every neighborhood К of 0 in U there exists a continuous linear
mapping u: U-»U of finite rank, such that u(x)-xeV for all xeM.
2 We use, as before, the notation SF = {yeF\ \\y\\^ 1}.
3 See [89], p. 170, proposition 37.
4 See [89], p. 164, proposition 35.

172
I. The Basis Problem. Some Properties of Bases in Banach Spaces
with bases, has a basis. We shall use essentially the terminology and
notations of R. Schatten [223], assuming that the reader is familiar
with some elementary notions and results of the theory of normed
tensor products.
Theorem 18.1. Let E, F be two Banach spaces with bases {xn}, \yn\
respectively, let {/„}<=?*, {#„}<= F* be the respective a.s.c.f. and let a.
be a uniform crossnorm on E® F. Then the system of all products x{ ® jy,
arranged into a single sequence {zk} by the numbering1
for k = n2
, 2
i-i far k = n2
(«=1,2,...),
A8.1)
n
is a basis of E®XF, having as a.s.c.f. the system {fi®gj} arranged in
the same way into a single sequence {hk}. Hence, in particular, {zk} is a
basis of E®yF and of ? ® AF.
Proof. Let us first show that [zk\ = E®xF. For any xeE,yeF we have
n
Rn = x®y- Y, ft(x)gjb>)xi®yj = x®y-s1n(x)®s2n(y)
where s^,s2 denote the partial sum operators associated to the bases
{xn} and {yn} respectively. Consequently, since a is a crossnorm on ? ® F
and {xn}, {yn} are bases,
0 as и^оо.
This argument extends by linearity to finite sums of elements of the
form x®y, and hence the linear subspace spanned by {zk} = {xi®yj}
is dense in E®F, whence also in E®XF. Thus, \_zk~\ = E®xF.
Furthermore, for the partial sum operators sn associated to the
biorthogonal system {zk,hk) ({zk} с E ®XF, [hk] <= (? ®^)*) we have, by
A8.1),
sn2 = sln®s2n, A8.2)
^ + l = sln®s2n+sl®(s2n+1-s2n) (/=l,...,n+l), A8.3)
sn2 + n+i+, = sln+l®s2 + l-(s1n+1-s1)®s2_l (l=\,...,n;s20 = 0). A8.4)
Indeed, it is sufficient to prove that these equalities hold for all
elements of the form x®y, where xeE,yeF. Now, we have
1 Note that this numbering is different from the classical diogonal numbering
(used e. g. in the proof that the set of all rationale is countable).
18. Bases of tensor products 173
n2 n
= Y,hk(x®y)zk= ? (fi®gj)(x®y)xi®yj
whence we infer A8.2). The proof of A8.3) and A8.4) is similar, taking also
into account A8.1).
Finally, since a is a uniform crossnorm on E®F, for any positive
integers n,m we have ||^ ®^|| < ||^|| ||^||, whence, since {х„}, {уп} are
bases of ? and F respectively, it follows that {si ® sj,} is uniformly
bounded. Hence, by A8.2), A8.3) and A8.4), {,?„} is uniformly bounded,
which, by § 4, theorem 4.1, proves the first assertion of theorem 18.1.
The second assertion follows now from the equalities f ® gj(xk ® y{)
=fi(xk)gj(y,) = eikejl (i,j,k,l=l,2,...), which completes the proof of
theorem 18.1.
The sequence {zk} = {xt®yj} above may be called the tensor
product of the bases {х„} and {yn}.
Let us give now some applications of the particular cases on = y and
<х = я of theorem 18.1.
Corollary 18.1. Let Tbe a locally compact space, ц a positive Radon
measure on T such that the space L^iT,^) has a basis {xn} and Fa Banach
space with a basis {у„}. Then the Banach space Lf(T,/j.) (of all equivalence
classes of F-valued v-integrable functions on T) has a basis consisting of
the system of all products zk(t) = xi(t)yj (teT), numbered as in A8.1).
Proof. This follows from theorem 18.1 and from the well known1
canonical isometry Lp(T,n)= Ll{T,fx)®yF'.
Corollary 18.2. Let T, S be locally compact spaces and fi,v positive
Radon measures on T and S respectively, such that the spaces L1(T,/j),
L}(S,v) have bases {xn} and {yn} respectively. Then the Banach space
Ll(Tx S, n®v) has a basis consisting of all equivalence classes of functions
of the form zk(t,s) = xi(t)yj(s) (te T, seS), numbered as in A8.1).
Proof. This follows from theorem 18.1 and from the canonical iso-
isometry2 L1(Tx 11
Corollary 18.3. Let Q be a compact metric space, {х„} a basis of the
space C(Q) and F a Banach space with a basis {}>„}. Then the Banach
space CF(Q) (of all F-valued continuous functions on Q) has a basis
consisting of the system of all functions of the form zk(q) = Xi(q)yj iqeQ),
numbered as шA8.1).
1 See [89], p. 59, theorem 2.
2 See [89], p. 61, corollary 4.

174
I. The Basis Problem. Some Properties of Bases in Banach Spaces
Proof. This follows from theorem 18.1 and from the canonical iso-
metry1 CF(Q)=C(Q)®XF.
A similar result is also valid for the space (co)F of all sequences of
elements in F converging to 0.
Corollary 18.4. Let Q, P be compact metric spaces and let {xn}, {у„}
be bases of the spaces C(Q) and C(P) respectively. Then the Banach
space C(QxP) has a basis consisting of all functions of the form zk(q,p)
= xi(q)yj(p)(qeQ,peP), numbered as in A8.1).
Proof. This follows from theorem 18.1 and from the canonical iso-
metry2 C(QxP)=C(Q)(S),C(P).
Corollary 18.5. Let E, F be Banach spaces with bases {xn} and {yn}
respectively. Then the Banach space 4>(E*,F) of all compact linear mapp-
mappings of E* into F has a basis consisting of all linear operators of rank 1 of
the form uk = xi®yj (i.e. uk{f)=f{x?)yj for all feE*), numbered as in
A8.1).
Proof. This follows from theorem 18.1 and from the canonical
isometry3 #(?*,F) = E®^.
A similar result is also valid for the space 4>(E,F), if E* has a basis
{hn}. These results may be considered as sharpenings of corollary 17.7
of § 17 on the approximability in the norm of L(E, F) of the operators
ue^(E,F) by linear operators of finite rank when F has a basis.
It is natural to ask whether theorem 18.1 remains valid if a is not
necessarily uniform, or, more generally:
Problem 18.1. Let E,F be two Banach spaces with bases and let a be
a crossnorm on E ® F. Does E®XF have a basis ?
It is clear that for the sequence {zk} defined by A8.1) we have [zj
= E0CIF. The difficulty lies in establishing that {sn} is uniformly bounded
with respect to a.
§ 19. Best approximation in Banach spaces with bases
We recall that if G is a set in a Banach space E and xeE, then any
element yoeG with the property
\\x-yo\\=inl\\x-y\
ge G
A9.1)
1 See [89], p. 90.
2 See [89], p. 90.
3 See e.g. [89], p. 165, condition (A4) or [223], Ch. Ill, theorem 3.5.
19. Best approximation in Banach spaces with bases
175
is called an element of best approximation of x (by means of the elements
of G). In the particular case when G is an и-dimensional linear subspace
of E, where «<oo, i.e.
G=[xl,...,xn\ (xu ...,х„ linearly independent), A9.2)
the elements of G are called polynomials and it is well known1 that every
xeE has at least one polynomial of best approximation. We shall also
consider the particular case when G is a closed linear subspace of finite
codimension n, i.e.
G@[_xl,...,xn] = E (xl5 ...,х„ linearly independent); A9.3)
in this case the elements of G are called polynomial complements.
Even in the concrete cases in which the existence and uniqueness of
the elements of best approximation is ensured (e. g. in the case of best
approximation of continuous functions on a segment \a,b~\ by algebraic
polynomials of degree <«), the effective computation of the elements
of best approximation presents difficulties, since the mapping nG: x->j0
(where yo = nG(x) is the element of best approximation of x) is, in
general, non-linear. In the present section we shall see that in every
Banach space E with a basis {xn} one can introduce a new norm, equi-
equivalent to the initial norm of E, such that in this new norm the mapping
nG of best approximation by means of the elements of the subspaces
A9.2) and A9.3) corresponding to the basis {xn} becomes linear and
the computation of the elements of best approximation becomes easy.
From this it will follow for the original norm of E that in spaces with
bases one can "replace" the respective mappings nG by continuous linear
mappings of E onto G which give the same order of approximation.
Definition 19.1. The norm in a Banach space E with a basis {xn}
is called a T-norm (with respect to the basis {х„}), if
a) for every xeE and «=1,2,... there exists a unique polynomial
yo=7iP(n)(x)eP(n) = [xl,...,xn] of best approximation of x;
b) this polynomial coincides with the «-th partial sum of the ex-
expansion of the element x with respect to the basis {*„}, i- e.
"pjx) = sn(x) (xe?,«=l,2,...). A9.4)
We shall denote the T-norms by (( ((.
Definition 19.2. The norm in a Banach space E with a basis {xn} is
called a K-norm (with respect to the basis {*„}), if
a) for every xeE and «=1,2,... there exists a unique polynomial
complement уо=лр(„>(х)еР(п)=[хп + 1,х„ + 2,...~] of best approximation
of x;
1 See e. g. [246], Ch. I, § 2, corollary 2.2.

176
I. The Basis Problem. Some Properties of Bases in Banach Spaces
00
<
00
X «,*«
i = / -1
b) this polynomial complement coincides with the «-th rest of the
expansion of the element x with respect to the basis {xn}, i.e.
¦n . {x\— r (x\ — x—<t (x\ (xpF n— 12 ) П9 S)
We shall denote the K-norms by )) )).
Definition 193. The norm in a Banach space ? with a basis {х„}
is called a TK-norm (with respect to the basis {х„}) if it is simultaneously
a T-norm and a K-norm with respect to this basis.
We shall denote the TK-norms by (( )).
Let us first give a useful characterization for each of these norms.
Proposition 19.1. Let E be a Banach space with a basis {х„}. Then
a) The norm in E is a T-norm if and only if
A9.6)
for which the
A9.7)
for any scalars a.l,...,a.n + 1eK with <хп+1ф01.
с) // the norm in E is a TK-norm, we have
n+l
X «,*, A9.8)
for any scalars <*,_!, a,,...,а„, аи + 1еК with |a,_1| + |an + 1|#O.
GO
Proof, a) Assume that the norm in ? is a T-norm and let ? o?,x,-
be convergent. Then ? а;х; has a unique element of best approxi-
approximation in P(/_i)=[x1,...,x,_1], namely
for every sequence of scalars {хп}Г-1 <= К with a
series in A9.6) are convergent.
b) The norm in E is a K-norm if and only if
n
X XiXi
i = l
<
n + l
X ««*,
i=l
Кро-Л X atxt ) = si-A X <*;*!=
\ill J \i/l /
whence, since ОеР(,_ц,
1 The bases {х„} satisfying A9.7) (i. e. such that the norm in ? is a K-norm
with respect to {х„}) are called strictly monotone (see Ch. II, § 1 for other results
on such bases).
19. Best approximation in Banach spaces with bases
177
X ai*i
X «¦•*.—Яр,-,, X *iXi
X XiXi
1 = 1-1
i. e. we have A9.6).
Conversely, assume now that we have A9.6). Then for every
CO n
x=Y, 0?,-*,-еЕ and p= ? ftx,.GP(n) with p^sn(x) (where 1<«<оо),
we have
\\x-sa(x)\\ =
X a;*i
= \\x-p\\,
i = n + 1 i = 1
and thus the norm in ? is a T-norm.
b) Assume that the norm in ? is a K-norm and let txl,...,txn + l be
n + l
scalars such that an + l #0. Then ? a,-*; has a unique element of best
approximation in P(n)=[xn + 1,xn + 2,...], namely
\ n + l /n + l \
*.*. - L <*.•*.• 4 2, a.-*i - an + i*n + i,
X <*ixi
X a.-
whence, since OeP(n\
n I n + 1
i.e. we have A9.7).
GO
Conversely, assume now that we have A9.7) and let x= ? о?,х,-е?
and /?= ? PiXieP(n) be arbitrary, with рфг„{х)= ? о?;х;. Then
i = n + 1 i = „ + 1
there exists a smallest index, say и + m, such that ^5n + m#an+m. Hence,
applying A9.7) successively, we obtain
x-rJx =
n + m — 1
««*«- X -
n+m + 1
«,*,- X -
<¦
У я-х- - У (б- —а-)
i = 1 i = r, + 1
n a,
Z_< i" i Z_< ^' i i''
i = 1 i = n + 1
and thus the norm in ? is a K-norm.
c) is an immediate consequence of the necessity parts of a) and b),
which completes the proof of proposition 19.1.
There exist T-norms which are not K-norms, as shown by
12 Singer, Bases in Banach Spaces I

178 I. The Basis Problem. Some Properties of Bases in Banach Spaces
Example 19.1. The numbers
(«(= max A ? |?,|+ sup \Ш (x={Qec0) A9.9)
1 =S П < CC уП j-l Л + 1 =S J < 00 J
define a norm on c0, equivalent to the initial norm of c0. This norm
((x(( is a T-norm, but not a K-norm, with respect to the unit vector
basis {х„} of c0.
Indeed, ((x(( is a T-norm equivalent to the initial norm of c0, by
theorem 19.1 below. On the other hand, we have
whence, by proposition 19.1 b), ((*(( is not a K-norm.
There exist K-norms, even satisfying A9.8), which are not T-norms,
as shown by
Example 19.2. For every integer rc^2 let Пи„ denote the collection
of all permutations of the set {2,3,...,n-l,n + l,n + 2,...}. Then the
))x))= sup sup
2<n<oo ffellun
(x={Qsc0) A9.10)
define a norm on c0, equivalent to the initial norm of c0. This norm
))x)) is a K-norm, satisfying A9.8), but not a T-norm, with respect to
the unit vector basis {xn} of c0.
Indeed, it is obvious that ))x)) is a norm on c0. Let us show that
this norm is equivalent to the initial norm ||x|| = sup |<y of c0. Since
and
and
obtain
)MKf
On the other hand, we have
for
(*ec0).
we
A9.11)
2-2
+ > ~zr
з-2з ¦ 22 т,е4
and choosing, for each J>2, a positive integer n>2 with пф} and
a permutation ajelll n with (jj{2)=j, we obtain
|Z
19. Best approximation in Banach spaces with bases
179
whence
8 11*11 ^ »x)) (xec0), A9.12)
which, together with A9.11), proves that the norms ))x)) and ||x|j are
equivalent.
Let us show now that for every pair du d2 of finite sets of indices
with di с d2 and every finite sequence of scalars {o?,}ied2 with
^ |о?,| Ф0 we have
A9.13)
iedi
ied2
whence, in particular, ))x)) is a K-norm, satisfying A9.8). Since dy is
finite, the norm )) ? a^jY) is attained for some «0>2 and ffo6^i и-
Since ^ |о?,-| ^0, a larger sum must be obtained when this same n0
isd2di
and a0 are used for ^ a.x,, whence we obtain A9.13).
ied2
Finally, let us show that
1
A9.14)
whence, by proposition 19.1 a), ))x)) is not a T-norm. We have, obviously,
Furthermore, let «^2 be fixed. If for a pair i, i+je{2,3,...,n—l,
и+l,...} and а аеП1 „ we have —j-<—, , then for the permuta-
permutation а' еЯ,„ defined by ffW a(i+J)
'('+y) = g(J), o-'(fe) = tr(fc)
we have, taking into account that ~i
? > ?
Consequently, for every «^2 and
for a>
we have
L , у
2"
«2"
= sup
sup X
m= 2
тФп

180
I. The Basis Problem. Some Properties of Bases in Banach Spaces
which, together with A9.15), implies A9.14), completing the proof of the
assertions of example 19.2.
Theorem 19.1. Let E be a Banach space with a basis {х„} and let
{/„}(=?* be the a.s.c.f. Then
a) One can introduce on E a T-norm equivalent to the initial norm
on E, by the formula
1
X fi(x)xt
i = n + 1
and also another equivalent T-norm on E, by the formula
A9.16)
(W(= YJ-\\fi(x)xi\\+™xoc
i=i 2
E/<(*)*.
A9.17)
b) One can introduce on E a K-norm equivalent to the initial norm on
E, by the formula
У -
' i~i 2;
X M
A9.18)
c) One can introduce on E a TK-norm equivalent to the initial norm
on E, by the formula
1
X Mx)x,
A9.19)
Proof. at) Let us first observe that the max in A9.16) is indeed attained
for some n, since {xn} being a basis, both summands in A9.16) tend to
0 as n -> oo.
Let us show now that ((x(( defined byA9.16)isanormon?. Obviously
((x((^0and(@(( = 0. If we have ((x(( = 0, then by
i = 2
X fiWXi
(xeE) A9.20)
we have x = 0. Finally, the inequality ((x + X(s?(W( + ((>'(( is obvious
from ((x + y(( = —fi\\Mx + y)xi\\+
X fi(x + y)x,
(for a suit-
able «0), and {(ax([ = \a\({x{{ is obvious directly from A9.16).
Furthermore, ((x(( is equivalent to the initial norm on E by A9.20)
and by the relations
19vBest approximation in Banach spaces with bases
181
1 ?
max ||/;(х)х;|| +
is; к n0
max
3 sup
X /,(*)*,
X /*(*)*,
X
; - x /*(*>*
X fM)xj
J"=l
X fi(x)x,
\\x\\^CvfXni+\)\\x\\ (xeE).
Finally, let us prove that ((x(( is a T-norm. Let {«.AfLi-i be a se-
00
quence of scalars with ol,_^0, such that X аЛ converges. We have
then, for a suitable «0 with /^«0<oo, ' = '
f=i
1 T-^
=-X ll«i^il
«П i =
X O?i-X;
и ,=Г-1
<- X
oo
У 0? X
i = n + 1
X
X «.:
whence, by proposition 19.1a), ((x(( is a T-norm.
a2) Consider now the ((x(( defined by A9.17). Since {*„} is a basis,
the max in A9.17) is indeed attained for some n and it is also obvious
that ((x(( is a norm on E. This norm is equivalent to the initial norm
on E, since for every xeE we have
ОС
X Ш
max 1
3 max
1 ^ П < 00
OO
2^ fj(x)Xj
00
X
i =
1
^ max i|/;(x)x(||
l s; i < oo
*
<
ОС
V f(v\x
I—t * Г- ^ j
j=i + l
11 \\x\\ + sup
V 1«B<
)
ОС
max
1 «n< oo
00
X
i = n
+ max
1 ^ n < oc
')>
X /iW*.-
i = n
)
Let us prove, finally, that ((x(( is a T-norm. Let {d.i}^=,-1 be a
sequence of scalars with <*,_!#(), such that ^ a;x, converges. Then
Xa.-*i = X ^7lla.^ill + max
= l \\ i = i 1 l«n<oc
X <*iXi
X ^ll°?;^ll
max
X a;x;
= (( X «,*, ,
whence, by proposition 19.1a), ((x(( is a T-norm.

182
I. The Basis Problem. Some Properties of Bases in Banach Spaces
b) It is immediate that ))x)) defined by A9.18) is a norm on E. This
norm is equivalent to the initial norm on E, since for every xeE we have
max
X Wxj
X /<(*)*<
X fj
+ sup
1 ^ n< oo
X /,(*)*!
Finally, for any scalars
Л 1
with
we have
И + 1
whence, by proposition 19.1b), ))x)) is a K-norm.
c) can be proved similarly to a2) and b) above, and thus the proof
of theorem 19.1 is complete.
We have given each of the above equivalent norms (instead of giving
only A9.19)) since they are useful for applications; in particular, the
norm A9.16) has been already used in example 19.1. Let us mention
that one can also prove that in the conditions of theorem 19.1 it is
possible to introduce another equivalent TK-norm on E, by the formula
((*))= sup
1 =Sn< oo
!/«(*)*«+
X №xi
i = n + l
A9.21)
where ({x(( is the T-norm A9.16) and ))x)) is the K-norm A9.18).
Remark 19.1. Theorem 19.1 admits, in a certain sense, a converse.
Namely, the above definition of T-norms also has sense when (х„,/„)
is a biorthogonal system with [х„] = ?, and if for such a system one can
introduce on ? a T-norm with respect to {xn}, equivalent to the initial
norm on ?, then {xn} must be a basis of ?. Indeed, even a more general
result is also true. Namely, for an arbitrary sequence {х„} <= ? with
хпф0 (и =1,2,...) and [х„] = ?, if one can introduce on ? a weak
T-norm with respect to {*„}, equivalent to the initial norm on ?, i.e.
к
an equivalent norm (x{ such that for every polynomial p= ? o?;X; and
every positive integer n^k the polynomial 'Y^a.ixieP^n)=\_x^...,x^\
i=l
is in this new norm a (not necessarily unique) polynomial of best
19. Best approximation in Banach spaces with bases
183
approximation oip by means of the elements of P{n), then {х„} is a basis
of ?¦ For if d.u...,ixn, о?„ + 1,...,о?„+т = о?(, are arbitrary scalars, then, since
(x( is a weak T-norm,
n + m
x«
( = 1
iXi
" (
- X *А
; = i V
/n + m
<2 X
\i = i
whence, by § 7, theorem 7.1, {х„} is a basis of ? endowed with the norm
(x(, and therefore also of ? with its initial norm. Defining similarly
weak K-norms, weak TK-norms, the same argument shows that if one
can introduce on E such a norm with respect to {xn} (where [х„] = ?,
хпф0 for «=1,2,...), equivalent to the initial norm on ?, then {xn\
is a basis of ?.
Let us now turn our attention to ? endowed with its initial norm.
We shall denote by en(x) the best approximation of xeE by means
of the elements of ^„^[x!,...,^], i.e.
en(x)= inf ||*-
A9.22)
From theorem 19.1 it follows
Corollary 19.1. Let E be a Banach space with a basis {х„}. Then there
exists a constant c, 0<c^ 1, depending only on the basis {xn}, such that
c\\x-sn(x)\\^en(x)^\\x-sn(x)
A9.23)
where sn(x) are the partial sum operators associated to the basis {xn}.
Proof. Let xeE and n be arbitrary and let poeP(n) be an element
of best approximation of x; such an element exists, since dimP(n) = «< oc.
Then en(x)=\\x-p0\\. By virtue of theorem 19.1, let ((x(( be a T-norm
on ?, equivalent to the initial norm of ?, and let Cb C2 > 0 be constants
such that C1||x||^((;c((^C2||.x|| for all xeE. Then
en(x)=\\x-p0\\
1
С
and thus we have the first inequality of A9.23) with c = —^. On the
C2
other hand, the second inequality of A9.23) is obvious by A9.22), which
completes the proof.
Corollary 19.1 shows that if \xn\ is a basis of ? and P(n)=[xb...,xn],
then en(x) and \\x — sn(x)\\ are "of the same order". Consequently, if
we want to classify the elements xeE by the rapidity of the convergence
to 0 of the sequences {e,,(x)}, then we can use the sequences {\\x — л'„(х)||},
which can be computed more easily.

184 I. The Basis Problem. Some Properties of Bases in Banach Spaces
Remark 19.2. Corollary 19.1 also admits, in a certain sense, a con-
converse. Namely, if for a biorthogonal system (*„,/„) with [*„] = ? we
have the first inequality of A9.23), then
\\sn(x)\\
~ \\en(x)
whence, by § 4, theorem 4.1, {xn\ is a basis of ?. Moreover, introducing
for a sequence {xn} <= ? with х„#0 («=1,2,...) and [*„] = ? the
operators
o?,x; for n=l,...,k
к
i = l
a,-*,- for n = fc+ 1, fe + 2,...
the same argument with 5„ instead of 5„ and with § 7, theorem 7.1, shows
that if for a sequence {xn} <= ? with [xn] = ? and х„#0 («= 1,2,...)
there exists a constant c, 0<c^l, such that
P№),n= 1,2,... ,
k=l
A9.24)
then |*J is a basis of ?.
§ 20. Polynomial bases. Strict polynomial bases.
Г systems and Л systems
Definition 20.1. Let {*„} be a finitely linearly independent sequence1
in a Banach space ?, with [х„] = ?. We call polynomial basis of the
space ? (with respect to {xn}) any basis of ? of the form
V = У 0?<n)r. (П=\ 2 )
B0.1)
and 5?r/c? polynomial basis of ? (with respect to {х„}) any basis of ?
of the form B0.1), where
mn = n, а(„"»#0 (и =1,2,...). B0.2)
Obviously, every strict polynomial basis is a polynomial basis (with
respect to the same sequence) and every basis is a strict polynomial
basis with respect to itself.
1 I. e. such that every finite subsequence of {х„} is linearly independent
(see § 6).
20. Polynomial bases. Strict polynomial bases. Г systems and Л systems 185
Definition 20.2. Let {xn} be a finitely linearly independent sequence
n
in a Banach space ? and let p= ? a.x,- be a polynomial with respect
; = i
to {х„}. The order of the polynomial p is the number
ord p = ord У otjXj = max у.
B0.3)
By virtue of condition B0.2), a strict polynomial basis {у„} with
respect to {xn} contains polynomials of all orders, namely, ordyn = n
(и =1,2,...), and we have
bi>-OVl = l>i>->*J (и=1,2,...)- B0-4)
We have seen in § 10, corollary 10.2 of the Krein-Milman-Rutman
theorem, that if the space ? has a basis, the problem of existence of
polynomial bases with respect to any finitely linearly independent
sequence {xn} с Е with [*„] = ? has an affirmative answer. On the
contrary, we shall see below that the problem of existence of strict
polynomial bases with respect to a finitely linearly independent com-
complete sequence {xn} in a space with a basis has, in general, a negative
answer.
Let us first give some necessary and sufficient conditions for the
existence of strict polynomial bases with respect to a finitely linearly
independent complete sequence {xn}.
Theorem 20.1. Let {xn} be a finitely linearly independent sequence in
a Banach space E, with [*„] = ?. The following statements are equivalent:
l°-10°. There exists a sequence of endomorphisms {tn}<=L(?, E)
such that
a) vn(x)ePM=[x1,...,xn] (xeE, «= 1,2,...),
b) vn(p)=P (peP(n),n=\,2,...),
c) lim vn(x) = x (xeE),
n-* oo
and satisfying any one of the following conditions:
dx) orduB(*)^ordt;n+1(*) (xeE, n= 1,2,...),
where we make the convention ord 0 = 0;
d2) vi[vn(x)] = vi(x) (xeE, i= 1,...,«;«= 1,2,...);
d3) lim rkjk2...rkm(x) = 0 (xeE),
where rn(x) = x-vn(x) (xeE, «= 1,2,...);
4-) SUP \\Гк\ '"/C2"' ''km II "^ "^ '
<km<
d5) there exists an equivalent norm \\\x\\\ on E such that
B0.5)

186
I. The Basis Problem. Some Properties of Bases in Banach Spaces
d6) there exists an equivalent norm |||x||| on E such that
llk(*)IKIIk + i(*)lll (*е?,и=1,2,...);
d7) there exists an equivalent norm \\\x\\\ on E such that
M X atxi
X at
n+ 1
X 0?;
for any scalars а1;...,аи+1 with а„+1#0;
d8) there exists an equivalent norm \x\ on E such that
\h-vn\= sup \x-vn(x)\^\ («=1,2,...);
||Sl
B0.6)
B0.7)
B0.8)
d9) there exists an equivalent norm \x\ on E such that for every xeE,
vn(x)eP(n) be a polynomial of best approximation of x in this new norm;
d10) there exists an equivalent norm \x\ on E such that for every xeE,
vn(x)ePin) be the unique polynomial of best approximation of x in this new
norm.
By statement n° we mean here that there exists a sequence {vj\ с L(E,E)
satisfying a), b), c) and dn) (и = 1,..., 10).
11°. E has a strict polynomial basis with respect to {xn}.
Proof. We shall prove the implications
6°
9
The implication 11°=>1° is obvious, since if {yn\ is a strict poly-
polynomial basis with respect to {х„}, one can take i^ = the partial sum
operator Sj associated to the basis {>'„} (/'=1,2,...).
Assume now that we have Iе. Then by a), b) we have
vn [x - vn(x)-] = vn(x) - vn [г„(х)] = 0 (x e E, n = 1,2,...),
whence, by dt) (with the convention ord 0 = 0), we obtain
0 ^ ord vt \_x - vn(x)] s$ ord vn \x - vn{xj\ = ord 0 = 0
(xeE, i=l,...,n;n= 1,2,...),
20. Polynomial bases. Strict polynomial bases. Г systems and Л systems 187
whence
i.e. d2). Thus, 1°=>2°.
Assume now that we have 2°. Put
1
X =
sup ||i;.(jc)|| (xeE). B0.9)
1«J<00
By c) we have |||*||| < oo and |||*||| is a norm on E. This norm is equiv-
equivalent to the initial norm of E, since by c) and the principle of uniform
boundedness we have sup ||^||<оо and
||x||=lim||i>B(x)||< |||*|||O jup ||r,.(*)Kfasup ||i>y|fi||*|| (xeE).
Finally, for any scalars a1;...,an+1 with <х„+1#0 we have, by d2)
and a), b),
1
+
sup
у ]
= 1
j=l
n+ 1
Ml?y
n+i\L!Xjxj - L ы vi + i\L!Xjxj 4 L ai
+ sup
"* X «jxj
whence, since ил+11 X ^^ - ^J X <*jxj) = X ajxj- X
V / \ /
хп+1фр(п)), we infer B0.7). Thus, 2J^7°.
Assume now again that we have 2 . Put
|||x|||= sup \\Vj(x)\\ (xeE).
Then, by d2), for any xeE and и= 1,2,... we have
IH^WIII = sup ||г;;г;п(х)||= sup ||^(*)||^ sup \\vj(
i-e. B0.6). Thus, 2° =*6°.
Assume now that we have 6°. Then, by c), we have
B0.10)
i-e. B0.5). Thus, 6°

188
I. The Basis Problem. Some Properties of Bases in Banach Spaces
Now assume again that we have 2°. Then for any xeE and any
positive integers n, m with n^m we have, by d2),
rn rm(x) = rm(x) - vn \rm(x)] = x-vm(x)- vn{x) + vn vm(x) = x-vm{x) = rjx),
whence, by c), we get
lim rkrkrk (x)= lim r\ (x) = 0,
i.e. d3). Thus, 2°=>3°.
The implication 3°=>4° is obvious (the set {rklrk2...rkm(x)\l!iki< ... <km
is countable, and the limit in d3) is the limit of this sequence in the usual
sense), by the principle of uniform boundedness.
Assume now that we have 4°. Put г;о(х) = 0, ro(x) = x (xeE) and
\x\= sup
¦¦ <к„
(xeE).1
B0.11)
Then, by d4), |x|<oo. Furthermore, \x\ is a norm on ?, which is
equivalent to the initial norm of E, since by d4) we have
||xK|xK( sup \\rk,rk2...rkj)\\x\\ (xeE).
\0iki< ¦ ¦¦ <km I
Since for any xeE and any positive integers n,m with n^m we
have, by a) and b),
rm rn(x) = rn(x) - vm [rn(x)~] = x - vn(x) - vjx) + vm vn(x) = x - vm(x) = rm(x),
B0.12)
it follows that for any xeE and any positive integer n,
\x-vn(x)\=\rn(x)\= sup \\rkjk2...rkmrn(x)\\
0 <кi< ¦ • • < km
^ sup \\rkirk2...rkjx)\\ = \x\,
whence, again by b), we get
\x-vn(x)\=\x-p-vn(x)+p\=\x-p-vn{x-p)\^\x-p\ (peP[n)),
i.e. d9). Thus, 41^9\
Assume now again that we have 2 . Put
W=llfiWII+iil|i'/+iW-i'.-WII+ sup |1л--ь-(х)|| (xeE). B0.13)
2 i«,<<x
1 In particular, if we also have d2), this norm reduces to
M= sup ||гу(л:)||= sup (|M|, ||г/х)||) (xeE).
20. Polynomial bases. Strict polynomial bases. Г systems and Л systems
189
Then by c), \x\ < ас. Furthermore, \x\ is a norm on E, which is equiv-
equivalent to the initial norm of E, since by c) and the principle of uniform
boundedness we have sup ||uj<oo and
1 ^ n< oc
||xKKMII + !lx-^iMKM^ sup K(x)||+2 X Г7 sup \\vn(x)\\
+ ||x||+ sup \\vn(x)\\^D sup Ikll+ 1^11*11 (xeE).
1« \ «
Finally, let xeE and peP{n) be arbitrary. We have then, by d2),
-vn(x)\ = \\v1[x-vn(x)
vi + l[x-vn(x)']-vi[x-vn(x)
sup ||x-rnU)-
« i < oc
sup ||х-^(
J
\x-p\ = \\v,(x-p)\\ + X ^i\\vi + l(x-p)-vi(x-p)\\
1 = 1 L
+ sup \\x-p-Vi(x-p)\\
1 « ! < ПО
whence, since by peP[n) we have vi + 1(p) = vt(p)=p (; = «,«+1,...), it
follows that
\x-vn(x)\^\x-p\,
where the equality holds only if
vl(x-p) = 0, vi + l(x-p)-Vi(x-p) = 0 (г=1,...,и-1),
which gives 0=ь1(х-/7)=г2(х-/7)=---=г;„(х-/?)=г;„(х)-^»=^М-/',
i.e. p = vn(x). Thus, 2°^10c.
The implication 10'"=>9~ is obvious1. The implication 9°=> 8" is
also immediate, since if we have d9), then, by 0eP(n),
|x-t-n(x)|^|x| (xeE, n= 1,2,...).
Thus, it remains to prove the implications 511=>1Г', 7' =>113 and
8°^> 11". Let us first prove that if we have any one of 5 or 7\ then the
sequence2
1 Let us observe that the implication 1O=>1J is also immediate, since if
ordi>,,(*)>ord!;,,+1(*), then, by d10), en(x) = \x-vn{x)\<\x-vn+l(x)\ = en+1{x)^en(x),
which is impossible.
Obviously, the sequence {xn} itself is not necessarily a basis of ? (take e.g.
? = '2> х„ = e1 + ¦ ¦ ¦ + е„, where {е„} is the unit vector basis of /2, and in = the partial
sum operator sn with respect to {<?„}).

190
I. The Basis Problem. Some Properties of Bases in Banach Spaces
(« = 2,3,...)
B0.14)
is a strict polynomial basis of E. By a) it is sufficient to prove that {jn}
is a basis of E.
Assume first 5°. Then, since vn(yn + l) = vn{xn + l)-vn(xn + l) = 0, we
have
м x wi
n+1
X
; = i
X «iJi
for any scalars а1,...,аи + 1 and hence, by §7, theorem 7.1, {у„} is a
basis of E endowed with the norm |||x|||, whence also of E endowed
with its initial norm.
Assume now 7°. Then, again since vn(yn + l) = 0, we have
n
i = l
=
<
/n + 1 \
/n + 1
=
')
=
/n+1
\>=1
n + 1
X PiXi
i = l
)
=
n + 1
for any scalars a!,...,an+1 with an + 1^0 (and suitable j81,...,j8B + 1),
whence, by § 7, theorem 7.1, {jn} is a basis of E endowed with the
norm |||x|||, whence also of E with its initial norm.
Assume, finally, that we have 8°. We shall prove that in this case
the sequence
Ji = *i, yn = rlr2...rn_l(xj (« = 2,3,...) B0.15)
is a strict polynomial basis of E, which will complete the proof. By a)
it is again sufficient to prove that {у„} is a basis of E.
By virtue of d8) we have
ki-.-'-J^KI-knl^i (« = i,2,...). B0.16)
On the other hand, by B0.12) we have, for any « = 2,3,...
and obviously r1(y1) = yi — i\{y1) = xl—vl(xl) = 0. Consequently,
X
(h-ri---rn){ X Wi
n + m
X'
i = l
for any scalars al,...,an+m, whence, by § 7, theorem 7.1, {у„} is a basis
of E endowed with the norm |x| and hence also of E endowed with its
initial norm, which completes the proof of theorem 20.1.
20. Polynomial bases. Strict polynomial bases. Г systems and Л systems 191
By the duality between vn and IE — vn and between polynomials
and polynomial complements, one can also give other equivalent con-
conditions, corresponding to 6°, 7° and 9°, 10°.
Problem 20.1. Is it possible to omit in theorem 20.1 the conditions
dx) —d]0)? In other words: if there exists a sequence of endomorphisms
{vn} с L(?,?) satisfying a), b) and c), does ? have a strict polynomial
basis with respect to {xn} ?
For any closed linear subspace G of the Banach space ?, we shall put
A(G) =/l?(G) = inf ||u|| B0.17)
where 3?(E, G) denotes the set of all continuous linear projections of ?
onto G (i.e. the set {ueL{E,E)\u2 = u, u(E)=G}).
Proposition 20.1. Let {xn} be a finitely linearly independent sequence
in a Banach space E, with [xn\ = E and let P(n) = [xb...,xn] («= 1,2,...).
The following statements are equivalent:
1°. There exists a sequence of endomorphisms {vn} <= L(?,?) satisfy-
satisfying a), b) and c) of theorem 20.1.
2°. There exist a sequence of endomorphisms {un}<=L(?,?) satisfy-
satisfying a) and a constant С 3= 1 such that
\\x-vn(x)\\^Cen(x) (xg?,«=1,2,...), B0.18)
where, as before, en(x)= inf \\x~p\\.
3°. There exists a sequence of endomorphisms {vn} с L{E,E) satisfy-
satisfying a), b) and
c') sup KH<go.
l=Sn< oo
4°. We have
sup ;.(Р(Я))< oo.
B0.19)
Proof. The implication Г=>30 is a consequence of the principle of
uniform boundedness.
Assume now that we have 3° and let xeE and « be arbitrary. Then
for any polynomial of best approximation pn{x)eP^t of x we have
11*-1>„(*)|| = \\x-pn(x)-vn{x-pn(x))\\^\\x-pn(x)\\ + \\vj \\x-pn(x)\\
= (l + \\vj)en(x)^(\+ sup UllW)
V
i.e. B0.18) with C=l+ sup ||^.||. Thus, 3°=>2°.
l«j<no
Assume now that we have 2°. Then by en(p) = 0 (pePM) we have b)
and by (>„] = ? we have \imen{x) = 0(xeE), whence c). Thus, 2° => Г,
«-¦ос
which, together with the above, shows that Г<=-2°<=>3° with the same
sequence {vn}.

192 I. The Basis Problem. Some Properties of Bases in Banach Spaces
The implication 3°=>4° is obvious, since if we have 3°, then
Щ
Assume, finally, that we have 4°. Then, since by B0.17) there exist
vne0>{E,PM) such that1 \\vj <Л(Р(И)) + 1 (n= 1,2,...), we have 3°. Thus,
4°=>3C, which completes the proof of proposition 20.1.
A natural way to obtain an affirmative answer to problem 20.1
above would be to show that for any given sequence of endomorphisms
{vn} <= L{E,E) satisfying a), b), c), we also must have dn) for some «
with l^w^lO. In particular, for d9) this conjecture seems also to be
supported by proposition 20.1 (implication 1°=>2° proved above with
the same {vn}). However, the following example disproves this con-
conjecture:
Example 20.1. Let {xn} be a normalized basis of a Banach space E,
with the a. s. с f. {/„}. Put
B0.20)
Then {!)„}<= L(?,?) satisfies a), b), c), but none of the conditions
di)-d10).
Indeed, obviously {;;„} satisfies the conditions a), b). Furthermore,
since ||xj = l and \fn(x)\ = \\fn{x)xn\\^>0 (xeE), we have
\\x-vn(x)\\ ^ \\x-sn(x)\\+2\fn + l(x
i. e. {vn} also satisfies c).
(xeE),
?, 1
However {г„} does not satisfy d9). Indeed, for xo= ? yxte^ we
have i = 1

(«=1,2,...)
(where 5o(xo) = 0), and therefore from d9) it would follow that in the
norm |x|ofd9) we have е„(хо) = е„__1(хо)(« = 1,2,...), whence ||xo|| = eo(xo)
= limen(xo)=0, which contradicts /1(x0) = |#0.
It is also easy to see that {<;„} does not satisfy d8), since d8) implies
d9) with the same sequence {vn} (and same norm |jc|). Indeed, if we
have d8), then, by b),
1 Since dim^n) = «<oc, it is also easy to show, by a compactness argument,
that actually there exist vne&(E,Pw) such that \\vn\\ = MPin))(n= 1,2,...) (see Ch. II,
§!)•
20. Polynomial bases. Strict polynomial bases. Г systems and Л systems 193
\x-vn{x)\ = \x-p-vn{x)+p\ = \x-p-vn(x-p)\
= \{IE-vn)(x-p)\^\x-p\
for all p^P[n), which proves our assertion.
Moreover, {;;„} does not satisfy any one of the conditions dn) for
я= 1,2, 3,4,10, since in the above proof of theorem 20.1 the implica-
implications Г => 2° => 3° => 4° => 9° and 2° => 10° => 9° have been actually proved
for the same {vn} (and same norms).
Finally, {;;„} does not satisfy any one of d5), d6), d7). Indeed, in the
above proof of theorem 20.1 we have seen that if any one of d5), d6),
or d7) holds, then the sequence {yn} defined by B0.14) must be a basis
of E. However, since now vn_1{xn) = sn_1(xn)-2fn(xn)xn_1= -2xn_u
we have
Л=^ь У„ = х„-2хп_1 (« = 2,3,...),
but this sequence is not even minimal. For if a functional1
B0.21)
satisfied A1(j1)=l, А1(уи) = 0 (« = 2,3,...), then by h1(y1) = fi1, ^(уп)
= Pn-2fin_1 (« = 2,3,...) we would obtain
pn = 2»-1 («=1,2,...),
00 1 00 1 00 1
whence, for the element xo= ? -lxieE,h1{x0)= ? 2' -j= ? -= oo,
1=1 2 ( = 1 2 ;=1 2
an absurdity. This completes the proof of the assertions of example 20.1.
Let us observe, however, that this example does not give a negative
answer to problem 20.1 above, since there exists another sequence of
endomorphisms of E, namely {sn} <= L(E,E), having all properties a),
b), c) and dn) («=l,...,10) and {xn} is a strict polynomial basis with
respect to itself.
We shall turn now our attention to finitely linearly independent
complete sequences {xn} <= E, with respect to which there exists no
strict polynomial basis.
Definition 20.3. A finitely linearly independent sequence {xn} in a
Banach space E is called a
a) sub-Г system, if sup Я(Р(П)) = ос, where P(n) = [x !,...,;*:„]
1 ^ П < GC
(n=l,2,...) and where А(Р(И)) are the numbers defined by B0.17);
b) sub-Л system, if lim/(P(n))=oc;
Since {х„} is a basis of E, every functional h^sE* can be written in the form
B0.21), with pi = hi(xi)(i=l,2,...).
13 Singer, Bases in Banach Spaces I

194 1. The Basis Problem. Some Properties of Bases in Banach Spaces
c) Г system, if it is a sub-Г system and [х„] = ?;
d) Л system (or Lozinsky-Harsiladze system) if it is a sub-Л system
and [х„] = ?.
By virtue of theorem 20.1 and proposition 20.1, in order that there
exist a strict polynomial basis with respect to {х„} it is necessary (and,
if the answer to problem 20.1 would be affirmative, then also sufficient)
that {х„} be a non-Г system with [х„] = ?.
By definition 20.3 we have obviously the implications Л=>Г' => sub-Г
and Л => sub-Л => sub-Г. It is easy to see that none of the converse im-
implications is true. Indeed, any Г-system (Л-system) {х„} of a hyperplane F
in a Banach space ? is a sub-Г but non-Г (respectively, sub-Л but
non-Л) system of the whole space ?, since inf ||u|| ^ inf \\v\F\\
¦¦€#(?,?(„,) i-e#(?,P(,,))
^ inf IIvf|| —»¦ oo as n—>oo and since [x_] = F#?. On the other
we#(F,P(n))
hand, it is well known' that in each l\ one can choose a subspace Gkn <= /n
in such a way that &(GkJ—>oo as n—>oo. Hence, choosing a basis
x\,...,x"n of l\ such that x\,...,xnkn be a basis of Gkn, the sequence
?1 rr2 tz2
ХХХ
B0.22)
where x]= {О^^Д^ОД...}(/= 1,...,«;«= 1,2,...), will be а Г (whence
also sub-Г) system in E = (l\ x l\ x ••• x l\ x ¦¦¦),l = l1, which is, however,
a non-sub-Л (whence also non-Л) system in ?. In fact, if there existed
projections vn of ? onto [x},xf,x|, ....x", ...,x"lj,x",..., x^J
= (/11x/^x---x/n1_1xGt x{0}x ¦¦•),! of norms ||у„||^М<оо (n = l,2,...),
^- ^ -:-*:— -"-I _., I _ Would in-
A^nn
then the restrictions v
llwJI = k.
duce projections wn of /^ onto Gkn, of norms
<||t)n||<M (и=1,2,...), in contradictions with our choice of the Gkn.
Thus, X{{x\,...,xnnZ\,x'[,...,xnk^)^arj i. e. B0.22) is а Г system. However,
the natural projections E^[x\,...,x"Z[,x",...,x"] = {l\ х---х/„х {0}х••¦),•
defined by {хь...,х„,...} -> {хь..., х„,0,0,...}, are of norm 1, whence
Я([х!,...,х^:},х5,...,хД)=1 (и=1,2,...), and thus B0.22) is not а Л
system.
1 Seee.g. [170], [252]. We recall that /'is the space of all «-tuples x={^,...,U
with the norm ||x||= Y, l^-l an<^ Cl x/Jx ¦¦¦),! is the space of all sequences {х„}
with хпе11„ (и=1,2,...) for which {^„HJs/1, endowed with the norm ||{х„}||
= S ll-*nlli'- There exists a natural linear isometry (/[ x /2 x •••),, = /1, given by
n= 1
the formula
{«.}•
20. Polynomial bases. Strict polynomial bases. Г systems and Л systems 195
Now it is natural to ask whether Л systems exist at all. The answer
is affirmative, namely, it is a classical result of S. M. Lozinsky and F. I.
Harsiladze' that for the sequence
we have, in the space ?=C([0,1]),
(«=1,2,...),
B0.23)
B0.24)
whence, since (by the theorem of Weierstrass) [х„] = ?, the sequence
{xn} is а Л system in ? (consequently, in ? = C([0,1]) there exists no
basis consisting of algebraic polynomials of all degrees). On the other
hand, in ? = I2 there exist no Л systems, since for any sequence {х„} <= ?
we have Я(Р(И))=1 (n=l,2,...) (because there exists a projection of
norm 1 onto any subspace G of I2, namely, the orthogonal projection).
Therefore it is natural to ask, which separable Banach spaces possess
Л systems. We shall give below some characterizations of such spaces.
Lemma 20.1. Let E = G@Y, where dimG<oo and let2
sup Я?(Г)<оо. Then sup ae(F)< oo.
F<Y f<E
dimF<oo dimF<oo
Proof. Since ? = G© Y, there exists a projection и of ? onto G,
such that m(F) = 0.
Now let Го be an arbitrary subspace of ?, with dimr0<oo. Then
(IE — u)(F0) is a finite-dimensional subspace of Y, whence there exists a
projection v1 of Fonto (IE—u){F0), such that
IKI
B0.25)
We claim that v = u+v1(IE-u) is then a projection of ? onto its
finite-dimensional subspace
D = G®(IE-u)(F0).
Indeed, for any xe? we have u(x)eG and v1{IE — u){x)e{IE — u)(F0),
whence v(x)eD and for any xeD we have v1(IE — u)(x) = (IE — u)(x),
whence г;(х) = м(х) + (/? — м)(х) = х, which proves the claim.
Observe now that Г0<=Д since for any хеГ0 we have x = u(x)
+ {IE-u)(x)eG@(IE-u)(F0) = D and that, by dim?)<oc,
1 See e.g. [174], Appendix 3.
2 We recall (see § 11) that the notation F<X stands for "F is a (closed linear)
subspace of the Banach space X".

196 I. The Basis Problem. Some Properties of Bases in Banach Spaces
codimB Fo = dim D — dim Fo ^ dim D — dim [(/? — u) (Fo)] = dim G,
whence there exists a projection w of D onto Fo, of norm1
||и>|| <g2")dimr>FosS2dimG. B0.26)
Consequently, wv is a projection of ? onto Fo, of norm2
\\wv\\^\\w\\ \\v
NI + (i + NI
sup A?(F)+1\ = M<oo,
F<Y
dim F < oo
which, since Fo has been an arbitrary subspace of ? with dim Fo < oo and
since M is independent of Fo, completes the proof of lemma 20.1.
Theorem 20.2. Let E be a Banach space. The following statements are
equivalent:
1°. There exists a sequence of subspaces {Gn} of E with dimGn<oo,
such that lim A(Gn)= oo.
n-> oo
2°. ? has a sub-Г system.
3°. ? has a sub-Л system.
IfE is separable, these statements are equivalent to each of the following
statements:
4°. ? has а Г system.
5°. ? has а Л system.
Proof. Assume that we have 1°. Let E1 be a finite dimensional subspace
of ?. Choose фи...,фП1еЕ*} with ||ф;|| = 1 (/= 1,...,щ) so that the re-
relations xeE1,\<f>i(x)\<:l(i=\,...,n1) imply ||jc||<2. This can be done
e. g. by taking a finite s-net {z1,...,zni} of S={xeE1\ \\x\\ = 1}, where
0<?^, and functionals 0;e?f such that ф,(г;) = 1, \\ф(\\ =1 (i=\,...,nx),
since then for any xe?j with |ф;(х)|^1 (i=i,...,n1), ||x||>l, and any
x
index^with l^:j^n1 such that г < е, we have
1 Indeed, this follows from the fact in any reflexive (in particular, in any finite
dimensional) Banach space В there exists a projection of norm < 2 onto any hyper-
plane H = {x\f(x) = 0} (take x^B with \\Xl\\ = l, f(x1)=\\f\\ and put w1(x) = x
fix)
-щ*> fora11 *sB>
2 We use here the fact that for any F-< Y we have /,y(F)= inf ||wj
20. Polynomial bases. Strict polynomial bases. Г systems and Л systems 197
ФЛ*,-
1
л
X \
x\\ )
X
Jl*ll
J
1
X
2
i
11*1
(
x)
1*11 /
< ^2. Let /i,...,/„ e?* be Hahn-Banach extensions
1 —e
whence ||x
of the 0,'s and let У1 = {л:е?|у;(л:) = 0 (i= 1,...,«!)}. Then
< oo, ?i n ^ = {0}, and the natural projection u1: x + j->x of E1 ® Y1
onto El has norm ||и1||<2, since for any xeE1, yeY1 with j|x+j||^l
we have \ф,{х)\= \ft{x)\= \f(x+y)\^ \\f\\ \\x + y\\^ 1 (i=l,...,ni), whence
ll*ll<2.
By 1° and lemma 20.1, choose a finite dimensional subspace E2 of У[
with Я(?2)^2. Furthermore, as above, choose /ni + i,---,/n2e?* with
||y;.|| = l so that the relations xeE1©E2, |/i(x)|<l (i=l,...,n2) imply
||x||<2. Then for Y2 = {xeE\ft{x) = 0 (/= l,...,n2)} we have Г2сКь
codim? J$^и2<oo, and the natural projection u2 of E1@E2®Y2 onto
?x©?2 has norm ||и2||<2.
Continuing in this way, we obtain two sequences of subspaces
{?„}, {Yn} of ? such that dim?n<oo, codim?^<oo, Я(?„)^и, Еп+1,
Yn+1<=Yn and that the natural projection un of E1 ©••¦©?„© Yn onto
?t ® •••©?„ has norm ИmJsJ2(«= 1,2,...).
Observe now that for any n, any subspace Bc?Itl and any projection
и of ? onto ?!©•¦•©?„©?, we have j|m|| >—. Indeed, then
6
(IE — un_l)unu is a projection of ? onto ?„, whence и^л(?„)
< ||(/? — mb_i)mbm|| <3.2 ||u||, whence the assertion follows.
Consequently, if x\,...,x"kn (where fen = dim?n) is any basis of ?„,
the sequence
Х\,...,х1,х\,...,х1г,...,х\,-,^,- B0.27)
is a sub-Л system in ?. Thus, Г => 3°.
The implications 3°^>2°^Г are obvious.
Assume now that ? is separable and that we have 3°, i.e. that there
exists in ? a sub-Л system {xn} and let us prove that in this case we have
5°. If [х„] = ?, then {*„} is already а Л system and thus we have 5C.
Assume that [*„]#? and let {yn} be a sequence in ?\[xB], such that
[xi>Jj] = ? and that {хьу3) is finitely linearly independent1. For each
It is easy to see that such a sequence exists. Indeed, since E is separable, let
{zn} be a dense sequence in ?. Let ii be the smallest index for which zf[ ?[*„], г2
the smallest index for which zti?[xn,zit~\ — [xn]@ [г(]] and continue in this way.
Then take yJ=zi.(J= 1,2,...).

198
I. The Basis Problem. Some Properties of Bases in Banach Spaces
к choose a projection uk of ? onto [>'i,.-,Vt] such that uk(Xj) = 0
(/'= 1,2,...)' and put rk = IE — uk. Then for any projection vn,k of ? onto
[х1,...,х„,у1,...,ук] the operator rkvn<k is a projection of ? onto
with Я([x, х„1)< ||rtj;n J| ^ ||rj| \\vn k\\, whence
IKJ
<kun,k\\ =* ll'tll \\un,k\\
/t([x ,,...,*„])
B0.28)
Now, since {xn} is a sub-Л system, for each к there exists a positive
integer nk such that и^ик implies X{[x1,...,x^])'^\\rk\\k. We claim that
the sequence {zn} <= ? defined by
i-t+i f°r "*-i +k^n^nk + k— 1 (fe=l,2,...)
iv, for n = nk + k {k=l,2,...)
(where no = 0), i.e. the sequence
Хл,...,Х ,У\,Х„ i i ,. .., Х„ ,\2,... \Z-\).D\)f
is а Л system in ?. Indeed, {zn} = {xt,yj} is finitely linearly independent
and [zn] = [Xi,j;] = ?. Furthermore, if nk _ x + к ^ n ^ nk + fe — 1 and г? is
an arbitrary projection of ? onto [z,,...,zn] = [x1,...,xn_k + 1,j1,...,j'k_1J,
then i; is a i;B_)k+ljt_1, whence by B0.28), n-k+\^nk_1 + \ and the
choice of «t _!, we infer
I — ll"n-*+l,*-ll
Similarly, if n = nk + k and г; is an arbitrary projection of ? onto
,...,zn] = [x1,...,xn_k,j1,...,jk], then г; is a vn-kik, whence by B0.28),
k = nk and the choice of nk, we infer
v =
vn-k,k\\
which proves that {zn} is a /1 system in E. Thus, 3°=>5".
Finally, the implications 5°=>4°=>Г are obvious, which completes
the proof of theorem 20.2.
Corollary 20.1. If a separable Banach space E has a subspace F
possessing а Л system2, then the space E itself has а Л system, namely,
every Л system ofF can be extended to а Л system ofE.
1 Such a projection exists. Indeed, extend the coefficient functionals h{ (i = 1,-. -,k)
associated to the basis {yi,...,yk} of [>1,...,.ук] to [xj]@ [yu-..,л] by 0 on [x,]
and linearity and then to the whole space E by the Hahn-Banach theorem. Denoting
these extensions again by hb put uk(x)= ? hi(x)yi (xeE).
i=l
2 In analogy with the term "basic sequence", one may call "Л sequence" (in E)
a sequence {xn}czE such that {xn} is а Л system in [j;,] =
20. Polynomial bases. Strict polynomial bases. Г systems and Л systems 199
Proof. Observe that every Л system {xn} in FczE is a sub-Л system
in E, since for any projection v of ? onto P(n) = \xl,...,xn\ the restriction
i;|F is a projection of F onto P(n), of norm ^ ||j;j|. Hence, Ьу theorem 20.2
(implication 3°=>5°) and its proof given above, {х„} can be extended to
а Л system B0.29) of ?, which completes the proof.
Since the spaces lp A ^p<co) and c0 satisfy1 condition 1° of theorem
20.2, we obtain
Corollary 20.2. The spaces LP(T,v), where 1 ^p<со,рф2 and where
(T,v) is a positive measure space such that LP(T,v) is separable and the
spaces C(Q), where Q is a compact metric space, have Л systems.
The converse of the second statement of corollary 20.1 is not true, i.e.
a subsequence of а Л system need not be а Л system in its closed linear
span, it may be even a basic sequence. Moreover, we have
Proposition 20.2. Every finitely linearly independent complete sequence
{xn} in a separable Banach space F (hence, in particular, every basis {xn}
of F) can be extended to а Л system of a suitable separable super space E.
Proof. Let G be a separable Banach space having а Л system {у„}
and let E = FxG, endowed with the norm ||{jc,j>}|| = ||jc|| + \\y\\. Then
the sequence {zn}<=? defined by
г2„={0,>-„}
B0.31)
is а Л system in ?. In fact, {zn} is obviously finitely linearly independent
and complete in ?. Furthermore, if v2n is an arbitrary projection of ? onto
[z1,...,z2n], and и is the natural projection of E = FxG onto {0} x G,
then the restriction мг;2„|@(хО induces a projections of G onto [yi,...,yB],
with
^g([>i>•¦-,>'*])< INI = IIm^JiojxgII «S N1 \\v2n\\ = \\v2n\\ ¦
Since a similar relation also holds for any v2n+1 and since {yn} is
а Л system in G, it follows that {г„} is а Л system in ?, which completes
the proof.
For an arbitrary Banach space ?, introduce now the functions2
ф„{Е)= sup A(G)= sup inf ||u|| (n=l,2,...). B0.32)
G-<E G < E ue^(f.G)
dimG=n dim G — n
1 See e.g. [170], [252].
2 For comparison, let us recall that in § 11 we have considered the functions
?„(?)= inf k{G)= inf inf ||u||
dim G = n dim G = n

200
I. The Basis Problem. Some Properties of Bases in Banach Spaces
We deduce from theorem 20.2 the following property of this sequence
of functions:
Corollary 20.3. If for a Banach space E we have sup ф„(Е)= оо,
then lim фп(Е)= оо.
Proof Take Gn<E with dimGn = w, such that A(Gn)^\jjn(E)-1
(и=1,2,...). Then, by our hypothesis, sup A(Gn)=oc, whence there
exists a subsequence {G } such that lim A(G )=oc. Consequently, by
theorem 20.2, E has a sub-Л system {*„}. Putting Pin) = [_x1,...,xn]
(n=l,2,...), we have then ф„(Е)^л(Р(п))^со as w->oc, whence
lim фп(Е)= оо, which completes the proof.
Notes and remarks
§ 1. The notion of a basis of a Banach space (see definition 1.1) was
introduced by J. Schauder [224]. The basis problem (see problem 1.1)
was raised explicitly in the book of S. Banach ([10], p. 111-112 and p. 245)
and it remains one of the important unsolved problems of functional
analysis (see e.g. I. Kaplansky [130] and G. Kothe [134]). The fact
that this problem is equivalent to problem 1.2 was observed by S. Banach
([10], p. 238). Problem 1.3 was also raised by S. Banach ([10], p. 238 and
p. 245).
For the terms "restriction of the field of scalars" and "extension of
the field of scalars", see N. Bourbaki [30], § 5 and [31], § 2. In [30] it is
also shown that the complexification ? of a real space G is "smallest"
in a certain sense, and that this "smallest" E is uniquely determined, up
to an isomorphism. The notation ?(r) for the real Banach space associated
to a complex Banach space E is used in the monograph of M. M. Day
[43], while in Bourbaki [30] the notation ER occurs. For the charac-
characterization, given in § 1, of real Banach spaces which admit a complex
structure, see J. Dieudonne [45]. For the term "complexification" and
for the characterization, given in § 1, of complex spaces which are
isomorphic to the complexification of a suitable real Banach space,
see e.g. A. Lichnerowicz [147], Ch. V1. For the involution on the space A,
1 Actually, both in Bourbaki [30], [31] and Lichnerowicz [147] these problems
are considered only for linear spaces without topology, and the corresponding
mappings are required only to be linear. However, the additional assumption of
continuity of these mappings can be made without alteration of the corresponding
results and proofs. On the other hand, it is easy to see that on every complex linear
space there exists an involution (without assumption of continuity) whence every
such space is the complexification of a suitable real linear space (without topology).
Notes and remarks
201
mentioned at the end of § 1, see e.g. M. A. Naimark [171]. Ch. Ill,
§ 14, p. 196.
§2. Example 2.1 was given by J. Schauder [224] (see also S. Banach
[10], p. 112). Example 2.2 was constructed by J. Schauder [224]. As
we have mentioned in § 2, the basis B.3) is called the Schauder basis of
C([0,1]). Various other bases of C([0,1]) and bases of the spaces C(Q),
where б is a metric compact space, will be studied in Vol. II.
Example 2.3 was also given by J. Schauder [225]. The fact that the
Haar functions constitute a basis in separable Orlicz spaces, was proved
by W. Orlicz ([185], p. 122-126); see also the book of M. A. Krasnoselskil
and Ya. B. Rutickil [138], p. 122-126).
The spaces Lx with respect to a levelling length function were intro-
introduced by H. W. Ellis and I. Halperin [55], who also constructed general-
generalized Haar bases for separable spaces of this type. Problem 2.1 was raised
by I. Halperin ([103], p. 247). For a study of Haar bases in modulared
function spaces LM(i>0 see J. Ishii and T. Shimogaki [111].
Problem 2.3 was raised by S. Banach ([10], p. 238). Problems 2.2
and 2.3 have been studied by F. S. Vaher, who has claimed (verbal
communication, International Congress of Mathematicians, Moscow,
1966) to have proved that the space A has no basis, thus solving the
basis problem in the negative. However, the proof was not complete
and to our knowledge it has not been given to print so far. On the other
hand, at the same time, E. Akutowicz [1] published the opposite result,
that the space A, and also the space Я1, has a basis, but it has since
been discovered that the proof is not correct. Thus, the problem whether
these Banach spaces of analytic functions have a basis or not, still remains
open.
S. Banach ([10], p. 238) also raised the following problem: Let В be
the space of all real-valued continuous functions on the unit square
O^t^l, O^s^l, admitting continuous partial derivatives of order 1,
endowed with the norm 11x11= max |x(;,,s)| + max |x,'(?,s)|
+ max \x's(t,s)\; does В possess a basis? This problem has recently been
solved in the affirmative by S. Schonefeld [226] and Z. Ciesielski [275].
§ 3. In his definition of a basis in a Banach space, J. Schauder [224]
separately required the continuity of the coefficient functional /„.
S. Banach ([10], p. Ill) proved that this condition is satisfied by all
bases in a Banach space, i.e. that we have theorem 3.1. The idea of the
proof of theorem 3.1 (through propositions 3.1 and 3.2), due to S. Banach
[10], has become a useful tool in various extensions of this theorem.
E.g. this theorem has been extended with a similar method to bases in

202
I. The Basis Problem. Some Properties of Bases in Banach Spaces
locally convex complete metric linear spaces (i.e. Frechet spaces) by
F. Newns [176], in complete metric linear spaces (without condition of
local convexity) by V. N. Nikolskil ([180], p. 125 and p. 135), С Bessaga
and A. Pelczynski ([24], theorem 1), M. G. Arsove ([4], theorem 2;
however, all these extensions were already known by S. Banach, as shown
by a remark of [10], p. 239), in certain inductive limits of Frechet spaces
by M. G. Arsove and R. E. Edwards ([5], theorem 12); more generally,
B. S. Mityagin observed ([169], p. 92) that whenever {х„\ is a basis of a
topological linear space for which the open mapping theorem or the
closed graph theorem is valid, the corresponding coefficient functionals
are continuous. For other extensions of theorem 3.1 (e.g. to T-bases), see
Vol. II, Chapter III and the corresponding Notes and remarks.
Theorem 3.2 was communicated to us by С Bessaga.
§4. Example 4.1 of an ?-complete biorthogonal system (х„,/„) such
that {х„} is not a basis of E, is well known, e.g. it is given as such in a
paper of A. I. Markushevich [156]. Theorem 4.1, which gives useful
characterizations of bases, is essentially due to S. Banach ([10], p.
107-108, theorems 2 and 4). Corollary 4.1 was proved by S. R. Foguel
[60]. Propositions 4.1 and 4.3 are due to B. R. Gelbaum ([71], §2,
theorem 1 and § 4, theorem 1 (d)).
Basic sequences (see definition 4.5) were already considered by S.
Banach [10]. The term "basic sequence" was introduced by С Bessaga
and A. Pelczynski [22]. A detailed study of basic sequences will be
made in Vol. II, Ch. III.
Problem 4.1 and the conjecture concerning the sequence D.7) are
due to A. Pelczynski [194]. Proposition 4.2 is well known (see e. g.,
A. Pelczynski [195], lemma 11).
The notion of block-perturbation of a basis (see definition 4.7) and
proposition 4.4 were given in [196] (see [196], § 2, definition 4 and lemma
2); let us mention that V. G. Vinokurov [261] considered perturbations
of the form z2j_1=x2j_1, z2j = <x2j^lx2j_.1+<x2jx2j (/=1,2,...), where
sup |a2j_!|<oo, inf |a2j-|>0. and established that they constitute a
1 < J < 00 1 < j < 00
basis of E ([261], theorem 4).
§ 5. The sufficiency part of proposition 5.1 was observed by S. Banach
([10], p. 106). Proposition 5.2 is due to S. R. Foguel [60] (for proposition
5.2b) and 5.2c) see [60], theorems 1 and 2).
The equivalence ГоЗ° of theorem 5.1 was proved by M. M.
Grinblium ([85], theorem 3) and the equivalences 1°<*4°, lco5°
of this theorem were proved by S. R. Foguel ([60], theorems 3 and 4);
the equivalence Го2° oftheorem 5.1 is nothing else but the equivalence
1°<*3C of § 4, theorem 4.1. The direct proof of the implication 4° => 1°
oftheorem 5.1, mentioned in remark 5.2, was given in [229].
Notes and remarks
203
Proposition 5.3 was proved by M. M. Grinblium ([85], theorems
1, 2, 4, 5).
Multipliers were used a long time ago in analysis (see e.g., W. Orlicz
[183], S. Banach [10]). The equivalence ГсЗ° of theorem 5.2 was proved
byM. I. Kadec ([126], theorem 1) and the equivalence Го2° was proved
independently, by S. Yamazaki ([267], § 5, theorem 1). In the case when
{*„} is a basis with the a.s.c.f. {/„}, proposition 5.4 is due to S. Yamazaki
([267], § 3, theorem 1 and § 2, lemma 2) and in the general case, essentially,
to R. J. McGivney and W. Ruckle ([166], corollary 3.3).
Theorem 5.3 and corollary 5.3 were given by S. Yamazaki ([267],
§ 5, theorems 2, 3 and § 6, theorem 1).
§ 6. The three types of linear independence of sequences considered
in § 6 were studied long-ago in infinite-dimensional Hilbert spaces (see
e.g. the book of S. Kaczmarz and H. Steinhaus [117]). Example 6.1a)
was given by V. N. Nikolskil ([180], p. 127).
Lemma 6.1 is essentially due to S. Yamazaki (see [265], the proofs
of theorems 1 and 2). The equivalences Го2°о4°о5° of theorem
6.1 were proved by A. I. Markushevich [156]. The. equivalence Го2°
can be also found in the book of S. Kaczmarz and H. Steinhaus ([117],
Ch. VIII, § 1). The equivalence ГоЗ° was proved in the paper [61]
(see [61], theorem 1). The equivalence 4°o 12° is due to V. N. Nikolskil
([180], § 8, theorem 1.2)).
§7. The equivalences Гс-5°о9° of theorem 7.1 were given by
M. M. Grinblium ([82], theorems A and A) under the additional hypo-
hypothesis that {xn} is a minimal sequence; however, this hypothesis is
superfluous. The equivalences Г о 6° о 7° <* 8° are essentially due to
S. Yamazaki ([265], theorems 1 and 2'). The equivalences Г о 10° о 1Г
were proved by V. N. Nikolskil ([180J, § 8, theorems 1.1) and 2.1)). The
equivalences Г<*3° and Г«^4° are also known (see e. g. M. Z. Solomiak
[253] and R. С James [113], respectively).
The notion of index of a sequence (see definition 7.1) was introduced
by M. M. Grinblium [82], while the norm of a sequence (see defini-
definition 7.1) was considered by С Bessaga [20]. The index Г(Е) of a Banach
space E (see definition 7.2) was introduced by M. M. Grinblium [82],
who also raised [82] the problem of the existence of Banach spaces
such that 0<Г(?)<1, mentioned after definition 7.2.
Corollary 7.1 is due to V I. Guraril (verbal communication). Corol-
Corollary 7.2 was communicated to us by С Bessaga. Corollary 7.3 was given
by V. I. Guraril ([93], theorem 2) and, independently, by M. Zippin
([273], lemma 2).
Block basic sequences have been considered e. g., by R. С James
[113]. For such sequences, С Bessaga and A. Pelczynski [22] have used

204
I. The Basis Problem. Some Properties of Bases in Banach Spaces
the term "block basis", while the term "block basic sequence" was
suggested in [237].
The fact that all (closed) hyperplanes of a Banach space are iso-
morphic to each other was observed by C. Bessaga and A. Pelczynski
[25], by considering the intersection of any two hyperplanes; lemma 7.1
was proved by M. Zippin [273], by using this idea of С Bessaga and
A. Pelczynski. The first part of theorem 7.2 was proved by M. Zippin
[273]; however, as we observed in remark 7.3, a theorem on extension
of block basic sequences of a particular form had been given already
in [196]. The second part of theorem 7.2 was observed in [44] and,
independently, by J. R. Holub [108].
§8. Pairs of sequences {х„}, {уп} satisfying (8.1) were considered by
S. Banach ([9], p. 1638; see also [10], p. 112). Fully equivalent sequences
were studied by L. A. Gurevich [99], who used the term "equivalent
sequences" for them. Equivalent bases were studied by M. G. Arsove
[3], who used the term "similar bases"; some authors still use this latter
term. The terms "domination" and "strict domination" were introduced
in [251].
The equivalences l°a)<s>2°a) and Гс)о2°с) of theorem 8.1 were
proved by M. G. Arsove ([3], lemma 1 and theorem 3). For the parti-
particular case of complete sequences {xn}, {у„} in E — F, the equivalence
I°d)o2°d) was proved by L. A. Gurevich ([99], theorem 2). Under
the additional hypotheses that [х„] = [vn] = E = F, that E is sequentially
weakly complete, and that the mapping Т:ф^>ф of [у„]* into [xj*
defined by (8.5) is an isomorphism of [у„]* onto [х„]*, the equivalence
l°d)<s>3°d) was proved by L. A. Gurevich ([99], theorem 3); however,
all these additional hypotheses are superfluous. In the particular case
when {*„}, respectively both {xn} and {yn}, are bases of ? and F respec-
respectively, the equivalences l°b)<=>4 b) and 1 d)<=>4°d) are essentially due
to M. G. Arsove ([3], theorems 5 and 4). The equivalence l°b)<s>5°b)
is essentially due to S. Banach ([10], p. 112, theorem 7; for a particular
case see also S. Banach [9], theorem II); actually, S. Banach assumed
that for every xe[xj the system of equations (8.6) has exactly one
solution уе[у„], but this is obviously equivalent to the assumption
that for every xe[xj the system (8.6) has a solution je[yn] and that
{(//„} is total on [jn] (indeed, assuming the uniqueness of the solution
of (8.6), take x = 0 in (8.6)). The equivalence l°b)«i=>6ob) is essentially
due to M. G. Arsove (see [3], the proof of theorem 5). Finally, for the
equivalence l°d)<s>6cd), see С Bessaga and A. Pelczynski ([22], pro-
proposition 1.2) and M. G. Arsove ([3], theorem 1). Various extensions of
parts of theorem 8.1 have been given, to more general spaces (see e.g.
M. G. Arsove [3]) and for other types of sequences in these spaces (see
e.g. M. G. Arsove and R. E. Edwards [5]).
Notes and remarks
205
Let us also observe that {xn}~{yn} (where {xn\ <= E, {yn} <= F) if
and only if
sup
Z aixi
sup
1 ^ n < oo
n
I
i —
i
<oo
(indeed, this is an immediate consequence of theorem 8.1 d), equivalence
\°o2°); under the (extraneous) assumption that {xn}, {yn} are basic
sequences, this remark was made by V. D. Milman ([167], lemma 2).
Proposition 8.1a) was given by С Bessaga and A. Pelczynski ([22],
lemma 1,2°).
Theorem 8.2 on characterization of bases is due to M. M. Grinblium
[86].
The fact that the full equivalence conserves the completeness of
sequences, their minimality and the property of being a basis, was
observed by L. A. Gurevich ([99], theorem 1). The notions of {an}-com-
pleteness and completeness of order p have been introduced by P. Davis
and Ky Fan [36 a].
The notion of affine equivalence (see definition 8.2) was introduced
in the paper [196] (see [196], definition 51). Problem 8.1 was raised in
the same paper (see [196], problem 1).
Permutatively equivalent bases have been considered by A. Pel-
Pelczynski [190], who called them "c-equivalent bases". The term "permu-
"permutatively equivalent" was suggested in [241] and recently used also by
A. Pelczynski [195].
Remark 8.2, proposition 8.3, definition 8.5, remark 8.3 and proposi-
proposition 8.4 were given by A. Pelczynski [195].
§ 9. In the particular case when ? is a Hilbert space, the implications
b) y) and b) <5) of theorem 9.1 were given by R. E. A. C. Paley and
N. Wiener [188]. R. P. Boas [27] observed that the proof given by
R. E. A. C. Paley and N. Wiener remains valid in an arbitrary Banach
space E. The implication [{*„} is minimal =>{у„} is minimal] (i.e. half
of the equivalence a) <5)) has been given by A. I. Markushevich [156];
a short proof of this implication was given in [61]. The implications
b) y) and b) <5) were reproved, using the technique of inversion of oper-
operators by means of geometric series, by K. I. Babenko [7]. Theorem 9.1 a)
and its equivalence a) <5), were proved (and b) y), b) S) reproved) by
F. W. Schafke [222], who also made remark 9.1, for the converse of
b) y) and b) 5) if OsSA<|. The implications b) a), b) P) of theorem 9.1
are due to P. Davis and Ky Fan ([36], theorem 4). Various extensions
1 Actually, the notion introduced in [196] is slightly different, since in [196] it
is assumed А„#0(п= 1,2,...) instead of (8.17).

206
I. The Basis Problem. Some Properties of Bases in Banach Spaces
of the Paley-Wiener theorem have been given: a) to more general spaces,
e. g. to complete metric linear spaces (see M. G. Arsove [4], where other
references are also given), and b) to generalizations of bases, e. g. to
T-bases (see Vol. II, Ch. III). On the other hand, theorems of Paley-
Wiener type are also known for special classes of bases in Hilbert spaces
(see Vol. II, Ch. V).
Example 9.1 was given by R. J. Duffin and J. J. Eachus [48]. For
proposition 9.1 see R. J. Duffin and J. J. Eachus [48], the footnote to
theorem B, where this result is given with the remark that it can be
proved by making use of an ergodic theorem.
In the particular case when ? is a Hilbert space, theorem 9.2 b) y) was
given by H. Pollard ([198], theorem 1.1), corollary 9.1, part 2°b) a), y) by
B. Sz. Nagy [173] and part l°b) a) by S. H. Hilding [107]. Theorem 9.2
for Banach spaces is due to J. R. Retherford [203], [209], who also
observed the equivalence of conditions (9.16), (9.17) of corollary 9.1 and
(9.5) of theorem 9.2. Theorem 9.3 was also given in the paper [203] of
J. R. Retherford. Example 9.2 is due to W. J. Davis (see J. R. Retherford
[209]). For some related results see also W. J. Davis [38].
§ 10. The fact that under the assumptions of theorem 10.1 the con-
conclusions of theorem 9.1 hold, was proved by M. G. Krein and L. A.
Liusternik [142] and reproved by С Bessaga and A. Pelczynski [22].
Recently, С W. McArthur and J. R. Retherford ([162], p. 120) have
observed that, by the inequality
|Еа;х,- ^4sup|a,|sup ]T x{ ,
where /is any finite subset of Ж = {1,2,3,...}, {х;};б7<=? and {а;},-б;<=Х
(in the case of real scalars the constant 4 may be replaced by 2), the
conclusion of theorem 10.1 also holds under the weaker assumption
sup
/с Г
/finite
ti-yi)
4 sup ||/„||
where 0<A< 1.
The implication a) 2°=>aL° of theorem 10.2 was given by M. G.
Krein and L. A. Liusternik [142]; see also M. S. Altman [2] for
aK°=>aL°. The implication a)l°=>aLG is nothing else but the particular
case k = 0 of part b), while parts b) and c) are due to M. G. Krein ([140],
the footnote on p. 333).
The last part of remark 10.4, for the case of the implication aJ°=>4°
of theorem 10.2, is due to В. Е. Veic [259].
The fact that under the hypotheses of theorem 10.3 there exist
у„>0 (n= 1,2,...) such that for {jn}<=? satisfying A0.18) the conclusions
of theorem 9.1 hold, as well as corollary 10.2, were proved by M. G. Krein,
Notes and remarks
207
p. p. Milman and M. A. Rutman ([143], theorems В and A). Corollary
10.2 solves in the affirmative a problem raised by S. Banach, S. Mazur
and S. Ulam (cf. "Scottish book", problem 108). Both the above mentioned
part of theorem 10.3 and corollary 10.2 are called "the Krein-Milman-
Rutman theorem". This theorem was also reproved by C. Bessaga and
д. Pelczynski [22]; for a weaker form of this theorem see also M. M.
Grinblium [83], [84]. The fact that the Krein-Milman-Rutman theorem
is actually a consequence of the older Paley-Wiener theorem was proved
in the paper [61]. С Bessaga and A. Pelczynski [21], [24] proved that
the Krein-Milman-Rutman theorem cannot be extended to B0-spaces
(in the sense of S. Mazur and W. Orlicz [157]). Furthermore, this theorem
cannot be carried over to n>*-bases in conjugate Banach spaces (see § 14
and the corresponding Notes and remarks), nor to monotone bases
(Ch. II, §1, remark 1.1).
Theorem 10.3 suggests the problem of introducing suitable topologies
in the set of all bases of a Banach space. In connection with this problem
let us mention the following definition of the distance between two basic
sequences {xn}, {jn}<=?, due to S. Banach [11]:
dist({xn}, {yn}) = ? -
S. Banach proved ([11], theorem 1) that if ? is a Hilbert space and #"
the set of all orthogonal basic sequences of ? endowed with the above
metric, then the set Я of all orthogonal bases of ? is a dense Gs set of
the second category in #\
V. Ya. Kozlov [136] introduced the following distance between
two classes of related bases of a Banach space ?:
dist(#",<30= sup
sup
xeE
= sup
1 < n < oo
where {xn}e3C, {yH}e<& and {/„}, {sj,1*} and {#„}, {s[2)} are the sequences
of coefficient functionals and partial sum operators associated to the
bases {х„} and {у„} respectively; two bases {xn}, {х'„} of ? are called1
related, if fn(x)xn= f'n(x)x'n («=1,2,...), where {./„}, {f'J^E* are the
a.s.c.f. to the bases {xn} and {x'n} respectively. V. Ya. Kozlov proved
([136], theorem 2) that with the above metric, the set sd of all classes of
related bases of ? becomes a complete metric space. Furthermore, V. Ya.
Kozlov [136] also observed that if ? is a Hilbert space, the space s/ is
unbounded and all orthonormal bases of ? are situated on a sphere of
radius 1.
V. Ya. Kozlov [136] used the term "equivalent bases" ,which has in the
present monograph a different meaning. The term "related bases", in a slightly
more general sense, has been used by L. W. Baric [17].

208
I. The Basis Problem. Some Properties of Bases in Banach Spaces
Let us also mention the following problem, communicated to us by
G. Neubauer: what are the properties of the set of all subspaces with a
basis of a Banach space E, in the metric space consisting of the set of all
subspaces of E, endowed with one of the usual metrics? (for such metrics,
see e.g., E. Berkson [19]).
Theorem 10.4, in a slightly different form, is due to V. D. Milman
([167], theorem 1) and so are part of example 10.1 ([167], p. 402, example
3) and of corollary 10.3 ([167], theorem 5).
Lemma 10.1, with a different proof, was given by I. M. Gelfand
([73], p. 244, theorem 5). Theorem 10.5 is due, essentially, to V. D. Milman
([167], theorem 3).
For a part of the terminology introduced in definitions 10.1 and 10.2
see N. Bari [13], [14] and J. R. Retherford [203]. In these definitions
PW, PH, N and KL stand for Paley-Wiener, Pollard-Hilding, Nagy,
and Krein-Liusternik, respectively.
Theorem 10.6 is a particular case of [251], theorem 7.
§11. The functions фп(Е) (see formula A1.2)) were introduced by
H. F. Bohnenblust [29]. Theorem 11.1, suggesting a possible method
for constructing a separable Banach space having no basis, was given in
[243] (see [243], p. 722, theorem 1.11).
§ 12. Theorem 12.1 is essentially due to S. Banach ([10], p. 107,
theorem 3); it was also reproved by L. A. Gurevich [100]. Corollary
12.1 was given by S. Karlin ([131], theorem 3), who also raised problem
12.1 in [131]. Corollary 12.2 was proved by M. M. Grinblium and L. A.
Gurevich ([87], theorem II). Problem 12.2 b) was raised in the paper
[245] (see [245], problem 4).
Theorem 12.2 was given in [240], theorem 1. Remark 12.1 was made
by V. F. Gaposhkin and M. I. Kadec [67]. Remark 12.2 was made in
[240], remarks 2 and 3. Example 12.2 was constructed in [240].
Theorems 12.3, 12.4, examples 12.3, 12.4, remarks 12.3, 12.4 and
corollary 12.3 were given in the paper [249].
Proposition 12.2 a), c) in a slightly less "quantitative" form (namely,
with
sup
< oo, sup
< oo
instead of A2.42) and 12.46)), is due to A. Wilansky ([262], lemmas 2 and 1).
Theorem 12.5 was given in [239], theorem 4 and corollary 4 and
[240], theorem 1 and its corollary; actually, part a) was proved in [239],
corollary 4 under the hypothesis that {xn} is a shrinking basis of E and
the remark that this latter assumption is superfluous, was made by
J. R. Retherford [205].
Definition 12.1 and theorem 12.6, with a somewhat different proof,
are due to W. Ruckle ([214], theorem 3.1).
Notes and remarks
209
Theorem 12.7 has been given by W. Ruckle [213] and L. W. Baric
and W. Ruckle [18].
Theorem 12.8 is due to R. J. McGivney and W. Ruckle ([166],
theorem 6.1); the proof of the sufficiency part, presented here, was
communicated to us by W. Ruckle.
§ 13. The fact that the notion of a basis admits an obvious extension
to complete metric linear spaces, was mentioned in the book of S. Banach
([10], p. 239); the passage to general topological linear spaces (see
definition 13.1) was made by M. G. Arsove and R. E. Edwards [5].
S. Banach also remarked ([10], p. 239) that the space S([0,1]) has no
basis; the more general example 13.1 (including the spaces Lp([0,1])
with 0<p< 1) was given in the paper [245] (see [245], theorem 1).
Weak bases in Banach spaces (see definition 13.2) were considered in
the book of S. Banach ([10], p. 238). The term "w*-basis" occurs in the
book of M. M. Day ([43], Ch. IV, § 3, theorem 1) without a precise
definition; actually, M. M. Day calls w*-bases those sequences in con-
conjugate Banach spaces, which we have called w*-Schauder bases (see
§ 14). A study of w*-bases in conjugate Banach spaces was made in the
papers [234], [240], [245]. The bw*-ha.ses in conjugate Banach spaces
were considered by J. R. Retherford [204], who also gave lemma 13.1 b),
with the mention that it is due to R. D. McWilliams (see [204], lemme 5.2).
The implication а)Г=>аK° of theorem 13.1 is called ([5], p. 97)
"the weak basis theorem". The weak basis theorem was stated without
proof in the book of S. Banach ([10], p. 238). A proof was sketched by
S. Karlin [131]. However, in the book of M. M. Day ([43], Ch. IV,
§ 3, theorem 2), this theorem is stated only under the additional hypo-
hypothesis that either E is weakly complete, or the coefficient functionals /„
associated to the w-basis {xn} are continuous on E for the norm topo-
topology. A complete proof of the weak basis theorem in its full generality
was given by C. Bessaga and A. Pelczynski [24], where it is also mentioned
that this theorem is actually due to S. Mazur; see also B. R. Gelbaum
([69], p. 194-195) and A. Wilansky ([263], p. 212). There exist various
extensions of the weak basis theorem to more general spaces (see e. g.
С Bessaga and A. Pelczynski [24] and M. G. Arsove and R. E. Edwards
[5]). The implication b)lc=>bK° of theorem 13.1 and the more general
proposition 13.1 were proved in the same paper of С Bessaga and A. Pel-
Pelczynski ([24], theorem 3).
Example 13.3 of a basis in a conjugate Banach space, which is not
a w*-basis, was given in the paper [245]. In the same paper (see [245],
theorem 2) example 13.4 of a w*-separable conjugate Banach space hav-
having no w*-basis (and thus of a separable locally convex space having
no basis) was given and problems 13.1 and 13.2 were raised (see [245],
problems 2 and 3).
1* Singer, Bases in Banach Spaces I

210
I. The Basis Problem. Some Properties of Bases in Banach Spaces
In the present book we have considered only w*-separability of E*
in the usual sense of separability of topological spaces (i. e. the ad-
admitting of a countable dense subset; see e.g. [50], p. 21, definition 11).
However, A. Pelczynski called our attention to the interest of consider-
considering a stronger condition of "sequential" w*-separability, since ([10],
p. 124, theorem 4) the conjugate space E* of any separable Banach space
E is sequentially w*-separable (i. e. there exists a countable sequence
{/„} in E* such that every feE* is the w*-limit of a suitable subsequence
{fnk} °f {/n})> while the space m* is w*-separable but not sequentially
w*-separable. Obviously, every topological linear space with a basis is
sequentially separable. In the paper [234] there was given an example
of a sequentially w*-separable conjugate Banach space E* having no
w*-Schauder basis (see § 14, definition 14.2), but we don't know of any
example of a sequentially w*-separable conjugate space E* having no
w*-basis (and more generally, of any example of a sequentially separable
locally convex space U having no basis).
§ 14. The notion of Schauder basis for a general topological linear
space (see definition 14.2) was considered by M. G. Arsove and R. E.
Edwards [5]. The problem: in which spaces does theorem 3.1 remain
valid (i.e.: in which spaces is every basis a Schauder basis?) has been
studied earlier by several authors (see the Notes and remarks to § 3).
The idea of the proof of proposition 14.1 is similar to one of С Goff-
man and D. Waterman concerning topological linear spaces on which
the only continuous linear functional is / = 0 ([76], the proofs of
lemma 2 and corollary 1).
Example 14.1 of a w*-basis which is not a w*-Schauder basis, was
given in the paper [234] (see [234], § 1). The basis of I1 occuring in
example 14.1 was considered previously by B. R. Gelbaum ([69], p. 188,
example 3) for a slightly different purpose. The equivalence of the
notions of w*-Schauder basis and 6w*-Schauder basis was established
by J. R. Retherford ([204], theorem 5.2). Theorem 14.1 and corollary
14.1 were proved in the paper [234] (see [234], theorem 3 and corol-
corollary 3).
The Krein-Milman-Rutman property (see definition 14.3) for Bo-
spaces was defined by С Bessaga and A. Pelczynski (see the Notes and
remarks to § 10) and for general topological linear spaces it was defined
in the paper [240], where example 14.2 of a conjugate space with a
n>*-basis, which does not possess the Krein-Milman-Rutman property
in the w*-topology was also given (see [240], § 2, section 3).
§ 15. The notion of (e)-Schauder basis of a topological linear space
(see definition 15.2) and proposition 15.1a), b) were given by Ch. W.
McArthur and J. R. Retherford [161].
Notes and remarks
211
§ 16. Although the study of bases in general normed linear spaces
might present some interest, we don't know of any bibliographical re-
reference concerning this subject.
§17. Part of theorems 17.1 and 17.2 (namely, the general forms
A7.4), A7.10) of continuous linear mappings m:?->F and v:F-*E,
where E has a basis {*„} and the equalities A7.5), A7.11) for ||и||, \\v\\
in the case when {xn} is monotone) were given by I. A. Ezrohi [56].
For the particular case when F = K (the field of scalars), mentioned
in remark 17.1, see the Notes and remarks to § 12, theorem 12.5 a). See
also W. Ruckle [216].
Corollaries 17.1 -17.6 are well known (see e. g., I. M. Gelfand [73]
for corollaries 17.1 and 17.2) and so are theorem 17.3 and corollary 17.7
(see e. g., L. A. Liusternik and V. I. Sobolev [152]).
Remark 17.6 was made in [245], p. 455.
§18. In the particular cases when a = y and a=A, theorem 18.1
has been proved by B. R. Gelbaum and J. Gil de Lamadrid ([72], theo-
theorems 1 and 2') and in the general case when a is an arbitrary uniform
crossnorm on E®F, by J. Gil de Lamadrid [75]. For results of the
type of corollaries 18.1 —18.4 see L. A. Gurevich [101] and Z. Semadeni
[227]. Corollary 18.5 is due to B. R. Gelbaum and J. Gil de Lamadrid [72].
Problem 18.1 was raised by B. R. Gelbaum and J. Gil de Lamadrid
[72].
§ 19. T-norms, K-norms and TK-norms with respect to a basis were
introduced by V. N. Nikolskii [179]; here T stands for "Cebysev" and
К for the Russian word for "canonical".
Proposition 19.1c) and the necessity parts of proposition 19.1a)
and b) were proved by V. N. Nikolskii ([180], §4 and §5); the suf-
sufficiency parts are due to J. R. Retherford and R. С James ([210], p. 111).
Example 19.1 is due to J. R. Retherford and R. С James ([210],
example C.2)).
Example 19.2 is a slightly changed version of an example due to
J. R. Retherford and R. С James ([210], example B.5)), namely, in their
example the second sup of formula A9.10) was taken over the set
Л(ЛЛ{1}) of all permutations of .4/\{l} = {2,3,...}; however, their
proof was based on the inequalities
1
«2"
(аеП(.А\{Ц))
тФп тФп
which are not correct (take e. g., n = 2 and take the permutation
<теЛ(Ж\{1}) defined by ffB)=100, a(n) = n-l for 3<n<100 and
<r(ri) = n for n ^ 101).

212
I. The Basis Problem. Some Properties of Bases in Banach Spaces
Theorem 19.1, with the norm A9.16) in part a), was proved by
V. N. Nikolskii ([180], § 4 and § 5). The norm A9.17) was given in [247];
in the particular case when E = c0 and {xn} = the unit vector basis of E,
this T-norm was also found, independently, by R. O. Davies [36]. For
a related T-norm, which is also strictly convex, see V. Istratescu [112].
The proof of theorem 19.1 presented here, based on the sufficiency parts
of proposition 19.1a) b), is somewhat simpler than the original proof of
V. N. Nikolskii; a similar proof has been also found by J. R. Retherford.
Remark 19.1 and corollary 19.1 are due to V. N. Nikolskii ([180],
§ 6 and § 7). Remark 19.2 was given in [243], theorem 5.4.
§ 20. Polynomial bases occur e. g., in the Krein-Milman-Rutman
theorem (§ 10, corollary 10.2) and strict polynomial bases already inthe
classical Gram-Schmidt "orthogonalization" procedure of finitely line-
linearly independent complete sequences in separable Hilbert spaces. The
term "polynomial basis" was used in [61] and the term "strict poly-
polynomial basis" was introduced in [243], definition 5.8.
The equivalence 1°<=>11° of theorem 20.1 was proved by V. N.
Nikolskii ([180], § 7) and that paper also contains, implicitly, the equiv-
equivalence l°<s>2°. The equivalences 3°<s>ll° and 9°о10°<=>1Г are also
due, essentially, to V. N. Nikolskii ([181], theorems 2, 3 and 4). The
implications 5°=>1Г and 6°=>1Г were observed in [243], proposi-
proposition 5.4.
Problem 20.1, for the implication Г=>11" of theorem 20.1, was
raised by V. N. Nikolskii ([180], § 7), with the comment that the answer
is probably negative.
The equivalence l°<s>2° of proposition 20.1 was observed by V. N.
Nikolskii [182]. The part of example 20.1 concerning property d,), is
a simplified version of an example due to V. N. Nikolskii [182].
As we mentioned in § 20, S. M. Lozinsky and F. I. Harsiladze proved
(see [174], Appendix 3) that the sequence B0.23) in E = C([0,1]) satis-
satisfies B0.24) and therefore it is а Л system. The term "Л system" (or
"Lozinsky-Harsiladze system") was introduced by M. I. Kadec [125],
who also proved in [125] that in the spaces Lp([0,1]), where 1 <рф2,
there exist Л systems. The problem, whether the non-existence of
Л systems characterizes spaces isomorphic to I2 among separable Banach
spaces, was raised in [247] and was studied in [40].
Sub-Г systems, sub-Л systems and Г systems were introduced in
[40]. Lemma 20.1, theorem 20.2, corollaries 20.1, 20.2 and proposi-
proposition 20.2 were proved in [40].
Corollary 20.3 was communicated to us, as a remark to the results
of [40], by S. B. Steckin.
Chapter II
Special Classes of Bases in Banach Spaces
In the present chapter we shall study various particular classes1 of
bases in Banach spaces.
Along with every new class of bases in Banach spaces which we
introduce, it is natural to consider also the set of all bases which do
not belong to this class; for the sake of brevity, we shall denominate
the bases of this set by the prefix "non-" followed by the name of the
original class, e. g. the bases which are not monotone, will be called
non-monotone bases (we shall make only one exception, namely, the
non-unconditional bases will be called conditional bases).
One of the main problems which we shall consider for each special
class of bases, is the existence problem, i. e., the following problem: does
there exist in every separable Banach space a basis belonging to the
respective class? In finite dimensional Banach spaces the solutions of
these existence problems are known and, with a few exceptions (e. g.
monotone bases or normal bases) they are obvious; therefore, in most
cases, we shall not mention them separately. In infinite dimensional
Banach spaces the answer to the corresponding existence problems is
either negative or unknown (an affirmative answer would also imply
an affirmative answer to the basis problem). In the first case, we shall
give an example of a separable Banach space which has no basis belong-
belonging to the respective class. In the second case, there arises the more
restricted problem of the existence of bases of that class in infinite
dimensional Banach spaces with bases. If this latter restricted problem
has an affirmative answer, then the original existence problem is ob-
obviously equivalent to the basis problem and we shall make no mention
of it; moreover, if the answer to the restricted existence problem is
trivial, we shall not mention it either (however, in some cases the solution
of the restricted existence problem may be difficult, see e. g. § 23, theo-
We use here the word "class" in the sense: family, collection (not in the sense:
equivalence class).

214
II. Special Classes of Bases in Banach Spaces
rem 23.2). Finally1, if the restricted existence problem is also unsolved,
we shall state it as a separate problem.
We shall also study other problems concerning the special classes
of bases, such as characterizations of bases belonging to these classes,
duality properties, etc.
Due to the importance of unconditional bases, we have divided the
present Chapter into the following two parts: I) Classes of bases not
involving unconditional convergence. II) Unconditional bases and some
classes of unconditional bases.
Finally, let us mention that one can obtain other classes of bases
by taking intersections of the classes considered in Ch. II, in the case
when these intersections are non-void (e. g., normal monotone bases,
shrinking unconditional bases, etc.). With a few exceptions, we leave
to the reader the study of the corresponding problems (e. g., existence
problems) for these classes of bases.
I. Classes of Bases not Involving Unconditional
Convergence
§ 1. Monotone and strictly monotone bases
Definition 1.1. A basis {xn} of a Banach space E is said to be mono-
monotone, if we have
5>.
!«,*,
A.1)
for all finite sequences of scalars txl,...,txn + meK. The basis {xn} is said
to be strictly monotone, if we have
A.2)
for all finite sequences a!,...,an+meК with ? |а;|/0.
i = n + 1
For instance, the natural basis of lp (p^l) is strictly monotone,
while the natural basis of c0 is monotone, but not strictly monotone.
1 Obviously, under the above hypotheses the restricted existence problem
cannot have a negative answer, since otherwise the original existence problem
would also have a negative answer and thus the answer to the original problem
would be known.
1. Monotone and strictly monotone bases
215
From Ch. I, § 7, theorem 7.1 \ it follows that a basis {xn} of a Banach
space E is monotone if and only if it is of norm v{Xn}= sup \\sn\\ = 1.
Hence: a) The Schauder basis of C([0,1]) is monotone (but not strictly),
n
because sn(x) = ? Л(х)хг is a polygonal function which interpolates
; = o
xeC([0,l]) in the points 0,al,..,an^ul of [0,1] (see Ch. I, §2);
b) The Haar basis of Lp([0,1]) (p^ 1) is monotone, by Ch. I, § 2, for-
formula B.17) (actually, it is strictly monotone for p>\, but not so for
p=\). In Vol. II we shall also see that every space C(Q) (Q compact
metric) has a monotone basis.
The notions of monotone- basis and strictly monotone basis admit
a geometrical interpretation. In fact, let us recall that an xeE is said
to be orthogonal (strictly orthogonal) to a yeE, and we write x ly
(x 1 ly), if we have ||x + aj|| ^\\x\\ for all scalars a, or, equivalently,
dist(x,[>]) = dist(x,0) (respectively, Цх+а^Ц >|М| for all а#0). А
subspace F of E is said to be orthogonal (strictly orthogonal) to a sub-
space G of E, and we write F 1G(F 1 1G), if we have x ly for all
xeF, yeG (respectively, x 1 ly for all xeF, yeG, j>#0). Hence, а
basis {xn} of E is monotone (strictly monotone) if and only if we have
\_xl,...,xn'\ l[xn+l,...,xn+m~\ for all positive integers n,m (respectively,
[Х1,...,х„]1 l[xn + l,...,xn+m~\ for all positive integers n,m).
Let us now consider the particular case when ? is a Hilbert space H.
We recall
Lemma 1.1. Let H be a Hilbert space and let x,yeH, уфО. The
following statements are equivalent:
Г. x ly.
2°. x 1 ly.
3°. (x,y) = 02.
Proof. Г^ЗС. If (х,у)фО, then for a= -
(x,y)
we have
,у)
iy,y)
whence x non ly.
By means of Ch. I, § 7, theorem 7.1, we have at once several characterizations
of monotone bases.
2 We denote by (,) the scalar product in the Hilbert space H.

216
II. Special Classes of Bases in Banach Spaces
3°=>2°. If (x,y) = O, then for every scalar j#0 we have
\\x + ay\\2 = (x + ay,x + ay)= \\x\\2 + \a\2\\y\\2>\\x\\2,
whence x _L Ly.
Finally, 2° => 1 ° is obvious, which completes the proof.
An immediate consequence of lemma 1.1 and of the preceding remark
is the following:
Proposition 1.1. Let H be a Hilbert space with a basis {х„}. The
following statements are equivalent:
1°. The basis {xn} is monotone.
2°. The basis {xn} is strictly monotone.
3°. The basis {xn} is orthogonal in the usual Hilbert space sense, i. e.
(Xj,Xj) = 0 for all /#7B,7=1,2,...).
Let us turn now our attention to existence problems. We shall first
show that the answer to the problem of existence of monotone and
strictly monotone bases in finite dimensional Banach spaces is negative,
by constructing a 3-dimensional Banach space which has no monotone
(and hence no strictly monotone) basis. Let us recall
Lemma 1.2. Let E be a Banach space and G a closed linear subspace
of E. If there exists a projection и of E onto G with \\u\\ = 1, then for
every xeG there exists a "maximal functional" fxeE* (i.e. an fxeE*
with J|/J| = 1, fx{x)=\\x\\) contained in the subspace
(KeruI={feE*\f(y) = O for all ye^evu} A.3)
of E, where Kerw= {yeE\u(y) = 0}.
Proof. Let xeG. Then for all jeKerw we have
ЦхЦ = \\u(x+y)\\ <||x+j||,
whence dist(x,Kerw)=||x||. Consequently, there exists an fxeE* such
that ||/J = 1, fx(x)= ||x|| and f(y) = 0 for all >>еКеги, which com-
completes the proof.
Actually, we shall apply this lemma only in the particular case when
?=a real Banach space, dim? = 3, dimG = 2. In this case, for any
projection и of E onto G we have dimKerw=l, whence dim(Ker u)L = 2,
and a maximal functional fx (for ||x|| = 1) is nothing else but the normal
vector to the support plane {yeE\fx(y)=\} at x of the unit cell
SE={yeE\ ||j||<1}. Thus, if there exists a projection и of E onto G
with ||и|| = 1, then by lemma 1.2, the following condition must be
satisfied:
1. Monotone and strictly monotone bases
217
Condition (A): Through each point x with ||x|| = l of the central
section GnSE of the centrally symmetric convex body SE one can con-
construct a support plane Hx of SE in such a way that the normals to all these
planes Hx be coplanar.
Now we can prove
Theorem 1.1. Let D be a regular dodecahedron. Then one can cut the
vertices of D by planes, so as to obtain a centrally symmetric convex
body S with the following property: the 3-dimensional real Banach space E
in which the unit cell {xeE \ \\x\\ < 1} is the convex body S, has no mon-
monotone (and hence no strictly monotone) basis.
Proof. The regular dodecahedron D has six pairs of faces parallel
two by two and the normals to any three of these pairs are non coplanar.
Now, let us consider the central sections GnD of D, where G is any
plane through the center of symmetry. They have, clearly, the following
properties:
a) If a central section GnD of D passes through a pair of parallel
edges of D, then this section satisfies condition (A) (see figure 1.1).
Fig. 1.1
b) If a central section does not pass through any pair of parallel
edges of D, then this section intersects the interiors of at least six faces
of D, whence it does not satisfy condition (A).
c) If a central section intersects the interiors of two adjacent edges
of D, then it intersects the interiors of at least eight faces of D. Indeed,
assume e.g. that the section intersects the interiors of the edges a, b of
the face P; then it also intersects the interiors of the edges a', b' symmetric
to a, b. Now, going along this section from a towards b', we must intersect
the interior of the faces Q and R (see figure 1.1), because a is an edge of
Q and b' is an edge of R. Since Q and R have no common edge, we must

218
II. Special Classes of Bases in Banach Spaces
intersect the interior of at least one more face, and thus the section
intersects the interior of at least eight faces of D.
Let us cut the vertices of D by planes H so that the following con-
conditions be satisfied:
(i) The polyhedron S which we obtain remains centrally symmetric.
(ii) Each plane H cuts only one vertex, but in such a way that all new
faces which we obtain (i. e. all the intersections of these cutting planes H
with D) are disjoint two by two.
(iii) The normals to any three pairs of symmetric faces of S are non-
coplanar.
(iv) No plane G passing through the center of symmetry can contain
a pair of non-symmetric edges of S.
We shall now show that no central section GnS of S (G being any
plane through the center of symmetry) satisfies condition (A). We have to
consider three cases:
a) If GnS does not pass through any pair of parallel edges of S,
then, by (ii), GnD does not pass through any pair of parallel edges of D,
whence, as we have observed in b), Gr\D intersects the interiors of at
least six faces of D and thus the polygon which bounds GnD has at
least six sides. However, from (ii) it follows that for any plane G' through
the center of symmetry, the number of the sides of the polygon which
bounds G' nS is ^ than the number of the sides of the polygon which
bounds GnD. Thus the polygon which bounds GnS has at least
six sides and, by our hypothesis on G, none of these sides coincides
with an edge of S. Hence GnS intersects the interiors of at least six
faces of S and thus, by (i) and (iiiI, it does not satisfy condition (A).
p) If GnS passes through a pair of parallel edges of S which are
contained in edges of D, then, by (ii) GnS intersects, besides the inte-
interiors of two pairs of faces of D, the interiors of two pairs of new faces
of S. Hence, by (i) and (iii), the section GnS does not satisfy condi-
condition (A).
y) If G nS passes through a pair of parallel edges of S which are not
contained in edges of D, then, since any such edge of S intersects the
interiors of two adjacent edges of D, from c) it follows that the polygon
which bounds GnD has at least eight sides. Hence, as we have observed
in a) above, the polygon which bounds GnS has at least eight sides
and, by our hypothesis on G, one pair of these sides coincides with a
pair of parallel edges of S. However, by (iv), no other side of the polygon
which bounds GnS can coincide with an edge of S and thus this polygon
1 We use the following fact: if a plane H' supports S at an interior point of a
face Q of S, then H' contains the whole face Q and thus the normal to H' coincides
with the normal to Q.
1. Monotone and strictly monotone bases
219
has at least six sides which are not edges of S. Hence GnS intersects
the interiors of at least six faces of S and thus, by (i) and (iii), it does
not satisfy condition (A).
Thus we have proved that no central section GnS of S (where
dimG = 2) satisfies condition (A). Consequently, by lemma 1.2, we
have ||и||>1 for all projections и of ? onto any 2-dimensional sub-
space G of E. However, if {x1; x2, хъ} would be a monotone basis of E,
3 2
then, by A.1), the partial sum operator s2'- X aixi ~* 'YJa-ixi would be
;= l
i= 1
a projection of ? onto G=[x1,x2], with ||s2|| = l. This shows that ?
has no monotone (and hence no strictly monotone) basis and thus the
proof of theorem 1.1 is complete.
Now we are also able to show that the answer to the problem of
the existence of Banach spaces ? with index Г(?) satisfying 0<Г(?)<1
is affirmative, as announced in Ch. I, § 7.
Example 1.1. For the 3-dimensional Banach space ? defined in
theorem 1.1 we have 0<Г(?1<1 П 41
In fact, since ? has a basis, we have Г(?)>0. On the other hand,
assume that Г(Е) = sup y, , = = 1, the first sup, re-
\xji inf sup \\sn\\
{xj\ l«n«3
spectively the inf being taken over all bases {xj}j= i of ?. Then for
every ?>0 there exists a basis {x^?)}?=1 of?suchthat sup Ц^г)|| < 1+e.
We may assume, without loss of generality, that ||x'?)|| = 1 (j= 1,2,3).
Then, since the unit sphere aE = {xe E | ||x|| = 1} is compact, each of the
sets {x(j?)}e>0, {x2E)}c>0, {xC?)}?>0 admits at least one limit point,
say xx, x2 and x3 respectively. The corresponding partial sum
operators sn:
а(х,- («=1,2,3) are then of norm ||,sj| = l,
il il
whence {*,-}?= i is a monotone basis of E, which contradicts theorem 1.1.
Consequently, we have Г(Е)<1, which completes the proof of A.4).
Obviously, this argument remains valid for any finite dimensional
Banach space E having no monotone basis. <
Since for any Banach space E of dimension 3 we have1 ф2(Е) = ,
1 Indeed, for any G<? with dimG = 2 and ue2?{E,G) take a monotone
basis {x1,x2} of G and an x3eKeru\{0}; then {х,}?_! is a basis of E with
sup ||.5„||< ||«||. Conversely, taking, for any basis [хЛ?=1 of E,G=[x1,x2]
1 ^n$ 3
and u = s2, we have ue3?(E,G),
surj ||sn
Thus, inf surj |
= inf
G<E
dim G = 2
inf
Г(Е)
= ф2(Е).

220
II. Special Classes of Bases in Banach Spaces
where ф7(Е) = inf inf ||w|| is the function introduced in Ch. I,
G<E ue»(E,G)
dimG = 2
§ 11, it follows that for the space E constructed above we have
Ф2(Е)>\. A.5)
there exists an
One can also show that for every integer
n-dimensional Banach space En such that
фг(Еп)=[, ф}(Еп)>\ U =2,3,
A.6)
for instance, one can choose En to be a suitable subspace of a space 1%
with k>2Bn — 3) and рФ integer1. However, we shall prefer to give
another construction of such spaces ?„, in theorem 1.2 below.
Let us also remark that the iteration of the method of cutting the
vertices of a centrally symmetric polyhedron by suitable hyperplanes
might perhaps present some interest for the construction of an in-
increasing sequence of finite-dimensional Banach spaces having the
properties required in Ch. I, § 11, theorem 11.1, i.e. for the construction
of a separable Banach space having no basis.
In order to prove theorem 1.2 below we shall need several lemmas.
In the sequel we shall denote by E" the real «-dimensional euclidean
space /2.
Lemma 1.3. Let zx, ...,zneE" with ||z,|| = 1 (/= 1,...,«) be arbitrary.
Then for every e>0 there exist n elements VjeSfz^e) with ||j,-|| = l
[i = 1,..., ri) which are linearly independent.
Proof. We shall first show that for every g>0 there exist elements
z;,—J (i=\,...,n) which are linearly independent. Let xl,...,xn
be an arbitrary basis of E" and let
z'i(t) = txi + (l-t)zi
Since x!,..., xn is a basis, we can write
zi=T,<*ijXj, Ф)= YJbij{t)xj
j=i j=i
whence, by A.7),
A.7)
A.8)
1 See H. F. Bohnenblust [29]. We recall that
*={?,};= i endowed with the norm ||x||
is the space of all fe-tuples
1. Monotone and strictly monotone bases
221
and thus, since xl,...,xn is a basis,
Put
A.9)
A.10)
Then D(l) = det(<51J)= 1/0. Since D(t) is an algebraic polynomial
in t (and thus it has a finite number of zeros), it follows that for every
g>0 there exists a te with 0<rE < -^-7 such that
2]/n max \5ч
Then, by A.8) z\(te), ...,z'n(tE) are linearly independent and by A.9)
we have \aiJ-biJ{tE)\=\tl.5ij-tcaij\=\tc(8ij-aij)\ <——, whence
21/n
" e2
and thus z| = z,'(fe) (;'= 1,...,«) satisfy our assertions.
Now, put
A.11)
Then j!,...,jn are of norm 1 and linearly independent. Further-
?
more, by ||z;|| = 1, ||zj — z,-|| ^ — (/= 1, ...,/1) we have
'¦->'¦ II =
whence
y'
Z' l|z|||
1
i-Zf|| + || Zr
- -
e
~2
1
Nil
?
^ 2
e
f 2 =
(/=1,...,n),
(/=1,
which completes the proof of lemma 1.3.
Definition 1.2. A system of N elements zx,...,zN in a linear space is
said to be n-independent, where n^N, if every subsystem consisting of n
elements (shortly: every «-subsystem) is linearly independent.

222
II. Special Classes of Bases in Banach Spaces
Lemma 1.4. Let zu ...,zNeE" (n^N) be n-independent. Then there
exists an e = e(zl7..., zN)>0 such that every system z\,...,z'N with
z\eS(zh e) B = 1,..., N) is also N-independent.
Proof. Let Xi,...,xn be an arbitrary basis of E". Then we can write
Let zkl,...,zkn be an arbitrary subsystem consisting of n elements.
Then by our hypothesis zkl,...,zkn are linearly independent and there-
therefore
akil • ¦ ¦ ak
akn\ ¦ ¦ ¦ ак„п
#0.
Since D is a continuous function, on E, of the coordonates
,!' ак,2, ¦¦¦,акпП, there exists an e'^0 such that whenever
I Kj-4,jJ<4 we have
D' =
Ф0.
A.13)
Since all norms on E are equivalent, it follows that there exists an
e' > 0 such that whenever
max K-zJ =
we have A.13), i.e. the system z'kl,...,zkn is linearly independent1. Thus
for every «-subsystem S'={zki,...,zkr} of S={zu...,zN} we find an
e'>0. Since the number of all subsystems S' is finite, the number
e = mine'>0 will satisfy our requirements, which completes the proof
of lemma 1.4.
Lemma 1.5. Let zu...,zNeE" (n^N) with ||z,.|| = l (i=\,...,N) be
arbitrary. Then for every e>0 there exist N elements у{е8{гье) with
||j;|| = 1 (/= 1,..., N), which are n-independent.
Proof. Let us number all «-subsystems {zks,...,zkn}, denoting them
by S1 = {zl,...,zn},...,Sr. By lemma 1.3, for every e>0 there exist
n elements y^eSiz^e) with ||>>;|| = 1 (i=\,...,n) which are linearly in-
independent. Assume now that by an arbitrarily small displacement of
Actually, this also follows from Ch. I, § 10, theorem 10.3.
1. Monotone and strictly monotone bases
223
the whole system S={zl,...,zN} we have obtained that all «-sub-
«-subsystems S1,...,SP, where p<r, are of norm 1 and linearly independent.
By lemma 1.3, the «-subsystem Sp+l can be also made of norm 1 and
linearly independent, by an arbitrarily small displacement. By the
above proof of lemma 1.4, this displacement can be chosen in such a
way that the linear independence of the systems S,,..., Sp be conserved.
This completes the proof of lemma 1.5.
Lemma 1.6. Let zl,...,zNeE" (n^N) with \\zt\\ = 1 B= 1,..., N) and
?>0 be arbitrary. Then there exist N elements yu ...,yNeE" with i!>",|| = l
B= 1,...,N) and an e'>0 such that
and that every system y[,---,y'N with у[е8(уье) (i=\,...,N) is n-in-
n-independent.
Proof. By lemma 1.5, from the system zl,..., zN one can obtain, by
an —displacement, a system yu...,yN with \\yt\\ = 1 B= 1,..., N), which
is «-independent. For this system there exists, by lemma 1.4, an e with
0<e' < —, such that every system у\,---,у'ц with j|eS(j,-,e')('= 1, ...,N)
is «-independent. Furthermore, for any y'^Siy^e') (i=l,...,N) we have
\\у[ — Z;|| < llj; —>',-1| + || J,- — Z;|| <e' H— < e (i=l,..., N),
and thus we have A.14), which completes the proof of lemma 1.6.
Definition 13. A system RU...,RN of m-dimensional linear mani-
manifolds1 in E"(m<n) is said to be total, if for every linear manifold Q a E"
of dimension n — m there exists an R, such that Rtr\Q consists of at
most one element.
In the particular case when Rl,...,RN are one-dimensional linear
subspaces of E, this reduces, obviously, to the totality of any system
л:;еКД{0} (/=l,...,N) in E" in the usual sense (Ch. I, §5), or, equiva-
lently, to ]Г
Definition 1.4. If Gl ф {0}, G2# {0} are closed linear subspaces of a
normed linear space E, the number
0(G1,G2) =
sup dist(x,G2), sup dist{y,Gl
6 G
yeG2
1Ы1=1
A.15)
1 I.e. translated linear subspaces.

224
II. Special Classes of Bases in Banach Spaces
is called the opening of the subspaces GUG2. This is not a metric in
the set У{Е) of all closed linear subspaces G/ {0} of E (since in general
it does not satisfy the triangle inequality), but it is well known that there
also exists a metric §(GU G2) on 5^(?) such that
A.16)
A-17)
e.g. one can take
0(G1,G2) = max (sup dist(x,<rG2), sup dist{y,aGl)),
where aG={xeG ||x|| = l}. We leave to the reader the easy verifica-
verification that this 6 satisfies the triangle inequality and A.16).
We shall use in the sequel the opening 6(GUG2). We shall denote
for a linear subspace Gocz E and an e > 0,
V(G0,e)={G linear subspace of E \ 0(G,Go)<s}. A.18)
Lemma 1.7. Let Gl,...,Gn{n_-l)<-
n(n-l)
E" be an arbitrary system
of {n — 2y dimensional linear subspaces of E". Then for every e>0 there
exist (n — 2)-dimensional linear subspaces G\, ...,G'Mn_1} of E" with
which form a total system.
Proof. Choose in each subspace G, a basis у\,--.,у'„-2
n{n-\f
Let x1,...,xn be a basis of E" and let
n
I'fc — Zj akjxj
n(n-\)
2=1,...,
A.19)
n(n-l)
Consider the system of homogeneous linear equations
11
"л-2,1 • ¦ • "n-2,n
t>n ¦¦¦bitt
b2l ...b2n
= 0 2=1,...,
n{n-\)
A.20)
1. Monotone and strictly monotone bases
225
«(«-1)
with respect to unknowns
2q
n). Here the
coefficients are all subdeterminants1 of order n — 2 of the nx(n — 2)-
matrices
/=1,..., -*—Jl A.21)
n-2,l ¦ ¦
J,)
an-2,n
9
taken with their corresponding signs from the developments of the
nxn determinants A.20). We claim that for every e'>0 there exist
numbers a'ff with \a'$ — (Щ\<е' I k= 1,...,« — 2; j=\,...,«; /=1,...,
I such that the determinant of the system of equations obtained
from A.20) by replacing the сЩ by the a'ff, i.e. of the system of equations
l1 • • • "in
= 0
b2l...b2n
be #0. In fact, for /=1 consider instead of A.21) the matrix
h(n-\)
A.22)
A-23)
For r=l this matrix becomes (8kJ)k=l „_2, i-e. all minors of
j=l....'.n
order n — 2 are =0, except the first one, which is =1. Similarly, for
each г with 1 < г < one can replace the matrix A.21) by a matrix
of the form
where (Рк1])к=1 „_2 is obtained from {8kj)k=1 „_2 by a suitable
У = 1,..., n j = 1 n
permutation of the columns, in such a way that for t= 1 all coefficients
*i *i
p * in the corresponding г-th equation (ob-
u К
of the unknowns
tained from the г-th equation of A.20) by replacing the dk] by the
1 For each fixed i, these subdeterminants are nothing else but the Pliicker-
Grassman coordinates of the subspace Gt.
15 Singer, Bases in Banach Spaces I

226
II. Special Classes of Bases in Banach Spaces
become 0, except the ( — / )-th one, which becomes
± 1.
Now, let D(t) be the determinant of the system of equations ob-
obtained from A.20) by replacing the сЩ by the numbers d${t). Then
D(t) is an algebraic polynomial in t and D(l)=±l/0, whence there
exist arbitrarily small numbers г>0 such that D(f)#0. Since г>0
can be arbitrarily small, and since
\akj V) akj I — ' IP/tj akj I >
it follows that the numbers a'^(t) can be taken e'-near to d?] and so
that D(r)/0. This proves our assertion, taking dff to be these a'${t).
Now, since the determinant of the system of equations A.22) is #0,
it follows that this system has only the trivial solution
lq
b2p
= 0
A.25)
However, this implies that the system of subspaces G[ = \y'\l\ ..., y'i'l2]
, where
У к - L akj xj
n(n-l)
A.26)
is total. Indeed, assume the contrary, i. e. that there exists a linear manifold
QcE" of dimension 2 such that QnG't consists of more than one
element for each г. Translating this Q into the origin we obtain a linear
subspace Fc?" of dimension 2, such that
Now, let
Zl = Z
2 = 1,...
Z2 = Z
A.27)
A.28)
be an arbitrary basis of F. Then, since by A.27) there exist elements
n-2 2 n-2
x';eF nG'i\{0}, we can write x\ = Z ск]Ук = Z ^izi' W'tn
2
Zl
11 / n(n—1)\
dependent I г = 1,..., whence we have A.22). Since by the above
the system A.22) has only the trivial solution A.25), it follows that zx, z2
k=i 1=1 *=i
), and therefore the elements y'[,..¦, y1'-2, zn zi are linearly
1. Monotone and strictly monotone bases
227
are linearly dependent, in contradiction with the assumption that they
are a basis of F. This proves our assertion that the system G\,..., G'n(n_ 1(
is total. 2~
Now, by the definition A.15) of the opening and by the compactness
of spheres in finite dimensional spaces we have, for suitable elements
"
k=1
n(n-\)
\ with ||хЬ|| = ||у01| = 1 (/=1,...,
k=1
sup
,GJ), sup dist(y,G,
= max (dist (x'o, G[), dist (y'o, G,))
n-2
Z «i0)j
k= 1
n-2
Z K0)l
n-2
¦i- Z'
k= 1
A-y'kW,
Су
n-2
z
i
к
lA0)
n-2
Z /M
k=l
\
iw-^ii)
n-2
- z
t= 1
A.29)
Since by the above a'^f can be taken arbitrarily near to <$], from
A.19) it follows that ^' can be taken arbitrarily near to y'k. Furthermore,
since ||xo|| = ||_УоII = 1 and since all norms on E"~2 are equivalent, there
n-2
n-2
exists a constant М„_2 such that
°i0)l
> Z \PiO)\^Mn-2- These
k1 k1
facts, together with A.29), imply that G- can be taken arbitrarily 0-near
to G; for each г=1,..., , which completes the proof of lemma 1.7.
Lemma 1.8. Let Gl5..., Gn(n_1( be a total
«(«-1)
system of (« — 2)-
2
dimensional linear subspaces of E". Then there exists an e
= ? (Gl,...,G,_l\> 0 such that every system of\n —7)-dimensional linear
^ " I ( n(n-l)\
subspaces G\,..., G^(n_1( with G\eV{Gbe) 1г = 1,..., I is also
total. ~^T- \ 2 )
(
Proof. Choose у\,---,у'„-2 l'=l, •••,
n(n-l)
and xl,...,xn,
as in the proof of the
preceding lemma. We claim that since Gu..., Gn(n_u is total, the
2
determinant D of the system of equations A.20) is #0. In fact, assume

228
II. Special Classes of Bases in Banach Spaces
the contrary, i.e. that D = 0. Then the system A.20) has a non-trivial
i, b.
solution
lq
2q
n), whence the elements zx, z2eE"
defined by A.28) are linearly independent and the elements у\,...,у'„_2,
are linearly dependent (/=1,...,
. Consequently, for
the 2-dimensional linear subspace F=[z1,z2~] of E" we have then
FnG,.#{0} (г=1,..., ), which contradicts the assumption that
is total. Therefore D#0.
2
Observe now that if the subspaces Gh G\ are arbitrarily 0-near, then,
assuming also ||j?|| = l (fc= 1,...,« —2), one can choose in G\ a basis
П
y'1 = Y, a'kJXj (k= 1,..., n-2) such that the numbers a'$ are arbitrarily
near to the 4'j- Indeed, if 0(G;,G;)<e, then, by A.15), one can find in G[
elements z'(,..., z''_2 such that
G;)< sup dist(y,G;)<0(Gi,G;)<e (k=\,...,n-2),
and then, by lemma 1.3, one can find a basis y'\,...,y'n-i of G\ such
n
that y'l= X d$xjeS{zi,e){k=\,...,n-2), whence
\\A-4\\
114'-
and therefore (since all norms on E" are equivalent) the numbers d^f
can be assumed arbitrarily near to the d?].
Since the determinant D of the system of equations A.20) is #0 and
since D is a continuous function of the dp., it follows that for a suffi-
/ n(«
ciently small e>0 and for 0(G;,G;)<? |г=1,...,—-
, one can
choose a basis y'l= Y,a'k)xj (k=l,...,n-2) of G; I/= 1,...,——
j=i ' \
such that the determinant D' of the corresponding system of equations
A.22) be still /0. Then, as we have seen in the proof of the preceding
lemma, the system G'l,...,G'n(n_l) is total, which completes the proof
of lemma 1.8. 2
n(n-\)
Lemma 1.9. Let Gu ..., Gn(n_t) с En be an arbitrary — system
2 Z
of (n-2)-dimensional linear subspaces of E" and let e>0 be arbitrary.
1. Monotone and strictly monotone bases
229
n(n-\)
Then there exist — linear subspaces G\,..., G;(n_1( of E" and an
e'>0 such that 2
K(G,,e)
A.30)
and that every system of linear subspaces G[,..., G'n(n_ u with G[eV{G'he')
f. , n(n-l)\ .
i=l,..., ) is total.
Proof. By lemma 1.7, from the system Gl5..., Gn(n_ 1( one can obtain,
by an —-displacement (in the sense of в), a system G[,..., G'n(n_l} of
(n — 2)-dimensional subspaces, which is total. For this system there
exists, by lemma 1.8, an e with 0<e'<—, such that every system
Gj,..., G'n(n_1} of (n — 2)-dimensional subspaces with G\eV(G\,e')
г = 1,...,
is total. Furthermore, for any G;eK(G;,e') we have,
by A.16) and since в satisfies the triangle inequality,
0(G|, Gf) < 8(G'h Gf) < 0{G[, G\) + 0(G;, G.)
+?)<e'
and thus we have A.30), which completes the proof of lemma 1.9.
Lemma 1.10. Let Gu G2 be two linear subspaces of E" and let Qu Q2
be their orthogonal complements respectively. Then
0Fi>Q2) = 0(Gi, G2). A.31)
Proof. By a well known corollary of the Hahn-Banach theorem, we
have
dist(x, G,) = max \f(x)\ (xeEn, i= 1,2),
and thus, since in this formula the roles of G, and g; can be also inter-
interchanged,
0(G1,G2)= max (l/M
which completes the proof.

230
II. Special Classes of Bases in Banach Spaces
For a system of elements yx, ...,ykeE" we shall denote by 9>(yx, ¦.., yk)
the orthogonal complement in E" of the subspace [yv ..., jk].
Lemma 1.11. Let z,,..., zNeEn(N^n(n-\)) with ||Z;|| = 1 (/= 1,...,N)
a«J e>0 йе arbitrary. Then there exist N elements yieS{zhe) with
||j;|| = l (/=1,...,N), лис/г г/шГ
1°. The system Уи---,Уц is n-independent.
2°. Every system of subspaces (of dimension n — 2)
3)(уР1,Уц,),---,3!(Ур_к1^Л1,Уч^Л, A-32)
V г ""г у1
where Pi,---,pn(n-i), <7i, ¦••> <7n<n-i) are distinct indices taken from
2 2
{1,..., N}, is total.
Proof. By lemma 1.6 we may assume, without restriction of the
generality, that the system zu...,zN is «-independent and that for the
given g>0 every system y[,...,y'N with ^|eS(z;,e) (i=\,...,N) is «-in-
«-independent.
Let p{i\---,p(ril-n, cfi\--->cfn\)n-i) be the first choice of «(«—1) dis-
tinct indices taken from {1, ...,N}. Then for the
(n — 2)-dimensional subspaces
(" _
system of
A.33)
(which need not be total) there exist, by lemma 1.9, a total system of
(n — 2)-dimensional subspaces ®l5..., ®и(„„и and an e'>0, such that
«(«-1)
A.34)
and that every system of subspaces choosen from V(&i, e)
n(n-\)\
г=1,...,
is also total.
Now let Qi be the B-dimensional) orthogonal complement in E" of
/ nin-\)\
@i г=1,...,— . Then, by lemma 1.10 and A.34),
e
~2'
whence there exist elements z^
zJ(i)eQ,- such that
1. Monotone and strictly monotone bases
231
A.35)
Then the system z',1,,..., zbo , zbi>,...,zi(i> is «-independent,
J l>. fn[n- 1) У] Чп(п- 1) r
2 2
whence, in particular, each couple zp<", zp<d is linearly independent.
Thus u = l>J.o,4.)] f^l,...,^1^! whence
A.36)
and therefore the system of subspaces
^(Zp,,,,^,,),...,^^,,,^,,,
is total and every system of subspaces chosen from
A.37)
г=1,...,— -I is also total; replacing, if necessary, zp,i,,z^i) by
1 1 ее
-j ZpU), —; zlqO) and — by — in A.34), one may assume (see
the'end of the proof of lemma 1.3) that ||zp,i,|| = ||z^i,|| = 1 1г=1,...,
\ ' ' ^
. Since by lemma 1.10 we have
it follows (see the proof of lemma 1.7, formula A.29)) that there exists
an е'[>0 such that every system of subspaces ®(zpA),ze(,,),...,
zpMki.'z«W-.,) with
j-Zp,,,|<?), р,,.,-
; г= 1,...,
\
I is total; one may assume, without loss of generality, that e\<—.
Putting z\ = z{ for all i
thus obtained a new system of elements
..,^,, we have
A.38)
/ ? \ g
with || z/1| = 1, z?eSlzi,-l (i=\,...,N) and an e', with 0<е; <-,
having the property that the system of subspaces A.37), corre-
corresponding to our first choice of distinct indices Д", ...,/>^_1),

232 II. Special Classes of Bases in Banach Spaces
q[i\ ..., </„(,}_ i), is total and that every system of subspaces Si(zp^-,,zq{i)),...,
)>%П)) with ||zp(i, —z'(i,|[ <?'b pe(i) — z'(i)|| <?[ l/=l, ...,
is also total.
Let now pf\ ...,/>i,(^-i,, <7i2>, ¦¦-, 9^-i) be a second choice of и(и— 1)
2 2
distinct indices taken from {1,...,TV}. Then, starting from the new
system A.38) and from s\, one can find, by the above arguments, a
system of elements
_2
, ZN
with
A.39)
?i
(i=l,..., N) and an ?!'2 with 0<?2 <—,
such that the system of subspaces
m ,z2<2, \ A.40)
«/и - 1 » ' Qnfn -111 V '
is total and that every system of subspaces ®{zpBbzqm),..., S> Bp$.-i>'
with
2
z;
2
z^
„ Л
<e2 h=
is
also total. By virtue of zf eS(z}, ь\), the system of subspaces corre-
corresponding to the first choice of indices
A.41)
remains total. Furthermore, we have, by e\ <—,
Hzf-ZilKllzf-z/ll + llzJ-ZiHe^+^^e (i=l,...,7V), A.42)
whence zfeS(zf,?) (/= 1,..., TV).
Continue this procedure, starting from the third choice of n(n — 1)
distinct indices taken from {1,..., TV}, the new system A.39) and z.
Since the number of all possible choices of n(n — 1) distinct indices
taken from {1,..., TV} is finite, say M, repeating the above arguments M
times, we obtain a system of elements
with Hj;|| = l (i=l, ...,TV), such that for every choice of n(n-l) dis-
distinct indices ри.-.,рп{п-ц, <7i, ••-, gn(n-i) taken from {1,...,TV} the
1. Monotone and strictly monotone bases
233
n(n-l)
system of —-— subspaces(l.32)is total and that j,eS(z;,?)(/= 1, ...,TV),
whence y^,...,yN is «-independent. This completes the proof of lemma
1.11.
Now we can prove
Theorem 1.2. For every integer и^З there exists a Banach space E
with dim E = n such that for every couple of subspaces G, F of E with
dim G^2 we have
(g7f)<1. A.43)
Consequently, no subspace Eo of E with dim Eo ^ 3 has a monotone
basis and the space En = E satisfies A.6).
Proof For S=—j consider on Fr SEn = {xeE" \ \\x\\ = 1} a E-net
zx,...,zN with N^n(n — 1). Then, by lemma 1.11, there exist TV elements
jfeS(Zi,E) with ||jj|| = l (/=1,...,TV) such that we have 1° and 2° of
lemma 1.11.
For each yeE" we shall denote by ®{y) the orthogonal comple-
complement of у in E", i.e. ®(y)= {xeEn | (x,j) = 0}, and by Ty the strip
Put
•>-={x-Hj|xe®(j),
s =
A.44)
A.45)
i= 1
Then S is a centrally symmetric convex body in E" and therefore
one may consider the n-dimensional Banach space E whose unit cell SE
is S. We shall show that this space E satisfies the requirements of theorem
1.2. The idea of the proof is the following: Using the assumptions on S,
yi,---,yN and the totality property 2° of lemma 1.11, we shall show
that for any subspace G a E with dim G ^ 2 there exist at least n
elements xh ...,xineGr\FrS = FrSG such that each xik belongs only
to one of the sets FrTyj« (k=l,...,n), with )кф)х for кф1, i.e. only
to one of the sets
pi±)yjk=±yjk + &{y.j (/c=l,...,„), A.46)
with }кф]х for кф1, where the notation ( + ) means that one has to
take only one of these two signs. From this we shall easily deduce that
the existence of a subspace FcE with (G;F)=l would contradict
then-independence of the system yi,...,yN ensured by 1° of lemma 1.11,
which will complete the proof.

234 II. Special Classes of Bases in Banach Spaces
We shall proceed in several steps. Let us prove, firstly, that we have
1 " A-47)
We shall continue to denote by \\x\\ the euclidean norm. Since each
strip Ty< contains SEn (by ||j>;|| = 1), we have S=>S?,,. To prove the
other inclusion in A.47), take an arbitrary xeS. Then, since z1,...,zN
x
is a E-net for Fr SEn, there exists a z, such that zt < S, whence,
by yieS(zhS),
x
11^-^11 <2<5.
A.48)
Since E=[yi~]@3>(yi), there is a unique decomposition x = /?j;
where xoe®(y;). Since xeS<=Tyi, we have |j3|<l. Furthermore1,
since x-/?j; = x0 1 [yf], we have, taking into account A.48),
\\хо\\ = \\х-Ру{\\= min \\х-1У1\\^\\х-}\х\\уг\\<2д\\х\\,
— 00 < t < GO
whence, by x-xo = j8j; l&(ytKx0,
Ij8|= lljSjill = l|x-xo]| =]f\\x\\2 -\
, which, since xeS has been arbitrary,
and thus ||х„ .
>\-\b2
proves the second inclusion in A.47).
By A.45), every xeFrS belongs to at least one of the sets i*±>3"
= +J; + ®(Ji)> where the notation ( + ) means that one has to take only
one of these two signs. With the aid of A.47) we shall now show that if
xux2ePyinFrS or if xux2€P~yinFrS, then
\\Xl-x2\\<68. A.49)
In fact, since F=\_yl]@3)(yl), we have unique decompositions
x1 = J81j,+x<01>, x2 = p2yi + xlo\ where х@1},x^e&iy,). Since either
x1,x2ePyi or x1,x2eP~yi, wehaveeither /?1 = /?2 = 1, ог Р\=Рг^ ~*-
Taking into account Х;-0,-;>>, = 40 ![>,], xteS, |ft| = l 0'= 1,2) and
A.47), we obtain
1 Throughout this proof, we use the symbol 1 in the sense of the euclidean norm.
1. Monotone and strictly monotone bases
235
2S
<3S (/=1,2)
1
l-4<52
- 1
5 2
)
(because ]/l-4<52 = / 1 - —-^ > / 1 - - = -). Consequently, since
16n
)
which proves A.49).
Now let G be an arbitrary subspace of E with dim G = 2. Then its
unit cell SG=GnS is a centrally symmetric convex polygon which
contains, by A.47), a euclidean circumference. Divide an arbitrary
quarter of this circumference into n3 equal parts and denote the inter-
intersection points of Fr S with the rays starting from the origin and passing
through the corresponding division points, except the last one, by
s1,...,sni. Each of these intersection points may belong to several
faces p*1^* of FrS, but we shall show that in sufficiently small neigh-
neighbourhoods of these points one can find at least n other points xh,..., xin
of FrS such that each xik belongs only to one face A.46), with jk
for кф1
In fact, we have
1 and
: sfis: = — , whence
n 2
\\st\\ sin
- > sin
2
2и3 4и:
= 2пд
Consequently, there exists an e>0 such that for every ^
and s}€S(Sj,E),i^j, we have
<*№,¦<-.
A.50)
A.51)
Consider now the first elements sl,...,sn(n_l}. We claim
that there exists an element xheFrSG, xheS{sh,E), such that xhel*±}yj
only for one value j=j1 of the index/ Indeed, assume the contrary, i.e.
that each xieS(si,s)r^FrSG (/=1,...,— -| belongs to at least two

236
II. Special Classes of Bases in Banach Spaces
distinct faces
>^, i.e. to an intersection
pi ± )yp n pi ± )yq
A.52)
Then, since for each i the set S{si,s)nFrSG is infinite and the
number of all possible intersections A-52) is finite, it follows that for
each i= 1,...,— there exist a pair of indices p^q{ and a pair of
signs such that the (и —2)-dimensional linear manifold
A.53)
contains at least two distinct elements sl,sfeS(si,B)r)FrSG. Let us
show that here all indices />l5 ...,/>„,„_ 1M qu...,qn(n-i} are distinct. If
2 2
q—qj for 1Ф], we have to consider two cases:
a) s\eP(±)y», nPy"., 4eP(±)y". nРЧ, whence s',s)eРУя<. In this
case, by A.49) we must have \\sl -s]|| <6S, and by A.50) we must have
||sj — sj\\ >6S, which is impossible.
b) slel*±)ye, nPy«., sjeP(±)yi>, глР'у"; whence s'eF'., s]eP~y«<.
In this case divide the angle < sfOsj in the plane G with the aid of a
ray 05 from the origin parallel to РУч. nG and to P~y<lr\G, into two
angles <*!= «s'Os, a2= <sO,sj. We have then, by A.47),
sin a! =
16n6
and similarly, sina2 = —— > ——, whence ol1,ol2 > —. Hence
II^j II •*-
<sj0sj=ai+a2 > —,
which contradicts A.51).
We have thus proved that q^qj for г#/ in A.53). Similarly,
we have p^Pj for 1Ф]. Finally, from the same arguments it
follows that we also have Pt^qj for /#/ and thus all indices
/>i> ¦••'/'n(n-i)> <7i> ¦••> <7n(n-i) in A-53) are distinct.
2 2
Thus, the system of linear manifolds A.53) is not total
(since the intersection of G with each of these linear manifolds contains
n(n-l)
at least two distinct elements s}, sf), whence the system of
subspaces
linear
1. Monotone and strictly monotone bases
237
v v
obtained by translating these linear manifolds into the origin, is also
not total, which contradicts property 2° of lemma 1.11. This proves our
assertion that for the first elements 5b ...,5n(n_1) above there
2 2~
exists an element x^eSiSi^EJoFrSG such that xileP(±)yj only for
one value j=j\ of the index/ , ..
By a similar argument, for the second group of elements
.?„(„_!) ....^„(„.^ there exists an element xi2eS{sh,s)nFrSG which
belongs only to one Р(±Н. Since пъ>п , continuing in this
manner we obtain at least n elements xh, ...,xineFrSG, xikeS{sik,e)
{k=l,...,n), such that each xik belongs only to one P{± )y'« (k = 1,..., и).
Here all indices ju...,jn are distinct. Indeed, if jk=j, for кф1, then
xik,xheP(±)yj*, which, as we have seen above, is impossible by A.47)
-A.51).
This being established, the conclusion of the theorem follows easily.
Indeed, assume a contrario that there exists a subspace F a E with
(G;F)=1 and let xeF\{0} be arbitrary. Then, denoting by |||z||| the
norm in E, we have
|||j-Kx||| ^ 1 (jeG, |||j|||= 1, — oo<?<x>),
whence, in particular, for y = xik we obtain
Ill^ + ^lll^1 (-cc<f<oo), A-54)
and thus each line Lk = xik + {tx\ —co<t<co} (fe=l,...,n) supports
the cell S. Since by the above each xik belongs only to one face P( ± >>jk of
S, it follows that there is also a segment xik-\- {tx\ — tk^t^tk} of L
which belongs to that face P(±h4. Hence, taking into account that by
the definition of P(±)L we have (yjk,y) = 0 for all yePl±)y\, and con-
considering any t^=0 with —tk^t^tk, we get
A.55)
which, since all jk are distinct, contradicts the «-independence of the
system y1,.-.,yN, ensured by 1° of lemma 1.11. This proves that for
every couple of subspaces G,FcE with dimG = 2, whence also for
any G,F<=E with dimG^2, we have A.43).

238
II. Special Classes of Bases in Banach Spaces
Assume now that a subspace ?0 of ? with dim?0 = m^3 has a
monotone basis xi,...,xm. Then for G=[x1,x2], ?=[*з] we have
(G;F)=l, which contradicts A.43). Finally, the fact that the space
En = E satisfies A.6) follows from dim?<oo with a similar argument
to that used in example 1.1. This completes the proof of theorem 1.2.
We observe that the above proof of theorem 1.2 is, essentially, an
extension of the previously given proof of theorem 1.1. In fact, for
n = 3, in that proof of theorem 1.1 we have shown that if G is any plane
through the origin, GoS intersects the interiors of at least 6 faces of S,
or, equivalently, the interiors of at least 3 = n pairwise non-symmetric
faces of S (existence of xtl,xi2,xi3l) whence, by condition (iii) (which
corresponds to the 3-independence of yi,---,yN, since these yt are the
normals to the faces Pi±)yi of S), it has followed the non-existence of a
projection и of E onto G with ||и|| = 1 (which, since dim?=3 and
dimG = 2, is equivalent_to the non-existence of a subspace Fc? with
dimf= 1, such that (G; F) = 1). There has only been a difference in the
construction of S in the two proofs (condition (iv) versus the totality
of ®(ypi,yqi), @(yP2,y42), &iyP3>yqi))-
Since for any n-dimensional Banach space E and any k= 1,..., n — 1
we have, obviously,
1
= inf
г/„ -- -r М\> inf inf \\и\\=фк{Е), A.56)
I (t) (xj) lsSisSn G<E ue&(E,G)
dim G = ft
it follows that for n-dimensional space ? constructed in theorem 1
above we have
0<Г(?)<1. A.57)
This shows that for the numbers Г{п) defined by
Г(п) = inf Г(Вп
A.58)
where the inf is taken over all Banach spaces Bn of dimension n, we
have F(n)< 1 (и = 3,4,...). It is natural to ask how large is the number
sup Г(п)= sup inf sup y,x... If this number would be 1, then for
3 sS n < go 3 sS n < oo В„ [xj] = Bn '
each e > 0 every Banach space В of a sufficiently large finite dimension
n = n(e) would have a basis {xj} such that y{Xji>i —e. However, we shall
show that this is not the case; on the contrary, we have sup Г(п)< 1,
З^П < GO
i.e. there is a constant С< 1, independent of n (namely, С = sup Г(п%
З^П< GO
such that for every positive integer n there exists a Banach space Bn of
dimension n, with the property that for every basis {x;}"=1 of Bn we
1. Monotone and strictly monotone bases
239
have y{ )<C. This fact might perhaps also be of some interest for the
problem of constructing a separable Banach space having no basis.
For an arbitrary n-dimensional Banach space ? put
U(E)=Un.i(E)= sup
A.59)
where the sup is taken over all bases {*,}"= i of ?. We have then, for
every и-dimensional Banach space ?,
inf
A-60)
{xj}
On the other hand, since for any complementary subspaces G,F of
? we have (G;F) = , where ue0>(E,G), m(F) = O,' we can write
{/(?)= sup sup(G;[x]) =
dim G = n — 1
1
1
inf
G-<?
dim G = n — 1
inf ||m||
A.61)
and consequently, for the n-dimensional Banach space ?„ = ? con-
constructed in theorem 1.2, we have
U(En)< 1.
Define now the number U(n) by
U(n) = mW(Bn),
Bn
A.62)
A.63)
where the inf is taken over all Banach spaces В of dimension n. Then,
by A.60) and A.62), we have
Г(п)<Щп)<1 (n = 3,4,...) A.64)
and we shall prove below that sup U(n) = C<l, which, of course,
3*П< GO
will also prove the desired result on sup Дп).
3 ^ П < GO
Lemma 1.12. Let ВЩ,В„2 be two finite dimensional Banach spaces
and let Bni x Bni be their cartesian product endowed with the norm
Il{z1,z2}||=]/||z1||2 + ||z2||2. Then we have
I). A-65)
1 Indeed, (G;F)= inf
yeG
zsF
sup
yet
z eF

240
II. Special Classes of Bases in Banach Spaces
Proof. Let G be an arbitrary hyperplane in Bni x Bn2 and let
z={z1,z2}eBnixBn2. Put
= {yeBn2\{O,y}eG}. A.66)
Then, since the image of G, by the canonical isometrical embedding
Bnt^BnixBn2 is Gn(Bnix{0}), respectively Gn({0} х В„2), the co-
dimension of G; in Bn. is1 at most 1, whence, by the definition of U(Bn),
A.67)
Now, since dimG,, dim[z,]< oo, the value dist (aGi, [z;]) = (G,-; [zf])
is attained2 for a suitable pair у{еав., а,г;е[г,], and thus
A = 1,2),
1
whence forx,= Ji^G; (г=1,2) we obtain
a,-
\\xt\\ A=1,2). A.68)
Consequently, for x={xux2}eBni x Bn2 we have
whence
;;O]) = dist(<TG, [z]):
<max(I/(BB1),l/(BJ),
whence, since the hyperplane G in ВП1 х ВП2 and the element zeBni x Bn2
were arbitrary, we infer
U{BaixBai) = sup sup (G;[z])<max(l/(Bni),l/(Bn2)),
G<B S В B
G<Bnl
codim G = 1
which completes the proof of lemma 1.12.
1 We use here the following obvious remark: if H= {xeE\f(x) = 0} is a hyper-
hyperplane in a normed linear space E, then for any subspace F of ? the intersection
FnH= {xeF | /|f(x) = 0} either coincides with F (if /|f = 0), or is a hyperplane
inF(if /|F^0).
2 See e. g. [246], Appendix I, theorem 2.3.
1. Monotone and strictly monotone bases
241
Theorem 1.3. We have
U(nl+n2)^ma.x(U{nl),U{n2)) (nbn2 = 3,4,...). A.69)
Consequently,
sup Дп)< sup I7(n)<max(l/C),17D),17E))<1. A.70)
Proof. Let и1ги2^3 and e>0 be arbitrary. Then, by the definition
A.63) of U{nt), there exist Banach spaces ВеП1,Ве„2 of dimension nx and
n2 respectively, such that
ЩП1) + г (i=l,2). A.71)
Consequently, by lemma 1.12,
Щщ+п2)= inf
<max(I/(BB1),
whence, since e>0 and И!,и2^3 have been arbitrary, we obtain A.69).
The second inequality in A.70) follows now by induction from A.69)
and from the obvious remark that every integer и ^ 6 can be written
in one of the forms n = 3fc+3, n = 3fc + 4 or n = 3k + 5, where к is a
positive integer. On the other hand, the first and third inequalities in
A.70) are established by A.64), which completes the proof of theorem 1.3.
Let us pass now to the infinite dimensional case. We shall show that
the problem of existence of monotone bases in every infinite dimensional
Banach space with a basis, has also a negative answer.
For a function xeC([0,1]) we shall use the notation
Mx = {te[0,l]\\x(t)\ = \\x\\}. A.72)
In the sequel we shall consider only real-valued functions.
Lemma 1.13. For every xeC([0,l]) and e>0 there exists a
= S(e,x)>0 such that
(J (f-
A.73)
Proof. Let us denote the set in the right side of A.73) by Mx(e). We
claim that r, = \\x\\- max |x@l>0, A.74)
where СМДе)=[0,1]\Мх(е). Indeed, since CMx(e) is closed and
bounded, whence compact, and since x CM (El is continuous, max \x(t)\
leCMx(s)
is attained in at least one point ioeCMx(e). Now, if ^ = 0, then \x(to)\
= max |х(г)| = ||х||, whence toeMx, a contradiction which proves that
ГеСЛЫЕ)
>?>0, i.e. A.74).
16 Singer, Bases in Banach Spaces 1

242
II. Special Classes of Bases in Banach Spaces
Now take д = -. Then for every jeC([0,l]) with ||j>||<<5 = y
wehave, by A.74), 2
max \x{t)+y{t)\^ max \x(t)\+ max \y(t)\
1еСМх(с) ieCmx(e) te[O,l]
<IW-I/ + I = 11*11 -| = ||x||-<5
<MHIjKII*+jII,
whence we get Mx + ),c: Mx(e), i.e. A.73), which completes the proof of
lemma 1.13.
Lemma 1.14. Let xeC([0,1]) be differentiable on [0,1] and let
0<tl=mint^t2 = maxt<l. A-75)
Assume, furthermore, that there exists a differentiable function
jeC([0,1]) such that on some interval {a,b) эМ, we have y(t)>0 and
y'(t) has constant sign. Then there exists an a0 > 0 such that for 0 < a < a0
we have
Mv
:{t2,b) if y'{t)>0 on (a,b) and x{t2)>0, A.76)
Mx+xya{a,t1) if y'(t)<0 on (a,b) and x(?,)>0. A-77)
Proof. Let us consider the case when y'(t)>0 on (a,b) and x{t2)>0.
Since Mxc(a,b), by lemma 1.13 there exists an ao>O such that for
0<a<a0 we have Mx + xy<=(a,b). Now let t0e{a,t2) be arbitrary. Then,
by t2 e Mx we have
x(t2)>x(t0)
and by t2>t0,y'(t)>0 (te(a,b)) we have
whence for 0<a(<a0),
x(t2) + ay{t2)>x{
Since toe(a,t2) has been arbitrary, it follows that
Mx + xy<=[t2,b).
However, 12фМх+асу, since by t2eMx and a>0,y'(t2)>0 we have
Thus
(t2,b), which proves A.76). The case when y'(t)<0
y
on (a,b) and x(tl)>0 can be treated in a similar way, which completes
the proof of lemma 1.14.
1. Monotone and strictly monotone bases
243
Lemma 1.15. Let x,yeC([0,l~\). We have ([x];[y])<l if and only
if either
signx(?) = signj(?) (teMx) A.78)
or
signx(?)=-signj(?) (tsMx). A.79)
Proof. Necessity. Assume that we have neither A.78) nor A.79). Then
two cases are possible:
a) There exists a toeMx such that j(;0) = 0. In this case for every
scalar a we have
b) There exist tut2eMx such that signx(/1) = signj(/1), signx(/2)
= — signj(?2). In this case for every a^O we have
and for every «<0we have
||x + aj|| ^ \x(t2) + ccy(t2)\ ^ \x(t2)\ = \\x\\.
Thus, in all cases, for every scalar a we have ||x + aj|| ^ \\x\\, whence
Sufficiency. We may assume, without loss of generality, that \\x\\ = 1.
Assume that we have A.78). Then, since Mx is compact and \y\\Mx is
continuous,
mm\y(t)\ = n>0. A.80)
A.81)
For E>0 denote, as in the proof of lemma 1.13,
MX{S)= U(f-<5
that
Since Mx is closed and x,y are continuous, there exists a E>0 such
inf
A.82)
(teMx(S)).
Let
X= max \x(t)\,
re С Мх(»)
a =
1-Х
A.83)
A.84)
A.85)
By A.74) we have 0<A<||x|| = l, whence а>0. Let us estimate
\x{t)-ay{t)\.

244 II. Special Classes of Bases in Banach Spaces
If teMx(S), then by a>0, A.83), A.82) and A.85) we have
\x(t)-ay(t)\=\\x(t)\-a\y(t)\\^max{\x(t)\-a\y(t)\,-\x(t)\ + a
\ [i ) ( а/г \—л
^ max<l-a-, a||j>||> = max<l - —, —
while if teQMx{S), then by a>0, A.84) and A.85) we have
1-A .. .. 1 + A
Consequently, since a>0,
Г 2||y|| '"" 2
and O<A<1, we get
аи 1—я 1Ч
= max |
whence ([x];[y])<l, which proves the sufficiency of condition A.78).
Assume now that we have A.79). Then for the function — yeC([0,1])
we have A.78), whence, by the above,
which completes the proof of lemma 1.15.
Theorem 1.4. Let {xn} cz C([0,1]) be an infinite sequence of diferen-
tiable real-valued functions on [0,1], such that
1°. The set of all stationary points1 of any linear combination
n
Y, oLiXfif) has no limit points in @,1) and
i= 1
2°. There exists a function je[xn] such that both y(t) and y'(t)
have constant sign on @,1).
!<1- I1-86)
Then we have
Уи„!
Consequently, if a closed linear subspace Eo of C([0,1]), with
dim?0=oo, consists of non-constant analytic functions on @,1) and
contains a function у such that both y(t) and y'(t) have constant sign
on @,1), then Eo has no monotone basis.
Proof. Since the system of two homogeneous linear equations
1 We recall that a point t is said to be a stationary point of a differentiable
function z if z'(?) = 0.
1. Monotone and strictly monotone bases 245
always has a non-trivial solution <xb a2, a3, there exists a function
з
z, = ? aix,e[x1,x2,x3]\{0} such that z1@)=z1(l)=0, whence
i = i
0<?0 = max ?<1.
Assume now that Zi(ro)>O (since otherwise one can take the func-
function — z, instead of zj and that for the function je[xn] of condition 2°
we have y{t)>0, y'(t)>0 on @,1) (the other cases can be treated sim-
similarly).
We claim that there exists а т with t0 < t < 1, such that
(te(to,r)).
A.87)
Indeed, since Z!(?o)= max |z,(?)|, in every interval (?0>?i) there
te[0,l]
must exist a r^'e^o,^) with Zi(?20>)<0. Assume now that there exists
no т such that we have A.87). Then in every interval (?0,?20)) there
exists а Д0>е(г0,гB0>) such that z'^t^^O. Hence, by a well known
theorem of Darboux1, in every (tf\ ?B0>), where tf},tf} are as above,
there exists a fef/i",^0') such that z\{t) — 0, which contradicts the
assumption Г. Thus we have A.87).
Since zx is continuous and Z!(?0)>0, we may also assume that
(t€(to,r)).
A.88)
Since y(t),y'(t)>0 on @,1) and z,(fo)>O, by lemma 1.14 (with
t2 = t0, b = x) we have M.l+ay cr (to,t) for sufficiently small a>0.
Furthermore, since zl,ye[xk\, for every E>0 there exist an integer
n = n(S) andafunction z2 = z^e[xb ...,х„] such that \\z2 — (zl + oi.y)\\<S;
obviously, we may assume that n ^ 3. Since z2=(zl + a.y) + [z2 — (z, + a.y)~\,
by lemma 1.13 one can take e>0 and S = d(s,z1 + ay)>0 so small that
U
Put
t0 = mm t.
teM.
A.89)
A.90)
Then, by A.89) we have to<t'o<x, whence, by a>0, A.88) and
y{t)>0 (te{0,1)), we get z1(t'0) + a.y(t'0)>0, and therefore, taking 6 suffi-
sufficiently small, we may assume that
z2(t'o)>0.
A.91)
See e. g. G. Fichtengolz [58], p. 283.

246
II. Special Classes of Bases in Banach Spaces
Now let x e [х„ + ь..., xm] (n < m) be arbitrary. Then, by condition 1 °
and the theorem of Rolle, we have sign x(t) = const ^0 on some in-
interval (to,?o), where Toe(to,t'o) (indeed, otherwise x(t) would have an
increasing infinite sequence of zeros in (to,t'o) and between any two of
them there would exist a zero of x'(t), contradicting condition 1°). By
A.90), A.89), A.91), A.88), A.87) and lemma 1.14, the function
Z3 = z2+J8z1e[x1,...,xj A.92)
satisfies MZi cr (то,1'о) for sufficiently small /?>0. Furthermore, by the
continuity of z2 and by A.91) we may assume that
z2(?)>0 (?e (Wo)),
which, together with /?>0,(то,?о) <= Uo^) ап<* A-88), implies
z3(/)>0 (?eMZi). A-93)
Hence, since signx(t) = const^0 on М,л с(то,?о), by virtue of
lemma 1.15 we obtain _________
(Ы;И)<1- A.94)
Consequently, by A.92) and хе[х„+1, ...,xm],
i. e. we have A.86).
Assume now that ?0 is a closed linear subspace of C([0,1]), with
dim ?0 = oo, consisting of non-constant analytic functions on @,1) and
containing a function у such that both y(t) and y'(t) have constant
sign on @,1), and let {х„} be an arbitrary sequence in ?0, such that
n
[х„] = ?0. Then, since every linear combination ]Г а;х((г) is an analytic
i= 1
n
function, condition Г is satisfied; in fact, otherwise ? a;xj(?) would be 0
on a convergent sequence in [0,1], whence, by the uniqueness theorem
for analytic functions, it would be identically 0 on [0,1], contradicting
n
the hypothesis that ]Г a,X;@ is non-constant on [0,1]. Furthermore,
since [х„] = ?0, condition 2° is also satisfied. Hence, by A.86), {xn} is
not a monotone basis of ?0, which completes the proof of theorem 1.4.
We shall show now that there exist subspaces ?0 of C([0,1]), even
with a basis, which satisfy the conditions of theorem 1.4, and thus there
exist infinite dimensional Banach spaces ?0 with a basis, having no
monotone basis.
1. Monotone and strictly monotone bases
247
Lemma 1.16. Let x,yeC(\0,1]) with \\x\\ = \\y\\ = 1 and e>0. If
there exists a /0e[0,1] such that
M?o)l^l-?, ljUo)l«S?, A-95)
then for all scalars a we have
||x + ajj| ^1 -Зе = A -Зг)||хЦ. A.96)
Proof. If |a|<2, we have
while if |a|^2, we have
| - |a| || y\\ |= jl -
_3e.
With the aid of this lemma, we prove the following proposition on
selection of basic sequences in C([0,1]):
Proposition 1.2. Let {*„}<= C([0,l]), ||xj = l (n=l,2,...). If for
every couple e>0, ^>0 there exists a positive integer N = N(s,S) such
thatforeach n>N the set A(xn,?)={te[0,l~\\\xn{t)\<E} be a S-net for
[0,1], then there exists an infinite subsequence {хПк} of {xn}, which is a
basic sequence.
Proof. Let {/?„} be an arbitrary sequence of numbers such that
a;
0. Put
?„ = -—- (и=1,2,...).
A.97)
Take xni=xl. Since this function is uniformly continuous on [0,1],
thereexistsa ^!>0 such that \xni{t')-xni{t")\<?1 whenever \t'-t"\<61.
Take an arbitrary n2>N{Ei,6l) and consider xnr We claim that
(JAjT[xJ)^j8i. A.98)
Indeed, take an arbitrary t^eMXn . Since n2>N{Eu6l\ there
exists by the definition of N{?^6^, a ?2e[0,1] such that
Then \xni{t2)\^\xni{ti)\-Sl = l-sl and thus, by lemma 1.16 applied
to x = xni,y = xn2,t0 = t2 and ? = ?,, we get
for all scalars a, whence we infer A.98).

248
II. Special Classes of Bases in Banach Spaces
Now, the unit sphere a[Xn iXn] = {xe[xni,xn?] | ||x|| = l} is compact,
whence, by the theorem of Arzela, equicontinuous, and thus there
exists a S2>0 such that \x{t')-x(t")\<?2 whenever xecr^ Хп],
\t' — t"\<S2. Take an arbitrary n3>N{s2,S2) and consider х„з. We
claim that
(Dw^ijTI^J) > Pi • A.99)
Indeed, take an arbitrary xea[XniXn]. Then, repeating the above
argument for x, хПз, e2 and S2 instead of xni, хП2, ё! and <5t respectively,
it follows that
for all scalars a, whence, since хеа[Хп Хп ] has been arbitrary, we infer
A.99).
Continuing this process indefinitely, we obtain a subsequence {xnk}
of \xn\ such that
A.100)
Then, by Ch. I, §7, corollary 7.1, {х„} must be a basic sequence,
which completes the proof of proposition 1.2.
Corollary X.X.Let
xn(t)=t"
A.101)
Then one can select an infinite subsequence {xnk} of {х„}, which is a
basic sequence in ?=C([0,1]). Consequently, the subspace ?0 = (XJ
satisfies the conditions of theorem 1.4 and thus it has no monotone basis.
Proof The sequence A.101) satisfies the conditions of proposition 1.2;
indeed, ||xj = l and, since Л(х„,е)=[0,е") (и=1,2,...), one can take
Now, since xnk{t)=tnk is a basic sequence in C([0,1]), every ele-
00
ment x of the subspace ?0 = (XJ admits an expansion x{t) = ? akt"k
k=i
converging uniformly on [0,1] and thus it is an analytic function on
@,1). Furthermore, Eo consists of non-constant functions on @,1), since
by A.101) we have х„к@) = 0 for all fc=l,2,... Since any function
xn (t)=t"k can be taken as the у of theorem 1.4, the conditions of theorem
1.4 are fulfilled. This completes the proof of corollary 1.1.
Thus, the problem of existence of monotone bases in every infinite
dimensional Banach space with a basis, is solved in the negative. How-
However, the following problem remains open:
1. Monotone and strictly monotone bases
249
Problem 1.1. Does there exist an infinite dimensional Banach space
with a basis, such that 0 < Г(?) < 1 ?
Remark 1.1. The same argument as that used at the end of the proof
of theorem 1.4 also shows that the hyperplane
A.102)
of C([0,1]) has no monotone basis consisting of analytic functions on
@,1), whence, in particular, it has no monotone basis consisting of
algebraic polynomials. Since the sequence obtained from the usual
Schauder basis of C([0,1]) by deleting its first term is a monotone
basis1 of E^ and since the set of all algebraic polynomials satisfying
x@) = 0 is dense in El (by the theorem of Weierstrass), this shows that a
stability theorem of Krein-Milman-Rutman type (Ch. I, § 10) is no longer
valid for monotone bases.
In the footnote of Ch. I, § 11, a necessary condition was given for a
Banach space E to have a monotone basis. The following proposition
gives a necessary and sufficient condition:
Proposition 1.3. A Banach space E has a monotone basis if and only
if there exists a sequence {Gn) of closed linear subspaces of E with the
following properties:
a)
b)
n = n (n=l,2,...);
00
с) \J Gn is dense in E;
n= 1
d) there exists a projection и of E onto Gn, with ||ии|| = 1 (n=l,2,...).
Proof. HE has a monotone basis {*„}, then for Gn = [xbx2, ¦¦¦,xn]
(и =1,2,...) we have a), b), с) and d) with u = sn (n= 1,2,...).
Conversely, assume that E has a sequence {Gn} of subspaces satis-
satisfying a), b), c), d). Take an arbitrary element xt of Gl5 different from 0,
and for each и = 1,2,... take an element xn+1 of Gn +1, different from 0
and such that и„(хп+1) = 0. Then the sequence {xn} is complete in E
and we have
1 Indeed, we have already observed that the Schauder basis of C([0,1]) is
monotone and therefore, if we delete its first term, it remains a monotone basic
sequence. Furthermore, all these functions vanish in t = 0 (as second term of the
Schauder basis, i.e. as first term of our sequence, we take the function x(t) = t
instead of x(t)= 1 — t), i. e. they belong to El, whence they span the whole hyper-
hyperplane ?j.

250
II. Special Classes of Bases in Banach Spaces
E <*ixi
:=1
n+ 1
E<
for all finite sequences of scalars аь ...,а„+1еК, whence also A.1).
Consequently, by Ch. I, § 7, theorem 7.1, {х„} is a monotone basis of E,
which completes the proof.
Although there exist, as we have seen in the preceding, Banach spaces
which have no monotone basis, every basis of a Banach space E can be
"monotonized" by replacing the norm of E with a suitable equivalent
norm. Indeed, this follows from Ch. I, § 3, proposition 3.2b), since for
the equivalent norm |||x||| = sup
1 =Sn< on
quence of scalars ab...,an + m we have
and for any finite se-
n
E
a,
xi
= sup
к
I
i= 1
ai
Xi
sup
likin+m
к
E
=
n + m
E а.-х;
i= 1
Moreover, since |||xJ| = |l*J (п = 1,2,...), from this remark and Ch. I,
§ 1, theorem 1.2 it follows that for every bounded basis {х„} of a Banach
space E there exists an equivalent norm ||x||' on E in which the basis {х„}
is normalized and monotone.
As shown by the example of the unit vector basis in c0, the basis {х„}
need not be strictly monotone in the above norm |||x|||. However, by Ch. I,
§19, theorem 19.1b) and proposition 19.1b), every basis of a Banach
space E can be also "strictly monotonized".
Remark 1.2. Not only every basis {xn\, but also every sequence {xn}
such that хпф0> (п= 1,2,...) is equivalent1 to a strictly monotone basis.
Indeed, if 4, = {«„} <= К
«;*; converges > is the Banach space
of sequences of scalars introduced in Ch. I, § 3, proposition 3.1, then
? 1
= sup
2>«;
+
if, 2-
O.:X:
A.103)
is an equivalent norm on Аь in which the unit vector basis en={Snj}JL i
(n= 1,2,...) of Ax is strictly monotone. On the other hand, by Ch. I, §8,
proposition 8.1b), we have {х„}~{е„} and obviously this remains
valid for Al endowed with the equivalent norm )? a;x,-1 ). Thus
//¦=1 V
{xn} is equivalent to a strictly monotone basis.
1 For the equivalence of sequences see Ch. I, § 8.
I. Monotone and strictly monotone bases
251
From the remark that every basis can be "monotonized" and from
proposition 1.3 we infer
Corollary 1.2. A Banach space E has a basis if and only if there exist
a sequence {Gn\ of subspaces of E and a norm on E, equivalent to the
initial norm of E, such that we have a), b), c) and d) of proposition 1.3 in
this new norm.
Let us observe that there is an obvious connection between proposi-
proposition 1.3, corollary 1.2 and the results of Ch. I, § 20. In particular, problem
20.1 of Ch. I, §20 admits (by Ch. I, §20, proposition 20.1, equivalence
l°<t>3°) the following equivalent formulation: If a Banach space E has
a sequence of subspaces {Gn} with the properties a), b), c) of proposi-
proposition 1.3 and
d') there exists a projection vn of E onto Gn (и =1,2,...) such that
sup ||i'J < oo, then does E have a basis?
1 ^n < oo
The answer to the problem of existence of non-monotone (and hence
also of non strictly monotone) bases in Banach spaces with bases is
affirmative. In fact, let us first remark that for every normalized mono-
monotone basis {xn} we have
ll*.-*mll>ll*min(»,»)ll = l (п,т=\,2,...;пфт). A.104)
Now, if {xn} is a monotone basis of a (finite or infinite dimen-
f xn )
sional) Banach space E, then < > is a normalized monotone basis
(JI*JJ
of E, whence, by the above remark, for any yeaE n [xlrx2]
= {xe[xux2] | ||л:|| = 1} such that
0<
the sequence
x.
- У
II*! I
У2=У>
II*J
A.105)
is a normalized non-monotone basis of E. This proves our assertion.
We conclude this paragraph with the following proposition on
duality properties of monotone bases:
Proposition 1.4. Let {xn} be a monotone basis of a Banach space E
and let {/„}c?* be the a.s.c.f. to {х„}. Then
a) {/«} й а monotone basis of [/„].
b) We have
II/i 11 = 1, KII*JII/J<2 (и = 2Д...).
A.106)

252
II. Special Classes of Bases in Banach Spaces
Proof, a) is a consequence of Ch. I, § 12, theorem 12.3 and of the re-
remark made at the beginning of the present section, according to which
a basis is monotone if and only if it is of norm v = 1.
b) Assume that {xn} is a monotone basis of E. Then
,i= 1
fn
whence
a,
а„
=
=
2
* ll.v.11
и
п.
1
Kll
1
IKII
00
i= 1
f II <
/nil ^
l«l*lll
00
i= 1
\
/
' 00
\i = 1
П
i — 1
+
n- 1
Eaixi
L= 1
i= 1
(и = 2,3,...). On the other hand, we
have ||/n|| \xn\ ^ |/„(я:„)| = 1 (и=1,2,...), which completes the proof of
proposition 1.4.
The converse of proposition 1.4a) is not true, since e.g. the a.s.c.f.
{/„} to the basis {xn} of/1, given in Ch. I, § 12, example 12.2, is monotone,
but {xn} is not monotone; a similar statement is also valid for the basis
of c0 given in Ch. I, § 13, formula A3.15).
For strictly monotone bases a result similar to proposition 1.4a)
is no longer valid, since e.g. the natural basis {xn} of/1 is strictly mono-
monotone, while the a.s.c.f. {/„} to {*„} is monotone but not strictly mono-
monotone. Also, a result of converse type is again not valid, since e.g. the
natural basis {xn} of c0 is monotone but not strictly, while the a.s.c.f.
{/„} to {xn} is strictly monotone.
§ 2. Normal bases
Definition 2.1. A basis {*„} of a Banach space E is said to be normal,
if both {xn} and the a.s.c.f. {/„}<=?* are normalized, i.e. if
II*J=II/.II = 1 («=1,2,...).
B.1)
For instance, the natural bases of c0 and l"(p ^ 1) are normal bases.
The Schauder basis {xn}$ in C([0,1]) is normalized but not normal,
since the a.s.c.f. is
2. Normal bases
253
/o(x) = x@), .
'2/-1
1 B1-1
JL
B.2)
whence ||/0|| = 1, ||/J =2 (n=\,2,...). Indeed, /0 and /t have been
computed in Ch. I, §2, example 2.2, while for the other equalities it is
sufficient to verify that /I(xj) = Elj. Now, for any fixed pair (k,l) with
/2*, 0^/c<oo, it is obvious that
= 1, ..., 2 ) .
On the other hand, if h<k, then
tained in one of the 2Л + 1 segments
2/-2 2/ 1 .
k + l , fc+1 is entirely con-
'27-2 2/-l"| Г27-1 _2y_l
лЛ+l ' л/i + 1 Г лЛ+1 ' л/i+l
(j = 1,..., 2h). Since on each of these segments every x2h + m (m = 0,1,..., 2h)
is linear, it follows that every x2h+m (m = 0,1, ...,2h) is linear on
Г2/-2 2/ 1
fc+1 , -j^j , whence, by B.2),
which completes the proof of
Furthermore, the normalizedHaar basis {г(р)} =
i.e.1 the sequence {z[p)}, where
= l—^-> in Lp([0,1]),
Pr-v Г2/-2 2/-1
2/-1 2/
for te
2<c+1 '
0 for the other t
(/= 1,2,..., 2k; k = 0,1,2,...),
B.3)
See Ch. I, § 2, example 2.3.

254 II. Special Classes of Bases in Banach Spaces
is a normal basis of Lp([0,l]) ip>\), since the a.s.c.f. {hn} to {z[p)} is
), i=l,2—),
B.4)
where - + -= 1, whence ||й(|| = \\z<?)\\ = 1 (/=1,2,...). Indeed, it is suf-
P Я
ficient to verify that hi(zf)=SiJ. Since obviously hi{z\p))= 1 ((=1,2,...),
it remains to prove that A,-(zjp)) = O for /#/, and for this purpose it is
sufficient to observe that the Haar system {yn} is orthogonal, i.e.
(/J= 1,2,...;
B.5)
by virtue of Ch. I, § 2, formula B.19).
The notion of normal basis admits several geometrical interpreta-
interpretations. For this purpose, let us first recall
Lemma 2.1. Let E be a Banach space, feE*, H={yeE\f(y) = 0}
and xeE. Then
Н) = Щ. B.6)
Proof. For every ye Я we have
1
I/WI
whence dist(x, Я)
I/WI
11/11
. On the other hand, for 0<e<|
exists a ze? such that |/(z)|>(||/||-e)||z||. Multiplying with
and defining yeH by
fix)
there
/W
we obtain |/(jc)|>(||/|| -e)\\x-y\\, i.e.
№
B.7)
I/WI
I/WI
-?
. This
whence, since 0<e<||/|| has been arbitrary, dist(x^)
completes the proof. IIJ "
Let us mention that lemma 2.1 can be also derived from a corollary
of the Hahn-Banach theorem which we have already used in the pre-
preceding.
2. Normal bases
255
Corollary 2.1. Let Ebea Banach space and (*„,/„) ({*„} с Е, {/„} с ?*)
an E-complete biorthogonal system. Then
dist(xn,
(«=1,2,...),
B.8)
where, as in Ch. I, §3, formula C.9) are/ СЛ. /, §7, formula G.1), ?(n) is
the closedhyperplane of E spanned by the sequence {xu ...,xn_u xn+1,...},
i.e.
^я)={рс1,...,х,,.1,хя + 1,...-] (и=1,2,...). B.9)
Proof. Apply lemma 2.1 to Я = E(n) = {y e E | /„(j) = 0} (n = 1,2,...).
Now we can give some geometric characterizations of normal bases.
Theorem 2.1. For a basis {xn} of a Banach space E the following
statements are equivalent:
2°. We have
\\xn\\ = l and xnLE(n) (л= 1,2,...),
B.10)
where E(n) is defined by B.9).
3°. The unit cell SE={xeE\ \\x\\ ^ 1} contains xn on its boundary
<jE={xeE I \\x\\ = 1} and has at х„ a support hyperplane Hn parallel to
the hyperplane ?<л) (и= 1,2,...).
In this case, each hyperplane Hn is uniquely determined.
4°. The unit cell SE lies between the "hyperoctahedron"
B.11)
and the ''hyperparallelepiped"
max i
l=H<0O
B-12)
(as shown by B.11) and B.12), both terms are understood with respect to
the "system of coordinates" {*„}).
5°. We have \\xn\\ = 1 (n= 1,2,...) and
B.13)
Proo/. Го2°. By corollary 2.1, we have || /„|| = 1 = ||xj if and only
if dist(xn,?(n))=l = dist(xn,O), which is equivalent, by the definition of
orthogonality, to xnLE(n\ \\xn\\ = \ (n=l,2,...).

256
II. Special Classes of Basis in Banach Spaces
1°=>3°. If ||xnII =|| /J| = 1, the hyperplane Я„ defined by
Hn={yeE\fn(y)=l}
B.14)
supports SE at xn and it is parallel to E(n) (n= 1,2,...).
3°=>1° and the uniqueness of Я„. Assume that for a positive integer
n and a geE* with \\g\\ = 1 the hyperplane
H={yeE\g(y)=\}
B.15)
supports1 SE at xn and is parallel to E(n). Then, since xneH, we have
g(xn)=\ and, since Я is parallel to E(n\ we have #(х,)=0 0 = 1,...,
и-1,п+1, ...)¦ Consequently, since [*„] = ?, we have # = /./„, where
Я#0, whence, since Д = 1/„(л:„) = #(*„) = 1, we get # = /„, i.e. Я coin-
coincides with the hyperplane Я„ defined by B.14), which proves the unique-
uniqueness of Я„. Since Ы| = 1, it also follows that ||/J = 1. Furthermore,
since xneSE, we have ||xj^l. On the other hand, since х„еН, we
have l=g(xn)^\\g\\\\xj = \\xnl whence, finally, ||jcb|| = 1. Thus we
have 1°.
1°^4Q. If ||х„|| = ||/„|| = 1 («=1,2,...), for every x= ?с
have i=1
we
max |a,| = max
1 ^ i < oo 1 ^ i < oo
whence
X
max ||
1 ^ i < oo
M2.
B.16)
B.17)
4°=>5°. Assume that we have B.17). Then, by xneM1<=SE we have
00
\\xn\\ ^ 1. Furthermore, let ? а;х;е? be arbitrary. Then by
;= l
XjeSEcM2
1 We recall that every support hyperplane H of SE can be written in the
form B.15) with a suitable geE* of norm ||ff|| = l (see e.g. [246], Ch. I, § 1,
lemma 1.4).
2. Normal bases
257
we have max
i= 1
1, whence, since llxjl^l, we obtain
i= 1
(«=1,2,...).
In particular, for а1 = ---=ая_1 = аи+1 = ---=0,а„=1, we obtain
|| = 1 (и=1,2,...).
5°=>1°. Assume that we have 5°. Then
{Xj}J= 1 of
[]
whence, by corollary 2.1, ||/J = 1 (и = 1,2,...), which completes the
proof of theorem 2.1.
Remark 2.1. The hyperoctahedron Ml occurring in 4° above is
nothing else but the closed circled convex hull of the sequence {х„},
while the hyperparallelepiped M2 of 4J is nothing else but the inter-
intersection of all closed half-spaces determined by Я„ and SE, where Hn are
the support hyperplanes of SE occurring in 3°.
Remark 2.2. From definition 2.1 it is obvious that a basis {xn} of a
Banach space E is normal if and only if every finite section {
{xn} is a normal basis of the finite-dimensional Banach space
(m= 1,2,...). Thus, applying theorem 2.5 to each finite section {Xj}J=l,
we obtain several other geometric characterizations of normal bases {*„}.
In the particular case when ? is a Hilbert space Я, we have
Proposition 2.1. A basis {х„} of a Hilbert space H is normal if and
only if it is a normalized orthogonal basis of H.
Proof. By the equivalence 1°<=>2° of theorem 2.1, {*„} is a normal
basis of Я if and only if ||xj = l (n=l,2,...) and xn ±Xj for all )фп
(/,и=1,2,...). However, by virtue of §1, lemma 1.1, this happens if and
only if \xn} is a normalized orthogonal basis of Я.
Now we shall prove that the answer to the problem of existence of
normal bases in finite-dimensional Banach spaces is affirmative.
Theorem 2.2. Let E be an n-dimensional Banach space. Then E has
a normal basis.
Proof. Introduce a Cartesian coordinate system into E and let
D(y t
B.18)
where
(к=
П Singer, Bases in Banach Spaces I

258
II. Special Classes of Bases in Banach Spaces
Since dimE = n, the unit sphere oE={xeE | ||x|| = 1}, whence also
the product (aE)n = aE x • • • x aE (n times), is compact. Consequently,
since D(yi,...,yn) is continuous on (aE)", we can choose x1,..., xneE
with j|x,|| = l (j=\,...,n), which maximize \D(yl,...,yn)\ on (aE)n.
This being done, put
Then
D(xu...,xn)
and by our choice of xu ..., х„ we have
II/i 11= sup |/,(x)|= sup
1 = 1
= 1 (i=l,...,«),
which shows that {х;}"= t is a normal basis of E. This completes the
proof.
As shown by the above proof, the condition of maximizing
\В(У\, ¦¦¦,Уп)\ on (tffi)"is sufficient for {*,-}"=! to be a normal basis of E.
However, this condition is not necessary, as shown by
Example 2.1. The basis
*i = {U}, *2 = {-ii-} B.20)
of the space E=l\ (of all pairs of scalars х = {?и?2}, endowed with
the usual vector operations and the norm \\x\\ = |^il + |^2l) is normal,
since the associated coefficient functionals are f1 = {\,l},f2 = { — 1Д}-
However, for z1 = {l,0}, z2= {0, l}eaE we have
D(x1,x2) =
1
2
1
2
1
2
1 ,,
= 2 <
1
0
0
1
= D(zuz2),
and thus хъх2 do not maximize D(y1,y2) on c? x aE.
Remark 2.3. In the particular case of real scalars, the above con-
condition of maximization (and also the above proof) admits a geometrical
interpretation. Indeed, in this case 2"\D(y1, ...,yn)\ is the volume of the
max |a(| sg 1 > with axes y^...,yn and
parallelepiped M2n) =
thus the above condition requires to maximize the volume of MBB)
for |[y.|| = l (/=1,...,и).
Problem 2.1. Does every infinite dimensional Banach space with a
basis possess a normal basis? What about the space C([0,1])? What
about a reflexive infinite dimensional Banach space with a basis?
2. Normal bases
259
Although these questions are unsolved, it is known that every
bounded basis of a Banach space E "can be made normal" by replacing
the norm of E with a suitable equivalent norm. In other words, we have
Theorem 2.3. Let {*„} be a bounded basis of a Banach space E. Then
there exists an equivalent norm on E, in which the basis {xn) is normal.
Proof. By Ch. I, § 3, theorem 3.2, we may assume, without loss of
generality, that ||xj = 1 (n= 1,2,...). Put
= sup
1 ^ n,m < oo
m
i = n
(xeE),
B-21)
where {/„}<=?* is the a.s.c.f. to {*„}. Then |||x||| is a norm on ? and,
by Ch. I, § 3, proposition 3.2 b) and
i= 1
, it is equivalent to the initial norm of E. Furthermore,
= ||^|| = 1 . (/=1.2,...),
and hence also
sup
^ n,m < oo
m
Xai*i
=
00
X! ai*i
i= 1
Therefore, by theorem 2.1 (implication 5°=>1°), {xn} is normal in the
norm B.21), which completes the proof of theorem 2.3.
On the other hand, obviously, the answer to the problem of existence
of non-normal bases in Banach spaces with bases is affirmative, since
if {xn} is a normal basis of a Banach space E, then {2xn} is a non-normal
basis of E. Moreover, the answer to the problem of existence of nor-
normalized non-normal bases in Banach spaces with bases is also affirma-
affirmative. In fact, let us first remark for every normal basis {xn} we have
\\хя-хт
\fn(xn-xm)\ = \ (n,m= 1,2,...; пфт)
B.22)
where {/„} is the a.s.c.f. to {*„}. Now, if {xn} is a normal basis of a
(finite- or infinite-dimensional) Banach space E, let у be any element of
°еп [*1,*г] = {*6[*n*2] | 11*11 = 1} such that
Then, by the above remark, the sequence
Ji = *i, Уг=У, Уп = х„ (и = 3,4,...) B.23)
is a normalized non-normal basis of E. This proves our assertion.

260
II. Special Classes of Bases in Banach Spaces
We conclude this section with some remarks on duality properties
of normal bases.
Proposition 2.2. // {*„} is a normal basis of a Banach space E, the
a.s.c.f. {/„} is a normal basis of the space [/„].
Proof. By definition 2.1, we have ||/J = 1 (n=l,2, ...)¦ On the
other hand, the a.s.c.f. (in [/„]*) to the basis {/„} of [/„] is {ф(х„)},
where ф denotes the canonical mapping of E into [/„]* (see Ch. I, § 12).
Hence, by definition 2.1,
\\ф(хп)\\= sup
„|| = 1 («=1,2,...).
On the other hand, /Лб[/Д ||/J = 1, whence
\\ф(х„)\\ ^(хл)(/„)| = |/„(*„)| = 1 («=1,2,...),
and thus \\ф(х„)\\ = \ (n=l,2,...), which shows that {/„} is a normal
basis of [/„]. This completes the proof.
The converse of proposition 2.2 is not valid, as shown by
Example 2.2. Let E — c0 and let {*„} be the basis1 of the space E
defined by
хя={2,.. .,2,0,0,...} (и=1,2,...). B.24)
Then \\xj =2 (n= 1,2,...), whence {*„} is not a normal basis of E.
However, the a.s.c.f. {/„} is a normal basis of [/„]. Indeed, we have
(«=1,2,...),
B.25)
n-l
whence ||/J = 1 («=1,2,...). Furthermore, the a.s.c.f. (in [/„]*) to the
basis {/„} of [/„] is {ф(хп)}, where ф is the canonical mapping of E
into [/„]*. By B.25) we have
B.26)
whence, for any /={>?„}б[/„] with ||/||<1,
!>¦-
i= 1 i = n+ 1
1 We have xn= ]T 2et (n=l,2,...), where {е„} is the natural basis of
whence, by Ch. I, § 4, proposition 4.3, {xn} is a basis of E.
= c0
3. Positive bases
261
and therefore
\\ф(х„)\\= sup |/(х„)К1 (и=1Д...).
/?[/„]
11/11 «1
On the other hand, since /Яе[/Д ||/„|| = 1, we have
II0WII > l0W(/n)l = 1/„(*„I = 1 («=l,2,..¦),
and thus ||0(х„)|| = 1 (и=1,2,...), which shows that {/„} is a normal
basis of [/„]. This completes the proof of our assertion.
§ 3. Positive bases
Definition 3.1. A basis {х„} of a Banach space E is said to be positive,
00
if for every linear isometry T:E->E, with T(Xj)= ? fly*; (/=1,2, ¦••),
;= l
there exists a second linear isometry T+: E-+E such that
7V (*,-)= IX-K- (/-=1,2,....).
C.1)
Let us give some examples of positive bases.
Lemma 3.1. A continuous linear mapping T: c0—> c0, with
00
T(Xj)= Y aijxi (/= 1,2,...; {xn} = the natural basis of c0) is an isometry
i= 1
if and only if ^up |fly| = 1 y=l>2,...), C.2)
00
Xk,|^l 0=1,2,...). C.3)
GO
Proof. Let T: co^>co be a linear isometry, with T{xJ)= ? а^.дг,
(/= 1,2,...), where {*„} is the natural basis of c0. Then we have
sup \a{j\ = || Т{хД = \\Xj\\ = l (/'=1,2,...),
1 ^ I < 00
i.e. C.2). For a fixed / put e; = signa,-,- (/'= 1,2,...). Then
J"=l
sup
^ i < oo
J=l
Ц
J=l
__ 1
(«= 1,2,...), whence we infer C.3) and thus the conditions are necessary.

262 II. Special Classes of Bases in Banach Spaces
Conversely, let T:co^>co be a continuous linear mapping, with
00
T(Xj)= Y,aaxi (/=1'2' •••)' satisfying C.2), C.3) and let ^ a, be
;= l
arbitrary scalars. Then there is an index ^n, say j0, such that
n
\<x \ = sup |a-|= Y.*jxj ¦ By C.2) we have sup \aijo\\ajo\ = \ajo\ and
hence, for every e>0 there exists an index io = io(s) such that \aiojoajo\
n
>|aj-e. Then, by virtue of C.3), Z|aioi|<l, and therefore
KJ-8+ Z KJN< Z Ieiw-I l«il < l«iol Z kJ<l«J,
whence
Z ei<uaj
Z K«IK-
Consequently,
л
Та- -а- ЗН<
/ , IQJ J ¦
whence
V п
cjol — 2e =
Z a^i
-2e,
У а х =
=
=
п
У а.
sup
^ i < оо
00
V V
Z-* Ч i
л
7=1
=
00 Л
z z«;
i=l j=l
л
V
7=1
>
П
Г1 а x
= 1
-2e,
whence, since e>0 was arbitrary,
A Z «^i
Z ал
C.4)
On the other hand, by C.3) we have
—
л
sup
^ i< oo
00
Z «^ij ^^t
i= 1
- У У а а х
— Zj Zj "ij^J '
sup
Z«;
3. Positive bases
which, together with C.4), gives
263
П Z «л
Z «i^-
Since аь ...,а„ were arbitrary scalars and [*„] = ?, it follows that
T is an isometry, which completes the proof of lemma 3.1.
Proposition 3.1. The natural basis of c0 is positive.
Proof. Since the conditions C.2), C.3) remain invariant when pas-
passing from {ai}) to {|ау|}, the assertion follows from lemma 3.1.
Remark 3.1. A similar argument (with obvious simplifications) shows
that the natural basis of the finite dimensional space I™ is positive.
Lemma 3.2. Let l^p^2 and let y={rjn}elp, z={(n}el" be such that
Then
||z||"). C.5)
(/=1,2,...). C-6)
Proof. Let p>2. We shall first show that for any scalars n,( we have
, C.7)
with the equality sign holding if and only if r]C = O- Let \rj\ and |(| be
fixed and let us compute the minimum of
\ti-C\p. C.8)
If »7 = 0, this minimum is 2|?|p and if ( = 0 this minimum is
Assume now that ^?#0 and let |т/| = а>0, ( = -J?eie, j3>0. Then
а
а
d=dF) =
= (oc2 + P2+2a P cosOf + {a2+ p2-2ocPcosOf,
whence the minimum of dF) is 2(a2+ p2f = 2(\rj\2 + \C\2f, being at-
attained for в= + —. But, since W|>0, |CI>0, — > 1, we have
whence we infer C.7), with the equality sign holding if and only if ^? = 0.

264
II. Special Classes of Bases in Banach Spaces
Now let y={nn}elp, z= {С„}е/Р be arbitrary. Then, by the above
we have
\y+z\\p+\\y-z\\p=
i= 1
C.9)
with the equality sign holding if and only if we have C.6).
In the case when 1 ^p < 2, the proof is similar, with the difference
that for these values of p we have the opposite inequalities in C.7) and
C.9) (dF) attains its maximum for 0 = 0, я). This completes the proof
of lemma 3.2.
Lemma 3.3. Let \^рф2. If T:lp^lp is a linear isometry, with
cc
T(Xj)= ? dijXi (/= 1,2,...; {*„} = the natural basis of I"), then
;= l
eye,-» = 0 {i,j,m=l,2,...;jjtm). C.10)
Proof. Each pair х^,хт{]фт) satisfies C.5), whence each pair
T(Xj), T(xm) also satisfies C.5), whence, by lemma 3.2, we have C.10),
which completes the proof.
Remark 3.2. If we define the "support" of T(Xj) by
Supp7(je,.)={/|ay*0} (/=1,2,...), C.11)
then C.7) can be expressed in the equivalent form
Supp7(j<j)nSupp7(jeJ = 0 (/,m=l,2,...;7#ffi), C.12)
which may be called the disjoint support condition (for the pair (E, {*„}))•
Proposition 3.2. Let \^рф2. Then the natural basis of V is positive.
QO
Proof. Let T:/p->/p be a linear isometry, with T(Xj)= E аУх-
;= l
(/=1,2,...), where {*„} is the natural basis of lp and let <xl,...,an be
arbitrary scalars. Then, by lemma 3.3, for each / there is, among the
n numbers an
whence
00 / П
ЕЕ
.=i\j=i
,..., ain, at most one ф 0, whence
aij\aj)xi
/
/ 00
= ( ?
\i = 1
=
n
Л1 /
4
/ v
00 / П \
Lu \ Zj 4 3 I '
i= 1 \;=i /
n
El
00
E
i= 1
=
n
П
i^1 /
3. Positive bases
265
Consequently, by proposition 3.4 below, {*„} is a positive basis of E,
which completes the proof.
Let us consider now the case when p = 2.
Lemma 3.4. If {xn} is an orthonormal basis of a (finite or infinite
dimensional) Hilbert space E = H, then {*„} is not positive.
Proof. Put
X, +X-) Xl —X2
У г =
x, -
(и=ЗД...). C.13)
Then {у„} is an orthonormal basis of E, whence there is a linear
isometry T: ?->? such that
Т(хп)=у„
C.14)
Then
V (xi+x2)\\ =
X, —X,
x,+x2
x,-x,
' — v -I- V
- — KM 1-^2
i.e. T+ is not an isometry, which completes the proof.
Now we can show that the problem of existence of positive bases
in (finite or infinite dimensional) Banach spaces with bases has a nega-
negative answer, even in "very good" spaces.
Proposition 3.3. A (finite or infinite dimensional) Hilbert space E = H
has no positive basis.
Proof. Assume that E = H has a positive basis {xn}; we may assume,
without loss of generality, that {xn} is normalized (replacing, if necessary,
{*-} by
, the basis still remains positive). Let i,j be arbitrary
indices and put
x =
C.15)
Then *! and x may be considered as first elements of different ortho-
normal bases, whence there exists a linear isometry T: E->E such that
T(x1) = x. Then, since {*„} is a positive basis,
X:-X,

266
II. Special Classes of Bases in Banach Spaces
whence (х(,х^ = 0. Since i,j have been arbitrary, it follows that {xn} is
orthonormal, contradicting lemma 3.4. This completes the proof of
proposition 3.3.
Remark 3.1 and proposition 3.3 show that, similarly to monotone
bases, the notion of positive basis divides even the finite-dimensional
Banach spaces into two classes (according to the existence or non-
existence of a positive basis of the space).
We have the following characterization of positive bases:
Proposition 3.4. A basis {*„} of a Banach space E is positive if and only
00
if for every linear isometry T:E-*E with T(Xj) = ? ауя:( (/=1,2,...)
and any scalars al,...,an the series ? I E \aij\aj)xi converges
i=i \j=i
C.16)
Proof. If {*„} is positive, we have
I
I
i.e. C.16).
Conversely, assume that the condition is satisfied and let T: E->E
00
be a linear isometry with T(Xj)= Y,aijxi (/=1Д •••)• Then, taking
a1= •¦¦ =ak_1=ak+1= ••• =а„ = 0, ак=1, it follows that the series
00
У |aik|jC; converges (k= 1,2,...). Put
(*,) = I № (/=1,2,...),
C.17)
and extend T+ by linearity to the (dense) linear subspace of E spanned
by {*„}. Then for any scalars al5...,an we have, by C.16) and since T
is a linear isometry,
=
00 / П
/ " \
Til**)
n
^ с
00
(УааЬ
4. Л-shringking bases
267
and thus T+ is a linear isometry on a dense subspace of E, whence it
can be extended to a linear isometry T+ :?->?. This completes the
proof of proposition 3.4.
As shown by definition 3.1, the "positivity" of bases of a space E
depends very much on the form of the linear isometries T: E -> E. In
particular, if in a Banach space E with a basis the only linear isometries
T: E-> E are T— IE (the identical mapping of E onto E) and T= — IE,
then obviously every basis of E is positive. Since for each n ^2 there
exist и-dimensional Banach spaces with this property (e.g. for л^З one
can take aE={xeE \ \\x\\ = 1} to contain (/c + 2)-gons Ak (k=l,...,n)
such that Akr\Ak+l consists of one single point xk (fc=l, ...,n— 1),
with xl,...,xn_l linearly independent, and that except for the non-
extremal points of these polygons every other point of aE is an ex-
exposed point of SE), it follows that the answer to the problem of
existence of non-positive bases in finite-dimensional Banach spaces is
negative.
Problem 3.1. Does there exist in every infinite dimensional Banach
space with a basis a non-positive basis?
The difficulty consists in the fact that (by definition 3.1) we have to
consider also the linear isometries T of E into E.
Problem 3.1 is related to the following:
Problem 3.2. Is every positive basis necessarily unconditional1?
If the answer to problem 3.2 would be affirmative, then, since in every
infinite dimensional Banach space with a basis there exists a conditional
basis (by §23, theorem 23.2), the answer to problem 3.1 would be also
affirmative.
The classes of bases considered in § 1—§ 3 depend on the metric
properties of the Banach space E. All other classes of bases which we
shall consider in this Chapter (except orthogonal, hyperorthogonal,
strictly orthogonal and strictly hyperorthogonal bases in §20) depend
only on the topological linear structure of E, i.e. they are invariant
under an isomorphism of ? onto another Banach space El.
§ 4. ^-shrinking bases
We have seen in Ch. I, § 12, that for a basis {*„} of a Banach space E,
the a.s.c.f. {/„} с Е* is a basic sequence in ?*, but need not be a basis
of E*. Therefore it is natural to give
For the definition of unconditional and conditional bases see § 14.

268
II. Special Classes of Bases in Banach Spaces
Definition 4.1. Let к be a non-negative integer. A basis {*„} of a
Banach space E is called k-shrinking- if for the a. s.c. f. {/„} с E* we
have1
codim?.[/J = fc. D.1)
The O-shrinking bases, i.e. the bases {*„} such that [/„] = ?*, are
also called shrinking bases.
For instance, the natural bases of c0 and I" (/?>1) and the Haar
basis of Lp ([0,1]) (/?>1) are shrinking bases. We shall also give, for
arbitrary /c>0, an example of a /c-shrinking basis (see example 4.1
below).
Lemma 4.1. Let X be a Banach space and к a non-negative integer.
Then
a) A closed linear subspace G of X is of codimension ^k if and only if
a) for every (k+ 1)-dimensional linear subspace Fk + l of X, the inter-
intersection Fk+1nG contains a non-zero element x.
b) We have codim^G^k if and only if
fi) there exists a (k + l)-dimensional linear subspace Fk+1 of X such
that the intersection Fk' + lr\G contains a non-zero element x', which is
unique up to a homothety.
c) Consequently, codimx G = к if and only if we have simultaneously
a) and p).
Proof, a) Assume that codim^G^/c. Then there exist h linearly
independent functional ф1,...,фк, where h = codimx G^k, such that
ф](х) = 0 G=1,...,h)}.
D.2)
Let Fk+1 be an arbitrary (k+l)-dimensional linear subspace of X
and let y1,...,yk+l be a basis of Fk + 1. Then x = ? а,У;бС is equiv-
equivalent to i=1
Z«i№)=o t/=i Л),
i.e. to a homogeneous system of h^k linear equation with /c+1 un-
unknowns. Since this system always has a non-zero solution {al, ...,ak+i},
we have a).
The converse implication a) => codimx G^/c is obvious.
b) We have coding G^/c if and only if coding G^fc-1. By part
a) proved above, this happens if and only if there exists a /c-dimensional
linear subspace Fk of X such that Fk n G = {0}.
For the notation coding G see Ch. I, § 10, the footnote to formula A0.7).
4. ^-shrinking bases
269
Now, if Fk'nG = {0}, then for O#x'eG and Fk + 1=Fk®[x'], the
intersection Fk + lnG contains a non zero element x', which is unique
up to a homothety, i.e. we have /?).
Conversely, if we have /?), then for any /c-dimensional linear subspace
Fk of Fk+1, such that x'$Fk, we have Fk'nG={0}. This completes the
proof of lemma 4.1.
Let ? be a Banach space. For a biorthogonal system (*„,/„) ({*„} с Е,
{/„} с ?*) we shall use the notation
11/11п=11/1[х„+1,х„ + 2,...]!1 (feE*, n=l,2,...). D.3)
If E is и-dimensional A^и<оо), we shall put
ll/L=ll/L+i=-=0 (feE*). D.4)
Proposition 4.1. Let E be a Banach space and(xn,Q({xn} <=?, {/„}<=?*)
аи E-complete biorthogonal system. Then
||/1!„ = dist(/,[/!,...,./„]) (/e?*, и=1,2,...). D.5)
Proof. By the well known1 formula
dist(/,[/!,...,/„])= sup |/(x)| (feE*, n= 1,2,...), D.6)
where [/1,...,/n]1 = {
sufficient to prove that
The inclusion
(=1,...,«}, and by D.3), it is
-1,*„+2.-]- D-7)
is obvious by biorthogonality. Conversely, assume that xe[fl7 ...,/„]±,
i.e. that
Then, since E=\xj\\ ф[л:и+1;л:и + 2,...] (by Ch. I, §6, the impli-
n
cation 2°=>6° of theorem 6.1), we have x = Y. МХ)Х>+У=У' w^ a
suitable ye\_xn+l,xn + 2,...]. Thus we also have the converse inclusion
whence the equality D.7). This completes the proof.
1 See e. g. [246], p. 20. This formula also follows e. g. from the canonical iso-
metry ?*/[/,,...,/J = ?*A[//Ji ([/J

270
II. Special Classes of Bases in Banach Spaces
Proposition 4.2. Let E be a Banach space, (*„,/„) ({*„} <= E, {/„} с Е*)
a biorthogonal system, and
*"={/e?*| lim ||/J„ = ()},
V
D.8
V"= /e?* | lim/(у„) = 0 for all {yn} с Е such that
I n-* oo
sup ||vj<oo, Нт/;(_уп) = 0 (/=1,2, ...)[> D.9)
1 ^ n < oo n-* oo -*
V""= j/e?* | lim/(yB) = 0 /or а// {у„} с ? мс/г г/га?
t' n-* oo
sup \\yn\\ < со, f(yn) = O (n>i)\, D.10)
K<4)= /e?* | lim/(yB) = 0 /or a// {у„} с ? дас/г г/ia?
n-* oo
< oo, yn = У otfexfe#0 (w= 1,2,...),
sup
t=mn-i+l
where 0 = mo<m1<m2<
...I.
\_f.-]=V'=V"=V'"=Vw.
D.11)
D.12)
Proof By proposition 4.1 we have [/;] = V. The inclusions [/;] с F"
с V" с 1/D) being obvious, it will be sufficient to prove that
K<4)c|/'. D.13)
Let f$V, i.e. lim ||/||„#0. Then there exist a number eo>0, an
n-* oo
increasing sequence of positive integers {т„}, то = 0, a sequence of
scalars {а„} and a sequence of elements {yn} с ? such that
Уп
\\Уп\\
f(yn)
=
к =т
^ 1
>e0
2^ akX
rx- 1 + 1
k (и =1,2,.
(«=1,2,.
(и=1,2,.
••)>
¦¦),
¦•)•
D.14)
D.15)
D.16)
Indeed, since ll/I^IHH/llo^ll/lli^ll/lli^-, there exists an e>0
such that ||/||я^е(п = 0,1,2,...). Put mo = 0, choose z^lxj] with
1 8 ""
llzill ^ — > /(zi) > — and then choose wilr jt so that ji = X ^1>л:''
2 2 ;=i
4. ^-shrinking bases
271
\\yl — Zi
—,
with
8
^r-- Continuing by induction, choose
1 8
zj|^-, f(zn)>- and then mn,jn so that
m" 1
i=mn-i + l 2
8
-. Then, putting at = ^
(г = т„_! + 1,..., т„; п= 1,2,...) and eo = —, we have D.14), D.15) and
D.16).
Hence f? V(A) and thus we have D.13), which completes the proof.
From the above we obtain the following characterizations of k-
shrinking bases:
Theorem 4.1. Let {xn} be a basis of a Banach space E, with the a.s.c.f.
{/„}, and let к be a non-negative integer. The following statements are
equivalent:
1°. {xn} is k-shrinking.
2°. In every (k + \)-dimensional linear subspace Bk+l of E* there
exists a non-zero element feBk+l such that lim||/||n = O and there
n-> oo
exists a {k + \)-dimensional linear subspace B'k+l of E* for which this
element f is unique up to a homothety.
3°. In every (k + l)-dimensional linear subspace Bk+l of E* there
exists anon-zero element feBk + 1 such that \\mf{yn) = 0 for all bounded
n-* oo
sequences {у„} с Е satisfying \imf{yn) = 0 (/= 1,2,...) and there exists
n-* oo
a (k+ \)-dimensional linear subspace B'k+1 of E* for which this element f
is unique up to a homothety.
4°. In every (k + \)-dimensional linear subspace Bk+1 of E* there
exists a non-zero element feBk+l such that lim/(jn) = 0 for all bounded
n-* oo
sequences {у„} с Е satisfying f(yn) = O (n>i), and there exists a (k+\)-
dimensional linear subspace B'k+l of E* for which this element f is unique
up to a homothety.
5°. In every (k+ 1)-dimensional linear subspace Bk + l of E* there ex-
exists a non-zero element feBk + l such that lim f(yn) = 0 for all block basic
n-* oo
sequences у „= ]Г /хкхкф0 (n=l,2,...; mo = 0) with sup ||у„||<оо,
fc=mn-i + l l«n<oo
and there exists a (k + \)-dimensional linear subspace B'k+l of E* for
which this element f is unique up to a homothety.
Proof. This follows immediately from lemma 4.1 and proposition 4.2.
In view of the special importance of the particular case /c = 0, let
us give separately

272
II. Special Classes of Bases in Banach Spaces
Theorem 4.2. Let {х„} be a basis of a Banach space E, with the a. s. c.f.
{/„} с E*. The following statements are equivalent:
1°. {х„} is shrinking (i.e. [/„] = ?*).
2°. lim || /|| „ = 0 (/€?*).
n-»oo
3°. Every sequence {yn} <= ? w//A sup ||jn||<oo, lim/i(yII)=0
(i= 1,2,...), converges weakly to zero.
4°. isuery sequence {jn}<=? wi/A sup ||jn|| <co,/;(у„) = 0 (и>г),
converges weakly to zero. i«л < oo
ft/oc/c баягс sequence yn= ? afcxfc (и = 1,2,...; m0 = 0)
5°. isuery ft/oc/c баягс sequence yn=
и>г/й sup ||jj| < oo, converges weakly to zero.
6°. {/„} й a Aasis of E*.
7°. Тйе second conjugate space E** is isomorphic, by the mapping
(ФеЕ**)
to the Banach space of sequences of scalars
sup
1 ^ n< oo
n
I
]UjXi
1
< 00 ,
defined in Ch. I, § 5, remark 5.4.
8°. We have
( Ф(/()х; converges >,
D.17)
D.18)
D.19)
(where л denotes the canonical mapping of E into E**), and the image
of л(Е) in A2 by the isomorphism D.17) is the space
J
2^ af х( converges
i=l J
D.20)
defined in Ch. I, § 3, proposition 3.1.
// {х„} г'5 a monotone basis, the isomorphism D.17) й аи isometry.
Proof. The equivalences 1°<=>--<=>5° are nothing else but the par-
particular case /c = 0 of theorem 4.1.
The implication 6°=>1° is obvious. The implication 1°=>6° is a
consequence of Ch. I, § 12, theorem 12.1.
Assume now that we have Г. Then, by Ch. I, § 12, theorem 12.5c),e)
(or f)) and d), we have 7° and 8°.
Conversely, assume that we do not have 1°. Then, by the Hahn-
Banach theorem, there exists a ФеЕ**, Ф#0, such that Ф(/„) = 0
(и =1,2,...), whence т is not one-to-one. Thus 7°=>Г.
4. ^-shrinking bases
273
Finally, if we have 8°, then, by Ch. I, § 12, theorem 12.5f), we obtain
[/J1 = {0}, whence, by the Hahn-Banach theorem, [/„] = ?*, i.e. Г,
which completes the proof.
Other characterizations of shrinking bases will be given in §5, §6, § 12.
Now we can give the example announced at the beginning of this
section.
Example 4.1. Let J be the real linear space of all sequences of real
numbers {?„} with lim<Jn = 0, satisfying
D.21)
where the sup is taken over all positive integers n and finite increasing
sequences of positive integers kl, k2,..., k2n+i- Then J endowed with
the norm D.21) is a Banach space and the sequence {х„} <= J defined by
(«=1.2,-)
D.22)
is a shrinking monotone basis of J. The a.s.c.f. {/„} cj* is a 1-shrinking
basis of the space J*. The corresponding basis {(/„,0)}u {@,/„)} of1
J*xJ* is 2-shrinking, etc. Furthermore, the sequence {yn} aj defined by
yn=
D.23)
is a 1-shrinking basis of J, the sequence {(у„,0)} и {@,у„)} is a 2-shrink-
2-shrinking basis of J, etc.
We shall verify these assertions in several steps.
a) D.21) is a norm on J. In fact, this is obvious, except for the triangle
inequality of the norm. Let x={?n}eJ, y= {t]n}e J. Then for any
? > 0 there exists an increasing sequence of positive integers kl,k2,...,kn + 1
such that
whence ||x+j|| ^||xj| + ||y||.
1 For the definition of E x F see Ch. I, § 1. We recall that by {(/„,0)} и {@,/J)
we understand the sequence g2n-i = {(/*>0)}> 02n={(O'/n)} («=1,2,...).
18 Singer, Bases in Banach Spaces I

274
II. Special Classes of Bases in Banach Spaces
b) J is complete. In fact, this is an immediate consequence of D.21),
and also follows from the fact to be proved in step e) below that J is
isometric with a closed linear subspace of J**.
c) {х„} is a monotone basis of J. Indeed, for any x = {^„}eJ we have
Ik
where the sup is taken over all integers m ^ 1 and finite increasing
sequences of integers kuk2,...,k2m + -[ with fc^n+1. Assume that this
sup does nit converge to 0 as и->оо. Then there exist an increasing
sequence of positive integers {n}}, n0 = 0, integers ny_ i + 1 < fe'/' < ^2 < '
<k2j)m. + x < n,- and an e>0 such that
whence, by lim<Jn = 0 and D.21), Н!с„]|| = оо. This contradiction proves
n-* go
GO GO
that x= Y, ?ixi- This expansion is unique, since ? ?,oc; = 0 obviously
i = 1 i = 1
implies ?„ = 0 (n= 1,2,...). Thus {х„} is a basis of J. Furthermore,
for all finite sequences of scalars ab...,an+m and thus, by § 1, defini-
definition 1.1, {х„} is a monotone basis of J.
d) {х„} is shrinking. In fact, by the implication 2°=>1° of theorem 4.1,
it is sufficient to prove that
Нт||/||„ = 0 (feJ*), D.24)
П-* GO
where ||/||„ is defined by D.3). Assume a contrario that there exists an
/e J* for which D.24) is not satisfied. Then, as we have seen in the proof
of proposition 4.2, there exist an e>0 and a sequence {zn}<=? with
zn= Z piXi{mo = 0), \\zn\\^l (и=1,2,...) such that
Now put
D.25)
,= 1 n
D-26)
4. /t-shrinking bases
275
where
l
n
D.27)
Then zeJ, since, for any sum of type
i= 1
is true that for each term (at2j.,-at2.J either: a) mn_j
<k2i^mn for a certain и or b) mB_i + l<fe2i-i^'"H'/ии' 1;
where и'#и. The sum of the terms (at!l_,-«J2 falling in case b) is
less then or equal to
since for
i^mn (и =1,2,...) we have, by D.27) and D.21),
1 1
n ~~" n '
/ = т„ - i + 1
and since kl<k2--- <k2l+1.
On the other hand, the sum of the remaining terms (a)t2i_1-a/t2iJ
is less then or equal to
oo и _ N 2
since for m
and D.21),
n.1 + \^k2p^l<k2p<---<k2q.1<k2q^mn we have, by D.27)
Consequently,
|2 to
(°° 1 \^
5 У -= + 1 . However, by D.25) and D.26) we have
whence /(z)=oo. This contradiction proves D.24).
e) We have
codim,««n(J) = 1,
D.28)

276
II. Special Classes of Bases in Banach Spaces
where n denotes the canonical mapping of J into J**. Indeed, let
{/„} с J* be the a.s.c.f. to the basis {х„}. We shall prove that
<PeJ** => НтФ(/„) exists,
n-* oo
n(j) = {0eJ** | НтФ(/„) = 0},
п-юэ
and that there exists a <PneJ** satisfying
Фо(/.)=1 (и=1,2.-),
D.29)
D.30)
D.31)
whence J** = ti(J)© [Фо], which will complete the proof of D.28).
Let Ф be an arbitrary element of J**. Since by d) [/„] = ./*, by
Ch. I, § 12, theorem 12.1 we have
However, for n = 1,2,... we have
/
= ]| j II sup
where the sup is taken over all positive integers m and finite increasing
sequences of positive integers k^k2, ...,k2m+l with <P(fkj) replaced by
0 if kj>n. Hence, for и—>oo,
j
and thus
, D-32)
where the sup is taken over all positive integers m and finite increasing
sequences of positive integers kuk2,...,k2m+1.
On the other hand, since by с) {х„} is a monotone basis of J, by
Ch. I, § 12, proposition 12.2 c) we have
4. ^-shrinking bases
277
whence
D.33)
for any positive integer m and finite increasing sequence of integers
к1гк2,..., k2m+l. Combining D.32) and D.33) gives
ЦФ|| =
Гm
= sup X
Li=l
(Ф(Л2т+1))
D.34)
where the sup is taken over all positive integers m and finite increasing
sequences of positive integers kl,k2,...,k2m + 1.
Now we can prove D.30). If <Pen(J), then Ф = п{х) for some xeJ,
whence, by the definition of J, Ф(/п)=/п(х)-»0 as n->oo. Thus we
have the inclusion с in D.30). Conversely, let <PeJ** be such that
НтФ(/„) = 0. Then, by D.34) and the definition of J, the sequence
х={Ф(/„)} belongs to J. Hence, by c), x = ]Г Ф(/)х, = ]Г /(х)х( and
i= 1 i= 1
thus Ф(/)=/(х)(г'=1,2,...). This, together with d), proves that Ф = л(х),
whence the inclusion = in D.30), whence the equality D.30).
Furthermore, we have D.29). Indeed, otherwise for some fixed s>0
and for n arbitrarily large it would be possible to have |Ф(/„) — Ф(/т)| >?
for m>n, which contradicts D.34).
Finally, since by D.21) we have
= 1 (и=1,2,...),
D.35)
from Ch. I, § 12, proposition 12.2c) it follows that there exists a <PoeJ**
satisfying D.31), which completes the proof of D.28).
f) {/„} is a 1-shrinking basis of J*. In fact, by d) and Ch. I, § 12,
theorem 12.1, {/„} is a basis of J* with the a.s.c.f. {л{х„)}. Since1 by c)
[7t(xn)] = ?i(J), from e) it follows that {/„} is 1-shrinking.
g) The basis {(/„,0)} и {@,/„)} of J* x J* is 2-shrinking etc. Indeed,
by e) there exists а Ф0е7** such that every <PeJ** can be uniquely
written in the form Ф = л;(х) + лФ0, with xeJ and -oo<A<oo.
1 Indeed, since n(xn)en(J) (n=l,2,...) and since n(J) is a closed linear sub-
space of J**, we have [л(х„)] с n(J). On the other hand, if <Pen(J), then for
some xeJ we have Ф = п(х)=п{
7сG)с[тг(х„)]. Hence, finally, тг(
/;(х)х; 1=
and thus

278
II. Special Classes of Bases in Banach Spaces
Hence every (<P,'P)eJ** xj** can be uniquely written in the form
(n(x),n(y)) + (l<P0,[i<P0), with x,yeJ and — оо<Я,^<оо, and thus
codimJ,»)<J,,7i(J)x ti(J) = 2.
D.36)
However, for any pair of Banach spaces ?, F we have the linear
isometry1 ?* x F* = (?xF)*, by the mapping (/,#)->• 0, where
ф(х,у) = Дх) + д(у) (xeE,yeF).
Now, since {/„} is a basis of J* with the a.s.c.f. {n(xn)} (see f)),
{(/„,0)}u{@Jn)} is a basis of J*xJ* with the a.s.c.f. {(л{х„),0)}
и {@,л:(х„))} spanning the subspace 7r(J)x7i(J) (we identify (J*xJ*)*
with J**xJ** by the above isometry), whence, by D.36), {(/„,0)}
и {@,/„)} is 2-shrinking, etc.
h) By D.35) and Ch. I, § 4, proposition 4.3, the sequence {yn} cz J
defined by D.23) is a basis of J, with the a.s.c.f.
9n = f,~fn+i (и=1,2,.-.). D.37)
Since fne[gj] for и = 2,3,... and since fx$\_g^\ (because Фо(/1)=1,
?оЫ = ФоШ-Фо(/»+1) = 0 for и=1,2,... by D.31)), we have, taking
into account d),
codim,, 0 J = codim^, [#„] = 1,
which shows that {yn} is 1-shrinking. With an argument similar to that
used in g) it follows that {(у„,0)}и {@,jn)} is 2-shrinking, etc. This
completes the proof of the assertions stated in example 4.1.
The answer to the problem of existence of /с-shrinking bases is
negative, as shown by
Example 4.2. Let ? be a Banach space with a basis, whose conjugate
space ?* is non-separable (e.g. E-l1, or ? = Lx([0,l]), or ? = C([0,l])).
Then for any non-negative integer k, the space ? has no /c-shrinking
basis.
In the case when k = 0, the answer to the problem of the existence
of non-fe-shrinking bases is also negative, as shown by
Example 4.3. Let ? be a reflexive Banach space with a basis. Then
all bases of ? are shrinking.
In fact, this is an immediate consequence of Ch. I, § 12, corollary 12.2,
but can be seen also directly, as follows. If {х„} is a non-shrinking basis
of ?, then for the a.s.c.f. {/„} <= ?* we have [/„]#?*, whence there
exists а Фе?**, Ф#0, such that Ф(/„) = 0 (n=l,2,...). Since {/„} is
See e. g. [10], p. 192, theorem 14 and the remark to this theorem.
5. Retro-bases in conjugate Banach spaces
279
total on ?, it follows that Ффл{Е\ and thus ? cannot be reflexive.
This proves our assertion.
In Vol. II, Ch. IV we shall see that if к is a positive integer, then every
infinite dimensional Banach space with a basis possesses a non-fe-
shrinking basis.
§ 5. Retro-bases in conjugate Banach spaces
Definition 5.1. Let ?* be the conjugate space of a Banach space ?.
A basis {/„} of ?* is called retro-basis, if for the a.s.c.f. {Ф„} <= ?** we
have {Ф„} с: ti(?) (where л denotes, as before, the canonical mapping
of?into?**).
For instance, the natural basis of E* = lp(p^\) is a retro-basis. On
the other hand, the basis {/„} of E* = ll defined in Ch. I, § 14, example
14.1, formula A4.3), is a non-retro-basis, since for the a.s.c.f. {%„} cz ?**
we have /1фп(Е) (by Ch. I, §14, formula A4.4)). Furthermore the
natural basis Фои {л:(х„)} of J** (§ 4, example 4.1) is a non-retro-basis
of J**. In these two examples only the first coefficient functional фп(Е),
and all other coefficient functionals en(E). However, there also exist
non-retro-bases such that all associated coefficient functionals
Фпфл{Е) (и =1,2,...), e.g. the basis {hn} of ?* = /J defined in Ch. I,
§ 13, example 13.3, formula A3.14).
Some characterizations of retro-bases among bases of conjugate
Banach spaces are collected in
Proposition 5.1. For a basis {/„} of a conjugate Banach space ?* the
following statement are equivalent -.
1°. {/„} is a retro-basis of ?*.
2°. There exists a (shrinking) basis {х„} of the space E such that {/„}
is the a.s.c.f. to {х„}.
3°. {/„} is a w*-Schauder basis of ?*.
4°. All subspaces
Mn= Uu-Jn-u /»+i,/. + 2,-] E-1)
of E are closed for the w*-topology <?(?*, ?).
Proof. 1°=>2°. If{/n}isaretro-basisof?*,forthea.s.c.f. {Ф„} с ?**
we have Ф„ = л;(х„) with some х„е? (и=1,2,...), whence /((х;.) = с>и
{i,j= 1,2,...). Then, by Ch. I, § 12, corollary 12.1, {х„} is a basis of ?,
which is obviously shrinking and has the a.s.c.f. {/„}.
The equivalence 2°<=>3° is an immediate consequence of Ch. I,
§ 14, theorem 14.1.

280
II. Special Classes of Bases in Banach Spaces
2°=>4°. If we have 2°, then ^{Х]) = д^ (i,j=l,2,...) and by Ch. I,
§ 12, theorem 12.1, we also have
/=L
Consequently, for each n the subspace Mn of ?* defined by E.1) is
nothing else but the a(E*, ?)-closed hyperplane {feE* | /(х„) = 0}.
4°=>1°. If we have 4°, then /пфМп = Мп = the a(E*, Enclosure
of М„ (и =1,2,...), whence, by a well known corollary of the Hahn-
Banach theorem (on separation of closed subspaces from outside points),
applied in E* endowed with the topology a(E*,E), there exists an xneE
such that fn(Xj) = dnJ (nj'= 1,2,...). Consequently, for the a.s.c.f.
{<?„}<=?** to {/„} we have Фп = п(хп)еп(Е) (n=l,2,...), which com-
completes the proof.
The equivalence 1°<=>2° of proposition 5.1 above shows that the
study of retro-bases reduces to that of shrinking bases. Therefore in
the sequel we shall not study retro-bases of E*, but only shrinking
bases of E.
The equivalence 1°<=>3° of proposition 5.1 reduces the study of
retro-bases {/„} in conjugate Banach spaces E* to that of bases {/„} cE*
which are also w*-Schauder bases of E*. The implication 1°=>3°
shows that a retro-basis {/„} of E* is necessarily a w*-Schauder basis
of E*, but the converse of this statement is not true (if we do not assume
that [/„] = ?*), as shown e.g. by the sequence of coordinate functionals
{/„} in E* = {l1)* = m (see Ch. I, § 14, theorem 14.1).
Problem 5.1. a) Does every infinite dimensional conjugate Banach
space E* with a basis {/„} possess a retro-basis? b) If E has a basis and
E* is separable, does E* possess a basis? What about a retro-basis?
An affirmative answer to a) would imply an affirmative answer to
Ch. I, §12, problem 12.1.
The answer to the problem of existence of non-retro-bases is ne-
negative, as shown by
Example 5.1. Let ? be a reflexive Banach space with a basis. Then E*
has a basis and all bases in E* are retro-bases.
Indeed, by Ch. I, § 12, corollary 12.2, E* has a basis. On the other
hand, from E** = n{E) it follows that every basis {/„} of E* is a retro-
basis.
One may ask whether a normal basis of a conjugate Banach E* is
necessarily a retro-basis of E*. We shall now show that the answer is
negative.
Let us recall that a norm |||/||| on a conjugate Banach space E* is
said to be the dual norm of a norm |ll*||| on E if
5. Retro-bases in conjugate Banach spaces
= sup |/(x)| {feE*).
xeE
281
E.2)
Lemma 5.1. Let E be a Banach space and let |||/||| be a norm on the
conjugate space E*, equivalent to the initial norm on E*. If the set
= {feE*
E.3)
is closed for the weak* topology a{E*, E), then there exists on E a norm
\\\x\\\ equivalent to the initial norm on E, such that \\\f\\\ is the dual norm
of |||*|||.
Proof. Since HI/HI is equivalent to the initial norm ||/|| on E*, there
exist two constants Cl5C2>0 such that
CJ/KIH/IKC2 И/И {feE*). E.4)
Put
|||x|||= sup |/(x)| (xeE). E.5)
feE*
lll/IIKi
Then |||л:||| is a norm on E, equivalent to the initial norm ||*|| on E,
since by E.5) and E.4) we have
-J-1|*|| = jr sup |/(*)| ^ 111*111 ^ i- sup
C2 C2 feE* Cl feE
ll/ll« 11/11 «
sup |/(*)| = -J- II*» (xeE).
C
Furthermore, by E.5) we have |/(*)|< |||/||| |||*||| for all xeE, feE*,
whence
sup |/(*)|^ HI/HI {feE*). E.6)
je?
On the other hand, since by our hypothesis A is a a{E*, ?)-closed
circled convex set in E*, for every /oe?* with |||/0|||>l there exists,
by a theorem of S. Mazur1, an xoeE such that /0(x0)> 1, |||*0|||
= sup |/(*0)|<l. Consequently, in E.6) we have the equality sign,
feE*
i.e. HI/HI is the dual norm of |||x||| (since if there existed an feE* with
sup |/(*)|< HI/HI, then for /0=/ sup |/(x)|\-7 one would have
xeE
xeE
lll/olll>l, sup |/o(x)l = l, contradicting the above), which completes
xeE
111*111 «1
the proof of lemma 5.1.
1 See e.g. [270], p. 108, theorem 3.

282
II. Special Classes of Bases in Banach Spaces
Proposition 5.2. Let {х„} be a shrinking basis of a Banach space E,
with the a.s. с f {/„} с E*. Then
= sup
!/(*<)/,
(feE*)
E.7)
is an equivalent norm on E*, which is the dual norm of a suitable equivalent
norm on E.
Proof. Since {х„} is shrinking, {/„} is a basis of ?*, whence
Z f{Xi)fi
HI/IK2 sup
Z /(*/)/•
<2v(/nJ||/||(/eE*),
and thus HI/HI is a norm on E*, equivalent to the initial norm on E*.
Therefore, by lemma 5.1, it will be sufficient to prove that the set
A = {feE* | lll/HKl} is closed for the weak* topology <x(E*,E).
Let {gd}deA be a net in A, converging in the weak* topology <r(E*, E)
to an element geE*. Let s>0 be arbitrary and fix n, m with 1<и
oo. Then for any deA we have, by |||gj|| < 1,
m
Z 9(xi)fi
/Л /Л
m
Z&o
i = n
m
z
i — n
+
m
^ l +
Furthermore, since gd(xi)->-g(xi) (г =1,2,...), we may choose a
do = do{m,n)eA such that
Consequently, since e>0 and n, m have been arbitrary, we get
Z
; n=l,2,...),
whence |||gf|||<l, i.e. geA, which completes the proof of proposition 5.2.
Now we can give the example annouced above.
Example 5.2. Let J be the space of §4, example 4.1, let {х„} be the
unit vector basis D.22) of J (which is shrinking, as shown there), let
{/„} be the a.s.c.f. to {х„}, let |||/||| be the equivalent norm on J* defined
by E.7) and let E be the space J endowed with the equivalent norm |||x|||
5. Retro-bases in conjugate Banach spaces
283
of proposition 5.2. We claim that the sequence {д„} <= ?* defined by
ffi=/i. 9n=L-i-fn (и = 2Д...) E.8)
is a normal basis of ?*, but not a retro-basis of E*.
Indeed, from step h) of the proof of the assertions of § 4, example 4.1,
it follows that {gn} is a basis of J*, whence also of J* endowed with the
equivalent norm |||/|||, i.e. of E*. Furthermore, obviously the a.s.c.f.
{<?„} <= ?** to the basis {gn} of E* is
«Р„=-Фо+
E.9)
i= 1
where ФоеЕ** is the functional satisfying D.31) and where n denotes
the canonical mapping of E into ?**. Thus Ч>„фп{Е) {п= 1,2,...), i.e.
the basis {gn} of E* is "very non-retro".
Finally let us prove that the basis {gn} is normal, i. e. that
lllffJI= 1110=1 (и=1,2,-Х where |||"PJ| = SUP IW)!- For апУ
/eJ*
x={QeJ we have |/„(х)|= |с„|^ \\x\\, which, together with |/„(х„)|=1
= ||х„||, gives ||/„|| = 1 (и =1,2, ...)• Furthermore, for any x={QeJ
we have |/,.r/J(x)|= |^_i -^„|^ ||x|| and therefore ||/B_! —/„|| ^ 1
(и = 2,3,...). Hence
i = k
= max(!!/„_!||, ||/„_!-/„||, ||/J|)=
1 = 2,3,...),
and thus also |||!Р,||| = |||Ч1119,|||>|!Р,У = 1 (и=1Д-)- Now let
/eJ* with HI/HI <1 and fe, m be arbitrary. Then, by D.35),
Z
whence, taking into account D.31),
-*o(/)+ Z !>(*.•)](/)
i=/t
\ n- 1
-*o( I/(*»)/«)+ Z.
= 1 / i= 1
Z /W
1,

284
II. Special Classes of Bases in Banach Spaces
and thus IH^JKl («=1,2,...), which, together with the opposite
inequality observed above, gives |||Wn||| = 1 (и =1,2,...). This com-
completes the proof of the assertions of example 5.2.
Let us also mention, briefly, another interesting example of a normal
non-retro-basis of a conjugate Banach space.
Example 5.3. Let E = C(wa) be the space of all continuous functions
on the (compact metrizable) space of of all ordinal numbers «Sew™
endowed with the order topology (with "open intervals" as neighbour-
neighbourhoods). Then it is well known1 that E* = ll. Let {/„}<=?* be the
image in E* of the natural basis of I1 by this equivalence. Then obviously
{/„} is a normal basis of E*. Furthermore, the a.s.c.f. {Ф„} <= E** to
{/„} is equivalent to the natural basis of c0, but E=C(of) is not2
isomorphic to c0, and thus {/„} is not a retro-basis of E*.
§ 6. /г-boundedly complete bases
Definition 6.1. Let к be a non-negative integer. A basis {х„} of a
Banach space E is called k-boundedly complete, if
a) in every (k+ l)-dimensional linear subspace Pk + 1 of the Banach
space of sequences of scalars
К
sup
1 ^ Л < GO
п
у
i= 1
< oo
F.1)
defined in Ch. I, § 5, remark 5.4, there exists a non-zero element {a,} ePk +1
such that the series ? «;*; 1S convergent;
i= 1
b) there exists a (k+ l)-dimensional linear subspace P'k + 1 of A2 for
which the above element {a,-} is unique up to a homothety.
The O-boundedly complete bases, i.e. the bases {х„} such that the
< oo implies the convergence of
relation sup ? a.xi
l«n<co i=1
are also called boundedly complete bases.
For instance, the natural basis of lp (p^ 1) and the Haar basis of
Lp([0,1]) (p>l) are boundedly complete bases. We shall also give an
example of a /c-boundedly complete basis, for arbitrary k>0 (see
example 6.1 below).
1 See [217], theorem 6 and [26], corollary 3.
2 See [26], theorem 1.
6. ^-boundedly complete bases 285
Some characterizations of k-boundedly complete bases are collected
in
Theorem 6.1. Let {xn} be a basis of a Banach space E, with the a.s.c.f.
{fn}, and let к be a non-negative integer. The following statements are
equivalent:
1°. {х„} is k-boundedly complete.
2°. We have1
codim?.,Gi(?)©[/n]1) = /c,
where л denotes the canonical mapping of E into ?**.
3°. We have
j, ф(Е) = к,
F.2)
F.3)
where ф denotes the canonical mapping of E into [/„]*•
4°. In every (k + Vydimensional linear subspace Qk + 1 of ?** there
oo
exists a non-zero element <PeQk+1 such that the series Z ^(ft)xi is
convergent, and there exists a (k + \)-dimensional linear subspace Q'k+l
of E** for which this element Ф is unique up to a homothety.
5°. In every (k+\)-dimensional linear subspace Rk+1 of [/„]* there
00
exists a non-zero element ?/ei?t + 1 such that the series Z ?(fdxi is
i= 1
convergent, and there exists a (k+l)-dimensional linear subspace R'k + 1
of [/„]* for which this element ? is unique up to a homothety.
Proof. Г=>4°. Assume that we have Г, and let Qk + 1 be an arbitrary
(fc+l)-dimensional linear subspace of ?**. If Qk+1 n [/„]x/{0}, then
GO
for any Фебл+1п[/„]х such that Ф#0, the series Z ф(/;)х; *s
convergent (to 0). If 6* +1 n [/J1 = 1°}' then i=1
is a one-to-one linear mapping of Qk+i into the space A2 defined by
F.1) (we recall that by Ch. I, §12, proposition 12.2c), {Ф(/„)}еЛ2 for
every Фе?**). Hence Pk+1 = {№ifn)}\ Фе<2к + 1} is a (k+ ^-dimen-
^-dimensional subspace of A2 and thus by 1° there exists a non-zero element
00
{^(/n)}e^\+i> i-e- a non-zero $eQt+1, such that the series ? ^{fdxi
is convergent. ¦ =l
1 Since for any basis {х„} of E, n(E) ф [/„]х and <?(?) are norm-closed linear
subspaces of E** and [/„]* respectively (by Ch. I, § 12, theorem 12.5), the left hand
sides of F.2) and F.3) have meaning for any basis {xn} of E.

286
II. Special Classes of Bases in Banach Spaces
On the other hand, by Г there exists a (k+ l)-dimensional linear
subspace P'k+1 of A2 such that the non-zero element {<Xj}eP'k + 1 for
ОС
which the series ? а,х, is convergent, is unique up to a homothety.
'"!
Let {а*1»}, {а<2)},.'.".! K+1)} be a basis of P'k+1. Then by P'k+l^A2
and Ch.I, §12, proposition 12.2c), there exist ФиФ2,...,Фк+1еЕ**
such that
whence
is a linear mapping of [Ф(]- = / onto Ft + i> and hence an isomorphism.
Thus for e;+i = M=i we have dim&+1 = k+l and the non-zero
00
element 0eQ'k + 1, for which the series ? ФШХ< is convergent, is uni-
unique up to a homothety. '=1
4°o2°. By Ch. I, § 12, theorem 12.5 f), we have
ОС ~|
У Ф(/;)-Х; converges>.
Hence, by §4, lemma 4.1, we have 4°<=>2°.
2°=>3°. If ?** = 7i(?)©[/n]1©B, where dimB=k, then by re-
restriction to [/„],
[/„]* = ф(Е)®(В\ип]),
where dim(B|[/n]) = k (since Bn [/„]1= {0}).
3°o5°. By Ch. I, § 12, theorem 12.5e), we have
ф(Е) =
i converges
Hence, by § 4, lemma 4.1, we have 3°<=>5°.
5°ol°. By Ch.I, §12, theorem 12.5c), [/„]* is isomorphic to the
Banach space A2, by the mapping
(^ [/„]*).
Hence 5Dol°, which completes the proof of theorem 6.1.
From theorem 6.1 it follows that the k-boundedly complete bases
have the following relation of duality with k-shrinking bases:
Corollary 6.1. Let {х„} be a basis of a Banach space E, with the
a.s.c.f. [/„], and let к be a non-negative integer. Then
a) {х„} is k-boundedly complete if and only if {/„} is a k-shrinkmg
basis of [/„].
6. &-boundedly complete bases
287
b) {х„} is k-shrinking if and only if {/„} is a k-boundedly complete
basis of [/„].
Proof, a) The a.s.c.f. {?„}<= У„~\* to {/„} is nothing else but
?п=ф(хп) (n=l,2,...), whence1 ["Р„] = [ф(х11)] = ф(?)- Consequently,
by the equivalence Vo3° of theorem 6.1, {х„} is k-boundedly complete
if and only if codim[/n],['//n] = fe. However, by §4, definition 4.1, this
condition means that {/„} is a k-shrinking basis of [/„].
b) Since ф is an isomorphism of ?=[х„] onto ф{Е) =['?„] (by
Ch. I, § 12, theorem 12.2e)) and since any isomorphic image of a /c-shrink-
/c-shrinking basis is obviously a k-shrinking basis, {х„} is a k-shrinking basis
of ? if and only if {4>'„} = {ф{хп)} is a k-shrinking basis of [?*„] = ф(Е).
However, by part a) (applied to the basis {/„} of the Banach space
[/„]), this happens if and only if {/„} is a k-boundedly complete basis
of [/„]. This completes the proof of corollary 6.1.
Combining theorems 4.1, 6.1 and corollary 6.1, one can obtain other
characterizations of k-shrinking and k-boundedly complete bases.
Now we can also give the example announced at the beginning of
this section.
Example 6.1. Let J be the Banach space of sequences of real num-
numbers introduced in §4, example 4.1. Then the basis {х„} of J defined by
D.22) is 1-boundedly complete. The corresponding basis {(х„,0)}
и {@,х„)} of J x J is 2-boundedly complete, etc. Furthermore, if {/„} <= J*
is the a.s.c.f. to {xn}, then {/„} is a boundedly complete basis of J, the
sequence {#„}<= J* defined by D.37) is a 1-boundedly complete basis
of [#„], the corresponding basis {(#„,())} и {@,#„)} of 0„] х (>„] is
2-boundedly complete, etc.
Indeed, as shown in example 4.1, part f), {/„} is a 1-shrinking basis
of J* = [/„]. Hence, by corollary 6.1a), {х„} is 1-boundedly complete.
Furthermore, as shown in example 4.1, part g), {(/„,0)}u {@,/„)} is a
2-shrinking basis of J* xJ* = (J xj)*. Since this is nothing else but
the a.s.c.f. to the basis {(х„,0)}и {@,х„)} of J x J, from corollary 6.1a)
it follows that {(х„,0)}и {@,х„)} is 2-boundedly complete, etc. On the
other hand, as shown in example 4.1, part h), the sequence {yn) cz J
defined by D.23) is a 1-shrinking basis of J, having the a.s.c.f. {gn}
defined by D.37). Hence, by corollary 6.1b), {д„} is a 1-boundedly com-
complete basis of [#„]. Furthermore, as shown in example 4.1, part h),
{(yn,0)} и {@,у„)} is a 2-shrinking basis of J, with the a.s.c.f. {{д„,0)}
и{@,#„)}. Hence, by corollary 6.1b), {{gn,0)}vj {{0,д„)} is a 2-boundedly
complete basis of [д„] х [#„], etc. This completes the proof of the as-
assertions stated in example 6.1.
See the argument used in § 4, example 4.1, the footnote to part f)-

II. Special Classes of Bases in Banach Spaces
The answer to the problem of existence of fe-boundedly complete
bases is negative, as shown by
Example 6.2. Let к be a non-negative integer and let ? be a Banach
space with a basis, which is not isomorphic to any subspace of codimen-
codimension к of any conjugate Banach space (e.g. ? = c0, or ? = ^([0,1]), or
? = C([0,1])). Then the space ? has no fe-boundedly complete basis.
Indeed, assume that {х„} is a fe-boundedly complete basis of ?, with
the a.s.c.f. {/„}. Then, by the implication Г=>3° of theorem 6.1, we
have codim[fn],ф(Е) = к, where ф is the canonical mapping of ? into
[/„]*. Since by Ch. I, § 12, theorem 12.2e), ф is an isomorphism, it fol-
follows that ? is isomorphic to the subspace ф(Е) of codimension к of the
conjugate space [/„]*, which contradicts our assumption on ?.
In the case when к = 0, the answer to the problem of existence of
non-fe-boundedly complete bases is also negative, as shown by
Example 6.3. Let ? be a reflexive Banach space with a basis. Then
all bases of ? are boundedly complete.
Indeed, assume that {х„} is a basis of ?. Then, as we have seen in
§4, example 4.3, {х„} is shrinking, and thus the a.s.c.f. {/„} is a basis
of ?* = [/„]. Since ?* is also reflexive, again from § 4, example 4.3, it
follows that {/„} is shrinking. Hence, by corollary 6.1a), {х„} is bound-
boundedly complete.
In Vol. II, Ch. IV we shall see that if к is a positive integer, then
every infinite dimensional Banach space with a basis possesses a non-/c-
boundedly complete basis.
In view of the special importance of the particular case к = 0, let us
give separately
Theorem 6.2. Let {xn} be a basis of a Banach space ?, with the a.s. c.f.
{/„} <= ?*. The following statements are equivalent:
n
Г. {х„} is boundedly complete (i.e. the relation sup Z а'х- <G0
implies that Y^ ое(х,- converges).
2°. 7i(?)© [/„]1 = ?**, where n denotes the canonical mapping of E
into ?**.
3°. ф(Е) =[/„]*, where ф denotes the canonical mapping of E into
[/n]* (hence, in this case E is canonically isomorphic—and, if {х„} is a
monotone basis, E is canonically linearly isometric—to [/„]*j-
4°. For every ФеЕ** the series ? Ф(fi)xi converges.
i= 1
00
5°. For every "Pef/J* the series ? <P(fi)xi converges.
6. ^-boundedly complete bases
289
These statements are implied by the following statements, equivalent
to each other:
6°. For every number c>0 there exists a number rc>0 (independent
of n and {а,} с К), such that
n
Z aixt
i= 1
= 1,
00
У ax
i = n+ 1
imply
Z aixt
i= 1
F.4)
7°. For every e>0 there exists a 5 = 5(e)>0 (independent of n and
) с К), such that
Z atxi
Z aixi
= 1 imply
Z atxi
i = n+ 1
F.5)
Proof. The equivalences Го--о5° are nothing else but the par-
particular case k = 0 of theorem 6.1.
To show the equivalence 6°oT, assume that 7° is not satisfied, i.e.
there exists an e>0 such that for every c>>0 one can find (ijcK
and n with
Z a<x.-
00
У ax
i= 1
= 1,
00
Z a.xi
i = n + 1
>?.
F.6)
Then for с = (where v,Xn} is the norm of the basis {х„}) there
exists no rc>0 so as to have F.4). Indeed, let rc>0 be arbitrary, let
' -, whence rc = -—-, and let ft = — ~ (i= 1,2,...), where
{а,-} а К and n are as in F.6). Then
z ^
z /u-
= 1,
Z atxi
Z *jXj
Z *ixj
Z Рл
1
Z «ixj
]= 1
which proves that non-7° => non-6°.
= c,
19 Singer, Bases in Banach Spaces 1

290
II. Special Classes of Bases in Banach Spaces
Conversely, assume that 6° is not satisfied, i.e. there exists a c>0
such that for every rc> 0 one can find {а,-} с К and n with
Z aixi
= 1,
Z <*.¦*•
i = n+ 1
I«.X»
<\+rc. F.7)
Then for e = c(l —»/), where 0<f/<l is arbitrary, there exists no 5
so as to have F.5). Indeed, let 3>0 be arbitrary, such that1 <5<»/, let
rc == , whence д = c , let {a,} <= К and и as in F.7) and let
(i=l,2,...). Then
n
z
i =
GO
V « V
1
Z «Л
l+rc
Z ft*
= 1,
z /u
У а,х,
i—i J J
which shows that non-6° => non-7°.
Finally, assume that we have 6C and let {ое„} <= К be such that
sup
Z <*.•*
i= 1
= A. Choose a sequence of positive integers {nk} such
that lim \\уПк\\ = lim PvJi = B, where we have put yn = Y <xixi(n= 1,2,...).
00
If B = 0, then Y, aixi converges (to 0). If В/О, we shall show that {у„к}
i= 1
is a Cauchy sequence. In fact, otherwise there would exist a c>>0 and
subsequences [ynu}, {>'„,.} of {уПк} with nk >щ. (/'= 1,2,...), such that
Then, since
Упк, Уп\,
\\у4
(/=1,2,...).
— = c>0,
1 This is no restriction of the generality, since if for a <50>0 we have F.5),
then also for any 5 ^ <50 we have F.5).
6. &-boundedly complete bases
291
we would have, by 6°,
whence
and thus
\\Уп
В = lim
j-* со
rc) = B(l+rc),
which is impossible, since В/О. Consequently, {у„к} is a Cauchy
sequence, whence limy =xeE. Then for the sequence of coefficient
fe-* GO
functionals {/„} с Е* associated to the basis {х„} we have
fj(x) = lim fftj = lim /. ( X «/*) = «y 0= 12,...),
к —^ oo к —* ос \ /
whence x= ? /;(x)x,= ? а(х;, which shows that ? a,x,- converges
i = 1 i = 1 i = 1
n
whenever sup Z а«* < °°- This completes the proof of theorem
1 ^ n < oo
6.2. I = 1
The converse implications 1°=>6°, 1°=>7° in theorem 6.2 do not
hold even if {xn} is monotone and normal, as shown by
Example 6.4. Let ? = /x and let {х„} be the basis of E defined by
xl=\el+\e2, x2—— \e1-\-\e2, хп = е„ (и = 3,4,...), F.8)
where {en} is the natural basis of/1. Then {х„} is a boundedly complete
basis of E, which is monotone and normal, but {х„} does not satisfy
conditions 6°, 7° of theorem 6.2.
Indeed, {х„} is a boundedly complete basis of E, since so is {en}.
By § 2, example 2.1, {х„} is also normal. Since
at-a
\\а.1Х]_+а2х2\\ =
1-(*2
Z afxi
= max(|a1|,|a2|)+
i=3

292
II. Special Classes of Bases in Banach Spaces
is also monotone. Finally, for а1 = а2 = 1»аз = а4="=0> we have
X2
i=3
= 1,
X, +X-, +
1 ,
which shows that {xn} does not satisfy conditions 6°, 7°.
Since this basis {xn} is equivalent to the unit vector basis {en} of /\
which obviously satisfies 6°, 7°, example 6.4 also shows that conditions
6°, 7° are not isomorphic properties (i.e. they are not invariant under
an isomorphism of the space E onto another space Et). The following
problem arises naturally:
Problem 6.2. If \xn} is a boundedly complete basis of a Banach
space E, is it possible to introduce on E an equivalent norm, in which
{xn} satisfies conditions 6°, 7° of theorem 6.2?
§7. Bases of types wc0, (wcc)*, swc0 and (swc0)
For {zn}, z in a Banach space E, such that
to z, we shall use the notation zn^z.
zn} converges weakly
Definition 7.1. A basis {xn} of a Banach space E is said to be
a) of type wc0, if {xn} is a bounded basis1, and х„-^0;
b) of type(wc0)*, if the a.s.c.f. {/„} is a basis of type wc0 of [/„];
c) of type swc0, if {х„} is a bounded basis and there exists a sub-
subsequence {xin} of {х„} such that xin-^0;
d) o/ ?j/?e ($н>с0)*, if the a.s.c.f. {/„} is a basis of type swc0 of "[/„]•
For instance, the natural bases of c0 and F (p> 1) and the normalized
Haar basis2 of LP ([0,1]) (p> 1) are of type wc0, while the basis of c0
constructed in § 2, example 2.2, formula B.24), and the natural basis of I1
are of type non-wc0. Dual examples of bases of types (wc0)* and non-
(wc0)* follow from proposition 7.3 below (see also theorem 7.1).
Let us make the following remark concerning the hypothesis of
boundedness of {xn} in definition 7.1 above. We have assumed that
inf ||x„||>0, since for every basis {xn} the basis < х„\ conver-
l«»<=c (. Л II *„ II J
ges (whence also weakly converges) to 0 and we have also assumed
that sup |jxj< oo, since for every basis {х„} the a.s.c.f. to the basis
converges (whence also weakly converges) to 0.
n I
1 I.e. satisfying 0< inf j|xJK sup ||xj<oo (see Ch. I, § 3, definition 3.2).
1 ^n< <x 1 ^и< go
2 See § 2, formula B.3).
7. Bases of types wc0, (wc0)*, swc0 and (swc0)*
293
Similar remarks are also valid for the definitions which we shall
give in §§ 8, 9 and 10.
Proposition 7.1. A subsequence {x,J of a bounded basis \xn] of a
Banach space E satisfies xin Л 0 if and only if {xin} is a basis of type
wco °f [xij- Consequently
a) Every subsequence \xin] of a basis {х„} of type wc0 is a basis of
type wc0 of its closed linear hull [x;j.
b) A bounded basis {xn} is of type swc0 if and only if it contains a
subsequence {xin} which is a basis of type wc0 of [xf J.
Proof If {x,J is a subsequence of a basis {х„}, then, by Ch. I, § 4,
proposition 4.1, {x;J is a basis of [xfj. On the other hand, by the Hahn-
Banach theorem, xin -^ 0 if and only if xin -> 0 for the w-topology
of [x,J (i.e. for the weak topology с([х(п], [-?,„]*)). Hence xin^>0 if
and only if {x,J is a basis of type wc0 of [xin]. The statements a) and b)
being obvious consequences of this result, the proof of proposition 7.1
is complete.
Proposition 7.1b) makes possible to deduce certain properties of
bases of type swc0 from those of bases of type wc0.
Theorem 7.1. a) The Schauder basis of C([0,1]) is of types non-wc0,
swc0 and non-(swc0)*.
b) The normalized Haar basis of Ll{\Q,Y]) is of types non-swco,
non-(wc0)* and(swc0)*.
Proof a) Define /eC([0,l])* by
where
/(*)=*(*„) (xeC([0,l])),
and consider the subsequence {x2kn + l } of the Schauder basis1
of C([0,l]), where
kn = n2-l («=1,2,...),
Then
2/.-1 2/.-1
G.1)
G.2)
G.3)
G.4)
G.5)
1 For the definition of the Schauder basis {х„}™=0 of C([0,l]) see Ch. I, §2,
example 2.2, formula B.3).

294 II. Special Classes of Bases in Banach Spaces
whence, by G.2),
G.6)
On the other hand, since
i< I i = ^rW^i = ^T (я=1Д-Ь G-7)
we also have, taking into account G.2) and G.5),
t = 2/-cl + у — <-^=- (,
^ " j=n+l ^ ^
G.8)
Consequently, taking into account G.1), G.6), G.8) and the definition
Г2 /„ — 1 2/„ 1
of х2к„ + и on the segment k +1 , t +1 , we obtain
2/.-1
2/B-l
* 1
whence
G.9)
which proves that {х„}^=0 is of type non-wc0.
On the other hand, for the subsequence {xik]^=o={x2k+l}kX!=o of
{х„}„ж=0 we have \\xik\\ = 1 (/c = 0,1,2,...) and jlimxI.|[@ = 0 (?g[0,1]),
whence1 x,k-^0. Thus {х„}*=0 is of type swc0.
Finally, by §9, theorem 9.3 a) and the implication 1°^>5 of theorem
9.1, there exists a <2>eC([0,l])** such that
|ФШ1 = 1 (и = 0,1,2,...),
G.10)
1 See e.g. [10], p. 134.
7. Bases of types wc0, (wc0)*, swc0 and (swc0)*
295
where {/„}„ж=0 <= C([0,1])* is the a.s.c.f. to the Schauder basis {*„}„%
of C([O,lj). Thus {jcb}^°=0 isoftypenon-Ewc0)*.
b) Assume that the normalized Haar basis1 \zn) of Ll([0,1]) has
a subsequence zin A 0. Then, by a theorem of H. Lebesgue2, for every
g>0 there exists an
such that
Км л
< в (и =1,2,...) when-
whenever e с [0,1] is of Lebesgue measure <rj. However, this contradicts
the definition of the normalized Haar basis of ^([0,1]), according to
which
z2u+l(t)dt =
21-2 21-2
2k + l 2k + 1
Consequently, {zn} is of type non-swc0.
On the other hand, for the subsequence {z,.b}?L0= {z2k+1}?L0 of
{zn} we have
Y.K
k=0
= 2-l<2 (/1 = 0,1,2,...).
Indeed, by the definition of {zn} we have
G.11)
1X0 =
Г 1
y2" = 2"+1-l for te 0,—rr
k = 0 L z
я-1 Г 1 1
У2*-2п=-1 for fe —-г, —
Lz L
izik{t)=-\ for
whence
у -
k = 0
1
f
n
k = 0
1 See § 2, formula B.3) for p= 1.
2 See e.g. [10], pp. 7-8 and 136, or [175], Ch. VI, § 3, corollary 2.

296
II. Special Classes of Bases in Banach Spaces
Hence, by Ch. I, § 12, proposition 12.2c), there exists a <PeLl([0,1])**
such that
<P(h2k+1)=\ (fc = 0,l,2,...),
where {hn} <=Ll{[0,1])* is the sequence of coefficient functionals asso-
associated to the basis {zn} of ^([0,1]). This proves that {zn} is of type
non-(wc0)*.
Finally, the subsequence {гл}?°=1= {г2к + 2}?°=1 of {zn} is equivalent
to the unit vector basis of I1, since for any finite sequence of scalars
ab..., а„ we have, by the definition of {zn},
= I l
k= 1
Hence {zjk} is of type (wc0)* and thus {zn} is of type (swc0)*, which
completes the proof of theorem 7.1.
Problem 7.1.1 a) Does the space C([0,1]) possess a basis of type
wc0? What about a basis of type (wc0)* or (jivc0)*?
b) Does the space //([0,1]) possess a basis of type wc0 or swcol
What about a basis of type (wc0)* or non-(swco)*?
We shall see in §9, corollary 9.1b), that the space C([0,1]) has a
basis of type non-swc0.
The answer to the problem of existence of bases of types swc0 and
(swc0)* (and hence also for those of types wc0 and (wc0)*) is negative,
since all bases of any finite-dimensional Banach space and all bounded
bases of any subspace of I1 are of type non-swc0 (because, in I1, yn^>0
implies2 ||jn||->0, whence all bounded bases of c0areof typenon-Ewc0)*.
The answer to the problem of existence of bases of types non-wc0 and
non-(wc0)* (and hence also for those of types non-sivc,, and non-(swc0)*)
is also negative, since in any infinite dimensional reflexive Banach space
with a basis all bounded bases are of types wc0 and (wc0)* (this is a
consequence of §4, example 4.3 and theorem 4.2, but can be also de-
deduced from weak sequential compactness and biorthogonality).
The next theorem gives some characterizations of bases of type wc0.
Theorem 7.2. Let {xn} be a bounded basis of an infinite dimensional
Banach space E, with the a.s.c.f. {/„}. The following statements are
equivalent:
1 Recently, part of this problem has been solved in the affirmative (see the
Notes and remarks).
2 See e.g. [10], p. 137-139.
7. Bases of types wc0, (wc0)*, swc0 and (swc0)*
297
1°. {*„} is of type wc0.
2°. The set \xn} is weakly sequentially compact (i.e. every infinite
subsequence {x( } of {xn} contains a subsequence {x{) } such that
3°. {xn} is weakly countably compact (i.e. every infinite subsequence
{xin} of {xn} has a weak limit point1 in E).
4°. {xn} is weakly conditionally compact (i.e. the weak closure oj
\xn} is weakly compactJ.
5°. The relations {/?„} <= K, sup
i=l
< oo imply lim /?„ = 0.
Proof. The implication 1°=>2° is obvious. The equivalences
2°o30o4° are consequences of the Eberlein-Smulian theorem on weak
compactness3 and can be seen also directly, while the equivalence
Г«-5" is an immediate consequence of Ch. I, §12, proposition 12.2a).
Finally, the implication 2°=>Г is a consequence of the following more
general result:
Proposition 7.2. Let {yn} be a basic sequence in a Banach space E.
If xeE is a weak limit point of {>>„} (in particular, if yn^x), then x = 0.
Proof. By the Hahn-Banach theorem there exists a sequence {gn} <=?*
such that дг(у^ = 5ц (i,j= 1,2,...). Since x is a weak limit point of {}>„},
we have
gi(x)=\imgi(ykn(i)) = 0 (/=1,2,...), G.12)
П-> со
where {>"мо} is some subsequence of {>>„} depending on i. Since [у„]
is weakly closed4, we have xe[>>n], whence, by G.12),
x=
= 0.
i= 1
This completes the proof of proposition 7.2. Now, if a basis {х„}
does not satisfy Г of theorem 7.2, then there exist an feE*, an e>0,
and a subsequence {x,J of {х„}, such that
\f(xj\>e (и =1,2,...).
G.13)
1 I.e. a point xeE such that every weak neighbourhood of x contains an
element of the subsequence {xin}, different from x. Let us mention that for this
definition of a weak limit point there are weakly convergent sequences which have
no weak limit point; indeed, such is e.g. any stationary sequence.
2 Let us recall that we use the term "compact" in the sense: bicompact Haus-
dorff.
3 See e.g. [50], p. 430, theorem 1.
4 See e.g. [43], Ch. I, § 6, corollary 4, or [50], p. 422, theorem 13.

298
II. Special Classes of Bases in Banach Spaces
However, by 2° and proposition 7.2 applied to1 {у„} = {х;„}> there
exists a subsequence {xtjj of {xtj such that xh -^ 0, which contradicts
G.13) and completes the proof of theorem 7.2.
The bases of types wc0 and (wc0)* are in the following relation of
duality:
Proposition 7.3. Let {xn} be a bounded basis of a Banach space E,
with the a.s.c.f. {/„}. Then
a) {xn} is of type wc0 if and only if {/„} is a basis of type (wc0)*
of [LI
b) The following statements are equivalent:
Г. {*„} is of type (we0)*.
2°. {/„} is a basis of type wc0 of [/„].
3°. The relations {an}
: K, Sup
< oo imply lim аи = О.
Proo/. a) The a.s.c.f. {•?„} <= [/„]* to the basis {/„} of [/„] is noth-
nothing else but 4>п = ф(хп) (п= 1,2,...), where ф is the canonical mapping
of ? into [/„]*. Since by Ch. I, § 12, theorem 12.2e), ф is an isomorphism,
we have xn -^ 0 if and only if ф(х„) -^ 0. Consequently, {xn} is of
type wc0 if and only if {ф{х„)} is a basis of type wc0 of [0(х„)]. How-
However, by definition 7.1, this condition means that {/„} is a basis of type
(«of of [/„].
b) The equivalence I°o2° is nothing else but definition 7.1b).
2°o3°. Since the canonical mapping ф: ?-»[/„]* is an isomorphism,
we have sup
1 ^П< GO
< oo if and only if sup
<oo.
if and only if the relations {а„} <= К, sup
Hence, by the equivalence Г «-5° of theorem 7.2, {/„} is of type wc0
\a-iXi < oo imply
liman = 0. This completes the proof of proposition 7.3.
n-» oc
Let us mention that one can also prove the equivalence b) 2°«-b) 3°
in a manner similar to the proof of the equivalence Г«-5° of theorem 7.2,
i.e. directly with the aid of Ch. I, § 12, proposition 12.2b).
Combining theorem 7.2 and proposition 7.3, one can obtain other
characterizations of bases of type (wc0)*.
Let us now turn our attention to w*-convergence in conjugate Ba-
Banach spaces ?*. For \hn}<=E*, heE* such that {hn} converges to h for
the w*-topology <t(?*,?), we shall use the notation hn ^h.
1 See Ch. I, § 4, proposition 4.1 a).
7. Bases of types wc0, (wc0)*, swc0 and (swc0)*
299
Proposition 7.4. Let {xn} be a bounded basis of an infinite dimensional
Banach space E, with the a.s.c.f. {/„}. Then /„ -^ 0.
Proof. Since inf \\xn\\ > 0 and since а„х„->0 for У а;х;е?, we
1 < Л < 00 . ,
¦e. i= 1
have
which completes the proof.
Proposition 7.4 suggests the question, whether every basis {/„} of
?* or every w*-basis {/„} of ?* satisfies /„ -^ 0. The answer is negative,
as shown by
Example 7.1. Let {/„} be the basis of E* = ll considered in Ch. I,
§ 14, example 14.1, formula A4.3). Then, as we have seen in that example,
{/„} is a basis of ?* and a w*-basis of ?* (but not a w*-Schauder basis
of ?*, or equivalently1, not the a.s.c.f. to any basis {xn} of ?). However,
for ?>! = {!,0,0,...}e? = c0 we have
) = (-1)"
(« = 2,3,...)
G.14)
which shows that {/„} does not converge for the w*-topology <r(?*, ?).
Proposition 7.4 and example 7.1 suggest the question, whether the
condition of being a w*-Schauder basis of ?* (or equivalently, the con-
condition of being the a.s.c.f. to a suitable basis {xn} of ?) or, at least a
w*-basis of ?*, is necessary for a basis {/„} of ?* to satisfy /„ -^ 0.
The answer is negative, as shown by
Example 7.2. Let {xn} be the basis Фои {п{хп)} of ?*=J**, where
the notations J, {х„} and <?oeJ** are those of §4, example 4.1. Then
Х„-^0, since {xn}^=2 = {K(xn-i)}^=2 is tne a.s.c.f. to the natural basis
{L-i}n = 2 of E = J* (see §4, example 4.1). However, {х„}„°°=1 Js not a
w*-basis (whence also not a w*-Schauder basis) of ?* = J**, since by
GO
/= ? /(*;)/; (JeE = J*) and §4, formula D.31), we have
;= l
i = 1
1 See Ch. I, § 14, theorem 14.1.

300
II. Special Classes of Bases in Banach Spaces
§ 8. Some properties of the set of all elements of a basis.
Weakly closed and (weakly closed)* bases
In § 7, theorem 7.2, we have given some characterizations of bases
{х„} of type wc0 in terms of w-topological properties of the set of all
elements of the basis \xn}, which we denote again by {xn}. It is natural
to ask about other properties of this set for an arbitrary basis {xn}, both
in the norm-topology and w-topology of E.
Proposition 8.1. Let {х„} be a basis of a Banach space E. Then
a) Both in the norm-topology and w-topology of E all points of the
set {xn} are isolated points.
b) The set {xn} is closed in the norm topology of E if and only if
inf
(8.1)
с) The set {х„} is closed in the w-topology of E if and only if 0 is not
a weak limit point of {xn}.
Proof, a) If {/,.}
= {xeE
= {xsE
is the
1
a.s.c.f. to {*,-}, the cell S[x,
{*,-
\\x — х„
1
2 II/„I
and the w-neighbourhood Vfn.i(x)
2 II f II
Г
У
since \\fn\\\\xj-xn\\>\fn(xj-xn)\=\ iorj^n.
|| xn || = 0, then in every e-cell S@,e) there exists an ele-
\Jn{X — Xn)\ < —
of х„ contain no point of the set {Xj}, dif-
ferent from xn
b) If inf
Hn< со
ment of {xn}, i.e. 0е{х„} = the closure of \xn} in the norm-topology of
E. However, since {xn} is a basis, 0ф{хп}, and thus {х„} is not closed.
Conversely, if {xn} is not closed in the norm-topology, there exists an
xe{xn\\{xn\. Then, by §7, proposition 7.2, x = 0, whence inf ||xn|| = 0.
1^П< CO
c) If 0 is a w-limit point of {xn}, then 0е{х„} = the closure of {xn}
in the w-topology of E. However, since \xn} is a basis, 0${xn}, and
thus {xn} is not w-closed.
Conversely, if {xn} is not w-closed, there exists an xe{xn}\{xn} in
the w-topology. Then x is a w-limit point of {xn}, whence, by § 7, propo-
proposition 7.2, x = 0. Thus 0 is a w-limit point of {х„}, which completes the
proof.
Definition 8.1. A basis {xn} of a Banach space E is said to be weakly
closed, if {xn} is a bounded basis and the set \xn} is weakly closed; or,
equivalently1, if {х„} is a bounded basis and 0 is not a weak limit point
See proposition 8.1 c) above.
8. Weakly closed and (weakly closed)* bases
301
of the set {xn}. If the a.s.c.f. {/„} is a weakly closed basis of [/„], we
shall say that {xn} is a (weakly closed)* basis.
For instance, the basis of c0 constructed in § 2, example 2.2, formula
B.24), and the natural basis of I1 are weakly closed. On the other hand,
every basis of type swc0 is non weakly closed, whence, in particular,
the natural basis of lp(p> 1), and the normalized Haar basis of Lp([0,1])
(p> 1) are non weakly closed. Dual examples of (weakly closed)* and
non (weakly closed)* bases follow from the fact that for weakly closed
bases we have a duality proposition similar to § 7, proposition 7.3 (the
proof is similar, using again that the canonical mapping ф: E-> [/„]*
is an isomorphism and that the a.s.c.f. to the basis {/„} of [/„] is {ф(х„)}).
Theorem 8.1. a) The Schauder basis of C([0,1]) is non weakly closed
but (weakly closed)*.
b) The normalized Haar basis of 1^([0,1]) is non weakly closed and
non- (weakly closed) *.
Proof, a) From § 7, theorem 7.1a) it follows that the Schauder basis
{xn}n = o °f C([0,1]) is non weakly closed. On the other hand, from § 7,
formula G.10) it follows that the w-neighbourhood Уфу j;i@) of 0 in [/„]
corresponding to that Ф contains none of the elements /„ (n = 0,1,2,...)
of the a.s.c.f. {/„}*=0 t0 {xn}f=o- Hence 0 is not a w-limit point of
{/„}*=0 and thus {х„}„°°=0 is (weakly closed)*.
b) Let F9l,92,...,9m:E@) (where gug2, ...,gmeLl([0,l~\)*, e>0) bean
arbitrary w-neighbourhood of 0 in Ll([0,1]). Then there exist essentially
bounded measurable functions yi,y2> ¦¦¦¦> lm on [0>l] such that
gi(x)=\x(t)yi(t)dt
Since 7; is essentially bounded and measurable, it is also integrable,
whence, by a theorem of Lebesgue1, the derivative of
exists and coincides with у,-(?), almost everywhere on [0,1](г= 1,2, ...,m).
Consequently, there is a point t0 with 0<;0<1, such that all /?-(г0)
exist and satisfy
Я('о) = "Л('о) (i=l,2,...,m).2 (8.2)
1 See e.g. [211], Ch. II, §23, or [175], Ch. DC, §4, theorem 2. For a simple
proof, which makes use also of the essential boundedness of yt, see [257], Ch. 11,
§11.53.
2 Indeed, since the sets {/e[0,1] | jSj(O=Vi@} ('= 1-2,..., m) are each of meas-
measure 1 and c:[0,1], their intersection must be of measure 1.

302
II. Special Classes of Bases in Banach Spaces
Let {z2kn + i } be any subsequence of the normalized Haar basis
-^-l of ^([0,1]), such that
lljJI J
^ («=1,2,-).
(8.3)
Then, putting
c-
and taking into account that 26„-а„-с„ = 0 (n = l,2,...), we obtain
= 24 U
= 2k" [2 «..(/i^»)/!^ - <х№у) hf - а г(А?>) /гCи)].
?
Furthermore, by (8.2), there exists an r] = rj(s)>0 such that |a;(A)| < —
^
- 2 "~2 = -i_ («=1,2,...), it follows that for N = N|)/(e)] such that
2kn+l 2kn
(i=l,...,m) whenever |A|<»;. Since by (8.3) №B)|, |«Bn)|, l^
2 "~2 = -i
2kn+l 2
—^<n (n>N) we shall have
el ? 1
(n>N; «=1,...,«),
and thus
Since КЙ1>в2 ...,emi?@) has been arbitrary, it follows that 0 is a w-limit
point of the set {zn}7i.e. {zn} is a non weakly closed basis. On the other
8. Weakly closed and (weakly closed)* bases
303
hand, from § 7, theorem 7.1 b) it follows that {zn} is also non-(weakly
closed)*, which completes the proof.
Remark 8.1. Theorems 7.1 b) and 8.1 b) show that there exist normal-
normalized bases {х„} such that 0 is a weak limit point of the set {х„} but no
subsequence of \xn} converges weakly to 0.
The answer to the problem of existence of weakly closed bases is
negative, since in any infinite-dimensional reflexive Banach space with
a basis all bounded bases are of type wc0 (see § 7) and hence non weakly
closed. By duality (see the remark after definition 8.1), a similar asser-
assertion is also valid for (weakly closed)* bases.
Problem 8.1. Does the space ?=^([0,1]) possess a weakly closed
basis? What about a (weakly closed)* basis?1
We shall see in §9, corollary 9.1a), that the space C([0,1]) has a
weakly closed basis; on the other hand, we have seen in theorem 8.1a)
above that it also possesses a (weakly closed)* basis.
Obviously, the answer to the problem of existence of non weakly
closed bases in finite-dimensional Banach spaces is also negative.
Problem 8.2. a) Does every infinite dimensional Banach space with
a basis possess a non weakly closed basis? In particular, does the I1
possess such a basis?
b) Does every infinite dimensional Banach space with a basis
possess a non-(weakly closed)* basis? In particular, does one of the
spaces c0 or C([0,1]) possess such a basis?
In connection with problem 8.2 a) it may present some interest to
observe that the space I1 has a subspace with a non weakly closed basis.
In order to give such an example, let us recall the following weakened
version of a theorem A. Dvoretzky2: For each positive integer n, every
infinite dimensional Banach space E has an n-dimensional subspace Gn
and an euclidean norm |||x||| in it (i.e. a norm in which Gn will be an n-di-
n-dimensional inner product space), such that
\х\\<\\\х\Ы2\\х\\
(xeGn).
(8.4)
Applying this theorem of Dvoretzky for every positive integer n
and for E = ll, it follows that the space3
1 Recently this problem has been solved in the affirmative (see the Notes and
remarks).
2 See [52].
3 We recall that if E is one of the spaces co,ll or m and Ea (n=l,2,...) are
Banach spaces, (E^ x E2 x • • ¦ x Е„ х • • -)? is the Banach space of all sequences
У={у„} with у„еЕп (n=l,2,...) and {||>"„||е„}??, endowed with the usual vector
operations and with the norm ||>>|| = || {HyJcJ ||?. For the definition of the space
/2 see § 1, the footnote to the example given after formula A.6).

304
II. Special Classes of Bases in Banach Spaces
/„2
х/„2х ¦••)
(8.5)
is isomorphic to a closed linear subspace of
Indeed, if for each n we take in the и-th factor of Bl a subspace Gn
whose existence is assured by the above theorem of Dvoretzky, the
closed linear subspace G of Bt defined by
С = (Сг x G2 x ••• x Gn x •¦•),!
can be mapped isomorphically onto В by mapping suitably Gn onto /2.
Since B1 is isometric to I1 (see § 18, lemma 18.5b)), it follows that В
is isomorphic to a closed linear subspace F of I1.
Now we can give the example announced above:
Example 8.1. Let F be the closed linear subspace of I1 constructed
above. Then F has a non weakly closed basis.
Indeed, since F is isomorphic to the Banach space В defined by (8.5),
it will be sufficient to prove that В has a non weakly closed basis. We
claim that the sequence {Zj} с В defined by
where {ykn)}"k=i is the natural basis of /2 (n=l,2,...), is a non weakly
closed basis of B.
Indeed, let us first show that {zn} is a basis of B. Let y={yn}eB be
arbitrary. Then, since jne/2 and since {ykn)}l=l is the natural basis
of/2 (n= 1,2,...), there exist scalars akn) (k= 1,...,«; n= 1,2,...) such that
n
yn= ? a^jjt"' («=1,2,...), whence, by (8.6),
?
J=l t=l
k= 1
Consequently, we have
" j
7
0,...,0,
= ( "z 14?
n+ 1
к = й+ 1
(n+ 1)|2
ak Ук 1Уп + 2> Уп+31 ¦
i=n+2
• 0 as n -> со,
8. Weakly closed and (weakly closed)* bases 305
QO
which proves thatj has an expansion of the form ? a.-z,-. This expansion
is unique, since У У 4я)гя(я_п =0 implies \ У al^jM =0
, , , ^—^ + k \ L-' \
„ n--l k=l 2 U=l Jn=l
whence ? 4и)^я) = 0 (« = 1,2,...), whence, since {j[n)}^=1 is a basis
of/2, we obtain 4n) = 0 (fc= 1,2,...,«; n= 1,2,...). Thus \zn} is a basis
of B. Alternatively, this fact can be also deduced from Ch. I, § 7, corol-
corollary 7.3.
Let us prove now that {zn} is non weakly closed. Let K(/] ,@)
(where gl,g2,...,gjeB*, e>0) be an arbitrary w-neighbourhood of 0
in B. Then, by the isometry B* = (/2 x l\ x ••• x/2 x •••)m, there exist
l^1'}, {Й2'}, •¦•, {^J/*}e(/f x/|x •¦¦ x/2x •••)m such that
Ii0ill= sup
1 $П< CO
Hence, if 0(B°={ft""l)K=1 (/1=1,2,...; i=l,...,y), we obtain
0<(W±) + .)= ^({О^^О, j4">, 0, 0, ...}) = ^>(yi">) = ft--') (8.7)
(/с=1,...,и; И=1,2,...; i=\,...,j),
Put M=
дЛ= sup
0,-ll= Z ,sup
(8.8)
i = 1 i = 1
Iя'''I2 ) . Then
Z Zl/T'l2 * Z Zl/ГЧ2 ^м (/,=1,2,...)
4i= 1 *= 1 / i=l \k= 1 /
whence for each и = 1,2,... at least one of the numbers
i= 1
J
i= 1
M2
must be ^ , and thus
n
min max I ft"
min
<"" 2
M
20 Singer, Bases in Banach Spaces I

306
II. Special Classes of Bases in Banach Spaces
Consequently, taking into account (8.7), we have
inf
max
9i(Zntn-
= 0,
(8.9)
whence z.,._,
« and k. Thus 0 is a weak
limit point of {zn}, i.e. \zn} is non weakly closed. This completes the
proof of the assertions stated in example 8.1.
Remark 8.2. Actually, one can avoid the use of the above theorem of
Dvoretzky, embedding B = (/^x/|x ••¦),, into I1 with the method ap-
applied in § 18, the proof of theorem 18.3 (i.e. using Rademacher functions
and the Khinchin inequality to embed l\ into l\n, etc).
Dually, one can show that the space c0 has a subspace with a non
(weakly closed)* basis. Indeed, since B2 = {1\ x l\ x ••¦ x l\ x •¦¦)co is iso-
morphic to a closed linear subspace of c0, it is sufficient to prove that
B2 has such a basis. However, the "natural" basis of B2 has this property,
since its a.s.c.f. is nothing else but the image of the basis {zj} of В defined
by (8.6) under the canonical isometry B= Bf.
Remark 8.3. Again, one can embed B2={1\ x l\ x •••)co into c0 with-
without using the above theorem of Dvoretzky, by embedding directly ll
into l2i». For this purpose, observe first that one can map isometrically
ll onto the subspace [rl5...,rn] of //"([0,1]), where ru...,rn are the
first « Rademacher functions1, by the mapping
ei^ri (i=\,...,«), (8.10)
where {eJ-L i is the unit vector basis of /*. Indeed, we have obviously
(8-11)
for any scalars au...,an. On the other hand, for every pair к1фк2
with l^kl, k2^2n there exists at least one i with l^/<« such that
/2k2~l); for if r,\
2"+l
2/c!-l
2"
= Г,
, then
both '4 / and —2—— must belong to the same dyadic interval
¦"}/! + 1 ^/1+ 1
—, —1, where
and continuing in this way, by induction,
we obtain finally that if rl J~ | = r,-( J^, ] (/= !,...,«), then both
and
must belong to the same dyadic interval
See §14, formula A4.1).
8. Weakly closed and (weakly closed)* bases
307
:-l k\ . 2/c,-l (k:-\ k,\ , , ,
, — , whence, since obviously —~-e[ — , — (/=1,2), we
2" 27 2n+1 \ 2" 27
infer kt = k2. Consequently, for k= 1,2, ...,2" we obtain 2" distinct
^2k-l\ /2/с-Г,,
. Since there are exactly 2" dis-
«-tuples у
tinct «-tuples of signs e,-= ±1, there exists a (unique) к with
such that гЛ д+1 \ = Et (/=1,...,«). Now, if {a;}"=1 is arbitrary (we
restrict ourselves to real scalars), then applying this remark for e, = signa,
Bk— 1
if af#0 and 1 if af = 0, we find a fe with l^fc^2" such that r,( n+1
= signa, (/= 1,...,«), whence V
(8.12)
which, together with (8.11), proves that (8.10) is an isometric linear
mapping of /„* onto the subspace \_rl,...,rn] of 1/°([0,1]). Since the
subspace G2n of L°°([0,1]) spanned by the characteristic functions
Zr*-i к \ (^=1, •••, 2") is obviously equivalent to /"„, the subspace
[ru ..., rn] of G2n is equivalent to an «-dimensional subspace of /"„ and
hence we have an isometric linear embedding of l\ into /"„, whence, in
particular, of l\n into /22». Composing this embedding with the embed-
embedding ll^>l\» of remark 8.2, we obtain the desired embedding /^->/^".
Let us now turn our attention to w*-topological properties in con-
conjugate Banach spaces. Similarly to the proof of proposition 8.1a) it
follows that if {/„}<=?* is the a.s.c.f. to a bounded basis {xn} of E,
then both in the norm-topology and w*-topology of ?*, all points of the
set {/„} are isolated points. Furthermore, from §7, proposition 7.4 and
Ch. I, §12, theorem 12.1, it follows that in this case the set {/„} is not
w*-closed. However, if {/„}<=?* is a bounded basis of E*, which is
not the a.s.c.f. to any bounded basis of E, the situation is different. In
fact, in this case the fj are not necessarily w*-isolated points of the set
{/„}, as shown by
Example 8.2. Let {/„} be the basis of E* = ll considered in Ch. I,
§ 14, example 14.1, formula A4.3). Then
/2.-1-/1 =«2.-1 (« = 2,3,...),
where {<?„} is the unit vector basis of /\ Since by §7, proposition 7.4,
en -^ 0, it follows that
/2.-1^/1, (8-13)
and thus /, is not a w*-isolated point of the set {/„}.
20»

308
II. Special Classes of Bases in Banach Spaces
Furthermore, if {/„} is a bounded basis of ?*, then1 E is separable,
whence2 {/„} contains a subsequence {fin} which w*-converges to an
feE*. However, it may also happen that /#0, as shown by the pre-
preceding example, formula (8.13).
§ 9. Bases of types P,P*,aP and aP*
Definition 9.1. A basis {xn} of a Banach space E is said to be
a) of (УРе P, if {xn} is a bounded basis and
sup
I
<OO;
(9.1)
b) of type P*, if the a.s.c.f. {/„} is a basis of type P of [/„];
c) of type aP (of type aP*), if there exists a sequence of scalars
{?,„} with |в„| = 1 (и=1,2,...), such the basis {епх„} of E is of type P
(respectively, of type P*).
For instance, the natural basis of c0 is of type P, but not aP*, the
natural basis of I1 is of type P*, but not aP, while the natural basis of
F (p> 1) is of types non-aP and non-aP*.
Let us observe that in definition 9.1a) it would have been sufficient
to assume that inf Цх„||>0, since (9.1) implies sup ||xj<co.
1 $n< go 1 =Sn< oo
Similarly, in definition 9.1b) it would have been sufficient to assume
that sup||xn|| < oo.
1 in< 00
Some characterizations of bases of type P are collected in
Theorem 9.1. Let {xn} be a basis of a Banach space E, with
inf ||xJ>0, and let {/„}<=?* be the a.s.c.f. to {х„}. The following
1 !gn< go
statements are equivalent:
1°. {*„} is of type P.
2°. sup ||xj<oo and the sequence {>>„} с Е defined by
(9.2)
is a basis of E.
3°. There exists a constant M>0 such that for every monotonic se-
со
quence {&.„} tending to zero, the sum ? a,X; exists and satisfies
i= 1
(9.3)
1 See e.g. [10], p. 189, theorem 12.
2 See e.g. [10], p. 123, theorem 3.
9. Bases of types P, />*, a Panda P*
309
4°. There exists a constant M > О .тас/j ;/ш; we have, for every finite
sequence of scalars a u ..., а„
" n
X«i ^M J]a./. . (9.4)
i=l i=l
5°. T/iere exwfj a ФеЕ** such that
Ф(/„)=1 (и =1,2,...). (9.5)
6°. There exists a Ve[/J* лис/г г/гаг
«"(/»)= 1 (и =1,2,...), (9.6)
7°. The sequence {fi-f2,f2-fi,...} w not complete in [/„].
8°- inf II /„ II > 0 and {j\-f2,f2-fi,...} is a basic sequence.
1 ^ П < CO
9°. The sequence {gn} <= E* defined by
9i=fi, 9n=f»-i-f« (« = 2,3,...) (9.7)
is a basis of [/„].
Proof. The equivalence Го 2° is a consequence of Ch. I, §4, proposi-
tion 4.3.
Г => 3°. Assume that we have 1 °, and let a t ^ а2
tending to zero. Using the sequence (9.2), we have
be a sequence
f-а,-
i=2
(9.8)
Since {х„} is of type P, we have sup \\yn || = M < oo, whence, since
а„^0, we obtain anjn^0. On the other hand, from ||jj| ^M, а„^а„+1
00 OO
(и =1,2,...) and ? (a,- —ai+i) = a, it follows that ^] (а;-а(+1)>>( exists
and satisfies
i= 1
i= 1
laJ. Hence, by (9.8), we have 3°.
The implication 3°=>Г follows by applying (9.3) successively to
the sequences {^"'} (и = 1,2,...) defined by
<=-=а(")=1, а(-I = ао.J=...=о („=1,2,...).
The equivalence l°«-5° follows from Ch. I, § 12, proposition 12.2c).
Furthermore, the equivalence 4° о 5° is a particular case of a classical
theorem of E. Helly1, and the equivalence 5°o6° follows from the
1 See e.g. [10], p. 55, theorem 4.

310
II. Special Classes of Bases in Banach Spaces
Hahn-Banach theorem. The equivalence 6°oT is a consequence of a
well known corollary of the Hahn-Banach theorem1, since the relations
(9.6) are equivalent to
xpif\=l \pc-f _/) = 0 (и = 2,3,...). (9.9)
2° =>8°. Since sup ||xj<oo, by Ch. I, §3, corollary 3.1b) we
have inf ||/J>0. Furthermore, the sequence {fi-f2,f2—f$,-¦¦} is
nothing else but the a.s.c.f. to the basis {yn} of ?, since
(fi-i-fi)(yj) = Si-U (i = 2,3,...; y= 1,2,...).
Hence, by Ch. I, § 12, theorem 12.1, {fl-f2,f2-f3, •••} is a basic
sequence.
8°=* 9°. If we have 8°, then
Indeed, otherwise there would exist a sequence {an} <= X such that
/i= Eai-i(/i-i-/i) = «i/i+ E(«/-«i-i)/i.
i=2 >=2
whence, since by Ch. I, § 12, theorem 12.1, {/„} is a basic sequence,
al = l, ca2 — al=a3 — a2 = '" =0'
and thus =o
Hence
as
which contradicts inf ||/J > 0. This proves (9.10).
Thus {gX=i isa basis of [g^=2 and в1фШп=2- Since obviously
Ш = Ш?= i. it follows that {вЛ is a basis of [/"]¦
Finally, the implication 9° =>7° is obvious, since every basis is a
minimal sequence (see Ch. I, § 6). This completes the proof of theorem 9.1.
The assumption sup ||xj<oo in 2° above is necessary for the
1 $Л< 00
validity of the implication 2°=>Г, as shown by
Example 9.1. Let ? = c0 and {xn} = {nen}, where {en} is the natural
basis of c0. Then inf ||х„|| >0 and | ?>¦{ is a basis of E, but {xn}
is not of type P. ^"<c0 l-i=1 J
1 See [10], p. 58, theorem 7.
9. Bases of types P, P*, aP and aP* 311
Indeed, applying Ch. I, §4, proposition 4.3 to {en} and а„ = и
(и =1,2,...), it follows that
n
i= 1
is a basis of ? (since
И+1
И+1
< 1). However,
i= 1
= и (и=1,2,...).
Similarly, the assumption inf ||/J| >0 in 8° above is essential.
1 ^ Л < GO
By the duality methods used already in the preceding sections,
i.e. taking into account that the canonical mapping ф: E-> [/„]* is an
isomorphism and that the a.s.c.f. to the basis {/„} of [/„] is {ф(х„}},
we obtain
Theorem 9.2. Let {xn) be a basis of a Banach space E, with sup ||х„ ||
1 !gn< GO
<oo, and let {/„} <= E* be the a.s.c.f. to {xn}. Then
a) {xn} is of type P if and only if {/„} is a basis of type P* of [/„].
b) The following statements are equivalent:
1°. {xn}isoftypeP*.
2°. {Q is a basis of type P of {Q.
3°. There exists a constant L>0 such that we have, for every
finite sequence of scalars a l,..., an
(9.12)
4°. There exists an feE* such that
/(*„)= 1 (/i=l,2,...). (9.13)
5°. The sequence \xl — x2, x2— x3,...} is not complete in E.
n
z«,
n
i — 1
6°. inf
{xt — x2, x2— x3,...} is a basic sequence.
7°. 77ie sequence {zn} c: ? defined by
zl=xi, zn = xn_l-xn (и = 2,3,...) (9.14)
й a basis of E.
Naturally, one can also prove the equivalences b) 2°«-ob) 7°
directly, similarly to the proof of the corresponding equivalences of
theorem 9.1.
Combining theorems 9.1 and 9.2 b), one can also obtain other
characterizations of bases of type P*. Obviously, all the preceding
characterizations of bases of types P, P* imply characterizations of
bases of types aP and aP* respectively.
Let us consider now the corresponding properties for the Schauder
basis of C([0,1]) and the normalized Haar basis of Ll([0,1]).

312
II. Special Classes of Bases in Banach Spaces
Theorem 9.3. a) The Schauder basis {х„}™=0 of C([0,l]) is of types
non-aP* and поп-Р, but aP. Namely, the sequence {г„х„}^=0, where
2"""
2/-
й a ?шю of /y/>e P o/ C([0,1]).
b) The normalized Haar basis {zn} of L}([0,1]) is of types non-aP
and non-aP*.
Proof, a) Assume, a contrario, that {х„}™=0 is of type aP*. Then,
by the implication Г=>4° of theorem 9.2 b), there exists an /eC([0,l])*
such that
|/(х„)| = 1 (/1 = 0,1,2,...)-
However, this contradicts §7, theorem 7.1a), according to which
{х„}"=0 is of type swc0. Consequently, {х„}*=0 is of type non-aP*.
Furthermore, define toe [0,1] by G.2) and kn, /„ by G.3), G.4). Then,
as we have seen in the proof of §7, theorem 7.1a), lim;>c2k,, + ,n(?0)=l.
Hence, since all xt(t0) are >0, we infer
i = 0
>oo as n—>oo,
which proves that {xn}^=0 is of type non-P.
Finally, we shall prove that for the sequence {у„} <= C([0,1]) defined
У„= i^iXi (« = 0,1,2,...), (9.16)
i = 0
where the е„ (и = 0,1,2,...) are defined by (9.15), we have
\уя\\ =
(9.17)
It is obvious that ||joll = IIJill = IIJ2II = 1- since Jn@)=l (и = 3,4,...),
we also have
\\yn\\>\ (n=3,4,...). (9.18)
Assume now that Hjv + i-iII = 1- Then, since by (9.16), (9.15) and
by the definition of {х„}"=0 we have
9. Bases of types P, P*, aP and aP*
313
г n
,'2/-2 2/ .
J'2t + ,_1@ for /d—^-, ,
2/—1\ 2/-1
for t = -j^T
2/-l
linear for the other /,
it follows that ||j2k + ill ^1, whence taking into account (9.18), ||j2fc + ill =1-
Consequently, we have (9.17), which proves that {г„хп}™=0 is a basis
of type Pof C([0,l]).
b) Assume, a contrario, that {zn} is of type aP. Then, by the impli-
implication 1°=>5° of theorem 9.1, there must exist a <Pel}([0,1])** such
|Ф(«„I = 1 (и =1,2,...), (9.19)
where {hn} с l}(\0,1])* is the a.s.c.f. to {zn}. However, this is impos-
impossible, since {zn} is of type (swc0)* (by § 7, theorem 7.1 b)). Consequently,
{zn} is of type non-aP.
Finally, from §8, theorem 8.1b) it follows that in every w-neigh-
bourhood of 0 of the form J^.,:@) (/?^([0,1])*, e>0) there exists at
least one zn, whence
JnfJ/fzJ^O (/6LH[0,l])*),
and thus there exists no fel}{\Q, 1])* satisfying
Consequently, by the implication 1°=>4° of theorem 9.2 b), {zn} is
of type non-aP. This completes the proof of theorem 9.3.
Remark 9.1. The following "Schauder basis" of C([0,1]) is sometimes
also considered, because of its symmetry with respect to t — \:
xo(t) = t, Xl(t)=l-t (/?[0,1]),
(9.20)
and x2k+l(/) = as before. For this basis, theorem 9.3a) remains valid,
with the same proof, provided that we replace in (9.15) et=—\ by
e1 = l.
Let us also mention the following corollary of theorem 9.3:
Corollary 9.1. a) // {xn}^=0 is the Schauder basis of C([0,l]), then
the sequence (9.16), where the en are defined by (9.15), is a weakly closed
basis of type P* of C([0,l]).

314 II. Special Classes of Bases in Banach Spaces
b) If {zn} й the normalized Haar basis of ^([0,1]), then for any
sequence of scalars en with |г„| = 1 (n=l,2,...), the sequence
is not a basis of Lx([0,1]).
Proof, a) By theorem 9.3a), {г„х„}„°°=0 is a basis of type P of C([0,1]).
Hence, by the implication 1°=>2° of theorem 9.1, the sequence {>>„}"= 0
defined by (9.16) is a basis of C([0,1]), and it is, obviously, of type P*.
Furthermore, since for the functional /oeC([0,1])* defined by
fo(x) = x
(9.21)
we have /0(yn)=l (и = 0,1,2,...), it follows that 0 is not a weak limit
point of the sequence {yn}™=0 and thus {jn}"=0 is weakly closed.
b) By theorem 9.3b) and the implication 2°=>1° of theorem 9.1, for
any sequence of scalars {г„} with |г„| = 1 (и = 1,2,...), the sequence
is not a basis of Lx([0,1]). This completes the proof of corol-
corollary 9.1.
Problem 9.1.1 Does the space 1>([0,1]) possess a basis of type P?
Or, equivalently-. does it possess a basis of type P*?
The answer to the problem of existence of bases of types aP, aP*
(and hence also for bases of types P, P*) is negative, as shown by
Example 9.2. Let E be an infinite dimensional reflexive Banach space
with a basis. Then E has no basis of type aP or aP*.
Indeed, every basis {*„} of E is of type w c0 (see § 7), whence, by
the implication 1°=>4° of theorem 9.2 b), {xn} is of type non-aP*. Con-
Consequently, by the implication 1°=>2° of theorem 9.1, every basis of E
is also of type non-aP.
The answer to the problem of existence of bases of types non-P,
non-P* (and hence also for bases of types non-aP, and non-aP*) in
finite-dimensional Banach spaces is obviously negative. On the other
hand, the answer to the problem of existence of bases of types non-P,
non-P* in infinite dimensional Banach spaces with bases is affirmative.
Indeed, this follows from the implication 1°=>2° of theorem 9.1 and the
implication 1°=>7° of theorem 9.2b), since (9.2) is of type non-P and (9.14)
of type non-P* (since e.g. for (9.2) we have
1
II/«II
/
J=l
II /J
sup
5
> oo
la l
as n->oo).
1 Recently, this problem has been solved in the affirmative (see the Notes and
remarks).
10. Bases of types / + ,(/ + )*, a l+ and (fl / + )*. The cone associated to a basis 315
Problem 9.2. a) Does every infinite dimensional Banach space with
a basis possess a basis of type non-aP?In particular, does the space
c0 possess such a basis?
b) Does every infinite dimensional Banach space with a basis pos-
possess a basis of type non-aP*? In particular, does the space Z1 possess
such a basis?
Let us mention the following related problem. The unit vector basis
{xn} of E = c0 is of type P (whence aP), and the basis {jn} =
i= 1
of E = c0 is again of type aP, since e.g. for ?„ = ( — 1)" the basis
2n- 1 и
{Е„У„} is °f type pi indeed, we have z2n_1 = Y (~^УУ]= Z xu-\,
In n j=l i=l
z2n = Z (—^УУ}= — Y X2i- Again, it is easy to see that the basis {zn}
of E = c0 is of type aP, and that for the signs r\n= ± 1 for which {t]nzn}
is of type P, the basis < Y 4izi\ is again of type a P. Can this process
u=t J
of iteration be continued indefinitely? If not, then in a finite number
of steps one would arrive to a basis of type non-aP of E = c0, obtaining
thus a negative answer to problem 9.2 a).
In connection with problem 9.2 let us also mention that the space I1
has a subspace with a basis of type non-aP*. Indeed, such is e.g. the
subspace F of I1 defined in §8, example 8.1, since by (8.9) and the
implication Г=>4° of theorem 9.2b), the basis (8.6) of (8.5) is of type
non-aP*. Dually, the space c0 has a subspace with a basis of type non-aP.
§ 10. Bases of types /+,(/+)*» al+ and (al+)*. The cone
associated to a basis
Definition 10.1. A basis \xn} of a Banach space E is said to be
a) of type l+, if {х„} is a bounded basis and there exists a constant
r]>0 such that we have, for all finite sequences au ...,an>0,
A0.1)
b) of type (/+)*, if the a.s.c.f. {/„} is a basis of type /+ of [/„];
c) of type al+ (of type (al+)*), if there exists a sequence of scalars
{е„} with |е„|= 1 (и= 1,2,...), such that the basis {е„х„} of E is of type
/+ (respectively, of type (/+)*).
For instance, the natural basis of Z1 is of type /+, but not (a/+)*,
the natural basis of c0 is of type (/+)*, but not al+, while the natural
basis of lp (p> 1) is of types non-a/+ and non-(a/+)*.

316
II. Special Classes of Bases in Banach Spaces
Let us observe that in definition 10.1 a) it would have been sufficient
to assume that sup j|xj<oo, since A0.1) implies inf [|х„||>0.
1 ^ И < OO 1 ^ Л < OO
Similarly, in definition 10.1b) it would have been sufficient to assume
that inf ||х„||>0.
Some characterizations of bases of type /+ are collected in
Theorem 10.1. Let \xn} be a basis of a Banach space E, with
sup ||xj <oo. The following statements are equivalent:
1°. {х„} is of type I+.
2°. For а„^0 (и = 1,2,...)
00 00
Z<X(Xj converges о Za.<0°- A0.2)
i= i ;= l
3°. There exists a constant r]>0 such that we have, for every sequence
00
а„^0 (и =1,2,...) with Z а,х;е?,
A0.3)
i= 1
4°. There exists a constant rj>0 such that we have, for every1
хесо{х„\,
\\x\\>ri. A0.4)
5°. There exists an feE* such that
Re/(xB)>l (#1=1,2,...). (Ю.5)
6°. {xn} admits a "contraction bounded from below" of type P*, i.e.
there exists a sequence of scalars {/?„} satisfying
0< inf j3n^ sup /*„<1, A0.6)
1 ^ Л < 00 1 ^И< GO
such that {pnxn} is a basis of type P* of E.
7°. There exists a sequence of scalars {/?„} satisfying
sup
со
(и = 1,2,...),
(Ю.7)
such that {pnxn} is a basis of type P* of E.
8°-9°. The same as 6°—7° respectively, with P* replaced by l+.
10°. For every sequence of scalars {/?„} satisfying
0< inf j3ns? sup 0„<оо, (Ю.8)
1 ^ n < oo 1^и<оо
{Р„хп} is a basis of type l+ of E.
1 We denote by со {х„} the convex hull of the set {х„}.
10. Bases of types / + , (/ + )*, a l+ and (a/ + )*. The cone associated to a basis 317
Proof. 1°=>3°. Assume that we have Г and let а„^0 (и =1,2,...),
. Then
n + p
i = n + 1
1
n + p
i = n + 1
<e
OO
whence ^а;<0°- Furthermore, from A0.1) we infer
00
z
;= l
a
X;
= lim
П-* 00
n
z
i= 1
<X;X;
i.e. A0.3).
The implication 3°=>2° is obvious, taking also into account the
hypothesis sup ||х„||<оо.
1«П<СО
2°=>Г. Assume that {xn} is of type non-/+. Then there exist an
increasing sequence of positive integers {mn} and numbers a|n)^0
nn; n= 1,2,...; mo = 0) such that
E a'");ci
i =mn- 1+1
l
A0.9)
Since {х„} is a basis, there exists, by Ch. I, § 7, theorem 7.1, a constant
such that
Z a|">.
Z аРЧ
Hence, by A0.9), Z aixi converges but Z a.-=00> where we have
put а( = а|п) («!„_! + ! ^i^mn; n =1,2,...). Thus, 2° is not satisfied,
which proves that 2°=>Г.
1°=>4°. Assume that we have Г and let xeco{xn}. Then there exist
n
a positive integer n and scalars аь ...,а„^0 with ? а,= 1, such that
x= Z a;-xi. whence, by 1°,
\\X\\ =
4°=>Г. Assume that we have 4° and let ab...,an^0 be arbitrary.
П
Then for y. =-jj-!— (i=l, ...,n) and x= Z lixi we have хесо{х„},

318 II. Special Classes of Bases in Banach Spaces
whence, by 4°,
п
z
= 1
n l
Л
Z У,*,
i= 1
which is nothing else than 1°.
The equivalence l°<s-5° is an immediate consequence of a well
known theorem of S. Mazur and W. Orlicz1, if ? is a real Banach space.
In the case of complex scalars, the implication 5°=>Г is again obvious,
since for (Xj^O and Re/(x;)^l (/=!,...,и) we have
1
Z «</(*«)
On the other hand, if we have Г for a complex Banach space ?,
then, considering ? as a real Banach space ?(r) in the usual way, we can
find, by the above, a functional geE?r) such that д(х„)^\ (и =1,2,...).
Then the functional /(x) = gf(x)-/gf(/x)(xe?), where / = ]/^T, satisfies
A0.5).
5°=>6°. Assume that we have 5°. For /e?* satisfying A0.5), define
(*) by
A0.10)
and put
</(x) = Re/(x) (xe?),
1
A0.11)
Then,by sup ||x-||<oo and д(хп)>1 (и =1,2, •••) we have
0<
sup
whence also A0.6). Furthermore, by A0.11),
g(pnxn)=l A1=1,2,...),
whence, by the implication 4°=>1° of §9, theorem 9.2 b), {р"„х„} is a
basis of P* of ?.
See [158], p. 147, theorem 2.41; for other proofs see [228], [200], [57].
10. Bases of types / + , (/ + )*, al+ and (a/ + )*. The cone associated to a basis 319
The implication 6° => 7° is obvious.
7° => 1°. If we have 7°, then sup
1
= C<oo, where
z -Si
i=l Pi
{/„} cz E* is the a. s. c. f. to {х„}. Hence, for any finite sequence ab...,an:
we have
Z ал
1
1
С jf t /?f С sup
The implications 1°=5>8°=>9° are obvious.
9°=>Г. If we have 9°, then for any аь...,а„>0 we have
Z ал-
Z т&*
у а,
SUP
with a suitable f/>0 corresponding to the basis {/?„х„}.
1 ° => 10°. Assume that we have 1 ° and let {/?„} be an arbitrary sequence
of scalars satisfying A0.8). Then
sup
sup
sup ]|х„]|<со.
Furthermore, since <— р„хп[ = {х„} is of type /+ and since
0 < = inf —
sup pn 1«„<оо /3„
sup ~-=
inf
< oo,
applying the implication 9°=>1° above to the basis {finxn} and the
sequence of scalars < — > we obtain that {Р„х„} is of type /+.
\.rnJ
Finally, the implication 10°=>Г is obvious. This completes the
proof of theorem 10.1.
Remark 10.1. In contrast with 3°-5°, 7° and 9°, conditions 2°, 6°, 8°
and 10° imply that sup ||х„|| <оо. Indeed, if we have 6° or 8°, then
sup
sup ||/?„х„|| <оо, and it is obvious that 10°
Is5j<oo
also implies sup ||х„|| < oo; for 2° see Ch. I, § 3, proof of lemma 3.1 b).
1^П<00
Combining §9, theorem 9.2b) and the equivalences Г«.6°«-7° of
theorem 10.1, one obtains other characterizations of bases of type /+,
e.g. the following: Л basis {xn} of a Banach space E, with sup ||xjj < oo,

320
II. Special Classes of Bases in Banach Spaces
is of type l+ if and only if there exists a sequence of scalars {/?„} satisfying
A0.6) such that the sequence
zi=/*i*i> ^п = Ря-1Хя-1-Ряхя (и = 2,3,...) A0.12)
is a basis of E.
Now we shall give some other characterizations of bases of type /+
in terms of the "associated cone". Throughout the sequel by "cone" we
shall understand "closed convex cone having the origin as extreme
point", i.e. a closed set Ж such that Ж+Ж а Ж, "а. Ж а Ж (Я^О)
and Жп(-Ж)={0}.
Definition 10.2. Let {xn} be a basis of a real1 Banach space E. The set
A0.13)
is called the cone associated to the basis \xn}.
It is easy to see that Ж{Хп) coincides with the cone <ё{Хп) generated by
{xn} (i.e. the smallest cone containing the basis {*„}). Indeed, the inclu-
inclusion Ж{Хп] => <?{Хп} is obvious since Ж{Хп] is a cone containing each xi
(/'= 1,2,...); on the other hand, for any cone 4> containing each Xj we
П 00 П
have У otiXiE^ (al9 ...,an^0), whence also ? oiixi= lim У а^^е^
!, whence the
а;Х,-е?, а„^0, и=1,2,... , and thus JfJXn,
equality.
We recall that a subset 38 of a cone JT is said to be a base of the
cone JT if 38 is closed and convex and if every хеЖ\{0} has a unique
representation of the form x = A,y, with l>0, ye38. Furthermore, we
1 We shall only consider cones in real Banach spaces. Whenever a complex
Banach E with a basis {х„} is isomorphic to the complexification of the real Ba-
an = real (n= 1,2,...)[ (or, equivalently, the map-
nach space G = •
ping ^.а.3х3-* ]? (Rent;)*, is continuous—see Ch. I, §1), one can define the
j=i j=i
associated cone by
Г со I
= 0 (i=l,2,...)
and this may be regarded as the cartesian square of the cone
of the real Banach space G.
10. Bases of types / + , (/ + )*, a l+ and (o/ + )*. The cone associated to a basis 321
recall that a set J( in a topological linear space i? is called an extremal
subset of a closed convex set s/, if it is a closed convex subset of s/ and
if together with every interior point of a segment in л/ it contains the
whole segment (i.e. the relations х,уел/, Хх-\-(\— л)уеЛ and 0<Я<1
imply х,уеЖ); an extremal subset of л/ consisting of a single point is
called an extremal point of л/. Finally, we recall that a cone Ж is called
solid, if it contains an interior point.
Some properties of cones associated to general bases \xn) are col-
collected in
Proposition 10.1. Let {xn} be a basis of a real Banach space E and let
Ж{Хп] be the cone associated to the basis {xn}. Then
a) Ж{Хп] has an unbounded base, if dim?=oo.
b) Ж{Хп) has no weakly compact (hence also no compact) base, if
dim?= oo.
c) Each ray r~ \J.Xj 10^i<oo} is an "extremal ray" (i.e. an
extremal subset) of Ж{Хп] and these are the only extremal rays of Ж~{Хп].
Consequently, the rays rj intersect any base 3§ of Ж{Хп] in extremal points
of @.
d) Ж{Хп] is not solid, if dim?= oo.
Proof, a) Define feE* by
-Mx) (xeE),
/w= I
where {/„} <= E* is the a. s.c.f. to {xn}. We claim that the set
A0.14)
1} A0.15)
is an unbounded base of Ж{Хп]. Indeed, 38 is convex and closed, and for
every хеЖ{Хп1\{0} we have x = ky, where A = f(x)>0 and y = -j.— xej$.
J \X)
This representation is unique since the relations x = Alyi = A2y2,
Al,/.2>0,y1,y2?& imply,byA0.15), f{x) = Xl = A2, whencealso У\=у2;
therefore 38 is a base of Ж{Хп). Furthermore, we have
i.e. 2n\\fn\\xne3S(n = \,2,...), and this sequence is unbounded, since
II2"II/„||xn|| =2"||/J| \\xj>2»\fn(xn)\=2» (й=1,2,...).
b) Assume that 38 is a weakly compact base of JfJXni. Since 38 is a
base of JfJXnl, there exist /1„>0 and Jne^ such that xn = inyn{n=\,2,...).
21 Singer, Bases in Banach Spaces I

322
II. Special Classes of Bases in Banach Spaces
Then, since Si is a weakly compact set, whence also sequentially weakly
compact1, {yn} has a subsequence {уПк} converging weakly to an element
yoe&. For nk>j we have then 0 = fj(ynk)-*fj(yo) as fc-юо, whence
?(yo) = O (/=1,2,...), whence, since {/„} is total on E, yo = 0, and thus
0e&. However, this is impossible for a base S6 of a cone (since if
je^\{0}, then, Й? being convex, the whole segment <0,j> belongs to Si,
whence the element yeJf{Xn}\{0} has two representations y=\-y and
у = 2—), which proves that J^Xni has no weakly compact base Si.
c) Let us prove first that each ray r}= {Xx}- | 0^Я<оо} is an ex-
extremal ray of Ж{Хп]. Assume that y,zeJf{Xn}, aoy + (\— ao)z = XoXj,
where 0<ao<l,Ao>0. Then
0 for /#/,
Яо for i=j,
whence, since /j(y),/;(z)^O (by у,геЖ{Хп>) and ao>0, l-ao>0, it
follows that we must have f(y)=f{z) = Q for i^j. Consequently,
('=1,2,-),
whence, since {/„} is total on ?, j=y\(y)Xj-. Similarly, we have also
z=fj(z)xj, which proves that r} is extremal.
Now let Si be an arbitrary base of Ж{Хп). Then x}eЖ{Хп)\{0} admits
a unique representation х}-Ху}, with Я>0, д^еД whence j,- = — Xj
is the unique point of intersection of the ray r-; with Si. To prove that
this is an extremal point of Si, assume that there exist y,ze2fi, уфг and
aowith 0<ao<l, such that aoj + (l -ao)z = — x}. Then, by the above
1 1
(with Ao = —), we must have y = fjiy)xj, whence Xj = -—у = ЛУр
Я j JjW
which, since Я# (because otherwise aov + (l-ao)z=j, whence
Ш
Ш
у = z), contradicts the definition of a base Si of Ж{Хп}. Thus, ^ intersects ^
in an extremal point of 06.
Now let r={Xy \ Я^О} be an arbitrary ray in Ж{Хп), different from
00
all rays г.. Since yeЖ{Хп), we have у = ? a,*,-, with а„^0 (и= 1,2,...),
and with some a/0, whence J = -iBa-x.) + i|2 X aixi| , but 2ajxj$r
1 See e.g. [50], p. 430, theorem 1.
10. Bases of types /+, (/+)*, al+ and (al+)*. The cone associated to a basis 323
since by our hypothesis г-3Фг. Hence r is not an extremal ray of Ж,Хп,,
which proves that the r}- (/'= 1,2,...) are the only extremal rays of Ж{х\.
d) To prove that Jf,Xnj is not solid if dim?= oo, let x0 = ^ о^х(еЖХп]
i= 1
and c>0 be arbitrary; it will be sufficient to prove that the cell S(xo,c)
= {xeE | \\x — xo\\ ^c} is not contained in Ж{Хп]. We may assume, with-
without loss of generality, that || xn \\ = 1 (n = 1,2,...)." Since a° ^ 0 (n = 1,2,...),
а„->0 (и->оо) and dim?=oo, there exists an index N = N(c) such
that a%<c. Put
а„° for
а„°-с for n = N.
Then for x= ? а,ос,=
i = 1
i * JV
we have
I («*-
whence xeS(xo,c), but a,y = a^-c<0, i.e. хфЖ{Хп], which completes
the proof of proposition 10.1.
We recall that two cones Ж' and Ж" in Banach spaces E and F
respectively, are said to be locally isomorphic, if there exists a one-to-one
linear mapping т of Ж' onto Ж" such that for {zn} с Ж', zoeJf',
lim zn = z0
A0.16)
Now we shall give some characterizations of bases {xn} of type /+
in terms of properties of the cone Ж{Хп) associated to {xn}.
Theorem 10.2. Let {xn} be a bounded basis of a real Banach space E,
with the a.s.c.f. {/„} and let Ж{Хп] be the cone associated to the basis {*„}.
The following statements are equivalent:
Г. {*„} is of type I+.
2°. We have
i= 1
3°. There exists a local isomorphism т of the cone Jf{enj associated to
the natural basis {en} of I1, onto Ж{Хп), such that
т(е„) =
A0.18)

324
II. Special Classes of Bases in Banach Spaces
4°. There exists a constant M>0 such that for every xejf{Xn} the
OD OO
series ? fi(x)xi is absolutely convergent (i.e. ? ||/Дх)л:(|| <ooj and
i=l i = 1
A0.19)
5°. Ж{Хп) has a bounded base.
Proof. The equivalence 1°<=>2° is an immediate consequence of the
equivalence 1°<=>3° of theorem 10.1.
2°=>3°. Assume that we have 2° and define x: 3f{enj-+ 3f{Xri) (where
{е„} is the natural basis of Z1) by
Z *tei = Z *i*i «»>0, 11 = 1,2,...; X af<oo . A0.20)
\
Then т is linear, satisfies A0.18) and, by 2°, т maps jf{en} onto Jf{Xnj.
Since {х„} is a basis, т is one-to-one. Let us prove A0.16).
If
OD OD
zn= Z a!n4--+z0= X «i^i as n^co,
00 00
where а|п),а,-^0, X а'п)> Zai<00> then Z Ип)-а(НО as и->оо,
i = 1 i = 1 i = l
whence
II Ф„)-Фо
as n-»co.
Conversely,
)ll =
CO
assume that
^ sup x7
1 ^S j < oo
t(zn) = ^ а((п)х( -^ t(z0) = X «¦*¦ as и -> со,
00 00
where а*п), af>0, X a'n)' Za;<00 and let г>0 be arbitrary. Then
i=l i=l
there exists a positive integer /E such that
a, <
;Г1? f/+l+ SUP ||X-|
1 <J< 00
A0.21)
10. Bases of types /+, (/ + )*, al+ and (a/ + )*. The cone associated to a basis 325
(where rj>0 is as in A0.3)), whence
Z «л-
sup
sup
A0.22)
On the other hand,
i = Ic
^j v i if i
i= 1
Here the right side ->0 as n-^oo, because of the continuity of the
coefficient functional, respectively by the hypothesis т(г„)-> t(z0). Thus
Et]
rj+l+ sup H^ll
l«j<o
whence, taking into account A0.3) and A0.22),
Z «I"'**
1+ sup Ilxj-H
Г]+\+ SUP \\Xj
-г (и>ЛГ(г)). A0.23)
Now, from
J X J ?
i = 1 i = 1 i = /c i = /?
for n->co we obtain, taking into account A0.21) and A0.23),
lim X Ип)-
whence, since г>0 has been arbitrary,
CO
lim||zB-z0||=lim
3°=>2°. Assume that we have 3° and let Y, ^х^Ж^ ,. Then by
A0.18),
A0.24)
П ОС
Since lim ^ aixi = Z aix" and since т is a local isomorphism,
""*" i= 1 i= 1
/ n \ со
linn 4 X a"xi exists, whence, by A0.24), Уа,<оо.
и—* oo \ . „ / .

326 II. Special Classes of Bases in Banach Spaces
Conversely, if а„^0 (и = 1,2,...), Z ai < °°> tnen> by sup ||xj<oo,
00 00
the series ? щх{ converges, whence Z a-ixie^{xn>,-
i = 1 i = 1
1°=>4°. If [х„} is of type /+, then for every хе.Ж\Хп] and и =1,2,...
we have
sup
Z /«(*)*,
whence, taking n->oo, we obtain A0.19) with M = — sup ||x,-||.
4°=>Г. If {xn} is a bounded basis satisfying A0.19), then for any
n
al,..., а„^0 we have, setting x = Z aixi in (Ю.19),
i= 1
M
a, s?
1 III —¦
M ,.^1 ' '
i= 1
Г=>5°. Assume that {xn} is of type /+ and put
38= {ye.yf{Xn, | Ду)=1}, A0.25)
where feE* is any functional satisfying A0.5). Then ^ is a base of
-^|х„1 (see the argument in the proof of proposition 10.1a), taking also
into account that for any xeJf{Xnj\{0} we have f(x) = fl
4i= 1
= Y, aif(xt)^ Z ai>0) and f°r every >'= Z aixi<=-'% we have
i= 1 i= 1 i= 1
(и = 1,2,...), whence
sup
= sup \\xn\\f
«i^ = sup
i.e. ^ is bounded.
5°=>Г. Assume that the cone Jf{Xn, associated to the bounded basis
{х„} has a bounded base 38. Then sup||j|| = C<co. Since 0ф38 (see
the proof of proposition 10.1b)) and 38 is closed, we have also inf||j||
= >y>0. Furthermore, each х^еЖ{Хп] admits a unique representation
Xj = AJyJ, with ^>0,^е^, whence finxn = yne<M (n= 1,2,...), where we
have put /?„ = — (и= 1,2,...). We have then ||/SBxJ| ^C (и= 1,2,...)»
10. Bases of types / + , (/ + )*, a/+ and (o/ + )*. The cone associated to a basis 327
whence /?„ s?
С
inf
S j < 00 '
-, whence sup /?„< со. Furthermore,
if xeсо{/?„*„}, then, since ^ is convex, xe38, whence ||x|| ^ inf ||j]| =ц.
Consequently, by theorem 10.1 (implication 4°=>1°), {finxn} is a basis
of type /+, whence, again by theorem 10.1 (implication 9°=>1°), {х„} is
a basis of type /+, which completes the proof of theorem 10.2.
Remark 10.2. Let us also mention the following alternative proof
of the implication 5°=>1°: Assume that the cone Ж{Хп) associated to
the bounded basis {xn} has a bounded base 38. Then, since 38 is closed
and 0$3$ (see the proof of proposition 10.1b)), 38 can be1 strictly sep-
separated from 0 by a hyperplane, i.e. there exists a functional fsE* such
that inf/(y) = E>0, whence f(pnxn)^d, where /?„>0 is the (unique)
number for which finxne38 (n= 1,2,...). Since \\р„хп\\ ^sup||j|| =C<oo
С С
(и =1,2,...), we have В„ ^ ^ («=1,2,...), whence
6- inf
A1=1,2,...),
whence, by theorem 10.1 (implication 5°=>1°), {xn} is of type /+, which
completes the proof.
Remark 10.3. Condition 3° implies that {*„} is already a bound-
bounded basis of E. Indeed, assume that we have 3°, but sup ||xj
1 $П<00
= sup ||т(е„)|| = оо. Then there exists a subsequence {ein} of {е„} with
T(ein)#0 (n =1,2,...), Нт||ф(„)|| = со, whence
lim
1
he,-
1
= 1 (и=1,2,...),
which contradicts A0.16). Similarly, if we have 3°, but inf ||xj
1 $П < 00
= inf ||т(е„)||=0, then there exists a subsequence {et } of {en} with
1 ^ Л < 00 "
Нтт(е;п) = 0 = т@), which, since \\et \\ = 1 (n= 1,2,...), contradicts again
A0.16).
Let us recall that a cone Jf is said to be generating, if ? = JT— JT
= {y — z\y,zeJf). From theorem 10.2 we obtain, in particular, the
following characterization of bases equivalent to the unit vector basis
Of/1:
1 See e.g. [43], Ch. I, §6, theorem 5, or [133], p. 245, theorem A).

328
II. Special Classes of Bases in Banach Spaces
Corollary 10.1. A bounded basis {*„} of a real Banach space E is
equivalent to the unit vector basis of I1 if and only if the associated cone
Ж,Хп) is generating and has a bounded base.
Proof. The cone Ж{еп, associated to the unit vector basis {en} of I1
is generating and by theorem 10.2 (implication 1°=>5°) it has a bounded
base. Therefore, the cone Ж{Хп} associated to any basis {х„} equivalent
to {en} has the same properties.
Conversely, assume that {*„} is a bounded basis such that Ж{Хп] is
generating and has a bounded base 3&. Let xeE be arbitrary. Then,
since Jf{Xn} is generating, x=y-z, with у,геЖ{Хп}, whence, by theo-
theorem 10.2 (implication 5°=>4°),
Z I/,WI ^ Z 1/,-GOI + Z l/i(
i=l i=l i= 1
1
inf \\xn
Z!l/.-W*,ll+ Xll/«(z)x,.|| <oo,
i= 1
where {/„}<=?* is the a.s.c.f. to {х„}. Conversely, if Z |«;! <oo, then,
m '=1
since sup \\xn\\ < oo and since E is complete, Y^ а.{х{ converges, which
completes the proof.
Let us recall that a cone Ж induces a natural partial order relation
on E, namely x~^y if and only if х—уеЖ (in particular, x^O if and
only if хеЖ). The cone Ж is said to be normal, if1 the norm on E is
"semi-monotone", i.e. if there exists a constant L>0 such that
O^x^y => ||x|| ^L||j||. A0.26)
Proposition 10.2. // {xn) is a basis of type l+ of a real Banach space
E, then the associated cone Ж,х , is normal.
Proof. If
,- (;'=1,2,...), then, by theorem 10.1 (implication
Z aixi
sup ||xj Z«i^ sup ||xj
1
- sup \\xn\
Z Pl*i
1 Initially, normal cones have been defined (see [139], [137]) by the following
equivalent condition: there exists a constant 5 > 0 such that
(x,yeX,\\x\\=\\y\\=l).
10. Bases of types / + , (/ + )*, a/+ and (a/ + )*. The cone associated to a basis 329
(where ц>0 is the constant occurring in 10.3)), which completes the
proof.
Let us observe that proposition 10.2 can be also obtained as a
corollary of theorem 10.2, since one can prove1 that if a cone Ж in a
Banach space E has a bounded base, then Ж is normal.
The converse of proposition 10.2 is not valid, since e.g. the cone
Ж{Хп] associated to the unit vector basis {xn} of c0 is normal (and also
generating), but {xn} is not of type /+.
It is natural to raise the problem of characterizing other types of
bases {*„} of a Banach space E by geometric properties of the asso-
associated cone Ж[Хп}. We shall give below some results of this kind, in-
involving a special class of bases <% of Ж{Хп). Namely, a subset 3d of a
cone Ж in a Banach space E will be called a hyperbase of Ж, if there
exists a strictly positive functional feE* (i.e. f(x)>0 for all хеЖ\{0})
such that
l}. A0.27)
Lemma 10.1. For a subset 3% of a cone Ж in a Banach space E the
following statements are equivalent:
Г. & is a hyperbase of Ж.
2°. Every хеЖ\{0} has a representation x = ky with A>0, ye3S,
and there exists a functional feE* such that
.{yeJtT\f{y)=l}.
A0.28)
3°. Every хеЖ\{0} has a representation x = Xy with л>0, уеЗЗ,
and 0 does not belong to the closed linear manifold2 spanned by 38.
Proof. The implication 1°=>2° is obvious.
Conversely, if we have 2° and хеЖ\{0}, then, by 2°, x has a re-
representation x = Xy, with a>0, уеЗд, whence, by A0.28), f(x) = Xf(y)
= Я>0, and therefore / is strictly positive. If ye Ж and f(y)= 1, then
there is a representation y = Xz, with l>0, zeJ, whence, by A0.28),
/(z)=l, and thus 1 =/(y) = A/(z) = A and j = ze^, which proves A0.27).
Thus, 2°=>1°.
Furthermore, if we have 2°, then 39, whence also the closed linear
manifold spanned by $8, is contained in the hyperplane {yeE \ f(y)= 1},
which does not contain 0. Thus, 2 =>3°.
Finally, assume that we have 3° and let V be the closed linear mani-
manifold spanned by 39. Then for any fixed y0e3# the set V—y0 is a closed
linear subspace of E, not containing — y0, whence, by a well known
1 See [163], proposition 2.
2 We recall that a set F<=? is called a linear manifold if it is of the form
xo + G= {xo + g \geG}, where G is a linear subspace of E.

330
II. Special Classes of Bases in Banach Spaces
corollary of the Hahn-Banach theorem, there exists a functional feE*
such that f(y-yo) = 0 (yeV), /(yo)=l, whence we infer A0.28). Thus,
3°=>2°, which completes the proof of lemma 10.1.
From the proof of proposition 10.1a) we see that every hyperbase
is a base. However, the converse is not true, i.e. there exist (even com-
compact) bases which are not hyperbases, as shown by the following example:
Consider in the space E = l2 the circled closed convex hull Q of the
fundamental parallelotope of Hilbert, i.e. the compact set
4 (/=1,2,...)
A0.29)
Then the linear subspace G = [j n Q spanned by Q is dense in
n= 1
? = /2 (since it contains all almost zero sequences), but does not coincide
with E (since otherwise by the theorem of Baire some n0Q would have
an interior point, in contradiction with dim?=oo). Take an arbitrary
OO
xeE\G = E\\JnQ and put
n= 1
jf = {/.(y-x)\yeQ,^0}. A0.30)
Then Ж is a cone and 38 = Q-x={y-x\ yeQ} is a compact base
of Ж, but not a hyperbase of Ж. Indeed, for any y,zeQ, k,/x^0 we have
/ к Ц \ к
\у + г)Ж (
+
zeQ by the convexity of Q), whence Ж + Ж
(since
Ж. For any
yeQ and к,/л^0 we have obviously цк(у — х)еЖ, which shows that
цЖ cz Ж for all /i^0. Furthermore, if for some y,zeQ, к,ц>0 we
have k(y — x)-—u(z-x), then x = у-\ zeQcG (by the
к + k + fi
convexity of Q), in contradiction with our assumption xeE\G, and
thus Жп(-Ж)={0}. To show that Ж is closed, let yneQ, kn^0
(и =1,2,...) be such that кп(у„-х)-->zeE as n->oo. Then, since
xeE\GczE\Q = E\Q, we have inf ||jn-x|| =<5>0, whence
о
t(Ua»0'..-*)-zII
whence sup л„<оо. Since Q is compact, it follows that there exists
1 ^S n< oo
a sequence of indices {nk} such that Я„к-»Я^О, ynk-*yeQ as fc->co,
whence z = k(y — х)еЖ, which proves that Ж is closed. Thus, Ж is
a cone.
10. Bases of types / + , (/ + )*, a l+ and (a/ + )*. The cone associated to a basis 331
Furthermore, by the definition of Ж, every element of Ж\{0} admits
a representation z = k(y — x), with k>0, yeQ. To prove the uniqueness
of these representations, assume that k(y — x) = fi(z — x), where k,/x>0,
кфц, y,zeQ. Then x — у zeG, which contradicts the
к-ц к-ц
assumption xeE\G. Thus <ffl = Q — x is a compact base of Ж
Finally, since G is dense in E, it follows that 0 belongs to the closed
linear manifold G — x = E spanned by 0& = Q — x, whence, by lemma
10.1 (implication Г => 3°), i% is not a hyperbase of Ж.
However, let us mention that the cone Ж in this example, and more
generally, any cone Ж in a separable Banach space E, has a hyperbase,
since it is known1 that there exist strictly positive functionals feE*
whenever E is separable.
Returning now to the cones Ж{Хп} associated to bases \xn} of Banach
spaces, we observe that by the above proof of proposition 10.1a) Ж{Хп]
has an unbounded hyperbase. Some characterizations of other types of
bases {xn} by geometric properties of the hyperbases $8 of the associated
cone Жг ,, or of the hyperbases 0?ni of the cones Ж,, x , associated to
the bases {?„*„} of E, where ?„=±1 (и=1,2,...), are given in
Proposition 10.3. A bounded basis {xn} of a (real) Banach space E is
a) of type P*, if and only if there exists a hyperbase 38 of the cone
Ж{Хп) associated to the basis \xn}, containing all Xj (j= 1,2,...).
b) of type wc0, if and only if for every {sn}, Sj= +1 (/'= 1,2,...) and
every hyperbase @{?n} of the cone Ж{?пХп} associated to the basis {гпхп},
the (unique) numbers Д,->0 for which ^$1с"''эpjSjXj (J= 1,2,...) satisfy
lim/?n= oo.
И-+ОО
c) shrinking, only if for every {«„}, ?,-=±1 (/=1,2,...) and every
hyperbase 3Sl?n} of the cone Ж{ЕпХп] we have
dist(O, ^!)-^oo as n->oo, A0.31)
where
<%^ = ^'>n[xn,xn+l,xn + 2,...-] (и=1,2,...). A0.32)
Proof, a) If {xn} is of type P*, then, by § 9, theorem 9.2 b) (implication
Г=>4°) there exists an feE* such that /(*,-)= 1 (/=1,2,...). Then
OO GO
for any x = ? a1xi€JfJXnl\{0} we have f(x)= ? a,->0, whence
i = 1 i = 1
8)={уеЖ{Хп] | /(y)= 1} is a hyperbase of Ж[Хп] containing, obviously,
all х,.(/=1,2,...). Conversely, if 38 is a hyperbase of Ж{Хп} such that
Xj€<% (/=1,2,...), then there exists a strictly positive feE* such that
See e.g. [144], §2, theorem 2.1.

332
II. Special Classes of Bases in Banach Spaces
^={yeJf{Xn)\f{y)=\). Then /(*,)= 1 (/ = 1,2,...), and therefore, by § 9,
theorem 9.2 b) (implication 4° => Г), {х„} is of type P*.
b) If {xn} is of type wc0, then for any {?„}, e~ + 1 (/ = 1,2,...), we
have lim /(е„х„) = 0 (/е?*). Let &te"} be an arbitrary hyperbase of
n~*¦ oo
the cone Jt~{enXn). Then there exists a strictly positive1 /e?* such that
0Cn} = {yejf{CnXni\f{y)=l}, whence гухуе^|?"! and thus
f(Ejxj)
(/=1,2,...), whence lim/?n=oo. Conversely, if a bounded
1
basis {xn} is not of type wc0, then there exists an /e?* such that
lim f{xn) ф 0. Let {nk\ be the set of all indices for which f(x ) Ф 0 and
n-*oo
let {mk} be the complementary set of indices. Put
(k= 1,2,...), A0.33)
(fe=l,2,...), A0.34)
1
A
h(x) = f(x)+ ?
i —
Then ?,= ±1(/=1,2,...), heE* and
1
(xe?).
2" IIЛII
0
A0.35)
A0.36)
whence h is strictly positive on Ще„х„}\{®}, whence 8${е"]
= {yejf{EnXn} \h(y)=\) is a hyperbase of Jf{CnXn]- Furthermore,
^|E! (/=1,2,...), but lim ф oo (since by
/(Х)
,; /г(?„Х„)
inf ||/J>0 and A0.36) we have Нт/г(?„х„)= Нт/(?„х„)
< *
П00
c) If {*„} is shrinking, let ?,= ±1 (/=1,2,...) and let ^|?n! be an
arbitrary hyperbase of the cone Jf{tnXn}. Then there exists a strictly
positive2 /e?* such that @{с"} = {уеЖ[ЕпХп] | f(y)= l}. Since {х„} is
shrinking, for any
then =
'\ y=
(i = n,n+ 1,...) we have
\\y\\
for n>N{s). Therefore
> - for all
?
{nEl} whenever n>N(E), which completes the proof of proposition
10.3.
1 With respect to tf{ttl
2 With respect to Ж,
10. Bases of types /+, (/+)*, al+ and (al+)*. The cone associated to a basis 333
Let us turn now our attention to the other types of bases introduced
in definition 10.1. Some characterizations of bases of type al+ are
collected in
Theorem 10.3. Let {xn} be a basis of a Banach space E, with
sup ||х„ || < oo. The following statements are equivalent:
1°. {*„} is of typeal+.
2°. There exists an /e?* such that
inf \f(
A0.37)
3°. There exists a basis {yn} of type P* (of a Banach space F) such
that {xn} is affinely equivalent1 to {>>„}.
4°. There exists a basis {>>„} of type P* (of a Banach space F) such
that {xn} affinely dominates2 {>>„}.
5°. There exists a basis {yn} of type al+ (of a Banach space F) such
that {xn} affinely dominates {>>„}.
// ? is a real Banach space, these statements are equivalent to the
following:
6°. There exist a sequence {?„}, ?,-= ± 1 (/= 1,2,...) and a hyperbase
&{e"} of the cone Jf{enXn} associated to the basis {?„х„} such that the unique
numbers Pj>0 for which ^^aPjEjXjij^l,!,...) satisfy sup Д„<оо.
1 ^ П < 00
Proof 1°=>2°. If {?„*„} is of type /+, where |?„| = 1 (и =1,2,...),
then, by the implication 1°=>5° of theorem 10.1, there exists an /e?*
such that
|/(xn)|>Re/(?nxn)>l A1 = 1,2,...).
2°=>1°. If/e?* satisfies A0.37), put
f(xn) . ...
?„ = (И=1,2, ...).
I/WI
Then |cn| = l, !|?„х„|| = ||х„|| (и =1,2,...) and for heE* defined by
1
h =
we have
h(Enxn) = —-
(«=1,2,...),
jnfjfixj)]
whence, by the implication 5°=>1° of theorem 10.1, {е„х„} is of type /+.
1 See Ch.I, §8, definition 8.2.
2 See Ch. I, § 8, definition 8.2.

334
II. Special Classes of Bases in Banach Spaces
1°=>3°. If {?„*„} is of type /+, where |е„| = 1 (л=1,2,...), then, by
the implication Г=>6° of theorem 10.1, there exists a sequence of
scalars {/?„} satisfying A0.6) and such that {/?„?„*„} is a basis of type P*
of E. Since {х„} is affinely equivalent to {/?„?„*„}, it follows that we
have 3° with F = ?, {>>„}= {Р„е„х„}.
The implication 3°=>4 is obvious. The implication 4°=>5 fol-
follows from the fact that every basis of type P* is also of type /+ (take
/?„=1 in 7° of theorem 10.1 and apply the implication 7°=>1° of this
theorem).
5° => 1°. Assume that {xn} affinely dominates a basis {_>>„} of type al+
of a Banach space F. Then there exists a sequence of scalars {/.„} with
0< inf |AJ < sup |А„|<оо such that {х„} dominates1 the basis
1 =Sn< oo l=Sn<oo
{лпуп} of F, whence, by the implication b) 6=>b) 1 of Ch. I, §8,
theorem 8.1, there exists a continuous linear mapping ueL(E,F) such
that
u{xn) = Anyn (и =1,2,...).
Put sn = -~-(n=l,2,--.). Then |ej=l, \\enxj=\\xj (n=l,2,...)
I '"П I
and for all finite sequences a1;...,an^0 we have, taking into account
that {ё„у„} is of type /+ for suitable {ё„} with \Sn\ = l (n= 1,2,...),
E <*i?Axi
E a.-l^-l^^
ц inf | А -| n
1^/<оо г
i= 1
Thus
?„<5„л:„} is of
(with a suitable >?>0 corresponding to {^„
type /+, i.e. {х„} is of type a/ + .
Г =>6°. If {х„} is a basis of type al+ of a real Banach space E, then,
by the implication 1°=>2° proved above, there exists an feE* such
that |/(х„)|^1 (и = 1,2,...). Put ?n = sign/(xn) (и = 1,2,...). Then
/(?„х„)^1 (и =1,2,...), whence / is strictly positive on ЩЕпХг,}, whence
0c"} = {yejf{tnXJf(y)=l} is a hyperbase of Jf{tnXn]. Furthermore,
1^-rsjxJe^'U=l,2,...), and f}n =-L-< 1 (n=l,2,...).
6° => 1°. If {xn} is a basis of type non-a/+ of a real Banach space E,
then, by the implication 2° => Г proved above, we have inf |/(х„)| = О
1 See Ch. I, §8, definition 8.1; actually, we have replaced here the sequence
А„} of Ch. I, §8, definition 8.2 by }j-\.
10. Bases of types /+, (/+)*, al+ and (al + )*. The cone associated to a basis 335
(feE*). Let ?,= + 1 (j= 1,2,...) and let 0е be an arbitrary hyperbase
of the cone JflenXn}- Then there exists a strictly positive1 feE* such
that
6,- =
/(y)=l}, whence
?,x,
e^|?"!, and thus
(/=1,2,...), whence sup fi=<xi, which completes the
1«
f{SjXj)
proof of theorem 10.3.
Remark 10.4. The implication 5°=>1° of theorem 10.3 shows that
the property of being a basis {yn} of type al+ is conserved by all bases
{xn} with sup ||х„||<оо which affinely dominate {>>„}. Other such
1 <П< GO
properties are e.g. that of being a bounded basis of type non-wc0 or a
weakly closed bounded basis. Moreover, affine equivalence also con-
conserves the opposite properties, i.e. those of being a bounded basis of
type non-a/+ or wc0 or non-weakly closed. On the other hand, the
implication 3°=>1° of theorem 10.3 shows that the property of being
a basis {>>„} of type P* is not conserved by all bases {xn} which are
affinely equivalent to {>>„} (since there are bases of type al+ which are
not of type P*, as we shall see in § 12, example 12.4). Other properties
of bases which are not conserved by affine equivalence are e.g. that of
being of one of the types aP*, /+, P, aP or (/+)*.
By the duality methods used already in the preceding sections,
it follows that {xn} is of type l+ if and only if the a.s.c.f. {/„} с Е* is
of type (/ + )*. Furthermore, from definition 10.1b) and theorem 10.1
one can obtain various characterizations of bases of type (/+)*. Com-
Combining two of them (namely, those obtained from the equivalences
Г<*6°<*7° of theorem 10.1) with §9, theorem 9.1, one can obtain
other characterizations of bases of type (/+)*, e.g. the following: A
basis {xn\ of a Banach space E, with inf |[xJ>0, is of type (/+)*
1^Л< 00
if and only if there exists a sequence of scalars {/?„} satisfying A0.6) and
such that the sequence
1
Л=Е 7TXi
;=i Pi
A0.38)
is a basis of E.
Obviously, all the preceding characterizations of bases of types
/+,(/+)* imply characterizations of bases of types a/+ and (a/+)*
respectively.
Let us consider now the corresponding properties for the Schauder
basis of C([0,1]) and the normalized Haar basis of /^([0,1]).
With respect to

336
II. Special Classes of Bases in Banach Spaces
Theorem 10.4. a) The Schauder basis {х„}„°°=0 of C([0,l]) is of
types non-al+, non-(l+)*, but (al + )*. Namely, the sequence {епхл}, where
the ?„ are defined by (9.15), is a basis of type (/ + )* of C([0,1]).
b) The normalized Haar basis {!„} of Lx([0,1]) is of types non-al+
and non-(al+)*.
Proof, a) The proof of the assertion that {xn}*'=0 is of type non-a/+
is similar to the proof of the fact that it is of type non-a P* (§ 9, theorem
9.3)), using the implication 1°=>2° of theorem 10.3.
Furthermore, the assertion that {xn}™=0 is of type non-(/+)* is
equivalent, by the above remarks on characterizations of bases of
type (/+)*, to the assertion that for every sequence of scalars {р„}™=0
f 1
fl 1
satisfying 0 < inf /?„ < sup /?„ < 1 the basis < — х„ > is of type
0«n<co 0«n<oo (_/?„ Jn=o
non-P. Now, the proof of this latter assertion is similar to the proof of
the fact that {*„}„! 0 is of type non-P (§ 9, theorem 9.3 a)).
Finally, the assertion that {е„хл}лж=0 is of type (/+)* is a consequence
of the fact that it is of type P (§ 9, theorem 9.3 a)).
b) The proof of the assertion that [zn\ is of type non-a /+ is similar
to the proof of the fact that it is of type non-a P* (§ 9, theorem 9.3 b)),
using the implication 1°=>2° of theorem 10.3.
Finally, the proof of the assertion that {zn} is of type non-(a/+)* is
similar to the proof of the fact that it is of type non-a P (§ 9, theorem
9.3 b)), replacing (9.19) by
|Re<P(/O|>l (и =1,2,...).
This completes the proof of theorem 10.4.
From the implication 7°=>Г of theorem 10.1 with /?„=1 (и=1,2,...)
and §9, corollary 9.1a) it follows that the sequence (9.16), where the ?„
are defined by (9.15), is a basis of type l+ of C([0,1]).
Problem 10.1. Does the space Lx([0,1]) possess a basis of type /+?
Or, equivalently •. does it possess a basis of type (/+)*?
By the equivalence Го7° of theorem 10.1, this problem is equiv-
equivalent to § 9, problem 9.1.1
The answer to the problem of existence of bases of types al+, {al+)*
(and hence also for bases of types /+, (/+)*) is negative, as shown by
Example 10.1. Let E be an infinite dimensional reflexive Banach
space with a basis. Then E has no basis of type al+ or (al+)*.
In fact, every basis {х„} of E is of type wc0 (see § 5), whence, by the
implication 1°=>5° of theorem 10.1, {xn} is of type non-a/+. Since E*
1 Hence, since problem 9.1 has been solved in the affirmative (see §9, the foot-
footnote to problem 9.1), the answer to problem 10.1 is also affirmative.
11. Besselian and Hilbertian bases. Stability theorems
337
is reflexive, too, applying this result to E* it follows that the a.s.c.f. {/„}
to {xn} is a basis of type non-a/+ of [/„]=?*, and thus {xn} is also of
type non-(a/+)*.
The answer to the problem of existence of bases of types non-/+,
non-(/+)* (and hence also for bases of type non-a/+ and non-(a/+)*) in
finite-dimensional Banach spaces is obviously negative. On the other
hand, the answer to the problem of existence of bases of types non-/+,
non-(/+)* in infinite dimensional Banach spaces with bases is affirmative.
Indeed, if {xn} is a basis of type /+ of an infinite dimensional Banach
space E, then, as we have observed in the above, there exists a sequence
of scalars {/?„} satisfying A0.6) and such that the sequence A0.12) is a
basis of E. However, this latter basis is of type non-/+, since otherwise
there would exist an ц > 0 such that
sup
lin< oo
sup
1 in< oo
= sup \\2piXl-pnxJ
which is impossible because of sup ||/?nxj< sup ||jcJ<oo. Simi-
1<П<СО 1 ^ Л < OO
larly, if {xn} is a basis of type (/+)* of E, then there exists a sequence of
scalars {/?„} satisfying A0.6) and such that A0.38) is a basis of type
non-(/+ )* of E.
Problem 10.2. a) Does every infinite dimensional Banach space with
a basis possess a basis of type non-a/+? In particular, does the space I1
possess such a basis?
b) Does every infinite dimensional Banach space with a basis pos-
possess a basis of type non-(a/+)*? In particular, does the space c0 possess
such a basis?
By the equivalence Го7° of theorem 10.1, this problem is equiv-
equivalent to § 9, problem 9.2.
In connection with problem 10.2 let us mention that the space I1
has a subspace with a basis of type non-a/+. Indeed, such is e.g. the
subspace F of I1 defined in § 8, example 8.1, since by (8.9) and the impli-
implication 1°=>2° of theorem 10.3, the basis (8.6) of (8.5) is of type non-a/+.
Dually, the space c0 has a subspace with a basis of type non-(a/+)*.
§ 11. Besselian and Hilbertian bases. Stability theorems
Definition 11.1. A basis {xn} of a Banach space E is said to be
a) Besselian, if
Y,<*ixi is convergent =>?|<Х;|2<оо,
i = 1 i = 1
(П-1)
22 Singer, Bases in Banach Spaces I

338
i.e. if
where {/„} is the a.s.c.f. to {xn};
b) Hilbertian, if
00 00
Ew2<a>=> E
II. Special Classes of Bases in Banach Spaces
aixi is convergent,
A1.2)
A1.3)
i.e. if for every {<xn}
with ? |<х,|2<оо there exists an (obviously
unique) xeE such that i=1
/,(*) = a, A1=1,2,...). (П.4)
For instance, the natural basis of I1 is Besselian, the natural basis
of c0 is Hilbertian, and the natural basis of I2 is simultaneously Besselian
and Hilbertian.
Let us observe that by Ch. I, §3, lemma 3.1, for every Besselian
basis {xA we have inf ||xJ>0 and for every Hilbertian basis {xn}
1 ^ Л < 00
we have sup \\xn\\ <oo.
ИП< 00
The following theorem gives some characterizations of Besselian
and Hilbertian bases and shows their relations of duality:
Theorem 11.1. Let {*„} be a basis of a Banach space E and let
{/„}c?* be the a.s.c.f. to {*„}. Then
a) The following statements are equivalent:
1°. {xn} is Besselian.
2°. {х„}>{е„},1 where {en} is the unit vector basis of I2.
3°. There exists a continuous linear mapping и of E into I2 such
u(xn) = en (n=l,2,...). (П.5)
4°. There exists a constant c>0 such that we have
E aixi
for all finite sequences of scalars al5..., а„.
5°. We have
(Фе?**).
A1.6)
A1-7)
1. T
1 I.e. the convergence of ? а,х( implies the convergence of ? а(е( (see Ch. I,
§8, definition 8.1). i=1 i=1
11. Besselian and Hilbertian bases. Stability theorems
339
that
6 • {/„} й а Hilbertian basis of [/„].
b) The following statements are equivalent:
1'. {xn} is Hilbertian.
2'- {en}>{xn}, where \en} is the unit vector basis of I2.
3\ There exists a continuous linear mapping v of I2 into E, such
v(en) = xn (n=l,2,...). A1.8)
4°. There exists a constant C>0 (n= 1,2,...) such that we have
E aixi
<C
A1.9)
for all finite sequences of scalars au ..., an.
5°. We have
(feE*).
A1.10)
6". (/„} is a Besselian basis of [/„].
Proof. The equivalences a) lGoaJ" and b) l°<=>bJ° are obvious from
the definitions. The equivalences aJ°<=>aK°<=>aL° and bJ°<=>bK°<=>bLc
follow from the equivalences 6°<=>Г<=>2° of Ch. I, §8, theorem 8.1b).
a) 4°=>aM°. By Ch. I, § 12, proposition 12.2c), we have
sup
E
i= 1
<oo (ФеЕ**),
A1.11)
whence, by A1.6), we infer A1.7).
The implication aM°=>a)l° is obvious, by considering the canonical
embedding n of ? into ?**. Thus a)l"o-oaMa.
aJ°<=>aN". By Ch. I, §12, proposition 12.1, we have a) 2° if and
only if1 {en}>{/„}, which, by the equivalence bJ°ob)l° observed
above, happens if and only if we have aN°.
bL°=>bMn. By A1.9) we have
E.
(/б?*,и=1,2,...)
1 Here we identify canonically (/2)* with I2 and hence the a.s.c.f. to {е„} is
identified with \е„\.

340
II. Special Classes of Bases in Banach Spaces
and thus
(/е?*,и=1,2,...),
A1.12)
whence we infer A1.10).
bMc^bNo. The a.s.c.f. {У»} <=[/„]* to the basis {/„} of [/„] is
nothing else but 4>п = ф(хп) (и =1,2,...), where ф is the canonical
mapping of E into [./„]*• Since by bM° we have
E
¦<oo
[/,]),
A1.13)
from definition 11.1 a) we see that {/„} is a Besselian basis of [/„].
bN°=>bJ°. If we have bN°, then, by the implication a)l°=>aJ°
observed above, {/„}>{*>„}, whence, by Ch. I, §12, proposition 12.1,
we have bJ°, which completes the proof of theorem 11.1.
Corollary 11.1. a) The space c0 has no bounded Besselian basis.
b) The space I1 has no bounded Hilbertian basis.
Proof, a) Assume that {xn} is a Besselian basis of c0, with the a.s.c.f.
{/„} c(co)*^/1. Then, by the implication a)l°=>aM° of theorem 11.1,
/„^>0 in (со)* = /\ whence also in the norm-topology1 of/1. Conse-
Consequently (see Ch. I, §3, corollary 3.1b)), the basis {xn} is not bounded.
The proof of part b) is similar, making use of the implication
b)l°=>bM° of theorem 11.1. This completes the proof.
By the same arguments one can also show that no subspace of c0
can have a bounded Besselian basis, and no subspace of I1 can have a
bounded Hilbertian basis.
Remark 11.1. If we omit the requirement of boundedness of the
basis {xn}, then the situation is different. In fact, every Banach space E
with a basis has a Besselian basis and a Hilbertian basis, namely, if {х„}
is any normalized basis of E, then {пх„} is a Besselian basis of E and
— xn> is a Hilbertian basis of E.
\
Indeed, if ? <X;/X; converges, then Нтпа„х„ = 0, whence
whence
и|а„|<?
2 t.
n2
See e.g. [10], p. 137-139.
11. Besselian and Hilbertian bases. Stability theorems 341
X) GO
and thus Y, lail2> whence also ? !arl2> converges. The second
t = JV(e)+l i=l
assertion follows by duality.
Corollary 11.2. For a basis {xn} of a Banach space E, with the a.s.c.f.
{/„}, the following statements are equivalent:
Г. {xn} is simultaneously Besselian and Hilbertian.
2°. {х„}~{е„}, where {en} is the natural basis of I2.
3°. There exists an isomorphism и of E onto I2, such that we have
A1.5).
4°. There exist two constants c>0 and C^c such that we have
<C
A1.14)
for all finite sequences of scalars a.l,..., а„.
5°. {х„} is Besselian and {/„} is a Besselian basis of [/„].
6°. {xn} is Hilbertian and {/„} is a Hilbertian basis of [/„].
Let us consider now the corresponding properties for the Schauder
basis of C([0,1]) and the normalized Haar basis of Lx([0,1]).
Corollary 11.3. a) The Schauder basis {х„}„°°=0 of C([0,l]) is both
non-Besselian and non-Hilbertian.
b) The normalized Haar basis {zn} of Lx([0,1]) is both non-Besselian
and non-Hilbertian.
Proof. By § 7, theorem 7.1, both {xn}™=0 and {zn} are of type non-
wc0, whence, by the implication b) l°=>b) 5° of theorem 11.1, they are
non-Hilbertian.
Furthermore, by § 9, theorem 9.3 a), there exists a sequence of scalars
{?n}?L0 with ?„=+1 (n = 0,1,2,...), such that
sup
0«n<<x>
E ?ixi
i = 0
< oo,
whence, by the implication a) l°=>a) 4° of theorem 11.1, the Schauder
basis {xn}?L0 of C([0,1]) is non-Besselian.
Finally, since by § 7, formula G.11), we have
=2-!<2
2"
A1.15)
from the implication a) l°=>a) 4° of theorem 11.1 it follows that the
normalized Haar basis {zn} of ^([0,1]) is non-Besselian, which com-
completes the proof.

342
II. Special Classes of Bases in Banach Spaces
Problem 11.1. a) Does the space C([0,1]) possess a bounded Bes-
Besselian basis? What about a bounded Hilbertian basis?
b) Does the space L'([0,1]) possess a bounded Besselian basis?
What about a bounded Hilbertian basis?
By theorem 11.1 (implications a)l°=>aM° and b)l°=>bM°), an
affirmative answer to one of these questions would imply an affirmative
answer to the corresponding question of § 7, problem 7.1 (on the existence
of bases of type wc0 or (wc0)* in C([0,1]) and Lx([0,1])).
If there existed in C([0,1]) a basis \xn) with sup ||л:я||<оо,
1 « A < X
which constitutes an orthonormal system, then the answer to the first
question of problem 11.1a) would be affirmative (indeed, in this case,
GO GO
from ? а,х;еС([0,1]) с L2([0,l]) it would follow that ?|<х,|2<х).
i = 1 r=l
However, in Volume II we shall see that C([0,1]) has no basis {xn}
with these properties. Dually, if there existed in /^([0,1]) a basis {х„}
with inf ||xJ|>0, which constitutes an orthonormal system, then
1 г=п< oo
GO
this would be a Hilbertian basis of Ь:([0,1]) (since ^ |a,|2<oo would
imply Y, <X;X,eL2([0,1]) <= Ь*([0,1])), and so the answer to the second
i= 1
question of problem 11.1b) would be affirmative.
If we omit the requirement of boundedness of the basis {xn}, then
the situation is different. Namely, we shall see in Vol. II that there
exists in C([0,1]) a basis {xn} which constitutes an orthonormal system
and hence, by the above argument, {xn} has property A1.1) in C([0,1]).
It is easy to verify (see Vol. II) that any such sequence {х„} is also a
basis in Lx([0,1]) and hence, as observed above, {х„} has property
A1.2) in Lx([0,1]). One can also show (using the interpolation theorem
ofM.Riesz) that the same {xn} is also a basis in each Z/([0,1])A
having property A1.1) if/? ^2 and respectively property A1.2) if 1
However, for the spaces Lp([ — n, л]) with 1 <p< oo it is known more,
namely that they have a bounded Besselian basis if p^2 and a bounded
Hilbertian basis if 1 <p < 2, as shown by
Example 11.1. In the space ? = //([-я,я])A <р<<х>) the (orthog-
(orthogonal) sequence {х
„}=0,
where
x2n{t) =
is a bounded Besselian basis if
if
(/e[-7t,7t], n=l,2,...)
A1.16)
and a bounded Hilbertian basis
11. Besselian and Hilbertian bases. Stability theorems
343
Indeed, let us first show that {xn}™=0 is a basis of E. Obviously, the
sequence {/„} с Е* defined by
2„(х) = - \x{x)cosnxdx, f2n+i(x) = —
n J n
— n -я
(xeE, n = 0,l,2,...),
A1.17)
satisfies fk{Xj) = dkj (k,j= 1,2,...). Consequently, taking into account
that
sin(n + -)T .
\ 2 sinnt 1
— = + -cosni, A1.18)
2sin— 2tg —
2 B2
we have1, for the partial sum operators associated to the biorthogonal
system (*„,/„),
2" f (x) "
LS2nW_H'J — 2-i Jk\x)Xk\4 — - ' l_i \J2k-l\-'
k=0 Z k=l
dx
i f
= - \x(t +
2 sin —
2
1
2~%
A1.19)
where we have put
6.@ = - U + t)^*. (П.20)
2tg —
2
Now, taking into account that
A1.21)
1 We consider all functions extended beyond [ — я,я) by x(t + 2n) — x(t).

344
II. Special Classes of Bases in Banach Spaces
and putting
z1{t) = x{t)sinnt, z2(t) = x(t) cos nt (te[-n,n~\), A1.22)
we have, almost everywhere on \_-n,n\
Qn(t) = cosntzl(t) — sinntz2(t),
where z^t), z2(t) are the functions conjugate1 to zx(/) and z2(f) re-
respectively. Hence
ISnMI^ l^i
and thus
/ * \i /n
/ \ — n
/ я
J iz
However, by a well known theorem of M. Riesz on conjugate func-
functions2, there exists a constant Ap depending only on/?, such that
Applying this to zt and z2 defined by A1.22), we obtain
J mt)\'dlt>^ Ap( ] М01рл)' + Лр( ] \z2(t)\"dt
Consequently, taking into account A1.19), we get
\\s2n(x)\\^A'p\\x\\ (xeE),
with a suitable A'p depending only on p. Since
A1.23)
\\xj<Bny,
(n=l,2,...), A1.24)
1 We recall that for zeL2([-7i,7i]) and zez the function
z@= -Л z(
is defined for almost all t and it is called the conjugate function to z(t).
- See e.g. [274], Ch. VII, §2, theorem B.4).
11. Besselian and Hilbertian bases. Stability theorems
345
we have
1 i
PJI \\Щ <-Bя)"Bя) p||jc||=2||jc||
n
(xeE, и=1,2,...),
whence, by A1.23) and ||s2ll+ х(х)|| < ||s2ll(x)|| + ||/2ll+ l(x)x2n+11|,
sup ||sn||<oo.
0^n<oo
Since {xn}^=0 is complete in E = LP([-n,n\) (extended Weier-
strass theorem), it follows, by Ch. I, § 4, theorem 4.1, that {xn}^0 is a
basis of E.
Furthermore, assume that p^2 and let Jce? = Lp([-;t,7t]). Then
1
also JceL ([ —я,я]), whence, since
is an orthonormal
system, ^ |/,(x)|2<oo, and thus {х„} is a Besselian basis of E. By
i
duality (theorem 11.1), it follows that for l</?^2 the sequence {xn}
is a Hilbertian basis of Е = Ьр([-п,п~\), which, together with A1.24),
completes the proof of the assertions stated in example 11.1
We shall give now some stability properties for Besselian bases and
then some characterizations of Besselian bases in terms of stability
properties.
Theorem 11.2. Let {х„} be a Besselian basis of a Banach space E,
and let {>>„} be a sequence in E, such that
sup
11/11 si
=M<c,
A1.25)
where с is any constant for which we have A1.6). Then {>¦„} is a basis
of E, equivalent to {х„} (whence Besselian).
In particular, if {yn} a E is a sequence such that
i
II*,—J
i= 1
where с is any constant for which we have A1.6), then the same conclusion
holds.
Proof. Let a!,...,an be arbitrary scalars and take a gneE* with
ffj| = l such that gJ X al{xl-yl)) =
E «-Ах—Уд
i= 1
. Then

346
II. Special Classes of Bases in Banach Spaces
E ai(xi-yd
i= 1
M
= Z Ъ9я(х,-уд < E W2 E Ыъ-у
E aixi
A1.27)
M
Since — < 1, this shows that {yn} is P W-пеаг to {xn}, whence, by
the Paley-Wiener theorem (Ch. I, § 9, theorem 9.1), {>>„} is a basis of E,
equivalent to {х„}, which completes the proof1.
Definition 11.2. Let [хп], {у„} be sequences in a Banach space E.
The sequence {>>„} is said to be
a) quadratically near to {xn}, if we have
E Il*i-Jil
A1.28)
b) weakly quadratically near to {*„}, if E !Дх;~>'гI2<00 uniformly
with respect to feE*, \\f\\ < 1, i.e. if i=1
lim sup
A1.29)
Obviously, quadratically near => weakly quadratically near, but the
converse implication is not valid. Indeed, e.g. if p>2 and {zn} is a
normalized unconditional basis2 of ? = Lp([0,1]), then by § 14, corol-
corollary 14.3, {zn} is not Besselian, whence there exists a convergent series
со со со
E «iZ,e? such that E lla,zrll2= E larl2=0°- On the other hand,
i = 1 со i=l i = 1
since E atzi is unconditionally convergent, by § 16, lemma 16.1 (im-
i= 1 со
plication Г=>6°), the series E \f(aizdl whence also the series
со r= 1
E |/(<x,z;)|2, converge uniformly with respect to feE*, ||/||<1. Thus,
any {х„}, {у„} with xn-yn = anzn (n=\,2,...) are weakly quadratically
near but not quadratically near.
1 Let us observe that in the particular case when {*„} is an orthonormal basis
of a Hilbert space, we may take с = 1; in this case, the second statement of theo-
theorem 11.2 is also a consequence of Ch. I, § 10, theorem 10.4, since
b= sup sup
\?i\ - 1 1 ^n< oo
2 See § 14, definition 14.1.
11. Besselian and Hilbertian bases. Stability theorems
347
Now we can give the following corollary of theorem 11.2:
Corollary 11.4. Let {xn} be a Besselian basis of a Banach space E
and let {>>„} be a sequence in E, weakly quadratically near (or, in particular,
quadratically near) to {xn}. Then there exists a positive integer n0 such
that the sequence {xn}"°u {yn}™0 is a basis of E, equivalent to {xn}.
Hence, in this case
codimE[>B]^ = «0-l, codimE[>n] = fe<«o-l- A1.30)
Proof. If we have A1.29), then there exists a positive integer n0 such
/со \i
that sup E \f(xi—yi)\2 I = M<c, where с is any constant for
which we have A1.6). Hence, by theorem 11.2 applied to the sequences
{хл}, {х„}"°~x u {>'„}ад, the desired conclusion follows.
Definition 11.3. A sequence {yn} in a Banach space ? is said to be
12-Hnearly independent, if
{«„}e/2,
i= 1
Theorem 11.3. Let {xn} be a Besselian basis of a Banach space E
and let {>>„} be a sequence in E, weakly quadratically near (or, in particular,
quadratically near) to {xn}. Then
a) The following statements are equivalent -.
1°. [yn] is complete in E.
2°. {yn\ is I2-linearly independent.
3°. {>>„} is oj-linearly independent.
4°. {yn} is minimal.
5°. {>>„} is a basis of E.
6° lx 1 % ' v '
b) The sequence {>'„} can be modified to become a Besselian basis
of E, by changing suitable к elements of it, where /c = codim? [у„] < oo.
Proof, a) 2°=>6°. Assume that {xn} is Besselian and let {у„} be an
/2-linearly independent sequence in E, weakly quadratically near to
{xn}. Then by corollary 11.4 there exists a positive integer n0 such that
CO CO
for every xeE the series E fi(x)yi> whence also the series E fAx)}'h
i = no i = 1
converges. Consequently, the mapping
"W = E
A1.32)

348 II. Special Classes of Bases in Banach Spaces
is well defined on E and by the Banach-Steinhaus theorem it is continu-
continuous. Now, if m(x) = 0, then, since El/rMI2<00 (because {xn} i
IS
1= 1
Besselian), it follows by the /2-linearly independence of {>>„} that /,(x) = 0
(i = 1,2,...), whence, since {/„} is total on E, x = 0. Thus, и is one to one.
We shall prove now that и maps E onto E. Let xeE and n be arbi-
( °°
trary and take а дх,„еЕ* with || gxj\ = l such that g^J Е /iW(x.-X
\r=n+ 1
fi(x)(xt-yi)
. Then, taking into account A1.6),
E
< ( E I/.WI2) ( E
E
<-(l + v(J(J.)||x|
E
.-V.4I2
,,\|2
where v,x , is the norm of the basis {xj}. Hence for the continuous linear
operators
GO
s{x) = x-u(x)=Y.ftx)(xt-y,) (xeE), A1.33)
i= 1
и
v,(x)=Y.ft{x)(xt-yi) (xeE, n= 1,2,...), A1.34)
;= l
we obtain, taking again into account that {>•„} is weakly quadratically
near to {xn},
\\s-vj= sup \\s{x)-vn{x)\\= sup
д:е? xeE
E /,
E
as n->co,
whence .? is compact.
11. Besselian and Hilbertian bases. Stability theorems
349
Consequently, u = lE — s maps E onto1 E and hence, by the inversion
theorem of Banach, и is an isomorphism of E onto E, which satisfies, by
A1.32), и(х„)=у„(п =1,2,...). Therefore {*.}3-{>>„}•
The implications 6°=>5O=>4O=>3O=>2° and 6°=>Г are obvious.
Finally, the proof of the implication Г=>3° and the proof of b)
are similar to the proof of the corresponding statements in Ch. I, § 10,
theorem 10.2, using corollary 11.4 above. This completes the proof of
theorem 11.3.
Let us give now some characterizations of Besselian bases in terms
of stability properties.
Definition 11.4. A complete minimal sequence {х„} in a Banach
space E is called
a) quadratically stable, if every /2-linearly independent sequence
{yn\ a E, which is quadratically near to {xn}, is complete in E;
b) weakly quadratically stable, if every /2-linearly independent
sequence {>>„} с Е, which is weakly quadratically near to {xn}, is
complete in E.
Theorem 11.4. Let {xn\ be a complete minimal sequence in a Banach
space E. The following statements are equivalent:
1°. {xn} is a Besselian basis of E.
2°. {xn} is quadratically stable.
3°. {xn} is weakly quadratically stable.
Moreover, if one of these statements holds, then every I2-linearly
independent sequence j.v,,| с Е, which is weakly quadratically near (or,
in particular, quadratically near) to {х„}, is a Besselian basis of E, equiv-
equivalent to {xn}.
Proof. The implication Г=>3° and also the last statement of the
theorem, are contained in theorem 11.3 a) (implication 2°=>6°).
The implication 3° => 2° is obvious.
2°=>1°. Assume that {xn\ is quadratically stable, but not a
Besselian basis of E. Then there exists either an xoeE such that
00 CO
E fi{Xo)xi does not converge, or an xoeE such that E \А(хо)\2 = с°,
'•=i i=i
where {/„} с ?*, /,(ху) = й,7 (i,j= 1,2,...). In both cases there exists an
index n such that /„(xo)^O. Put
/„(*<)
A=1,2,...)-
A1.35)
1 See e.g. S. Banach [10], p. 154, theorem 14.

350
II. Special Classes of Bases in Banach Spaces
|2 oc
Then
l0l
X l/,(*i)l2 = ТУ7-ТГ2 < «5, i-e. {yn}
1 I f(r Ч12 ^ '
is quadratically near to \xn). Furthermore, we show that {yn} is /2-linearly
independent. Indeed, let {an}el2, ^ «;>', = 0- Then, applying /; to the
relations i=1
j=l j=l Jn\XO) j=l )n\XO)
we obtain
ft(x0)
(/=1,2,...).
A1.36)
Since ^ a,x, converges and E Kl2< °° and since either ]T /'i(x0)xi
i=l a, r=l r=l
does not converge or ? l./i(*o)|2 = oo, the equalities A1.36) are possible
;= l
only if а„ = 0, whence a; = 0 (/=1,2,...), which proves that {у„} is
/2-linearly independent.
Since by our hypothesis {х„} is quadratically stable, it follows
that {у„} must be complete in E. However, this contradicts the relations
У„(^о)
(/=1,2,
Thus, 2°=>1°, which completes the proof of theorem 11.4.
The answer to the problem of existence of Besselian and Hilbertian
bounded bases is negative, as shown by corollary 11.1 above. On the
other hand, obviously, the answer to the problem of existence of non-
Besselian and non-Hilbertian bounded bases in finite-dimensional
Banach spaces is also negative.
Problem 11.2} Does every infinite dimensional Banach space with
a basis possess a non-Besselian bounded basis ? What about a non-
Hilbertian bounded basis?
Some information concerning this problem is given by the implica-
implication 1°=>2° of corollary 11.2 above: every basis {х„} which is not equiv-
equivalent to the natural basis {en} of I2 is either non-Besselian or non-Hilbertian.
The answer to problem 11.2 in the usual concrete Banach spaces is
affirmative. In fact, this is true even for Hilbert spaces (which constitute
the most difficult case), as shown by
1 Apparently, in the mean time this problem has been solved in the affirmative
(see the Notes and remarks).
11. Besselian and Hilbertian bases. Stability theorems
351
Example 11.2. Le\ E = L2([-n,n\) and let 0</3<i. Then {Jcn}n°°=0,
where1
A1.37)
is a Besselian but non-Hilbertian bounded basis of E, and {у„}™= 0, where
b(')=l'l'«to> J'2«+i(')=l'l/le~'"' (/е[-я,я]; и = 0,1,2,...), A1.38)
is a Hilbertian but non-Besselian bounded basis of E.
Indeed, the sequence {QcE* defined by
e'"dz,
(JceL2([-7t,7t]), и = 0,1,2,...)
A1.39)
satisfies obviously fk(xj) = 8kj (k,j = O, 1,2,...). Consequently, taking into
account A1.18), we have2, for the partial sum operators associated to
the biorthogonal system (*„,/„),
= Ё fk(X)xk(t)=\t\-> ?
k=0 k=0
i
= -\t\-f
n
2 sin —
= \t\~"Qn(t) + —\t\-"
where we have put
sinni
dx.
A1.40)
A1.41)
Now, taking into account A1.21) and putting
t) = x(t)\tf sinn/, z2{t) = x{t)\t\i>cosnt (te
A1.42)
1 Here we use i as ]/— 1.
2 We consider all functions extended beyond [ — 71,71) by x(t + 2n) = x{t).

352 II. Special Classes of Bases in Banach Spaces
we have, almost everywhere on [ — я, я],
where z^t), z2(t) are the functions conjugate1 to zx(t) and z2(t) respectively.
Hence
and thus
J \t\~2'\Qn(t)\2dt^2 j \t\-2f\z1(t)\2dt + 2 J |гГ2'|г2(*)|2Л.
— re ~~ я ~ п
However, let us recall the following theorem on conjugate functions2:
If
re
j \t\*\z(t)\2dt<cc, where -1<<х<1,
— я
then there exists a constant Ax>0 depending only on a such that
re re
j \t\'\z(t)\2dt^Aa j \t\'\z{t)\'dt.
— re — re
Applying this theorem to zl andz2 defined by A1.42) and to <x= — 2/?,
we obtain
J
J
Consequently, taking into account A1.40), we get
Ь2„(х)\\^А'р\\х\\ (xe?),
with a suitable A'p depending only on /?.
Since
-2"Л = M,-
,,|| <Af!
A1.43)
we have
=M2< со
1 For the definition of conjugate functions see example 11.1.
2 See [8].
11. Besselian and Hilbertian bases. Stability theorems
353
whence, by A1.43) and ||s2n+1(x)|| sc ||i2n(x)|| + ||/2„+1(х)х2„ + 1||,
sup ||sj|<co. A1.44)
0 ^H< 00
On the other hand, let us show that {х„}^=0 is complete in ?. Let
ге? = Ь2([-я,я]) be such that (z,xn) = 0(« = 0,1,2,...), i.e. such that
z(t)\t\-fie'"'dt =
= 0,1,2,
Then, since z
(because []
and since {e~int}^=ou {e''"'}^°=1 is complete in ?
= L2([-rc,rc]), it follows that
z(t)\t\-f = O a.e. on [-я, я],
whence z = 0, which proves that [х„]^=0 = ?.
Thus, (*„,/„)"= о is an ^-complete biorthogonal system satisfying
A1.44). Hence, by Ch. I, § 4, theorem 4.1, {х„}„°°=0 is a basis of ?. From
A1.37) it is obvious that the basis {х„}^=0 is bounded.
Similarly, the sequence [у„}"=0 <= ? defined by A1.38) is a bounded
basis of ?. Let us prove now that: а) {х„}™=0 is Besselian but non-
Hilbertian and b) {jn}i?=o is Hilbertian but non-Besselian. Since by
A1.39) the image of {/„}^=0 under the canonical isometry Ь2([-я,я])*
= Ь2([-я,я]) is nothing else but {у„}^=0, it will be sufficient, by virtue
of theorem 11.1 above, to prove b).
Since the function \t\~p is not bounded above on [ — я,я], there
exists a zeL2([-n,n~\) such that z(t)\t\~^L2([-n,я]). Consequently,
we have x
where k = 0
2n
х(т)\т\-"eimdz
[-я,я]), /i = 0,l,2,...) A1.45)
is the a.s.сf. to {у„}"=0- Thus [у„}^=0 is non-Bessdian. On the other
hand, let {а„}™=0 be an arbitrary sequence of scalars such that
QO
X |at|2<co. Then there exists an yeL2(\_ — n, я]) such that
23 Singer, Bases in Banach Spaces I

354 II. Special Classes of Bases in Banach Spaces
Hence for the xeL2([ — n,n~\) defined by
x(t)=\tfy(t) (re[-7t,7t])
we have, taking into account A1.45),
and thus {yn} is Hilbertian. This completes the proof of the assertions
stated in example 11.2.
Since the results on stability given in the foregoing, and also their
proofs, are similar to those of Ch. I, § 10, it is natural to ask whether
both series of results can be obtained as particular cases of a more
general theory. We shall now show that this is indeed the case and that
this general theory also has other applications.
Definition 11.5. Let F be a Banach space and let {en} cfbea mini-
minimal sequence. A basis {*„} of a Banach space E is said to be
00
a) (F, {en})-Besselian, if {xn}>{en}, i.e. if ? fi(x)et converges for all
xeE, where {/„} is the a.s.c.f. to {*'„}; i=1
b) (F,{en})-Hilbertian, if {е„}>{х„}, i.e. if for every convergent series
00
Y, «,•?,- in F there exists an xeE such that /„(*') = «„ (n= 1,2,...).
In the particular case when F = lp A ^p< со) or c0 and {е„} = the
unit vector basis of F, the terms p-Besselian basis and p-Hilbertian basis,
respectively co-Besselian basis and co-Hilbertian basis, are also used.
The 2-Besselian B-Hilbertian) bases are nothing else but the Besselian
(Hilbertian) bases in the usual sense. By Ch. I, §3, lemma 3.1, a basis
(л-} is co-Besselian if and only if inf Цл-J >0 and {х„} is l-Hilbertian
1 € П < 00
if and only if sup |]л„|| < со. Furthermore, it is well known1 that {*„}
is co-Hilbertian if and only if ? xt is weakly unconditionally Cauchy
i=l
GC
(i.e. ? |/(л-;)| < со for all feE*).
Let us observe that every basis {* } of E is both (?, {xn})-Besselian
and (?,{xn})-Hilbertian. Furthermore, if {en} is a basis of F, a basis {*„}
of E is (F,{en})-Besselian (respectively, {F, {en})-Hilbertian) if and only
if {en} is (?, {xn})-Hilbertian (respectively, (?,{*„})-Besselian).
If {en} is a basis of F, theorem 11.1 (and also corollary 11.2) can be
extended to (F,{en})-Besselian and (F, {е„})-Hilbertian bases, replacing
See Ch. I, § 17, corollary 17.5.
11. Besselian and Hilbertian bases. Stability theorems
355
Y,aiei > deleting1 conditions aM°, bM° and
I2 by F and j/Xk|2 by
modifying conditions aN°, bN° as follows: {/„} is a ({/;„}, [/г„])-Нн-
bertian (respectively, ({hn}, [/;„])-Besselian) basis of [/„], where {hn} с F*
is the a.s.c.f. to {en}. Indeed, all proofs are the same, except that we
give another proof of bK°=>bN°, similar to the proof of aK°=>aN°):
if v.F^E is as in bK°, then [d* (/,.)] (^) = /,|>(^)] =/,(*,.) = 5y
(ij=l,2,...), whence v*(fi) = hi(i=l,2,...), whence, by the implication
aK°=>a)l°, we obtain bN°.
Definition 11.6. Let F be a Banach space, let je.Jcf be a minimal
sequence and let {*„}, {у„} be two sequences in a Banach space E. The
sequence {yn} is said to be
a) (F, {en})-near to {xn}, if
? Iki-
converges.
A1.46)
b) weakly (F,{en})-near to {*„}, if ? /(*;—J,-)^ converges uni-
uniformly with respect to feE*, ||/||< 1, i.e. if
lim
sup
feE*
ll/IKi
Z
= 0.
A1.47)
In the particular case when F = /p (l^p< со) or c0 and {en} = the
unit vector basis of F, the terms p-near and weakly p-near, respectively
co-near and weakly co-near are also used.
The 2-near (weakly 2-near) sequences are nothing else but the
quadratically (weakly quadratically) near sequences and the 1-near
(weakly 1-near) sequences are nothing else than the strictly (respectively,
weakly) near sequences. It is also immediate that {*„}, {yn} are weakly
co-near if and only if they are co-near, i.e. lim ||xn-jj =0.
п—> со
In general there is no relation of implication between the notions
(f > W)-near and weakly (F, {е„})-пеаг. If {en} is an unconditional basis2
of F, then (F, {е„})-пеаг => weakly (F, {en})-near, but the converse impli-
implication is not true.
1 They can be maintained (in the form a) 5° ? Ф(/,)е, converges for all
oo i=l
Фе?**, respectively bM° X /(JCi)A, converges for all feE*) e.g. in the case
i= l
when {е„} is a boundedly complete (respectively, shrinking) basis of E.
2 See § 14, definition 14.1.
23»

356
II. Special Classes of Bases in Banach Spaces
Definition 11.7. Let F be a Banach space and let \en} cFbea mini-
minimal sequence. A sequence {>•„} in a Banach space E is said to be (F, {<?„})-
linearly independent, if
X> ОС
X^^gF, ?а^; = 0 imply af = 0 (/=1,2,...). A1.48)
In the particular case when F = l" A ^p< oc) or c0 and {е„} = the
unit vector basis of F, the terms V-linearly independent and co-linearly
independent are also used.
If inf ||yJ>0, the sequence {yn} is co-linearly independent if
1 ^S n< cr
and only if it is co-linearly independent. Indeed, it is obvious from the
definitions that co-linearly independence implies (F, [en})-linearly inde-
independence for any F and {е„}. Conversely, if inf ||jj >0 and !>'„} is
1 ^ П < ОС
co-linearly independent, let (а„} be an arbitrary sequence of scalars
such that Yj ai>'i = 0- Then, since inf H.fJ>0, we must have
lim а„ = 0, whence, since {yn} is co-linearly independent, af = 0 (/ = 1,2,...),
n-> oc
which proves our assertion.
It is also immediate that for any infinite dimensional F and any
minimal sequence {en} in F, with sup ||ej| <oo, the (F, {en})-linearly
independence implies /'-linearly independence. Indeed, if {jn}c? is
OC OC
(F, {en})-linearly independent and ? |а,-| < ос, ?а(у; = 0, then, since
? |а,-| < ос, ?
i=1 i=l
by hypothesis sup \\en\\ < со, we have
1 ?
, whence, since {yn}
is (F, (en})-linearly independent, af = 0 (/= 1,2,...).
Definition 11.8. Let F be a Banach space, and let (en,hn)({en}cF,
{/;„} с F*) be a biorthogonal system. A complete minimal sequence {л'„}
in a Banach space E is called
a) (F,{en})-stable, if every (F, [en})-linearly independent sequence
{у„} с E, which is ([Л„], {А„})-пеаг to {*„}, is complete in E;
b) weakly (F,{en})-stable, if every (F, {en})-linearly independent se-
sequence {yn} с ?, which is weakly ([й„], {/г„})-пеаг to {*„}, is complete
in E.
In the particular case when F = l" A ^p< со) or c0 and {е„} = the
unit vector basis of F, the terms p-stable and weakly p-stable, respec-
respectively co-stable and weakly аэ-stable, are also used.
The 2-stable (weakly 2-stable) complete minimal sequences are no-
nothing else but the quadratically (weakly quadratically) stable ones. It
11. Besselian and Hilbertian bases. Stability theorems
357
is also immediate that a complete minimal sequence {х„} <= E with
inf \\xn\\ >0 is co-stable (weakly co-stable) if and only if it is strictly
1 ^ П< ОС
stable (respectively, weakly stable) in the sense of Ch. I, § 10, definition
10.3. Indeed, the "only if part is obvious even without the assumption
inf ||л'в||>0, since co-linearly independence implies co-linearly in-
1 ill< 00
dependence. Conversely, if a complete minimal sequence {л„} с Е with
inf ||xJ>0 is strictly stable (respectively, weakly stable), let \yn)
1 5SH< OC
be a co-linearly independent sequence in E, 1-near (respectively,
00
weakly 1-near) to {*'„}• Then, since ?](a'j—j,-) converges, we have
i=l
lim ||xn—yn|| =0, whence, since inf ||a'J >0, we infer inf ||jJ>0.
П-+0С l^fI<OO 1^П<00
Consequently, by the remark made after definition 11.7, {у„} is co-line-
co-linearly independent, whence, since {а„} is strictly (respectively, weakly)
stable, {yn} is complete in E, which proves our assertion. By the remark
made after definition 11.6 it is obvious that {л„} is 1-stable if and only
if it is weakly 1-stable.
The assertions on weak quadratic nearness and weak quadratic sta-
stability, and in the particular case when {en} is an unconditional basis of
F, all assertions, of theorem 11.2, corollary 11.4 and theorems 11.3 and
11.4 can be extended to (F, {en})-Besselian bases, replacing conditions
A1.25) and A1.26) by sup sup
= M<c and respec-
respectively, if [en] is an unconditional basis of F, by sup
1 $¦< 00
С
= M'< -r^-, where {AJcF*, h^e^S^, c>0 is any constant such
that с
И
Z a.ei
i= 1
n
Z aiA'.-
i= 1
for all finite sequences of scalars a.u...,a.n
and Vjg^j = the unconditional norm1 of the basis {en}. Indeed, the proofs
are essentially the same, using instead of A1.27) the inequalities
Z Qn{.xi-ydhi
i= 1
Z
м
Z a.A'i
A1.49)
and respectively, if {е„} is an unconditional basis of F, the inequalities
1 For these notions see § 14 and § 17.

358
II. Special Classes of Bases in Banach Spaces
i= 1
Zkl 11^,11 = Ell*-*
.1= 1
Z Иъ-уЛ
Z a.A'i
A1.50)
let us mention that the second result can be also obtained as a particular
case of the first, since
„)
Z
,--v,- ft,-
i= 1
(/e?*, Ц/И1).
In the particular case when F = l2 and {е„} = the unit vector basis
of F, from these extended results we obtain again the foregoing results
on quadratic and weak quadratic stability of usual Besselian bases. In
the particular case when F = c0 and {en} = the unit vector basis off,
from the extended results we obtain again the results of Ch. I, § 10 on
strict and weak stability of usual bases {л„} with inf \\xn\\ >0. In the
particular case when E — F = l1 and {л„} = {е„} = the unit vector
basis of I1, from the extended theorem 11.2 we obtain the following
result1:
Theorem 11.5. Let {xn} be the unit vector basis of E = l1, and let {yn}
be a sequence in E such that
sup ||xn-jn||=M<l.
Then {yn} is a basis of E, equivalent to {xn}.
A1.51)
In the particular case when F = /* and {en} = the unit vector basis
of/1, from the extended theorem 11.4 we obtain the following result:
Theorem 11.6. A complete minimal sequence {xn} in a Banach space E,
with sup ||xj<oo, is a basis equivalent to the unit vector basis of
1 $П<00
I1 if and only if every I1-linearly independent sequence {у„} <= E with
lim ||xn— jJ = 0 is complete in E. Moreover, in this case every such
П-* 00
sequence {yn} с E is a basis of E, equivalent to {х„}.
1 Let us observe that theorem 11.5 is also an immediate consequence of the
sufficiency part of Ch. I, § 10, theorem 10.4, since b= sup sup
= sup Ы. N~1
12. Relations between various types of bases
359
We leave to the reader to formulate the results obtained from the
extended theorems in the particular case when E = F, {х„} = {е„} and
in the case when F = /p A <p^2) and {en} = the unit vector basis of F.
§ 12. Relations between various types of bases
Theorem 12.1. a) For a bounded basis {х„} of an infinite dimensional
Banach space E we have the following implications:
Table 12.1
Hilbertian shrinking
V II
X I
II
I
swc0
II
1
поп weakly
closed P .
II II \
I J) \
non-al+ {l+f a
/ II У ii /
/ 1 / 1 /
non-a P* non-l + " (al+)*
V II II
X w *
non-P* (weakly closed)
||
1
non-(s w c0)*
II
"°То)ч
II
поп boundedly
complete
Table 12.2
Besselian boundedly
\ I
(wc0)*
(swc0)*
II
1
поп (weakly
closed) *
||
P non-(al+f
// II
/ ^ *
non-a P non-(l+)*
\ II
\ U
* non-P
\
non-
Besselian
complete
p*
К
u \
/+ aP*
У\\ //
al#
|| +
\
weakly closed
||
1
non-swc0
II
1
ПОП-WCq
II V
.Д
non-shrinking non-
Hilbertian
b) The converse implications are not valid, except perhaps non-al +
=>non weakly closed, (weakly closed)* =>(al+)*, non-(al+)*^non-
(weakly closed)* andweakly closed^al+, which are unknown1.
c) Between the 36 types of bounded bases considered in part a), the
only implications are the 36 trivial ones (shrinking => shrinking, etc.) and
the 144 given by tables 12.1 and 12.2 above. All the other 1296-36-144
1 After this monograph had been completed, we learned that this problem
was recently solved, namely, these four implications do not hold either (see the
Notes and remarks).

360
II. Special Classes of Bases in Banach Spaces
= 1116 relations between them, except perhaps the 4 considered in part b),
are: non-implication.
Proof, a) Shrinking ^-wc0. Assume that {х„} is a shrinking bounded
basis of an infinite dimensional Banach space E. Then, since
sup ||л-„||<оо, for the a.s.c.f. {/„} to {xn\ we have, by, Ch. I, § 3
l«n<x x
corollary 3.1, inf ||/„||>0. Hence, since /?„/„-> 0 for Y, Pifi
= ?*, it follows that '=1
and thus {х„} is of type wc0. Alternatively, the implication shrinking
^>wc0 also follows from §4, theorem 4.2, implication 1°=>2°.
The implication Hilbertian =>wc0 follows from the implication
b) l°=>b) 5° of § 11, theorem 11.1.
The implications wc0=>iwc0=>non weakly closed are obvious.
Non weakly closed =>non-a/+. Assume that {л„} is a non weakly
closed basis. Then, since 0 is a weak limit point of the set {л„}, in every
w-neighbourhood of 0 of the form Vf.E@) (where feE*, g>0) there
exists at least one xn, whence we obtain
inf |/(л-„)| = 0 {feE*),
^ n < со
2° of § 10, theorem 10.3, {*„} is of
and non-aP* =>non-P* are
and thus, by the implication 1
type non-a/+.
The implications non-a/+=>non-/
obvious.
The implications non-/+=>non-P* and non-a/ + =>non-aP* have been
observed in § 10 (taking /?„= 1 in T of theorem 10.1 and applying the
implication 7°=>1° of this theorem).
(/ + )*=> non-/ +. Assume that {xn} is a basis of type (/+)*. Then,
applying the implication 1°=>6° of § 10, theorem 10.1, to the a.s.c.f.
[/„} of {л„}, it follows that there exists a sequence of scalars {/?„} satis-
satisfying A0.6) and such that {/?„/„} is a basis of type P* of [/„], whence,
by §9, theorem 9.2 a), <— х„> is a basis of type P of E. Now, ^-w is
LPn J \.Pn J
of type non-/+, since otherwise one would have, with a suitable rj>0,
sup
sup
E :
< CO,
which is impossible. Consequently, by the implication Г
theorem 10.1, {х„} is of type non-/+.
10° of §10,
12. Relations between various types of bases
361
The other implications of table 12.1 follow from the implications
proved above, by duality (e.g. from the implication non-/+=>non-P*
we obtain P=>(/+)*, etc.).
Finally, the implications of table 12.2 follow from those of table 12.1,
either by duality (e.g. from the implication shrinking =>wc0 we obtain:
boundedly complete =>(wc0)*, etc.) or by a simple negation (e.g. from
the implication shrinking =>wc0 we obtain non-wc0=> non-shrinking,
i.e. the penultimate implication of table 12.2, etc.).
b) In order to prove that the converse implications are not valid,
it is sufficient to consider only successive properties of table 12.1 and
to give for them suitable examples.
wco?> shrinking, as shown by
Example 12.1. Let d be the real linear space of all sequences of real
numbers х={?„} such that
\\x\\ =sup X —г
оеЯ f=1 I
< CO,
A2.1)
where П denotes the set of all permutations1 of the set Jf= {1,2,3,...}.
Then d is a Banach space, and the unit vectors {л'„} are a non-shrinking
basis of d, of type wc0.
Indeed, let us first prove that dis complete. Let Zj = {Ci,j)}"= t (j= 1,2,...)
be a Cauchy sequence in d and let ? > 0 be arbitrary. Then there exists
an N = N{e) such that
llz_-z_
=sup
со Irtm) _ r(m + p)|
(m>N,p=l,2,...). A2.2)
Hence, taking /=1 and апеП with an(l) = n (и =1,2,...), it follows
that
(m>N, p=\,2,...; n=l,2,...),
and thus each sequence {C|,m)}m=i («=1>2,...) is a Cauchy sequence,
whence convergent, say to ?„ (n= 1,2,...). Now, by A2.2) we have
к !>¦(») _r(m + p)l
,p=l,2,...; fc=l,2,...;
i= 1
whence for /?—> со we obtain
* 1Г(т'— Г I
—-—;—— < ? (m>N, k=l,2,...; аеП),
1 I.e. the set of all one to one mappings of Ж onto Ж

362
II. Special Classes of Bases in Banach Spaces
whence, putting z={C}, we get zm — zed, \\zm — z\\ <? (m>N). Conse-
Consequently, z = (zm — z) + zmed and zm^>z as m->co, which proves the
completeness of d.
Observe that for every x={?n}ed we have Нт?„ = 0. Indeed, other-
П-* oo
wise there would exist an infinite subsequence {?„.} with inf |? |
0, whence, taking for each к a permutation океП with ak(i) = ni
, . л . . ,j , . • II I
(i=l,...,k), we would obtain \\x\
\F ¦ I * l<r I
— = L —r
4 L ~
i=\
(fc=l,2,...), whence ||x|| = oo, contradicting the assumption xed.
From this remark it follows that for every sequence x={c,1,?2>---}ed
the subsequence {?J1,?J2,...} consisting of all non-zero co-ordinates can
be rearranged into a (finite or infinite) sequence {??, ??2,...} such that
I?*J ^ l?*2| > •"" (note that if {?„} has an infinity of non-zero co-ordinates
and also at least one zero co-ordinate, then there is no permutation
аеП such that ?a(n) = ?jn (n = 1,2,...)). We claim that
\\x\\ =
{x={Qed),
A2.3)
where ?? are as defined above.
Indeed, let x={in\ed and ?>0 be arbitrary. Then there exists a
a = a (x,;;) e П such that
\\x\\ —
and an index N=N {x,e,a) such that
CO If I
l
—,
whence
JV
11*11 - Z
i= 1
< ?.
Now, if for an index i<N we have
permutation а'еП defined by
) = a(i), a'(k) = a(k)
« P P
we have, taking into account that —| > —
i i +1 i
/ /+1 / /+1
A2.4)
I, then for the
for
12. Relations between various types of bases
363
Thus, in a finite number of steps we obtain that for the permutation
! 6 Я defined by
ffl(k) = «7i
we have
whence
for k = N+\,N + 2,...,
у lg>.(ol , у
L ; > Z
JV
_ у
A2.5)
A2.6)
Since there exists a permutation <г2бЯ such that ?ff2(j) = ?*
(/=l,...,iV) and since by A2.5) and the definition of ?* we have
(»= !' ••' N)' k follows ЬУ AZ6)that
JV it I
IAII 2 < ?
JV ic* |
J
i= 1
whence, since xed and e>0 have been arbitrary, we infer A2.3).
Now we can prove that
- Z tixi
о
A2.7)
as n—>co (x={?n}ed).
Indeed, by the above remark we have, for any x={?n}ed,
where {tf*pi,tf*P2,...} is the rearrangement with \c**pi\^\^**P2\^ •••
of the subsequence {^„ + P1, ?n + P2, •••} consisting of all non-zero co-
coordinates of the sequence {?„+1, ?„ + 2> •••}¦ Let
/„ = max
Then 0</„<и, {ijn + pi, ?n + P2, •••} <=
|^ J (/=1,2,...), whence
||{0,...,(и„+1,?„+2,...}|| = Z
and
Z Jl: . A2.8)
Now let ?>0 be arbitrary. Choose N = N{x,e) such that
1^1 e
Z -^-<-.

364
II. Special Classes of Bases in Banach Spaces
Since limiJf = 0, for this N we can choose a positive integer M
i->00 ' g
such that |?*m+1| < — for all m>M, whence
Thus for m> M we have, taking into account that l?*m + iKI?*|,
CO I» I N IE* I CO It* I
'¦ ~~ Zj '¦ <~ l_, '¦
i = 1 ' i = 1 ' i = JV + 1 '
whence, taking into account A2.8) and that /„->oo as и->со, we infer
CO
A2.7). Thus, every x={?n}ed has an expansion x= ^ ^л';. Since this
expansion is obviously unique, {л„} is a basis of d. i = 1
Let us prove that the basis {х„} is non-shrinking. Put /.„= ^ —
(и=1,2,...) and define jnerf by i = 1 '
J>»H-,...,-,0,0,...> = — {1 1,0,0,...} (и=1,2,...). A2.9)
Then clearly
1Ы1=| I 4=1 (и=1,2,...),
A2.10)
and, if {/„} erf* is the a.s.c.f. to {*„}, we have
iOO = - (" = '. ' + 1' ' + 2, ••¦; '=1,2,...),
whence
Яу„) = 0 (/=1,2,...).
и-> oo
Define now a linear functional / on d by
A2.11)
A2.12)
Then by A2.1) we have \f(x)\ =
fed*, but
CO p
1 Д
/!•„ i = i
||x|| (x={(Jn}erf), whence
A2.13)
12. Relations between various types of bases
365
Consequently, by §4, theorem 4.2 (implication 1°=>3°), the basis {л„}
is non-shrinking.
Finally, let us prove that {xn} is of type wc0. Since \\х„\\ =1 (и = 1,2,...),
it remains to prove that xn A 0. Assume the contrary, i. e. that there
exist an fed*, an infinite subsequence {хПк} of {*„}, and an e>0 such
that
00 J
Then for л-= X -л-Иь={0,...,0,1,0, ...,0,^,0, ...}erf we have
« 1 П П\
k=\
k=l
contradicting fed*. Thus, xn^Q, which completes the proof of the
assertions of example 12.1.
Since the existence of non-shrinking bases of type wc0 is rather un-
unexpected, let us also mention, without proof, another example of such
bases.
Example 12.2. Let W be the real linear space of all sequences of real
numbers {?„} such that
t+i
-0 as fc->co,
A2.14)
where the sup is taken over all sequences of k+ 1 positive integers such
that .,- ,.,
Then W endowed with the norm
||{У11= sup Ak({Q)
l«t<0O
is a Banach space and the sequence {л„} <= W defined by
A2.16)
is a basis of И^, such that the a.s.c.f. {/„} c\V* is a non-shrinking basis
of W*, of type wc0.
w c0 *> Hilbertian, as shown by §11, example 11.2.
swco?>wco. Indeed, by §7, theorem 7.1a), the Schauder basis of
C([0,1]) is of type swc0, but non-wc0.
Non weakly closed?^swc0. Indeed, by §8, theorem 8.1 and §7,
theorem 7.1b), the normalized Haar basis of ^([0,1]) is non weakly
closed and of type non-swc0.

366
II. Special Classes of Bases in Banach Spaces
Non-/+ ?>non-al+ and non-P* ^non-aP*, as shown by
Example 12.3. Let ? = /* and let {hn} be the basis of /' considered
in Ch. I, § 13, example 13.3, formula A3.14). Then {hn} is of type non-/+
(and hence also non-P*) but aP* (whence also al+).
Indeed, {/;„} is of type P, whence non-/+. On the other hand, for
Ф= {1,0,1,0,1,0,...}ет = Aх)* we have
Ф(А1)=1, Ф(А2в)=1, Ф(А2в+1)=-1 (и=1,2,...), A2.18)
whence, by the implication 4°=>Г of §9, theorem 9.2 b), the sequence
{gn} с Z1 defined by
0i=Ai, g2n = h2n, g2n+l=-h2n+1 (и=1,2,...) A2.19)
is a basis of type P* of the space /\ and thus {hn} is of type aP*.
Non-P* ?5>non-/+ and non-aP*?^>non-a/+, as shown by
Example 12.4. Let ? = c0, and let {jn} с ? be the sequence defined by
2n-l 2n
У _ у л-. у _2 у л-. (и=1,2,...), A2.20)
i=i i=i
where {л„} is the natural basis of c0. Then {yn} is a basis of ?, of type
/+ (and hence also al+) but non-aP* (whence also non-P*).
Indeed, since {xn} is a basis of type P of ? (because of ||xj = l,
= 1, /i=l,2,...), the sequence < EA";f 's' by the implication
Г=>2° of §9, theorem 9.1, a basis of ?, and thus the sequence {yn}
defined by A2.20) is a basis of ?. Furthermore, if {/„} <=?* is the a.s.c.f.
to {*„}, then
/iCV2n-i)=l, /i(j2n) = 2 (и = 1,2,...), A2.21)
whence, by the implication 5°=>Г of §10, theorem 10.1, {)'„} is of type
/+. Finally, let us show that {yn} is of type non-aP*. Since the a.s.c.f.
Ш c E* to {yn} is
q2 , = f. ,-/•-, fl2B = i(/2B-/2B+1) (и=1,2,...), A2.22)
we have, for any sequence of scalars {?„} with |е„| = 1 (и= 1,2,...),
E e)9j= E
E (l^j—
12. Relations between various types of bases
whence, taking into account that |е„|= 1 (и= 1,2,...),
367
E ^^
| + у |i? ._e
n
V i|? — IP
/ i 2 \ 2j 1^2!— 1
E
?2j-l 2?2j-2
as
and thus {yn} is of type non-aP*.
Non-/+?^>(/+)*. Indeed, by §10, example 10.1, every basis of an in-
infinite dimensional reflexive Banach space is of types non-/+ and non-(/+)*.
The non-validity of the converses of the other implications of
table 12.1 follows by taking the a.s.c.f. to the bases of the above ex-
examples1 and applying the duality results proved in the preceding
sections (e. g. the basis {xn} of example 12.2 is non-boundedly com-
complete, but of type (wc0)*, etc.).
Finally, as we have observed in the above proof of part a), table 12.2
is equivalent to table 12.1, both by duality and by negation. Hence the
examples given above prove the non-validity of the converses of the
implications of table 12.2, too.
c) The examples given in the above proof of part b), together with
the natural bases of c0 and I1, are sufficient to show that all the other
relations between the 36 types of bounded bases considered in tables
12.1, and 12.2, except perhaps the 4 considered in the statement of
part b), are: non-implication.
Indeed, in part b) above we have proved 144 — 4=140 non-implica-
non-implications. Furthermore, the natural basis of c0 is Hilbertian, shrinking and
of type P, whence, by part a) proved above, it has all the 18 properties
of table 12.1 and none of the 18 properties of table 12.2. Thus, no prop-
property of table 12.2 is implied by any property of table 12.1, which gives
324 new non-implications. Similarly, the natural basis of I1 is Besselian,
boundedly complete and of type P*, which gives other 324 non-impli-
non-implications. On the other hand, the natural basis of I2 is Besselian, Hil-
Hilbertian, shrinking and boundedly complete, whence, by part a), it has
all the 18 properties of the left halfs of tables 12.1 and 12.2, and none
1 Let us remark that for this purpose, instead of the a.s.c.f. {Ф„} a(ll)* = m
to the basis [hn] of /' considered in example 12.3 it is more convenient to take
the basis < ? xi \ of c0, where {х„} is the natural basis of c0 (or, equivalently, the
O-i J
basic sequence {*i + *n+1} in m). This basis leads to the same results, since its
a.s.c.f. is {А„+1}.

368
II. Special Classes of Bases in Banach Spaces
of the 18 properties of the right halfs of these tables. Thus no property
of the right half of table 12.1 is implied by any property of the left half
of this table, and similarly for table 12.2. This gives 162 non-implica-
non-implications, but 4 of them (namely, non-/+?^> (/+)*, non-l+*>P and the dual
ones) have been proved in part b) and hence counted above, and thus
we obtain only 158 new non-implications. Furthermore, the basis {hn}
of /' considered in example 12.3 is of types P and aP*, and dually, the
basis
of c0 (where {xn} is the natural basis of c0) is of types P*
and aP, whence, by part a), no property of the left halfs of tables 12.1
and 12.2 is implied by any property of the right halfs of these tables,
except the 8 implications (P=>non-/+, etc.) given in these tables. This
gives 162 — 8=154 new non-implications. Since {hn} above is also a
basis of type non-/+ of/1, and dually, < ? xt > is a basis of type non-(/+)*
u = i J
of c0 (where {л„} is the natural basis of c0), none of the properties oc-
occurring in the couples (non-aP*, non-/+), ((/+)*, aP), (non-aP, non-(/+)*),
(l+,aP*) is implied by the other property of the respective couple,
which gives 8 new non-implications. Similarly, the bases of example 12.1
and § 11, example 11.2 and the natural bases of c0 and I1 show that
none of the properties occurring in the couples (Hilbertian, shrinking),
(non boundedly complete, non-Besselian), (Besselian, boundedly com-
complete), (non-shrinking, non-Hilbertian), is implied by the other property
of the respective couple, which gives again 8 new non-implications. In-
Indeed, we have only to show that the basis {*„} of d is Hilbertian (whence
its a.s.c.f. {/„} is Besselian). Let
со.
By A2.3) we may assume
i = i
(omitting, if necessary, those ?f which are 0 and rearranging the re-
remaining sequence) that |?t| ^ |?2| $s ••• Put
1
\Ь\ > \
Then — < - for ieJix and <\Q2 for ieJi2, whence
i i
whence, by A2.3), {Qed, which proves that {х„
is Hilbertian.
Thus, we have obtained finally
140 + 324+324+158+154 + 8 + 8= 1116
12. Relations between various types of bases
369
non-implications between the 36 types of bounded bases considered in
tables 12.1 and 12.2. This completes the proof of theorem 12.1.
Remark 12.1. a) The types of bases considered in § 1, §2 and §3, i.e.
those depending on the metric structure of the space E, have not been
included in theorem 12.1 above. It is immediate that all relations be-
between them and the types of bases considered in theorem 12.1, are:
non-implication.
b) Actually, several implications of tables 12.1 and 12.2 have been
already used in the preceding sections, especially in the proofs of
the theorems concerning the Schauder basis of C([0,1]) and the nor-
normalized Haar basis of ^([0,1]).
c) The implications of tables 12.1 and 12.2 establish certain con-
connections between some of the problems formulated in the preceding
sections (e.g. an affirmative answer to §9, problem 9.1, would imply
an affirmative answer to §8, problem 8.1, etc.).
By the duality results given in §8 and § 10, the 4 implications occur-
occurring in the statement of part b) of theorem 12.1 above are mutually
equivalent. Let us also formulate separately the problem of their validity:
Problem 12.11. Is every weakly closed basis {xn} of type al+1
By the equivalence \°o2° of §10, theorem 10.3, this problem is
equivalent to the following: If {л„} is a basis of a Banach space E, such
that 0 is not a weak limit point of the set {л„}, does there exist an feE*
such that inf |/(х„)|>0?
1 €n< 00
Let us observe that there exist, in finite-dimensional Banach spaces
E, sequences (but not bases) {х„} satisfying
inf |
^ It < 00
for which 0 is not a weak limit point; such is e.g. any dense subsequence
of the unit sphere aE= {xeE | ||л"|| = 1}.
In spite of the negative results of theorem 12.1 and of the unsolved
problem 12.1, one can also give some related positive results, e.g. the
following:
Theorem 12.2. Let {х„} be a basis of a Banach space E. Then
a) The following statements are equivalent -.
1°. {*„} is shrinking.
2°. {х„} has no block basic sequence {yn} of type21+.
1 Recently, this problem has been solved in the negative (see the footnote to
theorem 12.1b)).
2 I.e. such that {yn} is a basis of type /+ of [>¦„].
24 Singer, Bases in Banach Spaces 1

370
II. Special Classes of Bases in Banach Spaces
3°. {х„} has no block basic sequence {yn} of type P*.
b) The following statements are equivalent:
Г. {*„} is boundedly complete.
2". {*„} has no block basic sequence {у„} of type P.
Proof, a) 1 =>2°. If {xn} is shrinking, then, by §4, theorem 4.2 (impli-
(implication 1°=>5°), every bounded block basic sequence {у„} (with respect
to {л'и}) is of type wc0, whence non-/+.
The implication 2°=>3° is obvious, since non-/+=>non-P*.
3^ => 1°. Assume that {xn} is non-shrinking. Then, by the implication
2°=> 1° of § 4, theorem 4.2, there exists an feE* such that
HmU/L^O,
n~* со
where ||/||n is defined by D.3). Consequently, there exist a number
?>0, an increasing sequence of positive integers [mn\ (mo = 0), a
sequence of scalars {<х„} and a sequence of elements {zn} с Е such that
IU.II = 1
/(?„)>?
(«=1,2,...),
(и =1,2,...)-
A2.23)
A2.24)
A2.25)
By A2.23) and A2.24), {zn} is a block basic sequence with respect
to {xn}. Hence, by A2.25) and the implication 5°=>6° of § 10, theorem
10.1, there exists a sequence of scalars {/?„} satisfying A0.6) and such
that {у„} = {[Snzn} is a basis of type P* of [zj = (>„]. By A2.23), A2.24)
and A0.6), {)'„} = {finzn} is also a block basic sequence with respect
to {*„}.
b) 1°=>2°. Assume that {xn} admits a block basic sequence
Уп = Z aiA"i (и= 1,2,...; mo = Q) of type P, i.e. such that
inf
>0 and sup
Z Л
= M<co. Then for any positive
integer к and any n such that т„ > к we have
— Vi
but
I.v,
1 = 1
i= 1
«? V,
Z aiA'f
whence
sup
Z ал
,M< со,
<XjXj does not converge, since
inf
z
= inf ||jn|| >0, and thus {л„} is not boundedly complete.
1 !Sn< 00
12. Relations between various types of bases
371
2С=>Г. Assume that {л„} is not boundedly complete. Then there
exists a sequence of scalars {an} such that
sup
Z ал
< CO
A2.26)
00
and that ? afx, is not convergent. Since ?] а,л"( is not convergent,
i = 1 i = 1
there exists an increasing sequence of positive integers {mn} (wo = 0),
such that
inf
Put
Z
> 0.
jn =
а,-л-,- («=1,2,...).
A2.27)
A2.28)
Then, by A2.28) and A2.27), {у„} is a block basic sequence (with
respect to {*„}), which is, by A2.27) and A2.26), of type P. This completes
the proof of theorem 12.2.
Combining theorem 12.2 and part a) of theorem 12.1, it follows
Corollary 12.1. Every basis of a type T occurring in the right halfs of
tables 12.1 and 12.2, except non-Besselian and non-Hilbertian bases,
admits a block basic sequence {yn} of any of the types situated above T
in the same half-tables.
From corollary 12.1 we obtain, in particular, the following result
related to problem 12.1: every weakly closed basis {х„} of a Banach
space E admits a block basic sequence {yn} of type al+. However, we also
have the following slightly stronger result:
Proposition 12.1. Every weakly closed basis {xn} of an infinite dimen-
dimensional space E has an infinite subsequence {xin} of type al+.
Proof. Let {л„} be a weakly closed basis of a Banach space E. Then 0
is not a weak limit point of the set {х„}, whence there exists a w-neigh-
bourhood V=Vgt,...,9ral?@) (where g1,...,gmeE*, e>0) such that
К^л-„(и=1,2,...), i.e. such that
max \gj(xn)\^e (и =1,2,...).
1 ijim
Consequently, there exist aj0 with
sequence {xiri} of {xn}, such that
and an infinite sub-
subHence, by the implication 2°=>1° of § 10, theorem 10.3, {xin} is a
basis of type al+ of [ajJ. This completes the proof.

372
II. Special Classes of Bases in Banach Spaces
Proposition 12.1 suggests the question, whether we can replace in
corollary 12.1 above "block basic sequence" by "subsequence", at least
for the first type situated above T. The answer is negative, as shown by
example 12.1. Indeed, the basis {xn} <= d of this example is non-shrinking,
but every subsequence {xir} of {х„} is of type wc0.
It is natural to ask whether corollary 12.1 remains valid if we replace
in its formulation "the right halfs of tables 12.1 and 12.2" by "the left
halfs of tables 12.1 and 12.2". However the answer is negative. Indeed,
the basis {/?„} of I1 considered in example 12.3 is of type non-P*, but
admits no shrinking block basic sequence since the conjugate space of
every subspace of Z1 is non-separable. Moreover, {/?„} admits no block
basic sequence of type wc0 or swc0, since in every subspace of I1 weak
convergence of sequences implies their convergence in the norm-to-
pology. Dually, the basis
i= 1
of c0 (where {xn} is the natural basis
of c0), is of type non-P, but admits no boundedly complete block basic
sequence, since no subspace of c0 is isomorphic to any conjugate Banach
space, i.e. every subspace of c0 satisfies the condition of § 6, example 6.2
(with k = 0). Moreover,
admits no block basic sequence of type
{wcof or (swc0)*, by virtue of the above remarks concerning subspaces
of/1.
In spite of these negative results, one can also give some positive
results concerning the left halfs of tables 12.1 and 12.2, e.g. we have
seen in § 7, proposition 7.1 b) that every basis {xn} of type swc0 contains
a subsequence (x;j of type wc0.
Finally, let us also mention the following natural question-, if a
Banach space ? has a basis {xn} of a certain type T occurring in table 12.1
or table 12.2, what other types of bases does E possess? Theorem 12.1a)
can be also interpreted as a partial answer to this question. Another
result in this sense is the following:
Proposition 12.2. For a Banach space E the following statements are
equivalent -.
Г. Е has a basis of type P.
2°. E has a basis of type P*.
3r. E has a basis of type a P.
A". E has a basis of type aP*.
5°. E has a basis of type l+.
6е. Е has a basis of type (/+)*•
7°. E has a basis of type al+.
8°. E has a basis of type (al+)*.
13. Universal bases. Complementably universal bases. Block-universal bases 373
Proof. The equivalences ГоЗ°, 2соА\ 5°<^7r and 6'o8 are
immediate consequences of §9, definition 9.1 and §10, definition 10.1.
The equivalence 2°<^>5° follows from the equivalence ГоТ of
§10, theorem 10.1; the equivalence I°o6° follows from this by
duality. Finally, the equivalence 1°<^2° follows from the implication
Го2° of §9, theorem 9.1 and the implication Г о 7° of §9, theorem
9.2 b), since (9.2) is of type P* and (9.14) of type P. This completes
the proof.
§ 13. Universal bases. Complementably universal bases.
Block-universal bases
We have seen in Ch. I, § 4, proposition 4.1 a) that every subsequence
{xin} of a basis {х„} is a basic sequence.
Definition 13.1. Let Я be a family of bases. A basis {en} of a Banach
space F is said to be universal for Si if every basis in 8) is equivalent to
a subsequence of {en}.
For instance, every basis [xn] is universal for any family 09 of sub-
subsequences of {*„} (hence in particular, for the family of all subsequences
of {*„}).
Definition 13.2. A subsequence {xir} of a basis {х„\ of a Banach space
E is said to be complemented if for any sequence of scalars {а„} the
GO
convergence of the series ? а;х; implies the convergence of the series
cc i = 1 oc
Z ainxu (°r>in otner words, if Y, fdx)xtn converges for every xeE,
"=1 n=1
where {/„} с Е* is the a.s.c.f. to {*„}).
Remark 13.1. If [x,J is a complemented subsequence of a basis {xn}
of a Banach space E, then [x;j is a complemented subspace of E, namely,
E= [*;J © [*;„], where {/„} is the set of indices complementary to {/„},
since the mapping u-.E^E defined by
= Z /;»*.„ (xeE)
A3.1)
is a well defined continuous linear projection of E onto [x,- ] (by the
Banach-Steinhaus theorem). However, the converse is not true, since
e.g. for a conditional basis1 {*„} of the Hilbert space E = l2 there exists,
1 See § 14, examples 14.4 and 14.5.

374
II. Special Classes of Bases in Banach Spaces
by § 16, lemma 16.1, implication 1°=>3°, a convergent series ? а{х{еЕ
00 '= 1
having a non-convergent subseries ? %inxin, but the subspace [x,J
is complemented. "=1
Definition 13.3. Let ^ be a family of bases. A basis {е„} of a Banach
space E is said to be complementably universal for ^ if every basis in gg
is equivalent to a complemented subsequence of {en}.
For instance, by § 16, lemma 16.1, implication 1°=>3°, every un-
unconditional basis {*„} is complementably universal for any family Щ
of subsequences of \xn}.
Our next aim is to prove that the family Si of all bounded bases
contains a complementably universal element. For this purpose we need
some preparation. For the sake of simplicity we shall consider only
real scalars.
We shall use the following notations-.
[>¦,(')]('") =
™ = the linear space of all real sequences t={t(l), tB),...},
7tn = the mapping of Rx into R°° defined by
t(i) for i=l,...,n
0 for i = n+l,n + 2,...
" = nn(Rx) = the linear space of all sequences {/(l),...,?(n),0,0,...},
R = the real line,
A3.2)
Г = \teRx\ \t{i)\ ^ 1 (i= 1,2,...)},
In = rnR"={teRn\\t{i)\^\(i=\,...,n)},
dl"={tel" max|*@| = l},
en = the element of Rx defined by
( for /=l,...,n-l,n+l,...
1 for i = n.
A3.3)
A3.4)
A3.5)
A3.6)
Furthermore, we shall denote by В„ (respectively B) the set of all
oo
non-negative functions /?(•) on R" (respectively, on \J R") satisfying
n= 1
the following conditions:
(i) p(t) = O if and only if t = 0; p(ct)=\c\p(t);p(t + .
oc
for all t,seR" (respectively, t,se [j R") and ceR;
(ii) piet)= 1 for (= 1,..., n (respectively, for / = 1,2,...);
00
(iii) max \s(l)\ ^pis) for all se#" (respectively, se \J R");
n= 1
13. Universal bases. Complementably universal bases. Block-universal bases 375
(iv) pinm(s))^p(s) for se.R" and m=l,.,..,« (respectively, for
00
se [j K"andm=l,2,...).
n= 1
The elements of В„ and В will be called norms.
For n— 1,2,... andp,qeBn we shall put
p{t) _ qit)^
Lemma 13.1. dn is a metric on Bn. The metric space [Bn,dn) is
compact.
Proof. Obviously, dn(p,p) = 0. Conversely, if dnip,q) = O, then
= л = const. (teRn, ?#0) whence, by (ii), A=4^='. whence p = q.
,q) are ob-
obThe relations dnip,q) = dniq,p) and dnip,q)^dnip,r)
vious.
To prove the second statement, put
р=р\е1„ ipeBn), A3.8)
Вя={р\реВя}. A3.9)
Then Bn consists of functions which are uniformly bounded on the
compact space dl", since
p(t)=p(t) =
t(i)et
/
= t\t{f)\p{ei)=
and equicontinuous on dl", since
max k(O-f
i= 1
1
is,te8I").
Furthermore, Bn is closed in С(дГ), since if a sequence {pk} <= Bn
converges uniformly to a function feC(8l"), then f=p, where реВ„
is defined by
piO) = 0,pis) = max \s(i)\f ( S——\ iseR",s^0). A3.10)
max \s(i)
Indeed, since max \s{i)\ = 1 for sedl", we have p(s)=pis)= f(s)
for sedl", i.e. p=f. Furthermore, let us show that реВ„. Since pkeBn
and
max\pkis)-pis)\->0 as fc->oo, A3.11)
set?/"

376
II. Special Classes of Bases in Banach Spaces
it follows that/7E)^0 for all sedl", whence also for all seR". Assume
now that p{s) = 0. Then, by (iii) for pk and A3.11),
max \s(
1 ? i < 00
whence 5 = 0. By A3.11) and (i) — (iv) for pk, we have also (i) — (iv) for/7,
i.e. peBn. Thus, Bn is closed in C(8In).
Consequently, by the theorem of Arzela, Bn is a compact set in
С{дГ). We shall show that Bn is homeomorphic to В„, by the mapping
/>->/?, which will complete the proof. If p1=p2, i.e. Pi\di"=P2\di",
where pup2eB", then by the homogeneity of the norms pup2 we
have py—p2, and thus the mapping p—>p is well defined. Furthermore,
it is obviously one-to-one and maps В„ onto Bn. Finally, it is continuous,
whence a homeomorphism, since max\pk{t)— /?(/)|—>0 implies
tePI"
Pk[t)-\ <-^^^^ (tedr,k>N(e)),
Pit)
inf p{s)
sefl"
whence dn(pk,p)—>0 as k—>co. This completes the proof of lemma 13.1.
For a fixed index n and for peBm with m^n or peB, we shall put
Lemma 13.2. The restriction operator Jn has the following properties:
a) If m^n, then
dn(Jn(p),Jn{q))^dm(p,q) (p,qeBJ. A3.13)
b) If peBn + 1 and qeBn, then there exists a qeBn + l such that
q = JM, A3Л4)
dn(Jn(p),Jn(q)) = dn+l(p,q). A3.15)
Hence, in particular,
Jn(Bn + l) = Bn. A3.16)
Proof, a) If m^n, then 8Г => 8Г, whence, by the definition A3.7)
of dn, we infer A3.13).
b) Put ,A 1Л
¦ rPU) , РКП (П 17^
a= inf—, b = sup —. (I5.ii)
tEdi» q(t) ten» q{t)
Then, since by (ii) p(el) = q(el) = 1 and since all norms on R" are
equivalent, we have b^l^a>0. Furthermore, obviously
dn{Jn{p),q) = \ogb-\oga, A3-18)
(teR"). A3-19)
13. Universal bases. Complementably universal bases. Block-universal bases 377
Put
A3.20)
A3.21)
A3.22)
= co(Q,aP,±en + l),
where со means "convex hull". Let q be the Minkovski functional
of 6, i.e.
q(s)= mfo Л {seRn+l). A3.23)
We shall complete the proof by showing that q has the required
properties, i. e. qeBn+1 and satisfies A3.14), A3.15).
Obviously, A3.19) is equivalent to the inclusions
Consequently, we have
QnRn =
A3.24)
A3.25)
whence, in particular, Q is a neighbourhood of 0 in Rn+l. Since Q is
symmetric and convex, it follows that q satisfies condition (i) of the norm.
Furthermore, by A3.25), we have A3.14) and
A3.26)
whence
sup —¦ sup
q{)
whence, by A3.18), A3.14) and A3.13), we infer A3.15).
Since q and p satisfy (iii), we have QdIn(=In+l, aP<=PcIn + 1,
whence, taking into account ±en+lel" + l, it follows that 6<=/" + 1,
and therefore q satisfies (iii).
Since en+leQ and cen + l $Q for |c|>l (because Qczl" + l by
(iii)), we have q{en + l)=\. Furthermore, since q satisfies (ii), we have
g(e;) = <7(ei.) = 1 for i=\,...,n. Thus, q satisfies (ii).
Finally, let us show that q satisfies (iv). Put
hl|7r» = 0} (m=l,2,...). A3.27)
Then, since p satisfies (iv), we have1
1 We use the notation A + B={a + b\ aeA, beB}.

378
II. Special Classes of Bases in Banach Spaces
indeed, if seP, then p(s)^l and s = nn(s) + r, where nn(s)eR",reLn + u
and by (iv) for p we have p{nn(s))^p(s)^\, whence nn(s)ePnR",
whence se{PnRn) + Ln+1. Hence, taking into account A3.24),
Since obviously Q + Ln + l=>Q and Q + Ln+l3±en + l, it follows that
Q, whence the relations seRn + l, q{s)^l imply q(nn{s))^l.
Consequently,
+ l). A3.28)
Combining A3.28) with the assumption that q satisfies (iv), we
obtain, for any m with
q(s) > q(nn(s)) > q(nmnn(s)) = ZK{s)) = q(nm(s))
Thus, q satisfies (iv), which completes the proof of lemma 13.2.
Lemma 13.3. Let e>0. Then there exists a sequence {An} such that
for each n= 1,2,...
a) An is a finite subset of Bn;
" 1
P) А„ is an 8 Yj ^k'net (hence, in particular, an e-net) for Bn;
y) Jn-1{An) = An_l.
Proof. Put Al = Bl (observe that Bl is a one-point set, namely, the
norm p(a)=\a\(aeR)). Suppose that for some m^l we have already
defined m sets Al,...,Am satisfying a) — y). By virtue of lemma 13.1,
p
et for Bm+1. By lemma 13.2b), for each pair
let F be a finite
j
(p,q) with peF, qeAm there exists a norm q = q(p,q)eBm + l such that
Jm(q) = q and dm + l(p,q) = dm{Jjp),q). We shall show that the set
+ l = {q = q(p,q)\peF,qeAm}
A3.29)
has the required properties a) — y), which will complete the induction
and the proof of lemma 13.3. Since F and Am are finite, a) is obvious
by the definition of Am + 1. Furthermore, let p'eBm + 1 be arbitrary. Then,
since F is an +1-net for Bm + 1, there exists a peF such that
e m 1
dm+i(p',p) <^. Since JJp)eBm and since Am is an e ? --net for
*¦ k=l L
Bm, there exists a qeAm such that dm(Jm(p),q)<e
J
2
ВУ lemma
k=i
13.2b), there exists a q = q(p,q)eAm + 1 such that Jm{q) = q and
13. Universal bases. Complementably universal bases. Block-universal bases 379
(p,q) = dm(Jm{p),q). Hence
Ш+ 1 1
and thus we have fi) for Am + l. Finally, since for any qeAm we have
Jm{q(P,q)) = q with q(p,q)eAm + l (where /7eF is arbitrary), it follows
that Jm(Am + 1) = Am, which completes the proof of lemma 13.3.
i;)d (seR").
Then there exists a norm peBn+l such that Jn{p)=p and
Lemma 13.4. Let l^k^n and letpeBn, qeBk+l. Suppose that there
exists a sequence 1 ^(\ <i2< ¦¦¦ <ik<ik+l =n+ 1 such that
4it)=P\Y,tiJ)eij) iteRk), A3.30)
A3.31)
A3.32)
A3.33)
A3.34)
n+1)= inf F{r,s,c) {seRn,ceR). A3.35)
ГЕ Rk
We shall show thatp has the required properties, which will complete
the proof.
Observe that p satisfies (i), since it may be regarded as the natural
norm in the quotient space X/Y, where X = R"xRk + 1 endowed with
the norm \\{s,t}\\=p(s) + q{t) and
к
Proof. Put
(/Vi l
via the correspondence s + cen+l^{s,cek+l}. Indeed,
\,
inf
l)= mf F(rus,c)
rRk

380
II. Special Classes of Bases in Banach Spaces
Furthermore, we have Jn(p)=p, i.e.
p{s)=p{s) {seR").
A3.36)
Indeed, obviously p(s)^F{0,s,Q)=p{s). On the other hand, for any
reRk we have, by A3.30),
F(r,s,0)=p(s-
whence p{s)= inf F{r,s,O)^p{s), and thus we obtain A3.36).
reRk k
Let us show now A3.32). If л= ? /(/)e; for some teRk7 then
for any ceR we have, taking into account that ik+i=n+\,
p{U)e
On the other hand, for any reRk we have, by A3.30) applied to
the element t — reRk,
F\r, t '(/K-CW( Z {t-
whence pi ? r(/)e, +ce;k+i )^q{t + cek+l), and thus we obtain A3.32)
Vj=i ' * I
(with r(/c+l) = c).
Taking now in A3.32) t(l)= ¦¦¦ =t{k) = O, r(/c+l)=l, we obtain
p{en+l)=p(eiktl) — q(ek+l)= 1, because g satisfies (ii). Since p satisfies
(ii), we also have, by A3.36), p{em)=p{em) = 1 (m= 1,...,«), and thus
/7 satisfies (ii).
Furthermore, since q satisfies (iv), we have q{r + cek+l)^q(r) {reRk,
ceR), whence
p{s + cen+l)= inf F(r,s,c)^ inf F{r,s,0)=p(s)=p{s) {seRn, ceR).
reRk reRk
A3.37)
Combining A3.37) with the assumption that/5 satisfies (iv), we obtain,
for any m with 1 ^ m ^ n,
and thus p satisfies (iv).
Assume now that p{s + cen + l)^l for some seR", ceR. Then, by
A3.37), p{s)^l, whence, since p satisfies (iii), s e I". On the other hand,
13. Universal bases. Complementably universal bases. Block-universal bases 381
by the definition of p, for every e>0 there exists an r = rceRk such that
c/ ? r(j)ej + cek+1 h% 1+e, whence, since q satisfies (iii), |c|^l + e,
\j=i /
whence |c|^l. Thus, p{s + cen + 1)^l implies sel", |c|^l, whence it
follows thatp satisfies (iii). This proves that peBn+l.
Finally, let se#"+1 be arbitrary. Then for any reRk we have, by
A3.31), A3.30), ik+l=n+l and A3.32),
Fir,
V
i=l
>('Vi- Z r{f)e
i=l j=l
Z№)-<
к
n+l)ek+l)
= q
whence we infer A3.33), which completes the proof of lemma 13.4.
The next lemma gives essentially the construction of the com-
complementably universal basis.
Lemma 13.5. Let e>0. Then there exists a norm peB with the
property that for every q$B there is an increasing sequence of indices
i1<i2< ¦¦¦ such that
fCk(te\jRk), A3.38)
k=i /
A3.39)
Proof. By lemma 13.3 (with log(l+e) instead of e) there exist an
increasing sequence of indices l = Nl<N2<--- and a sequence {qn}
such that for each к = 1,2,...
qneBk for n = Nk,
A3.40)
Ak= \j {qn} formsalog(l+e)-netfor Bk, A3.41)
Jk(Ak+l) = Ak. A3.42)
Now we shall define inductively a sequence of norms {/?„} and a
sequence of finite increasing sequences of indices {a(«)} with the following
properties, for each n = 1,2,... -.

382
II. Special Classes of Bases in Banach Spaces
a) PneBn,Jn(pn+l)=Pn;
b) if Nk^n^Nk+l-l and a{n)={il{n),i2{n),...,ikn(n)}, then
c) if
d) if 1 ^ m ^ n, then
—1 and J[{qn) = qm, then
A3.43)
A3.44)
fll45)
*,„
Put Pi=qi and a(l)={l}; then a)-d) are obviously satisfied.
Assume that for some n ^ 1 we have already defined the norms pm
and the finite increasing sequences of indices a(m) so as to satisfy a)-d)
for each m^n. Choose к such that Nk+l^n+ 1 ^Nk + 2 — 1, whence,
by A3.41), qn+ieAk + 1 <=Bk+l, Then, by A3.42) there exists an m^n
such that <7те^ (whence, by A3.41), Nk^m^Nk+l-l) and Jk(qn+l)
= qm. By the inductive hypothesis (conditions b) and d)), we have then
and
Pn
j=i
(teRk),
and also A3.46). Hence, by lemma 13.4, there exists a norm p'eBn+l
such that Jn(p')=pn and that
'( 1 t(j)eiAm)+t(k+l)en+1) = qn+l(
Put
pn+1=p',x(n+l)={il(m), i2(m),...,ik{m), n + 1} = a(m)и {n + 1}.
Then obviously />„+1 and a(«+l) satisfy a)-d), which completes
the induction.
. A3.47)
13. Universal bases. Complementably universal bases. Block-universal bases 383
Then by a)p is well defined and peB. We shall show that p has the
required properties, which will complete the proof.
Let qeB be arbitrary. Then, by A3.41), there exists a sequence
{q'k} such that
q'k = qnkeAk, whence Nk^nk^Nk+l-l (k= 1,2,...), A3.48)
dk(q'k,Jk(q))<\og(l+e) (k= 1,2,...). A3.49)
We claim that without loss of generality we may also assume
JkWn) = q'k (/c=l,...,«;«=l,2,...). A3.50)
Indeed, let {</„} be an arbitrary sequence of norms such that
<tneAnin=\,2,-). Then, by A3.42),
whence the set
Fk=\J
A3.51)
is finite and therefore compact in the discrete topology. Consequently,
the cartesian product
(k= 1,2,...)}
A3.52)
is compact. Put
(«=1,2,...). A3.53)
Then each Zn is non-empty (since if we define q'k' = Jk{q'n) forfe=l,...,n
and q'k = arbitrary in Fk for к = n + 1, n + 2,... then {q'k'}e Zn), closed (since
the relations {q'k'
id)}
{}
{q'k}, {q'k(d)}eZn obviously imply
Jk{Cd))-—*Jt{<?), whence {tf}
deA
and Zt
Hence, by the
compactness of F, there exists a sequence {qk}eF such that {q'k'}e (| Zn,
n= 1
whence {^'} satisfies A3.50). Furthermore, by A3.52) and A3.51),
for every index к there is an index n = n(k)^k such that Jk{q'n) = qk
(where q'neAn by our hypothesis). Hence, if the sequence {q'j} satisfies
A3.49), then, taking into account A3.13),
dM.JM) = dk(Jk{q'n\ Jk(Jn(q))) < dn(q'n, J n{q)) < log(l + e).
Thus the sequence {q'k} satisfies the conditions A3.48)—A3.50),
which proves our claim on A3.50).

384 II. Special Classes of Bases in Banach Spaces
Now define the increasing sequence of indices il<i2< ¦by
ik = nk (fc=l,2,...), A3.54)
where nk is defined by A3.48). Let us show that for each к
a(nk)={iui2,.-,ik}. A3.55)
Since by A3.48) we have Nk^nk^Nk+l — l, by virtue of b) we have
Furthermore, if m</c, then nm^nk, whence, by A3.50) and A3.48),
Jm{4nk) = JmDk):=(}'m = (inm and hence, by с), ij{nk) = ij{nm) for j= l,...,m.
Thus, in particular, for j=m we obtain
U"t) = U«m) = nm = L (m = 1,..., k),
whence, by A3.56), there follows A3.55).
Consequently, by d), we get A3.39) and
qn{t) = q'iS.t) {teRk). A3.57)
On the other hand, by A3.49) we have
sup • sup ^ 1 + e,
whence, since qri(e1) = 9(e1)=l (by (ii)),
whence
l^sup — < A+e) inf — sjl + e,
к ?(/) f/" <?(/)
(teRk).
1 + e
A3.58)
From A3.57) and A3.58) it follows that we have A3.38), which
completes the proof of lemma 13.5.
Now we can prove
Theorem 13.1. The family 36 of all bounded bases contains a complemen-
tably universal element.
Proof. Let e>0 be arbitrary. Let p=pceB be as in lemma 13.5
and let E = Et be the completion of the linear space I \J Rk,p\; we
Jc=l
shall indentify \J Rk with its canonical image in E. The unit vec-
13. Universal bases. Complementably universal bases. Block-universal bases 385
tors {е„} form a (monotone normalized) basis of E, since their linear
combinations are dense in E and, by (iv),
i=i
;=i
for any finite sequence of real numbers al,...,an+m. We shall show that
this basis {en}e38 is complementably universal for the family 3d of
all bounded bases, which will complete the proof.
Let {xn} be an arbitrary bounded basis (of a Banach space, say X). Put
a= inf
b= sup
A3.59)
Then, since {х„\ is a bounded basis, 0<a^b<co. Furthermore, put
a(() = max ( — max
Ь l«<
Z
, max \t(n)\ [te{J Rk). A3.60)
Then obviously geB and hence, by lemma 13.5, there exists an
increasing sequence of indices il<i2<--- such that we have A3.38)
and A3.39). By A3.39), {ein} is a complemented subsequence of {en}.
Furthermore, since — ^ —, we have
b a
{teRn, m=!....,«; «=1,2,...)
and by Ch. I, § 3, formula C.8), we also have
\t(m)\ < —
Z
{teR", m= 1 n; «=1,2,...)
where v = v{Xn) is the norm of the basis {xn}. Consequently, by A3.60)
and A3.38),
1
1+e
+ e)
Z
{teRn,n=l,2,...
whence by Ch. I, § 8, theorem 8.1 d), the basis {xn} is equivalent to the
basis {ein}of[eij, which completes the proof of theorem 13.1.
Remark 13.2 Since the basis {е„} in the above proof is actually norm-
normalized, we obtain again that every bounded basis is equivalent to a
normalized basis (see Ch. I, § 3, theorem 3.2) and that the family 36
25 Singer, Bases in Banach Spaces I

386
II. Special Classes of Bases in Banach Spaces
of all normalized bases contains a complementably universal element.
Conversely, these two results imply theorem 13.1, since from these
results it follows that every normalized basis which is complementably
universal for the family of all normalized bases is also complementably
universal for the family of all bounded bases.
By remark 13.1 (made after definition 13.2) and by Ch. I, § 8, theorem
8.1 d), we have the following corollary of theorem 13.1:
Corollary 13.1. There exists a Banach space E with a basis such that
every Banach space with a basis is isomorphic to a complemented subspace
ofE.
This result suggests naturally the following problem:
Problem 13.1. Does there exist a separable Banach space Es such
that every separable Banach space is isomorphic to a complemented
subspace of?s?
By corollary 13.1, a negative answer to problem 13.1 would imply a
negative answer to the basis problem. This seems to be an "easier" approach
towards a negative answer to the basis problem, than constructing a
counter-example, since this approach is "purely existential". The answer
to problem 13.1 is believed to be negative, namely, it is conjectured that
there exists no separable Banach space Es such that every subspace of
C([0,1]) of the form [е2я""]„еМ. where M runs over all infinite subsets
of the set Л= {1,2,3,...}, be isomorphic to a complemented subspace
An affirmative answer to problem 13.1 would also have interesting
consequences. Indeed, if the space Es had no basis, we would have a
negative answer to the basis problem, while if the space Es had a basis,
then it would follow that every separable Banach space has the approxi-
approximation property (since it is obvious that every complemented subspace
of a Banach space with the approximation property has the approxi-
approximation property) whence also every Banach space has the approxi-
approximation property [89] and thus we would have an affirmative answer to
the "problem of approximation" [89].
Let us consider now the problem, "how many" complementably
universal bases and "how many" spaces with complementably universal
bases exist. The answer is given by
Theorem 13.2. Any two bounded bases which are complementably
universal for the family & of all bounded bases are permutatively equivalent .
Hence they span isomorphic Banach spaces and therefore the space E
constructed in the proof of theorem 13.1 is unique up to an isomorphism.
1 See Ch. I, §8, definition 8.3.
13. Universal bases. Complementably universal bases. Block-universal bases 387
Proof. Since every bounded basis is equivalent to a normalized basis
(see remark 13.2), it will be sufficient to consider only the case of two
normalized elements of Зй. By "equivalence class" or "class" of a basis
{*„} we shall mean the class of all bases which are permutatively equival-
equivalent to {*„}.
Let 3C be the equivalence class of a normalized basis which is com-
complementably universal for the family of all bounded bases. Then, by
remark 13.1 and Ch. I, § 4, proposition 4.2 and lemma 4.1, for the class1
2C00 and for the class <& of an arbitrary normalized basis there exist
classes Z and J^ such that
Hence, by Ch. I, § 8, proposition 8.4, we obtain
Now, assuming that ®J above is also the class of a normalized basis
which is complementably universal for the family of all bounded bases,
we get, by symmetry, that Щ = 9C x <W, and hence
Thus, since by Ch. I, §8, theorem 8.1 d), implication 6°=>Г, any
two equivalent bases span isomorphic Banach spaces, the proof of
theorem 13.2 is complete.
We shall denote by Eb the Banach space, unique up to an isomorphism,
which has a bounded basis complementably universal for the family
of all bounded bases. The following corollary shows that the property
described in corollary 13.1 also characterizes Kh up to an isomorphism.
Corollary 13.2. // E is a Banach space with a basis, such that every
Banach space with a basis is isomorphic to a complemented subspace of E,
then E is isomorphic to Eb.
Proof. Again, by "equivalence class" or "class" of a basis {*„} we shall
understand the class of all bases which are permutatively equivalent to
{xn}. Let <У denote the equivalence class of a normalized basis of Eb,
which is complementably universal for the family of all bounded bases
(again, we may assume that Eb has such a basis) and let '? be the class of
a normalized basis of E. Then, as observed in the proof of theorem 13.2,
Щ = <У x <& and <& = <& x SC, whence2 Eb x Eb^Eb and Eb x E^Eb. On the
1 SeeCh. I, §8, definition 8.5.
We write ?t = ?2 if the spaces El and E2 are isomorphic.
25»

II. Special Classes of Bases in Banach Spaces
other hand, by our assumption on E, there exists a Banach space F such
that E^EhxF. Consequently,
^Eb x F^(Eb x Eb
^Eb x (Eb x F)^
x E^
which completes the proof.
Remark 13.3. One can also prove, by an argument similar to that
used in the proof of theorem 13.2 (considering (?sx?sx ••¦)B and
applying § 18, lemma 18.5) that if the answer to problem 13.1 is affir-
affirmative, then the space Es of that problem is also unique up to an iso-
isomorphism.
We shall consider the problem, "how many" universal bases and
"how many" spaces with universal bases exist, together with the analogous
problem for block-universal bases.
Definition 13.4. Let @) be a family of bases. A basis {en} of a Banach
space E is said to be block-universal for 0S if every basis in Я is equivalent
to a block basic sequence with respect to {en}.
Obviously, every universal basis is block universal, since every
subsequence of a basis is a block basic sequence with respect to that basis.
For the proof of the next theorem we shall need
Lemma 13.6. Let E and f, be Banach spaces, such that E contains a
subspace F isomorphic to Fl. Then there exists an equivalent norm \\x\\^
on E such that the subspace F a E endowed with this new norm is linearly
isometric to Fl.
Proof. Let и be an isomorphism of F onto Flt such that
where C>1 is a constant. Put
= co {j;6F|||K(y)Kl}u^6?
IMl! = inf Я (*е?).
A3.61)
A3.62)
A3.63)
Then Hxll! is a norm on E (it is the Minkowski functional of A),
equivalent to the original norm on E, since by A3.62) and A3.61) we
have
-SE^AczSE={xeE\\\x\\^l}.
Furthermore, by A3.63), A3.62) and A3.61) we readily have
\Ш\\ = ILvll,
13. Universal bases. Complementably universal bases. Block-universal bases 389
i.e., и is a linear isometry of (F,||j||i) onto Ft, which completes the
proof of lemma 13.6.
We recall that a Banach space E is said to be isomorphically uni-
universal for a family (У) of Banach spaces if every space Хе(У) is iso-
isomorphic to a subspace of ?.
Theorem 13.3. Let E be a Banach space, isomorphically universal for
the family of all separable Banach spaces and assume that E has a basis.
Then every bounded basis {en} of E is block-universal for the family 08
of all bounded bases.
Proof. By hypothesis, ? contains a subspace isomorphic to C([0,1]).
Hence, by lemma 13.6, there exists an equivalent norm on ? such
that E endowed with this new norm contains a subspace linearly iso-
isometric to C([0,1]); we shall assume that ? is endowed with this new
norm and we shall identify C([0,1]) with its image in ? under this
linear isometry.
Let {/?„} be the a. s. с f. to the basis {en} of ? and let
ft. = *n|c<[o,i]) («=1,2,...). A3.64)
We claim that there exists a set A a [0,1] homeomorphic to the
Cantor discontinuum and such that
\цп\(А) = 0 («=1,2,.
Indeed, let Q1 =DA1)uDB1), where DUD2
A3.65)
[0,l~\ are two disjoint
Cantor sets such that |ju1|F1)<i, let Q2 = DA2)uDB2)uDC2)uDD2),
where D[2\ Dl22) с D[l\ Df\ DD2) с D{P are four mutually disjoint Can-
tor sets such that
?><23>
:-, let
where
: Dl42) are eight mutually disjoint Cantor sets
3 1
such that Y, К-1Fз)< q> and continue this process indefinitely; then
00
the set A = f\ Q{ has the required properties (A is compact, zero-
i=1 1
dimensional and perfect and by Ar^Qj we have \nn\(A)^\fin\(Qj)<—.
for n,j= 1,2,... with n^j, whence A3.65)). ^'
Now let {х„} be an arbitrary bounded basis of a Banach space X.
By the remark made after proposition 1.3 of § 1, we may assume that
{xn} is monotone (renorming, if necessary, the space X with a suitable
equivalent norm); furthermore, there exists1 a linear isometry w of X
into C{A).
See, e.g., [10], p. 93.

390
II. Special Classes of Bases in Banach Spaces
Let e>0 be arbitrary. Then, since ^ is regular and \fil\(A) = 0,
there exists an open set U^ A such that \fil\{U1)<~. Put F1 = [0,1]\G1.
Then Ft is closed, FlnA=$ and |ju1|([0,l]\Fi)<|- Put
] for teA,
0 for reFl7
linear for the other t.
Then we have, putting M = sup ||eJ,L = sup
A3.66)
\\fil(yl)el\\^M\hl(yl)\^M\\lyl(t)dfi1(t)+ J ^
Llfi [o, iufi
A3.67)
Furthermore, since yl = Z ^j(Ji)ej= Z fxjb;i)ej^ there exists a
positive integer щ such that
MLe
Z MjO'i)^
A3.68)
Similarly, since /J.i\Vl,---,nni\v, are regular and A a Uu \fil\(A) = ---
= \цП1\{А) = 0, there exists an open set U2^Ul with U2 => A such that
Y \Hj\{U2)<^. Put F2 = [0,l]\t/2. Then F2 is closed, F2=>Fl,
F2nzl = 0and
У lit) =
[w(x2)](t) for
0 for
linear for
Then we have

Z tijb;2)ej
< м Z 1^^2I
j-iL
J y2(t)d[ij(t)
II./ 2 II .
— in цл2||. --
?eF2,
the other r.
j y2(t)d[ij{t)
[O,H\F2
MLe
4
A3.69)
A3.70)
13. Universal bases. Complementably universal bases. Block-universal bases 391
Furthermore, since j2= ? Hj{y2)ej, there exists a positive integer
n2>nl such that j'=1
:^.
Continuing in this manner, we obtain an increasing sequence of
positive integers {nk}, a sequence of closed sets {F*} <= [0,1] with
Fk с Fk + l, FknA=0, Y, l^jlt[O>l]\Ffc)<^ (fc= 1,2,...), and a sequence
{jj с С([0,1]) such that
[w(xj](r) for teA
0
linear
for teFk (/c=l,2,...), A3.72)
for the other t
m. 1
Put
.MLe
~~ 2k
MLe
(/c=l,2,...;«0=l),
л-) = Z
л"= Z
A3.73)
A3.74)
A3.75)
We shall prove that и is a linear isometry from X into C([0,l]).
For this purpose it will be sufficient to prove that for any finite sequence
ofscalars а.и...,а.к we have
i= 1
к
Z ««*!
A3.76)
Formula A3.76) is obvious for /c=l, since Ца^Л = |at| HjJ
= !«[! IIa'JI = HajA'jII (by A3.72) and since w is an isometry). Assume
now that A3.76) holds for some m^l. Let а1,...,аП1 + 1 be arbitrary and
put x= Z а(л',ч a = am+i- Then, since by A3.72) м(л) + а>>т+1 is linear
i= 1
on each interval of the set [0, l]\(zl uFB + 1), we have
p |[
Fm+ l
However, by
sup |
m+i
A3.77)
+ 1) we have
= sup | [u(x)](t)\^\\u(x)\\, A3.78)

392
II. Special Classes of Bases in Banach Spaces
and by the induction hypothesis and the monotony of the basis {xn}
we have
|М*)|| = ||дгК||л- + ал-т+1||. A3.79)
On the other hand, since by A3.75) and A3.72) u(x) + aym+l\A
= w(x) + aw(xm+1) and since w is a linear isometry, we have
A3.80)
= \\w(x
From A3.77)-A3.80) we infer \\u(x) + aym+1\\ = \\x + axm+1\\, and
this completes the induction. Thus, и is a linear isometry of X into
C([0,1]), whence, in particular, {xk}~{u(xk)}.
Put ио = 0 and
A3.81)
Then by A3.73), A3.74), A3.75) and A3.64) we have
\\u(xk)-zk\\ =
whence, denoting by
00
I \\gk
MLs
A3.82)
the a.s.c.f. to the bounded basis {u(xk)} of u(X),
" MLs
sup Hfifjll X -^T= 2MLe suP llffjll <!
t=l l==j<oo д.= 1 Z l=Sj<oo
(for e sufficiently small), whence, by Ch. I, § 10, theorem 10.1, we infer
that {zk}~{u(xk)}~{xk}, which completes the proof of theorem 13.3.
Remark 13.4. One can also prove that under the assumptions of
theorem 13.3 every bounded basis {л„} of any infinite dimensional
Banach space X is equivalent to a normalized block basic sequence with
respect to {en}. Indeed, by the remarks made after proposition 1.3 of § 1,
we may assume that {xn} is normalized and monotone (renorming, if
necessary, the space X with a suitable equivalent norm). Then, taking
into account that и is an isometry, we get, by A3.82),
1 -
1= llzJI-1
MLs
lk-1
whence, as in the above, from Ch. I, § 10, theorem 10.1 we infer that
- {zk} ~ {xk}, which proves our assertion.
13. Universal bases. Complementably universal bases. Block-universal bases 393
Remark 13.5. Since C([0,1]) is isomorphically universal1, from
theorem 13.3 it follows, in particular, that every bounded basis {xn} of an
infinite dimensional Banach space X is equivalent to a block basic sequence
with respect to the usual Schauder basis of C([0,1]). Consequently, if a
separable Banach space X is not isomorphic to any block subspace2 with
respect to the Schauder basis of C([0,1]), then X has no basis, giving
thus a negative answer to the basis problem. It seems to be promising
to look for such subspaces X of C([0,1]), since e.g. in the spaces c0 and
1р{\^р<оо,рф2) there exist subspaces which are not isomorphic to
any block subspace with respect to the unit vector basis (such as, by
§ 18, proposition 18.1a), any subspace which is not isomorphic to the
whole space).
For this reason, we suggest
Problem 13.2. Describe, up to isomorphisms, all block subspaces
with respect to the various usual bases in concrete Banach spaces.
We conjecture that except for spaces isomorphic to /2, in every
Banach space E with a basis {en} there exists a subspace X not isomor-
isomorphic to any block subspace with respect to {en}.
Returning to spaces with universal and block-universal bases,
we shall now prove
Theorem 13.4. Let E be a Banach space with a basis. The following
statements are equivalent:
1 °. E has a bounded basis which is universal for the family of all bounded
bases.
2°. E has a bounded basis which is block-universal for the family of all
bounded bases.
3°. E is isomorphically universal for the family of all separable Banach
spaces.
Proof. 1°=>3°. Assume that we have 1°. Then the Schauder basis of
C([0,1]) is equivalent to some subsequence of the universal basis of E,
whence, by Ch. I, § 8, theorem 8.1 d), C([0,1]) is isomorphic to the sub-
subspace G of E spanned by this subsequence. Since C([0,1]), whence also
G, is isomorphically universal for the family of all separable Banach
spaces, it follows that the space E is isomorphically universal for this
family.
The implication 3° => 2 is an immediate consequence of theorem 13.3.
2° => 1°. Assume that we have 2° and let {en} be a bounded basis of E
which is block-universal for the family of all bounded bases. Then, in
particular, the universal basis of theorem 13.1 is equivalent to a bounded
1 See e. g. [10], p. 185, theorem 9.
2 I.e., subspace spanned by a block basic sequence.

394
II. Special Classes of Bases in Banach Spaces
block basic sequence {zn} with respect to {en}. However, by Ch. I, § 7,
theorem 7.2, {zn} can be extended to a bounded basis of E, which will
obviously be universal for the family of all bounded bases. This completes
the proof of theorem 13.4.
Remark 13.6. From theorem 13.4 it follows, in particular, that the
space C([0,1]) has a basis {en} such that every bounded basis {xn} of any
Banach space X is equivalent to some suitable subsequence {e,J of {en}.
Consequently, if a separable Banach space X is not isomorphic to any
subspace [ein~] spanned by a subsequence {ein} of the above basis {en} of
C([0,1]), then X has no basis, giving thus a negative answer to the basis
problem; however, observe that the universal basis {е„} of C([0,1]) is
not explicitly constructed, we have only proved its existence.
It is natural to raise
Problem 13.3. Describe, up to isomorphisms, the subspaces spanned
by all infinite subsequences of the various usual bases in concrete Banach
spaces.
Since every infinite subsequence of a basis is obviously a block basic
sequence with respect to that basis, this problem is a part of problem
13.2. We also formulate the following weakening of the conjecture made
after problem 13.2: Except for spaces isomorphic to I2, in every Banach
space E with a basis {en} there exists a subspace X which is not isomorphic
to any subspace spanned by a subsequence of {en}. Observe that this is
certainly true for ? = C([0,1]) and {en} = the usual Schauder basis of
C([0,1]), since every subsequence of the Schauder basis of C([0,1])
contains either a subsequence which is a basic sequence equivalent to
the unit vector basis of c0, or a subsequence which is a basic sequence
equivalent to the basis {1,1,1,...}, {0,1,1,...}, {0,0,1,1,...},... of the
space c; indeed, the proof of this latter statement is similar to that of
§ 18, proposition 18.4.
Corollary 13.3. There exist a continuum of mutually non-equivalent
normalized bases which are universal for the family of all bounded bases.
Proof. By theorem 13.4 and by Ch. I, § 8, theorem 8.1 d), it is sufficient
to prove that there exist a continuum of mutually non-isomorphic Banach
spaces which are isomorphically universal for all separable Banach spaces
and which have bases. For each p with 1 <p < oo let Ep = C([0,1]) x lp-
Then obviously each Ep is isomorphically universal for all separable
Banach spaces (since it contains a subspace linearly isometric to C([0,1])-
Furthermore, since C([0,1]) and each lp have bases, each Ep has a basis
as well (by Ch. I, § 4, proposition 4.2). Finally, let us show that if рфг,
then Ep is not isomorphic to Er. For this purpose it is sufficient to show
that if рфг, then E has no complemented subspace isomorphic to Г
13. Universal bases. Complementably universal bases. Block-universal bases 395
(clearly the subspace {0} x Г of Er is complemented in Г and it is iso-
isomorphic to Г). Assume the contrary, i.e. that there exists a bounded
linear projection и of Ep=C([0, l])x lp onto some subspace G iso-
isomorphic to Г and let v be an isomorphism of Г onto G. Furthermore,
let u1 and u2 denote the natural projections ?p->C([0,1]) x @} and
Ep—>{0}x/p respectively, and let lr'n={x = {c,j}elr\^,l= ¦¦¦ =c,n = 0}.
Then the mapping u2v\lr,n:lr'"-*{0} х lp has no bounded inverse1,
whence there exist elements х„еЛ'" such that ||ы2и(х„)|| <—, ||х„|| = 1
и
(и =1,2,...). Observe that by the well known2 characterization of weak
convergence in Г, we have х„^0. Furthermore, the mapping
Mlc([o,i])x (of- C([0,1]) x [0]—>G = lr is weakly compact and hence3 it
takes weakly convergent sequences into convergent sequences. Since
х„^0, we have u{v{xn)^0, where {ut и(х„)}сС([0,1])х {0}, and
hence \\ии^{хп)\\-*0. Since \\u2v(xn)\\ < — for n= 1,2,... we also have
n
||мм2ф:п)||->0. Consequently,
whence, since v is an isomorphism, we obtain ||л„|| —>0, which contradicts
the condition ||xj = 1 (n= 1,2,...) above. This completes the proof
of corollary 13.3.
Let us consider now the problem of existence of universal bases for
other classes of bases. We have the following negative result:
Theorem 13.5. There exists no shrinking basis which is universal for
the family of all shrinking bases.
Proof. Assume the contrary, i.e. that there exists a shrinking basis
{en} of a Banach space E, which is universal for the family of all shrinking
bases. Let o^ denote the first uncountable ordinal number. We define
inductively a family {A'(,}0!ca<mi of Banach spaces as follows-. Xo = I2;
if a = b+\, then X ={Хьх12\\; if a is a limit ordinal number, then
Xa = ( y[ ХЛ , i.e. the space of all families of elements {xb}0^b<a
W 'l2
with xbeXb{0^b<a), \\{xb}\\ =|/~X \\xh\\2<co. Then, since the unit
vector basis in /2 is a normalized monotone unconditional4 basis, from
1 See e.g. [10], p. 205, theorem 7.
2 See e.g. [10], p. 137.
3 See e.g. [50], p. 494, theorem 4.
4 See §14, definition 14.1.

396
II. Special Classes of Bases in Banach Spaces
Ch. I, § 8, proposition 8.3, it follows (by transfinite induction) that each
Xa has a normalized monotone unconditional basis. Since all Xa are
reflexive, the bases are shrinking (see § 4, example 4.3) and hence, by our
hypothesis, equivalent to subsequences of {е„}. Therefore, by Ch. I,
§ 8, theorem 8.Id), each Xa@^a<a>1) is isomorphic to a subspace of
E, whence1 E* is non-separable, which contradicts the assumption that
{е„} is a shrinking basis of E. This completes the proof of theorem 13.5.
Problem 13.4. Does a similar result to theorem 13.5 hold for the
family 0$ of all boundedly complete bases?
Problem 13.5. Does there exist a universal basis for the family & of
all bases?
Problem 13.6. Does there exist a basis in Hilbert space I2, which is
universal for all bases of/2?
II. Unconditional Bases and Some Classes of
Unconditional Bases
§ 14. Unconditional bases. Conditional bases
Definition 14.1. A basis {л„} of a Banach space E is said to be un-
00
conditional if every convergent series of the form ? а;л,- is uncon-
i= 1
00
ditionally convergent, i.e., if for every xeE the series ? Mx)xi (where
i= 1
{/„} is the a.s.c.f. to {*„}) converges unconditionally. The basis {*„} is
said to be conditional if it is non-unconditional, i.e., if there exists a
00
convergent series ? а;л( which is not unconditionally convergent.
i= 1
For instance, the natural bases of c0 and lp(p ^ 1) are unconditional
bases. We shall see in §15 that the Schauder basis of C([0,1]) and
the Haar basis of Lx([0,1]) are conditional bases. Now we shall prove
that for 1</><оо the Haar basis of Lp([0,1]) is an unconditional
basis (theorem 14.1 below).
We recall that the Rademacher functions rk(t) on [0,1] are defined by
rk(t) = sign sin 2k n t (k=l,2,...).
A4.1)
1 By [254], p. 57-60.
14. Unconditional bases. Conditional bases
397
By the definitions of the Haar functions1 y^f) and of the Rademacher
functions rk(t) we have, obviously, the following expression of the
Rademacher functions with the aid of Haar functions:
rk(t) =
1
A4.2)
By A4.1) (or by A4.2) and the definition of Haar functions) we have,
for each к =1,2,...,
rk(t) =
B1-2 2/-Г.
1 for te(^-,^-) (/=!,...,2*)
V 2*
-1 for te[
Bl-\ 21'
\ 2k '2'
A4.3)
(/=1 2*)
Since in the dyadic representation t = — + — 4—- + ••• of the
2 22 23
points te[Q, 1] the points of the open intervals (—-
2" ' 2k
2/-1 21'
(/=1,...,2* ') are characterized by tk = 0, the points of
(/=l,...,2k) by tk=\, and the division points —k (J=l,...,2k) by
the fact that they admit two dyadic developments, one with tk = 0 (in
this case tk + 1 — tk + 2= ¦¦¦ = 1) and one with tk=\ (and tk+l = tk + 2
= ¦ ¦ • = 0), it follows that we have, for each к = 1,2,...
Г l if tk=o
] iij=J _1 if ft=i A4.4)
1 [ 0 if tk=0 and rt=l are both possible.
Since /"fe_1B?) = signsin2/l~1rc2? = signsin2/lrcf for O^t^j and
for у<г^ 1, we also have the following recursive formula for the Rade-
Rademacher functions:
r-l) for !
A4.5)
1 We now slightly change the definition of Haar functions, putting
( I'

398 II. Special Classes of Bases in Banach Spaces
We recall that the Walsh functions Wj{t) on [0,1] are defined by
where
A4.6)
A4.7)
In particular, taking k = 2 (n1 =0,1,2,...), we see that the Walsh
system contains all the Rademacher functions, i.e., it is an extension of
the Rademacher system.
Since the Walsh functions are defined by means of the Rademacher
functions and since the Rademacher functions satisfy the recurrence
formula A4.5), we also have the following recursive formula for the
Walsh functions, for each / = 2m+l, m=l,2,...:
wt(t) =
wl+1Bt) for
2
и;шB*-1) for i
2
for
A4.8)
A4.9)
Wl±1Bt)
2
}-wl±±Bt~l) for ?,
2
Indeed, since /is odd, we have / = 2"'+2+••• +2"vl+2°, whence,
/+1 /-1
by A4.6), A4.5), and
we obtain
+ l =
= 2"l~l+2~l
+2"
l + ¦¦¦
rniBt)rn2Bt)...rl,y_iBt)=wl±1Bt) for O^r^i
2
lrn,B/-l)rn2B?-l)...rnv_iB?-l)=w/ + iBf-l) for ?
and
2
rniBt)rniBt)...rnv l
for
-rni{2t-l)rn2{2t-l)...rllv_lBt-l)=-wl^{2t-l)
for ^
Let us also mention the following useful recurrence relation between
the Walsh and Rademacher functions:
(/=l,2,...,2k;/c = 0,l,2,...). A4.10)
14. Unconditional bases. Conditional bases
399
Indeed, writing /-1 =2 + --- + 2"v, where «2>-->nv^0, we have,
by A4.6),
Since the Walsh functions are defined by means of the Rademacher
functions and the Rademacher functions can be expressed with the aid
of Haar functions (see A4.2)), it is natural that the Walsh functions
can be expressed directly with the aid of Haar functions. Indeed, by
A4.10) and A4.2) we have
A4.11)
whence, since for 1 ^/^2* the function wt(t) is constant on each inter-
interval [-^TT' 2^] = (-^Г> 2^) (/=l,2,...,2k) (because writing /-1
= 2И1+---+2"\ where и1>--->и„^0, we have, by A4.6), wt(t)
= rn, + i@-rnv + i@' and here «!^fc — 1 by l^/^2/l, and thus
2k '2*
c, whence w; is constant on each ( . , ^)), and since
,-r), it follows that we can write
A4.12)
or in other words,
7T
2*
where the coefficient of y2u+ = is —-=r щ I I —т~, -т I I.
]/2k \\ 2 2 JJ
With the aid of A4.12), A4.8) and A4.9) one can also establish a
recursive rule for the coefficients + occurring in A4.13). Firstly,
1/2*
from A4.12) we have w3 = (Уз+jA w4 = ——-(y3— j4), which we
1/2 1/2
shall express by saying that w3, wA = -—-(y3+y4), with the matrix of

400
coefficients
II. Special Classes of Bases in Banach Spaces
1
1/2
1
1
1/2
1
A4.14)
1/2 1/2,
Similarly, by A4.12) we have w5, w6, w7, w8 =i(y5 ±yb+y1 ±j8)
with the matrix of coefficients
A4.15)
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1\
м
1
2
1
2
\l _1 _1
\2 2 2
The matrix A4.14) can be obtained from A4.15) by the following
rule: Denote by Bt the first row and by B2 the second row of A4.14),
(B \
1 I. Then A4.15) will be the matrix
B2)
l±
1/2 1/2
1 1
]/2 ]Д
B2
B2 —
B7
I
Actually, this shows also the desired general recursive rule for the
coefficients in A4.13), namely, if w2k-i+1, w2t-i + 2,..., w2k
-(>'2k-i + 1± ¦¦¦±y2k) with the matrix of coefficients
then w2u + l,w:
of coefficients
2,...,w2k + i =
A4.16)
• + !+¦¦¦ ±J2k + <)> with the matrix
14. Unconditional bases. Conditional bases
I—в —в \
1/2 V^Bl
1 1
401
B,
1
\
1^
wB2ty
A4.17)
Indeed, from A4.12) for / and /+1 and from A4.8), A4.9) it follows
that for odd / with l^/<2*, l^fc<oo, we have (taking also into
account that j2k + j = 0 on [0,1] for 2k~1 + l^j^2k and on [?,1]
for ^1
i-\-2k
J
and, since w^^ is constant on I ^—[, -j—^ (because from
-1
26 Singer, Bases in Banach Spaces I

402
II. Special Classes of Bases in Banach Spaces
it follows that 1
^2* '), we also have
1 2^'
and thus it remains only to observe that the sums
can be also written in the form
' 1
above
'ч-2k-' + ,-•
This completes the proof of our assertion concerning the recursive rule
A4.17) for the coefficients occurring in A4.13), putting
Bm =
1
(m=\,2,...,<
J=l 2"
We recall that a real matrix у4 = (а^) is called symmetric, if A = A'
( = the transposed matrix (aj()), respectively orthogonal if ЛЛ' = /. We
shall now prove that all matrices of coefficients
above are
orthogonal and symmetric. \ B2k
Indeed, the matrix A4.14) is obviously orthogonal and symmetric,
which shows that the assertions are valid for к = 1. Let к be an arbitrary
integer >2 and assume that the first assertion is valid for k — \, i.e., that
is orthogonal. Let us show that in this case the matrix A4.17)
is also orthogonal. By our hypothesis we have
B,
B2B[
k-l B\
1
0
0
0
1
0
... 0
... 0
... 1
i.e.,
14. Unconditional bases. Conditional bases
403
whence
which proves that A4.17) is orthogonal. Thus all matrices A4.16) are
orthogonal (k = 2,3,...).
Let us prove now that all matrices A4.16) are also symmetric. For
this purpose we recall that A4.16) is the matrix (b^), where
and thus we have to prove that blj = bjl (/J= l,2,...,2k~l), i.e.,
7-1 j W ((I-
w,
Let us write
<1418)
/— 1 =2"' + ¦•• +2"\ 7-1=2"" + ¦¦¦ +2"'".
where nx
•¦• >т„^0, and put J =
r
/-1
^r,
1 / \
ry' ^П") • Then we have to prove that
26»

404
II. Special Classes of Bases in Banach Spaces
The midpoint of J is
[y-i-i".. /lY
+ I - I ¦ Thus by A4.4), we have
= — 1 if and only if nx + 1 coincides with
one of the numbers k— 1 — mb...,/c— 1 — тц,к, and otherwise we have
rnx + j (J) = 1 A = 1,..., v). Let us observe that here nx + 1 = к is impossible,
since by l^2k~1 we have 2"' + ••• +2"v = /— 1^2k-1 — 1 =2k~2
+ 2k~3+---+2, whence nx^k-2 and thus n} + 1 <fc-l (/.= l,...,v).
Similarly, rm_1 + 1(L)= — 1 if and only if mx+l coincides with one
of the numbers k — l—n1,...,k—l—nv (the case mx+l=k being
again impossible), otherwise rmx+l(L)=l. However, it is obvious that
we have ид+1 = &—1—ma if and only if ma+l=k—l — nx. Hence
wt(J)= Wj(L), which completes the proof of the symmetry of the matrix
A4.16).
We have seen in Ch. I, § 2, that the Haar system is orthonormal on
[0,1]. We shall prove now that the Rademacher and Walsh systems also
have this property. This will follow from the next lemma, which we shall
use also later (in the proof of lemma 14.4):
Lemma 14.1. Let m1>m2>-->mli^l and let аь...,ад be integers
We have
1 if all (Xj are even,
0 otherwise.
A4.19)
Proof. The first relation is obvious, since if all a,- are even, then each
r*> (t), whence also the integrand, is = 1 a.e. For the proof of the second
relation we may assume that al = ¦ ¦¦ =ад= 1 (since for any even number
[1 and any j we have rpm.(t)=l a.e.). The product rm2(t)...rmj(t) admits
(by m2> •• >mll) 2mi dyadic intervals of constantness (each of length
—), the union of which is [0,1]. Any such interval J is divided into
2"""2 parts (each of length —), in which rmi(t) is alternatively
+ 1,-1. Consequently, j rmi(t)rm2(t)...r (t)dt is decomposed into
о
a sum of 2m2 terms of the form const. jrmi(t)dt, and each term is 0
j
since j rmi{t)dt = O (because of the above decomposition of J). This
j
completes the proof of lemma 14.1.
14. Unconditional bases. Conditional bases
405
Corollary 14.1. The Walsh system {wn} (whence also, in particular,
the Rademacher system {rn}J constitutes an orthonormal system on [0,1],
i. e.,
I fl for m = n
wm(t)wn(t)dt=< (m,n=l,2,...). A4.20)
о (О for тфп
Proof. The first relation is obvious, since w%(t)= 1 a.e. Assume now
that тфп. Then there exists at least one mk such that rmk occurs as
a factor in wm, but not in wn, or as a factor in wn, but not in wm.
Consequently, § wm(t)wn(t)dt is of the form A4.19) with ocfe=l, and thus
о
by lemma 14.1 it is =0, which completes the proof.
As we have seen in Ch. I, § 2, the Haar system1 {у„} is complete in
the spaces Z/([0,1]) for 1^/xoo (being even a basis of these spaces).
We shall now show that the Walsh system {wn} is also complete in these
spaces. Since both {yn} and {wn} are (finitely) linearly independent
(being orthogonal systems), by A4.12) we have [w2x + 1, w2« + 2,...,w2k + i]
= Ь'2к + 1->У2к + 2>--->У2**'] (because we have by A4.12) the inclusion с
and both subspaces are of the same dimension 2k by the linear independ-
independence of {yj}, {wj}). Taking into account that y\ = wl and again the linear
independence of {yj}, {wj}, it follows that [wn]j° = [jn]i0
A ^/> < oo), which completes the proof of our assertion.
(,
by
Define now {д„\, [hn\ <= Lp(|_0,l])* = Z?([0,l]) I
V
gn(x)= j x(t)yn(t)dt
о
o,-+-=l
p ч
A4.21)
hn{x)=\x(t)wn{t)dt
A4.22)
Then, since >>„, wneL«([0,l]) A <q^oc), we have gn,hneLF{[O,l])*
(K/7<oo). Since [>•„}, {wn} axe orthonormal systems, it follows that
({Jn}. {вп}) and ({wn}, {hn}) are biorthogonal systems. Let us denote
respectively by {sn} and {Sn} the sequences of partial sum operators
associated to these biorthogonal systems, i.e.,
A4.23)
i= 1
i= 1
1 From now on, we shall write yn instead of у„ and x instead of x. This will
not lead to any confusion.

406 II. Special Classes of Bases in Banach Spaces
Lemma 14.2. With the above notations, we have
s2k(x)=S2k{x) (xeZ/([0,1]), /c = 0,l,2,...). A4.24)
Proof Take an xeL"{[0,1]) and a positive integer к and put
1 J''-'
Then, by A4.12) (for k— 1 instead of fc), we have
1 = 1
2k-i
1=1
= I JV-'+j I И2к-1+1(х)Ьц.
Since the matrix F/j)i,j=i,...,2><-i is orthogonal and symmetric
(as shown in the above), we can express y2k..l+l (/= l,...,2fe) by
w2k-i+j, (/= l,...,2fe^1) with the same matrix that expressed w2k~i + l
by JV-'+j in A4.12) (for /c —1 instead of k), whence
g2k-t+j(x)= \x{t)y2k-t+j(t)dt= \x(t)\ X btjw2k-i+l(t) \dt
J J L/=i J
0
2k-l
= I Ьц1г2к-1+1(х),
and thus
J=l
A4.25)
for all xeL"([0,1]) and fe= 1,2,... On the other hand, we have
1 1
1 = w1 $x(t)wi(t)dt = y\ \x{t)yi(t)dt
о о
Hence, by A4.25) applied successively for fc=l,2,... we obtain
A4.24), which completes the proof of lemma 14.2.
14. Unconditional bases. Conditional bases
Let us introduce the notation
ki
407
A4.26)
Lemma 14.3. For p = 2v (v= 1,2,...) there exist positive constants mp
and Mp depending only on p such that
i i i
k=l
\x(t)\"dt^Mp
ле/Л[0,1]), $x(t)dt = 0
A4.27)
whenever one of the members exists.
Accepting, for the moment, lemma 14.3, we shall prove
Theorem 14.1. For 1 <p< 00 the Haar system {yn} is an unconditional
basis of the space Lp([0,1]).
1
Proof Let p = 2v and ле//([0,1], \x(t)dt = 0. By lemma 14.2 we
have 0
[2k-
Z
(the other products are =0 because of the definition of Haar functions).
Replacing this in A4.27), we obtain
\x(t)\"dt^Mp ^gf(x)yf(t)\2dt. A4.28)
.1 = 2 J J J Ll = 2 J
0 0 0
Now let xeLp([0,1]) be arbitrary. Then the element z = x — g1(x)yl
l
eL"([0,l]) satisfies $z{t)dt = O, gl{z) = gl(x) (/ = 2,3,...), whence, by
A4.28) applied to z, °
mP | | I&fWjfWr^^
E
(jceL'([0,1])).

408 II. Special Classes of Bases in Banach Spaces
Consequently, taking into account that a" + b" ^ (a2 + b2)z
+ b") and that $\g1(x)yl(t)\pdt=\gl{x)\"^ j\x(t)\"dt and
r _
denoting the constant
2>-\
again by mp, we get
+ 2)
и
gf(x)yf(t)\2dt
A4.29)
Applying this for sn(x) instead of x and taking into account that
д^^д^ (whence gi[sn(x)] =
obtain
for
=0 for/>и), we
g?(x)yf(t) Ydt
.1=1
M
Mp
\x{t)\*dt\ =-^|WI
т„
(xeZ/([0, !]),«= 1,2,...),
and thus, by Ch. I, § 4, theorem 4.1, {у„} is a basis of Z/([0,1]). By virtue
of A4.29) the same argument remains valid for any permutation {ja(n)}
of {yn} and therefore, obviously1, {у„} is an unconditional basis of
Z/([0,l]), for any p = 2v (v= 1,2,...).
Now, for any p with 2^p<cc, consider sn as a linear operator from
Z/([0,1]) into itself and denote its norm by \\sjp. Then, by the above,
for every even/? there exists a constant Cp^\ such that
A4.30)
\\sn\\p<Cp
(«=1,2,...).
1 11,
Assume now that 2vs=»<2v + 2. Then < — <—, whence
2v + 2 p 2v
there exists a (unique) Я with 0<Я^ 1 such that
1 1 1
/? 2v + 2 2v
1 If every permutation of a basis {х„} is a basis, then {х„} is an unconditional
basis (and conversely); indeed, see §17, theorem 17.1.
14. Unconditional bases. Conditional bases
409
whence, since by the Riesz convexity theorem1 фп( —) = \og\\s \\
1 1 W
is a convex function of — for 0 < — < 1, it follows that
P P
1
1
"\2v
and therefore
Thus, we have proved that A4.30) holds for any p with
whence, by Ch. I, § 4, theorem 4.1, {у„} is a basis of L"([0,1]) for any p
with 2^p<oo. Again, the same argument remains valid for any per-
permutation {ja(n)} of {yn} and therefore {yn} is an unconditional basis of
Lp([0,1 ]) for any p with 2 ^p < oo.
Finally, assume that l<p<2. Then, by the above, {#„} is an un-
unconditional basis of I?([0,1]), where —|— =1, whence {yn} is an
unconditional basis2 of Z/([0,1]), which completes the proof of theorem
14.1, provided that we prove lemma 14.3.
In order to prove lemma 14.3, we shall need several lemmas.
Lemma 14.4. Let ^?^([0,1]) and let ml,m2,---,mv be positive
integers, with m2,...,mv not necessarily distinct, such that
т^тахЦ.тз,...,»!,,). Then for Ak{x) defined by A4.26) we have
\\_Ami(x)-](t)[Am2(x)](t)...[Amv(x)-](t)dt = 0. A4.31)
о
Proof. By the definitions of Ak(x) and wn we have
+ b2™*{x)rmk{t)rmk_l(t)...r1(t).
1 See e.g. [50], p. 523, theorem 8.
2 See §17, theorem 17.7 and Ch. I, §12, corollary 12.2 (or, alternatively, the
final part of § 17, theorem 17.7 and Ch. I, §2, example 2.3).

410
II. Special Classes of Bases in Banach Spaces
Hence, when we express \_Ami(x)\(t),.:.,\_Amv{x)\{t) as linear com-
combinations of Walsh functions, none of the terms contains rmi(t) as a
factor, while in the expression of [zlmi(x)](/) all terms contain rms as a
factor. Consequently, applying lemma 14.1, we get A4.31), which comp-
completes the proof.
Lemma 14.5. For any t,9e[0,l~\ and k= 1,2,... we have
Wm(t)Wm{e)=
m=\
Proof. Let us denote the left side by a2k and the right side by nk.
Let k= 1. Then A4.32) becomes
which is true since by definition w1 = l, w2 = r1.
Assume now that A4.32) is valid for a positive integer k. Since by
A4.10) we have
it follows that
2k
2k
wl(t)wl(e)=rk+l(t)rk+1(e)G2k.
Consequently, taking into account the induction hypothesis, we get
i.e., A4.32) for fc+1. This completes the proof.
Corollary 14.2. For xeLl([0,1]), k= 1,2,... we have
= \х{в)гкA)гк(в) П {l + rjt)rm(9)}d9. A4.33)
J m= 1
14. Unconditional bases. Conditional bases
Proof. We have, by lemma 14.5,
П
= \х(в) П
J
m(/)г
which completes the proof.
Lemma 14.6. For 2^p^co and xeL"([Q, 1]) we have
k= 1 0
411
A4.34)
where for p = cc we take esssup|-| instead of \\-\pdt and sup
(€[0,1] f) 1 ^~ fc < GC
X>
instead of ? .
)c= 1
Proof. Let us first observe that for r # fc_ t, 0 # fc_ t (/= l,...,2t)
the function
M^)= П
A4.35)
vanishes except on a dyadic interval J of length fc_1, containing
f, and in this interval it takes the value 2*. Indeed, since rm= +1
(for the points ^ -t^y), there are two cases •. a) There exists an index /,
2s J

II. Special Classes of Bases in Banach Spaces
c-1, such that r,{t)= —r,(9). In this case, 0,@) = 0. b) There
exists no such /, i.e., rm(t) = rmF) (m= l,...,/c— 1). Then, since i\(t)
= i\{0), t and 0 are both in the same (open) dyadic interval oflength|(i.e.,
either in @,y), or in (y, 1)). Furthermore, since we also have r2(t) = r2F),
it follows that t and в must be in the same dyadic interval of length \.
Continuing in this manner, we see, finally (since also rk_l(t)=rk_l(&)),
that t and в are both in the same dyadic interval J of length
-j—^-. Now in J all l + rm(t)rmF) = 2 (m=l,2,...,k-l), whence 0,@)
= 2*~' in J, which proves our assertion.
Consequently, by A4.33) we obtain
\х{в)гк{1)гк{&) \[{\+rJt)rje)}dQ
m=l
= 2*-*
\x@)rk(t)rk{e)d0
¦l\\x@)\de
^ ess sup \x@) | «S ess sup \x(t)\.
fe[O,
whence
sup esssup|[Jt(x)](f)|^esssup|x(?)|. A4.36)
l«)c<=o fe[O,l] t?[0,l]
On the other hand, since by corollary 14.1 and the remark made
after this corollary, {wn} is a complete orthonormal system on [0,1],
we have
and hence
=l О
\\lAk(xj\(t)\4t\ = П \h2,<-l+j(x)\2\ =
J J
\hn(x)\-
=\\\x{t)\2dt
A4.37)
Now. let Lo = Lo([0,1]) be the linear subspace of Z/([0,1]) con-
consisting of all finite linear combinations of the Walsh functions wn and
let #TP =/EP([o, I» be the Banach space of all /?-summable (respectively,
bounded) sequences {xk}, where ^eLp([0,l]) (k= 1,2,...), ||{^}||Гр
(respectively, xteL°°([0,l]) for /c=l,2,... and
^ilWIf.onl
L=i k L ([0' j
14. Unconditional bases. Conditional bases 413
ll{**}ll.*-x= SUP И^И^ао.п))- Then, by biorthogonality, for every xeL0
rXi
the sum ? ||/lt(x)||p has only a finite number of non-zero terms, whence
{Ak(x)}e?Ip. Consequently, we can define a linear mapping u: Lo—>Stp
by x^>{Ak(x)}. Let ||м||р be the norm of this mapping, i.e.,
p
xeLo
хФО
= sup
= sup
xeLo
хФО
LI шх)г1
1-1 =L
Then, by the convexity theorem of M. Riesz for vector-valued func-
functions1, i/m — ) = log ||m|L is a convex function of — for 0^ — ^ 1.
W P P
P P
^1, it follows that
. Indeed, for any/? with 0^ — ^— there exists a
1 1 p p-2
such that — = Я-0 + A-Я)— (namely, A = - ). Con-
ConP 2 P
Since by A4.36), A4.37) we have
Iwith
sequently,
whence \\u\\p^\\u\\i\\u\\
l. Thus we can extend и by continuity
to й: Z/([0, l])->$p and we have A4.34), which completes the proof
of lemma 14.6.
Lemma 14.7. Let \<p<ao and let nit) be an arbitrary non-negative
integer-valued function of te[0,1]. Then there exists a constant Cp>0
depending only on p, such that
Proof. Put
fix{t) = ess sup — \xF)\de,
se@,l) 2S J
t-s
A4.38)
A4.39)
with the convention that x@) = O for 0^[0,1].
1 See, e.g., [50], p. 536, exercise 39.

414
II. Special Classes of Bases in Banach Spaces
Then, by a "maximal theorem" of Hardy and Littlewood [104]1,
there exists a constant Ap>0 such that
A4.40)
о о
On the other hand, by lemma 14.5 we have, for any k= 1,2,...
= \x(B) П {l + rJt)rje)}dO,
0 m= 1
and, as we have seen in the proof of lemma 14.6, the product \\ as
m=l
a function of в vanishes except on a dyadic interval J of length —^,
containing t, and in this interval it takes the value 2k. Consequently,
by A4.39) and since J <= I t --k, t + —k I,
A4.41)
Since the right side of A4.41) does not depend on k, we can apply
A4.41) for k = n(t), whence, taking also into account A4.40), we obtain
2'|[?х@]рЛ^2Мр|Мг)|"Л,
which, by putting Cp = 2"Ap, completes the proof of lemma 14.7.
Lemma 14.8. For l^/<co there exists a constant At>0 depend-
depending only on I, such that we have, for any aj,...,an>0,
Proof. Dividing both sides of A4.42) by ( ? ak I and denoting the
new variable —^— again by otj (/= 1,...,и), we see that the existence
1 For a shorter proof see, e.g., A. Zygmund [274], p. 30 — 32.
14. Unconditional bases. Conditional bases 415
of At above is equivalent to the existence of a constant С, > О such that
inf
«*,«*,-«*,+ I «'*U С. • A4.43)
J
Since this is obvious for /=1, we may assume that /> 1.
Let ?>0 be such that (/-l)^-<y. Put
Let а.ь...,а.„^О, ? «*=1. Then there are two cases.
a) max at>?. In this case for a suitable k0 with 1 ^ko^n we have
n
b) max ak<e. In this case there exist positive integers n^
/=0,1,...,/) with the following properties:
0 = ио<и1<---<и,_1<и, = и, A4.45)
T-4< ?' a^T + T (/ = 0,1,--.,/-2), A4.46)
«*¦
A4.47)
1 e "'
Indeed, there exists an nl with 0<nr<n such that ^ ^ at,
"-1 1 e "It
since otherwise Xafe<T~^' whence 1 = ^] ak<у-^г + а„, and
t=i /2 t=1 / 2
t, 1 ? /—1 e ? /-1
thus ?>а„>1— — — —¦ = — —, whence ^г>~т~, and thus
j>(/-1)|->( which implies 1>/-1, i.e., /-1=0, 1=1, in
contradiction to our hypothesis. Now, take the smallest such nv Then
If "' 1 ?
i hi ^
If 1 ?
a*^T + ^' since otherwise ^ ak > — + ¦—, whence
/2 I 2
k=l
+ ,
I 2
k=l
1 ? 1ё
= ^ at — ani > — + — —? = — — —, i. e., и, would not be the smallest
12 II
k=l
+
12
II

416
II. Special Classes of Bases in Banach Spaces
1 ? "'
positive integer such that y — —¦ ^ Z at- Now, if 1 = 2, then 1 =
' 2 k=l ;
"' " lc" l?"
= У at+ У a* ^—I 1- У a*., whence ^
fr = 1 fr — rt i 4- 1 ^- ^ t = и. 4- 1 ^ -^ Ь ~
? ab i. e.,
we have A4.47) for 1=2. On the other hand, if />2, then there exists
It
an и2 with n1<n2<n such that y — —¦
аь smce otherwise
V1 1 e , "^r 1 ? 1 e 2
X at<y--, whence Zaft<T + T + T~T=T' whence
 2 1 e
1 = Z aft < т + a"' anc* l^us fi>an> 1 —г' whence — > (/— 1) —
k= 1
у
-2)
, and thus 1 >
> ( )
2)
, which
> 2 V l) 21 ' """ " ' 2
implies (/—1)(/ —2) = 0, i.e., 1=1 or / = 2, in contradiction to our
hypothesis. Take the smallest such n2, etc. Continuing in this manner,
we obtain n3,...,n,_l satisfying A4.45), A4.46). Finally, if A4.47) is not
" 1 ? A /1 ?
satisfied, i.e., ? ak < (/— 1) —, then 1 = *""
1 e"' ' 1 el ?
+ у - (/-1)— = 1 h (/-1)—I (/- 1)— = 1, an impossibil-
impossibility, which proves that we also have A4.47).
Consequently, we have
z «„«„...«„+?«i>f z «*V z «*V-f z ««
which completes the proof of lemma 14.8.
Now we can give the
Proof of lemma 14.3. Let p = 2v. For simplicity, put
= Z Z
t=1 J=1
= Z I
A4.48)
i
(we recall that hl(x)= $x(t)dt = 0 by the hypothesis of lemma 14.3).
14. Unconditional bases. Conditional bases
Taking into account lemma 14.4, we have1
417
.*= l
p-2
Now, for 3^/i^p— 1 and a = we have, by the Holder in-
inequality,
l
* / 2(p-u) \ /
K+iFpn "dt = I \Д„+1Л Fpn "J\A
о о
пФ 1 rn M nn+ 1 I"'
Lo
p-2 E-P-/J
+l Г
i
a
Г 1
f
J
0
. о
1 р—ц . a p—
where we have put в = — = (indeed, =
a p-2 a-1 p-2
р-ц
= /»)• Since
? —2
2(р — и)\р — 2
(
whence ц -
\
1 For the sake of simplicity, we don't write the arguments x and t; this will
lead to no confusion.
27 Singer, Bases in Banach Spaces I

418
II. Special Classes of Bases in Banach Spaces
p — и
we have 0 < в = < 1 (since ц ^p —\<p and р — ц ^p — 3 <p — 2).
p-2
Hence, since aebl~e^a + b for a,b^0, OsgflsS I,1 and since /? = 2v, it
follows that
\AU,F*-»dt
о
Consequently, we have
A4.49)
Lo
A>+ldt
p(p-\)(p-2)
+p
_o
A>+ldt
p(p-\) p(p-\)(p-2)
where we have put A = 1 V ¦¦¦ + p. Putting
2 6
Fo = 0, the above inequality is also valid, obviously, for и = 0 (since
Fi = A it-
Summing for и = 0,1,2, ...,N and applying the Holder inequality,
we obtain (taking also into account that p is even)
1 Indeed, for - > 1 we have (— | < - < — + 1, while for T ^ 1 we have
b \b I b b b
/
. In
\
a\" a /a\e a
— I < 1 <-r+ 1, whence always U- < т + !. and thus <^bl
6/ b \b I b
our case, we can put j zl;+, Fpn~2dt = a, | zl^+ lE?/ = Z? (because the integrands are
>0 by p = 2v). ° °
14. Unconditional bases. Conditional bases
419
FpN+ldt
max
dt + Ap
Lo
max
Now, for each ;e[0,l] let n(t) be an integer, 0^n(t)^N, such that
|[S2,,.,(x)]@|p=[FB(,)(x)]''@= max {[FB(x)]"@}. Then, by lemma 14.7
0 ^n^N
applied for FN+1(x) instead of x (taking into account that by biorthog-
l
onality FJFN+l(x)) = Fn(x) for n = 0,l,...,N) we have f max \Fv\dt
^Cp\FpN+ldt, whence
_o
2 Г 1
Lo
_o
Denoting the left side by a and the factor of Ap in the second term
of the right side by b, this inequality becomes
whence
a * +Apb,
a[\-
ApCpb"a

420
II. Special Classes of Bases in Banach Spaces
Put в = —. Then
a
(АрСр
for
(ApCp+ApH" for
whence
and thus in any case
в > — = mini -
whence
l
Г / GO
Consequently, if I ? A\ )dt exists, then we have
0
1
1
A4.50)
A4.51)
0 0 0
Applying A4.51) to FNl(x) — FNl(x) instead of x, we get
l i
{FN2-FNydt^Mp
{N1 < N2), whence ||Fjy2 — FJVi ||LP-> x, and thus there exists (since Lp([0,1])
is complete) an element zeLp([0,l]) such that ||FjV-"!Ilj>->-0 as N-юо.
Then, by A4.51), we also have
pJ |Х„| dt.
о о
Furthermore, taking into account the continuity of the functionals
/?„ and corollary 14.1 (biorthogonality of \wn},{hn}\ we have
hn(z)=hn\lim [S2W(x)]{ = limhn{S2»(x)}
= limhA Yhj(x)w,>= limhn(x) = hn(x) (и = 1,2,...),
14. Unconditional bases. Conditional bases
421
whence, since [hn] is total on Lp([0,1]) (because {wn\ is complete in
L4([0,1]), where —|—= 1), we obtain z = x, and consequently1
i l
A2} dt,
J ln=\ J
о о
which is nothing else but the second inequality of A4.27), in the case
Let us prove now the first inequality of A4.27), in the case p = 2v.
Let N — 1 >«!>n2> ••• >и„_l. Then we have
X л- I dt
n= 1
+ 2
By virtue of lemma 14.4, the last two sums vanish (the first, by put-
ting Fni = Y, &m and observing that n is greater than all other indices,
m=l
and the second by observing that in each summand either m or n is
greater than all other indices). Consequently, omitting also the first in-
integral in the right side, we obtain
2.3.
1 Alternatively, this follows also from A4.51), lemma 14.2 and Ch. I, § 2, example

422
II. Special Classes of Bases in Banach Spaces
Summing over all possible combinations of positive integers
nun2, -..,«v_i satisfying N —l^nl>n2> " >nv_1? we get
I
JV - 1 ^ Hi > • • ¦ > nv- i
iV "I v- 1
о
A4.52)
1
dt=\FU ?/1„2 2 Л.
Ln= i J
о о
On the other hand, we have, by the biorthogonality of {>>„}, {hn},
N 2k~ '
and thus
h,{x) for /=1,2,..., 2N
0 for l=2N+l,2N + 2,...
1+j{x)w2k-l+j = Ak(x) for /c=l,...,.
7=1
0 for /c = J
whence, applying lemma 14.6 for Fjy(x) instead of x, we obtain
Fldt.
A4.53)
Denoting by S the left side of A4.52) and taking into account lem-
lemma 14.8 (with a.k=[Ak(x)]2(t),n = N,l = -=v) and A4.52), A4.53), we get
S+ X \Aldt
l i
IV 'I P-2
k= 1
Lo
14. Unconditional bases. Conditional bases
423
P P
Applying the Holder inequality with the exponents — and
2
we see that this is
-2
p-2
p-2 1
N Лц р
+\F'Ndt
J
о
Hence, putting a= < ^ A\Ydt, b= Ffidt, we have
?
о
bpa p +b ,
and thus, similarly to the proof of A4.50), we obtain
where yp>0, whence, applying lemma 14.7 with n{t) = n,
(N=1,2,..
Consequently, the series ? zi^ converges almost everywhere and
k=i ,
satisfies the first inequality of A4.27), with mp = , in the case
Yp Cp
p = 2v. This completes the proof of lemma 14.3. Thus, the proof of
theorem 14.1 is now complete.
Let us now turn our attention to conditional bases.
In finite dimensional Banach spaces all bases {х„} are unconditional.
It is natural to ask whether the converse is also true, i.e., whether this
property characterizes finite dimensional Banach spaces among Banach
spaces with bases. We shall see in § 23 that the answer to this question
is affirmative. In the present section we shall show only that in some
concrete infinite dimensional Banach spaces which have unconditional
bases, there also exist conditional bases.
Example 14.1. In E = c0 the sequence
*„= {!,..., 1,0,0,...} («=1,2,
A4.54)
is a conditional basis of E.

424
II. Special Classes of Bases in Banach Spaces
Indeed, by Ch. I, § 4, proposition 4.3, {xn} is a basis of ?. Furthermore,
if {е„} denotes the unit vector basis of E = c0, we have
m + p m + p i
I *ix, = I «i I
i-m i=m j = 1
m + p
J=l
m + p
whence
= sup
1 + Р
Consequently, ? а;х; converges if and only if ? a; converges and
thus {х„} is a conditional basis of ? (e.g., by § 16, lemma 16.1, equivalence
This assertion also follows from the fact that {xn} is a basis of type P*,
taking into account § 17, corollary 17.1b).
Example 14.2. In ? = /* the sequence
А1 = {1,0Д-}. Ая={0,...,0,1,-1,0,0,...} (« = 2,3,...) A4.55)
п-2
is a conditional basis of ?.
Indeed, we have seen in Ch. I, § 13, example 13.3, that {hn\ is a basis
of ?. Furthermore, if {/„} denotes the unit vector basis of /', we have,
for any
m+p
m+p
m+p—1
whence
m + p
m + p— 1
= «„
Consequently, У а,/г, converges if and only if
— a,-|<oo.
i= 1
In particular, for <xn = — (и= 1,2,...) it follows that the series ? <X;A,-
converges but ? (—1)'а,-А; diverges and thus {An} is a conditional
i= 1
basis of ? (by § 16, lemma 16.1, equivalence l°<s>4°).
The assertion of example 14.2 also follows from the fact that {hn+l}
is the a.s.c.f. to the basis \xn} of c0 defined in example 14.1, taking into
14. Unconditional bases. Conditional bases
425
account § 17, theorem 17.7, or from the fact that {«„} is a basis of type P,
taking into account § 17, corollary 17.1a).
Let us consider now the spaces Z/([0,l]) and lp(l^p<ao).
Let l^p<co and let rn(t) be the Rademacher functions A4.1) on
[0,1]. We recall1 that by the Khinchin inequality there exist two con-
constants Ap>0 and Bp>0 such that we have, for any scalars аь...,ап
I °
in the particular case when p=\, one can take A l =|.
A4.56)
Lemma 14.9. a) Le; ? xt йе аи unconditionally convergent series in
k= 1
space Lp([0,1]), w/гете l^/?<oo. Then there exists a constant
C=CpjXni such that
k= 1
b) Le?
\xk(i)\ \dt^C (и=1,2,...). A4.57)
xk be an unconditionally convergent series in the space V,
*=i
I
C>0 such that
={^)}^'= l {k= 1,2,...). Г/геи
a constant
A4.58)
Proof, b) Since ? xt is unconditionally convergent, there exists,
k= 1
by § 15, corollary 15.1, a constant M>0 such that
I rk№k
I rk(t)xk
M (re[0,1], и=1,2,...),
A4.59)
where rk are the Rademacher functions. Hence, putting in the left side
of the Khinchin inequality A4.56) afc=^*' (k= 1, ...,и), summing over i,
and applying the theorem of Lebesgue, we get
1 Seee.g. [117], p. 131—132.

426
II. Special Classes of Bases in Banach Spaces
l
I #
dt
(AP)P
z<
k=l
dt
(А„Г
k= 1
dt
Mp
= С
which completes the proof of b). The proof of a) is similar, using the
theorem of Fubini.
00
Lemma 14.10. Let Z xk be an unconditionally convergent series in
k= 1
the space Z/([0,1]) or V, where l^p^2. Then
Z
A4.60)
Proof. It is sufficient to consider one of these spaces, e.g., /p, since
the proof for the other is similar. For any scalars pu...,Pn we have,
by the Holder inequality and by lemma 14.9,
z
k= 1
n oo
= Z Zl
k= 1 i= 1
00 П
2 \2-p
2
Consequently, for any sequence of scalars {$„} with Z |/У2~р< аэ
theseries ^ ||^llp/?)t is convergent, whence, by the theorem of Landau1,
k=l
we have ? |jxt||2= ? (||xt|]p)p<co, which completes the proof.
1 See, e.g. [10], p. 86.
14. Unconditional bases. Conditional bases
427
Proposition 14.1. In the spaces Lp([0,1]) and I", where l
every bounded unconditional basic sequence is Besselian. Consequently,
in the spaces Lp([0,1]) and I", where 2^p < oo, as well as in the space c0,
every bounded unconditional basis is Hilbertian1.
Proof. Assume that {xn} is a bounded unconditional basic sequence
in Lp([0,1]) or V, where l</?<2, and let {а„} be an arbitrary sequence
00
of scalars such that ? а(хг converges. Then, since {xn} is an uncondi-
i= 1
ОС
tional basis of [xB], ? <Х;Х,- is unconditionally convergent, whence, by
i= 1
lemma 14.10 and since {xn} is a bounded basis of [xn], we obtain
Zkl
i= 1
and thus {xn} is Besselian.
Assume now that
? = LP([O,1]) or /", where
rem 17.7, the a.s.c.f. {/„}
quence in ?*, where E* =
respectively. Hence, since
tion of proposition 14.1
[/„]<=?*. Consequently,
{xn} is a Hilbertian basis
sition 14.1.
inf
1 «n« oo
is a bounded unconditional basis of
2^p<cc, or of ? = c0. Then, by §17, theo-
<= ?* is a bounded unconditional basic se-
1?(Г0,П) or I", with - + -= 1, or E* = l\
P 4
\<q^,2, by the above proved first asser-
it follows that {/„} is a Besselian basis of
by § 11, theorem 11.1b) (implication 6°=>1°),
of ?, which completes the proof of propo-
Corollary 14.3. In the spaces Lp([0,1]) and /p, where l<p<2, there
exists no bounded unconditional Hilbertian basis. In the spaces Lp([0,1])
and I", where 2<p<ao, there exists no bounded unconditional Besselian
basis2.
Proof. If there existed a bounded unconditional Hilbertian basis of
Щ[0,1]) or I", where l<p<2, or a bounded unconditional Besselian
basis of Lp([0,1]) or I", where 2<p<ao, then by proposition 14.1 above
and §11, corollary 11.2, {xn} would be equivalent to the unit vector
basis of I2, whence Lp([0,1]) or V would be isomorphic to /2, which is
not true for рФ2. This completes the proof.
1 We mention that the spaces L\[0,1]) and C([0,1]) have no unconditional
basis (see § 15).
We recall that /' has no bounded Hilbertian basis and c0 has no bounded
Besselian basis (by § 11, corollary 11.1).

428 II. Special Classes of Bases in Banach Spaces
Now we can give
Example 14.3. In ?=1Р([-я,я]) A</>#2) the sequence {х„}^=0,
where
-тг,я],и = 1,2,...), A4.61)
is a conditional basis of ?.
Indeed, this follows from corollary 14.3 above, taking into account
§11, example 11.1, according to which A4.61) is a bounded Besselian
basis of E for p>2 and a bounded Hilbertian basis of ? for l<p<2.
An example of a conditional basis of ? = /p A<рФ2) can be ob-
obtained from §23, proposition 23.2 and its proof, taking into account
§21, proposition 21.5 and § 18, theorem 18.3.
Although Hilbert spaces have "the best" geometric properties among
all Banach spaces, the construction of conditional bases in separable
Hilbert spaces appears to be more difficult than in the other concrete
Banach spaces with unconditional bases. We shall give below two
different ways of constructing such bases.
Example 14.4. Let E = L2{[~n,n~\) and let 0<?<^. Then {*„}?,
where1
and {yn}o, where
-я,я],« = 0,1,2,...), A4.62)
-я,я],« = 0,1,2,...), A4.63)
are conditional bases of E.
Indeed, we have seen in §11, example 11.2, that {xn}l' is a non-
Hilbertian bounded basis of E and {yn}% is a non-Besselian bounded
basis of ?, whereas by proposition 14.1 above every bounded uncon-
unconditional basis of ? must be both Hilbertian and Besselian.
Since the proof of the assertions of § 11, example 11.2, whence also
the proof of the fact that the sequences A4.62) and A4.63) are conditional
bases of ? = L2([ — я, я]), leans heavily on analytic tools, it is natural
to raise the problem of finding a simple geometric construction of con-
ditional bases in E = l2 of the form х„=
Z "Ч:(и= 1,2,...), where {е„}
is the unit vector basis of I2. i=l
Moreover, with the Gram-Schmidt process we can obtain from any
conditional basis {х„} of I2 or L2([0,1]) an orthonormal basis {zn} of
the form zn= ? a.fxh aw#0(«= 1,2,...), whence *„= ^"'^Й'^О
Here we use / as ]/ — 1.
14. Unconditional bases. Conditional bases
429
(и =1,2,...). Since {zn} is orthonormal, there exists a linear isometry и
of I2 or L2([0,1]), respectively, onto /2, such that u(zn) = en (и =1,2,...),
whence the sequence
(«=1,2,...)
A4.64)
i= 1
is a conditional basis of I2. Thus, there arises the problem of finding ex-
explicitly a conditional basis of /2 of the form A4.64). (Let us observe that
one cannot obtain the desired basis by applying the above procedure
to the basis {х„}? or [yn}% of example 14.4, since in this case the «<¦"', $">
above involve integrals which are not computable.)
A solution to this problem is given by
Example 14.5. The sequences {х„} с I2 and {>>„} с /2 defined by
!,„_, =<
Z 'xi-n
x2n = e2
(и =1,2,...) A4.65)
i= 1
A4.66)
where {е„} denotes the unit vector basis of I2 and а„>0 (и=1,2,...),
со х- 1
Y, jtf < oc, Z a= oo (e.g., one can take <xB = ), are conditional
j=i j=i n\ogn
bases of I2.
Indeed, for any finite sequence of scalars fiu ...,P2n we have
" / J
GO / П
whence
2n
j = n+ 1 \k= 1
2 n
+ z
e2h
i
k= 1
A4.67)
k=l

430 И. Special Classes of Bases in Banach Spaces
Since by the Holder inequality we have
Z
Plk-l(x-j-k + 1
Z Z Kl
j = n+ 1 \k = j-n+l / i=\
7=2
i — 1
GO П
Ш«72 Zl
7=1 i=l
it follows that for any finite sequence of scalars jS^ ..., jSm with m>2«
we have
Z ft*;
7=1 / 7=1
7=1
7
z
7=1 / 7=1
Similarly, for any finite sequences of scalars jS^ ..., /?m with m > 2и -1
we obtain
Z PjXj
7=1
Z ^
7=1
Consequently, by Ch. I, §7, theorem 7.1, {xn} is a basis of E = l2
yGC
1 + X iaj2)- Let us also observe that {х„} is a bounded
oasis, since J~l
\\x2n-i\\ = ^1 + Z a72' 11*2.11 = 1 (и=1,2,...). A4.68)
' 7=1
On the other hand, by A4.67) we have
7=1
7=1
+ z
j = n + 1
z
7=1
n
2 + У
7=1
n
l^k
1 k= 1
7
Z fe-l«
= 1
7
У i3
k= 1
l«j-t+l
2
— *+ 1
1 a j - к + 1
2 n
^ Z l^
7=1
14. Unconditional bases. Conditional bases
431
1
whence, in particular, for P2j-i = (j=l> ¦¦¦>n) we obtain, taking
n 2
into account that by our hypothesis
z
7=1
Since
^,?
7=1
7=1
7
Z a7-t
oo as и—>oo,
oo as n—>oo.
*=
= 1 (и=1,2,...), from §11, theorem 11.1b)
(implication 1°=>4°) it follows that {xn} is not Hilbertian, whence, by
proposition 14.1 above, {xn} is a conditional basis of E = l2.
Observe now that {xn,yn) is a biorthogonal system, since for all
m, и =1,2,... we obviously have
\
_1 = 5n
(Х2п-иУ2т)=[е2п-1 + Z
Z ( ~ «m - i
,— i +e,
i+Ue2i-lT"e2m
0 if m < n
= 0 if
Hence, by Ch. I, § 12, corollaries 12.2 and 12.1, {>>„} is a conditional1
basis of E* = I2, which completes the proof of the assertions of example
14.5.
Note that {>>„} is obviously different from the basis of the form
A4.64) obtained by starting with the basis {xn} defined by A4.65) (i.e.,
applying the Gram-Schmidt process to {xn}, etc.).
The question answered by example 14.5 above also suggests the
following more general problem-.
1 We again use the observation that a basis is unconditional if and only if
every permutation of this basis is a basis. Alternatively, one could observe that {>'„}
is non-Besselian, whence, by proposition 14.1, conditional; or, alternatively, one
could use § 17, theorem 17.7.

432
II. Special Classes of Bases in Banach Spaces
Problem 14.1. a) Let {en} be an unconditional basis of a Banach
space E. Does there exist a conditional basis {х„} of E of the form
i= 1
b) What about a conditional basis of the form
A4.69)
A4.70)
Example 14.5 shows that the answer is affirmative if E = l2. The
answer to problem 14.1 a) is still affirmative if E = c0 or I1, by examples
14.1 and 14.2 above and by § 18, theorem 18.2.
§ 15. Some separable Banach spaces having
no unconditional basis
In the present section we shall show that some concrete separable
Banach spaces (among which are, in particular, the important spaces
C([0,1]) and ^([0,1])) have no unconditional basis. Some other
separable Banach spaces having no unconditional basis will be given
in § 17.
Let us first recall that a series ? xi in a Banach space E is called
i= 1
a(E, E*)-unconditionally Cauchy or weakly unconditionally Cauchy, if
00 00
E \f(xi)\<<X) f°r a^ feE*- A series ? /; in a conjugate Banach
¦=1 >=1 oo
space E* is called a(E*,E)-unconditionally Cauchy if ? \ft{x)\<co for
all xeE.
00
Lemma 15.1. For a series ? /; in a conjugate Banach space E* the
;= l
following statements are equivalent:
ОС
1°. Y, fi is o(E*,E)-unconditionally Cauchy.
i= 1
2°. There exists a constant C>0 such that
(xeE).
A5.1)
i= 1
3°. There exists a constant C>0 such that
(a; = 0 or 1, /=!,...,«; «=1,2,...). A5.2)
15. Some separable Banach spaces having no unconditional basis 433
4°. There exists a constant C>0 such that
n
? ?;/; <C (?,= ±1, *=1,...,«; «=1,2,...). A5.3)
i=l
5°. There exists a constant C>0 such that
E ^-//
С
1, i=l,...,л; л=1,2,...). A5.4)
00
6°. ? /f is o(E*, E**)-unconditionally Cauchy.
i= 1
Proof. The implication 1° => 2° follows applying the uniform bound-
edness principle1 to the sequence of continuous non-linear functionals
{р„}, where
PnW = t \fi(*)\
, /1=1,2,...).
Assume now that we have 2° and let j8f with
arbitrary. Then
E ft/i
i= 1
whence we obtain A5.4). Thus 2° => 5°.
The implication 5° => 3° is obvious.
The implication 3° => 4° follows from
n
E ?.x<-
i= 1
n
E «л
i= 1
+
n
E aixi
i= 1
and xeE be
where а;=1,а,' = 0 if ?;=1 and a, = 0, aj=l if e;=—1.
Assume now that we have 4° and let ФеЕ** be arbitrary. Then,
putting ei = signRe<P{fi) if ЯеФ(/,)^0 and ?;=1 if ЯеФ(/;) = 0, we
have ?;= + 1 (/= 1,2,...) and, by 4°,
?|Re*(/f)l= X'
i = 1 i=l
( ^
\i = 1
ф Е ^/i
.>= 1
ЦФЦ
E«./«
<С||Ф|| (« = 1,2,...).
1 See e.g. [50], p. 53, lemma 13.
28 Singer, Bases in Banach Spaces I

434 II. Special Classes of Bases in Banach Spaces
00 CC CO
whence ? |Re<?>(/,)|<oc. Similarly, ? |1тФ(/;)|<оо, whence ? |Ф(/;)|
i = l i =1 >=1
ОС ОС
«с ? |ЯеФ(/;)|+ X |1тФ(/;)|<оо. Thus, 4°^6°.
i = 1 i = 1
Finally, the implication 6°=>1° is obvious, which completes the
proof of lemma 15.1.
Corollary 15.1. For a series ? xf in a Banach space E the following
statements are equivalent: l = i
1°. Y, xi 's weakly unconditionally Cauchy.
i= 1
2°. There exists a constant C>0 such that
3°. There exists a constant C>0 such that
<. = 0 or 1, г=1,...,я; и=1,2,.
4°. There exists a constant C>0 such that
(?.= +l, i=\,...7n; n=\,2,...).
5°. There exists a constant C>0 such that
\, i=l,...,/I;/I=l,2,...).
A5.5)
A5.6)
A5.7)
A5.8)
Proof. This follows by embedding ? in ?** and applying lemma 15.1
in ?**.
Now we can give
Theorem 15.1. Let E be a separable Banach space containing a sub-
space Eo with ?§ weakly complete1 and поп-separable (e.g., a subspace
Eo isomorphic to C([0,1]),). Then E has no unconditional basis.
Proof. Assume that E has an unconditional basis {х„} with the a.s.c.f.
{/„}c?* and put
4>„ = /„к («=1,2,...). A5.9)
1 Actually, it is sufficient to assume that E* contains no subspace isomorphic
to c0 (by lemma 15.8 below).
15. Some separable Banach spaces having no unconditional basis
435
Let феЕ$ be arbitrary and let feE* be an extension of ф to ?.
GC
Then, since {х„} is an unconditional basis of ?, the series ? f(x^j\
i= 1
is <r(?*,?)-unconditionally convergent to/(by Ch. I, § 14, formula A4.8)),
whence, by lemma 15.1, there exists a constant M>0 such that
i = 1
CO
Consequently, by corollary 15.1, the series ? /(*,)</>; is cr(?J, ?g*)-
i = 1
unconditionally Cauchy, whence, since E% is weakly complete, this
series is <r(?g,?J*)-convergent to а феЕ*, i.e.,
On the other hand, as observed above, we have
= ?/(*,)/,(*) = /(*) =
(*eE0),
and hence, considering the functionals Фх(у) = у(х) (ye?J), we obtain
= ф. Thus, the sequence < X!/(xi)^i( converges to ^> for c(E%,E%*).
i= 1
Since феЕ% was arbitrary, this proves that E% is separable for <r(?g,?J*),
whence1 also strongly separable, which contradicts the hypothesis. This
completes the proof of theorem 15.1.
We shall say that a Banach space ?0 has property (p) if every sub-
subspace В of E* such that for each фоеЕ% there exists a sequence {/?„} с В
satisfying
х) (хеЕ0), A5.10)
ИтФ(Рп) exists for each ФеЕ%*,
A5.11)
(i.e., Рл^>ф0 for <r(?g,?0) and {j8n} is a Cauchy sequence for a(E%,E%*))
is non-separable.
Theorem 15.2. Let E be a separable Banach space containing a sub-
subspace ?0 which has property (p) (e.g., by lemma 15.6 below, a subspace
Eo isomorphic to 1/([0, l~\)). Then E has no unconditional basis.
1 See e.g. [10], p. 134, theorem 2.
28*

436
II. Special Classes of Bases in Banach Spaces
Proof. Assume that E has an unconditional basis {*„} with the
a.s.c.f. {./„}<=?* and put
Фп = /п\е0 («=1,2,...), A5.12)
B= [<?„]. A5.13)
Then В is a separable subspace of ?g and we shall show that for each
фоеЕ% there exists a sequence {/?„}сВ satisfying A5.10) and A5.11),
in contradiction with our assumption that Eo has property (p), which
will complete the proof of theorem 15.2.
Let фоеЕ% be arbitrary and let /oe?* be an extension of ф0 to E.
30
Then, since {*„} is an unconditional basis of E, the series ? fo(xi)fi
i= 1
is <r(?*,E)-unconditionally convergent to /0 (by Ch. I, §14, formula
A4.8)), whence the sequence {/?„} с В defined by
Pn= ?fo(x№i («=1,2,...) A5.14)
satisfies A5.10). Furthermore, by the above proof of theorem 15.1, the
да
series X!/o(xi)^i 's ff(?o,-E**)-unconditionally Cauchy, whence {/?„}
i= 1
also satisfies A5.11). This completes the proof of theorem 15.2.
Now we shall prove that a Banach space Eo isomorphic to L\[0,1])
has property (p) (lemma 15.6, below). We shall denote by v the Lebesgue
measure on [0,1].
Lemma 15.2. Let В be a separable subspace of L°°([0,l]). Then there
exists a perfect set Tc[0,l] with v(T)>0, such that every equivalence
class /?eB contains a function a with cc\T continuous on T.
Proof. Let {/}„} be a countable dense set in В and let 0<e<l. By
the theorem of Luzin there exists a decreasing sequence of closed sets
г1ЭТ2^Г3^---
such that
A5.15)
(«=1,2,...), A5.16)
and that the equivalence class /?„ contains a function а„ with an\Tn con-
continuous on Tn («= 1,2,...). Put
To=
A5.17)
15. Some separable Banach spaces having no unconditional basis
437
Then To is closed, and thus1 we can write T0=TuS, where T is
perfect and S is finite or countable. We shall prove that T also has the
other required properties. By A5.16) we have
2 4 2"
whence, by A5.15) and A5.17),
(/1=1,2,...),
A5.18)
Now let fie В be arbitrary. Then, since {/?„} is dense in B, it has a
subsequence {/?nJ such that ||/?Вк-/?||->0 as /c^oo. By the above,
there exist functions ot.nke[ink with anjr^ continuous; then, since ТсТЛк,
the function a |r is also continuous (fc= 1,2,...). Since2 ||а„к-/?||->0
as /c->co and since T is perfect, {а„к|г} is a Cauchy sequence in C{T),
whence it converges to an аоеС(Г). Putting а = а0 on T and a= any
function of the class fi on [0,1 ]\T, and taking into account that T is
perfect, we obtain
<тах(||ао-а„
к||С(Г),
as
whence a = j6, i.e., ae/?, which completes the proof of lemma 15.2.
Lemma 15.3. Let M be a v-measurable subset of [0,1], with v(M)>0.
// v(/nM)>0 for some closed interval /c[0,l], then there exists а
compact set QcM, nowhere dense in* M, such that v(InQ)>0.
Proof. Let {tn}™=l be a dense sequence in InM. Then, by the
regularity of the Lebesgue measure, there exists a compact set
Q с (/ n М)\{?„}„х= 1 such that
Since Q is compact, it is closed. Furthermore, since Q is disjoint
from the dense subset {tn}™=l of InM, it is nowhere dense in InM,
whence (since Q <= / nM) also nowhere dense in M, which completes
the proof.
1 See e.g. [175], Ch. II, § 6, theorem 4.
2 We denote by a the equivalence class of the function a.
J I.e., nowhere dense for the relative topology induced by [0,1] in M.

438
II. Special Classes of Bases in Banach Spaces
Lemma 15.4. Let T be a v-measurable subset of [0,1], with v(T)> 0.
Then there exists a compact set T*<^T with v(T*)>0, such that if
vGnT*)>0 for a closed interval I с [0,1], then v(Gn T*)\g)>0 for
every compact subset QcT* which is nowhere dense in T*.
Proof. Let v|r be the restriction of v to T and let
T* = S(v\T) = the carrier of
A5.19)
We shall prove that T* has the required properties. Obviously,
v(T*)>0.oAssume that vGnT*)>0 for^ a closed interval /c[0, 1].
Then v{In T*) = v(In T*)>0, where I denotes the interior of 7.
Hence 1г\Т*фф and consequently, GnT*)\Q^0 for any compact
subset QcT* which is nowhere dense in T* (because GnT*)\Q = 0
would imply 7 n Т* с Q, whence Q would not be nowhere dense in T*).
On the other hand, G n T*)\Q is open in 7n T*, whence also in T*,
and GnT*)\gcT* = S(v|7,)- Consequently, by the definition of the
carrier, v(Gn T*)\Q)>0, whence also v((/n T*)\Q)>0, which com-
completes the proof.
Lemma 15.5. Let T be a v-measurable subset of [0,1], with v(T)>0.
Then there exists a bounded measurable real function a0 such that ao(t) = O
for te\_0, l]\T and that ao\T is not equivalent to any function belonging
to the first Baire class on T.
Proof. Let T* be as in lemma 15.4. Obviously, it will be sufficient
to construct an a0 on T* which is not equivalent to any function of the
first Baire class on T*.
Let {/„} be the sequence of all closed intervals 7„с[0,1] with
rational endpoints, such that vGnn T*)>0.
By lemma 15.3 with M=T*,l = ll, we can choose a compact
set 4t с Т*, nowhere dense in T*, such that v{IlnAl)>0. Then,
since by hypothesis \>(IlnT*)>0, by virtue of lemma 15.4 we have
v((IlnT*)\A1)>0. Since 7! nCT*^) = G! n T*)\AU we can apply
lemma 15.3 with M=T*\AU I = IU to obtain a compact set BlcT*\Al,
nowhere dense in T*\Al7 whence also in T*, and such that vG1nB1)>0.
We can continue this procedure to obtain two sequences of compact
sets {А„}, {#„} in T*, with the following properties:
a) An and В„ are nowhere dense in T* (n= 1,2,...);
b) Ап
juBj)j = 4) («=1,2,...);
с) v(InnAn)>0, v(InnBn)>0 («=l,2,...).
15. Some separable Banach spaces having no unconditional basis
439
Indeed, assume that A1,B1,..., An, Bn have been already con-
n
structed. Then, by a), \J (A^Bj) is nowhere dense in T*. Hence, by
lemma 15.4 and the hypothesis vGn+1 nT*)>0, we have v (I,+ 1nP)
ijvjBj) l>0. Since 7„+,nI T*\у Ц-uВ,)j = G„+1nT*)\\J(AjVjВД
л
we can apply lemma 15.3 with M=T*\ {J(Aj<jBj), 7 = 7„+1, to obtain
a compact set Л„ + 1 с Т*\ У (/4,-и ВД nowhere dense in T*\ (J (/4,-иВД
whence also in T*, and such that v(In+i nintl)>0. The construction of
Bn + 1 is similar, starting with the nowhere dense at /4n + 1u IJ(^ uB,-) .
This completes the induction. v=i /
Put
4 = \jAr A5.20)
Then Л с Т* is v-measurable and v(^)>0. We claim that for every
closed interval 7 с [0,1] with I n T* Ф 0 we /гаие
v(/n4)>0, v(In(T*\A))>0. A5.21)
Indeed, since vGn T*)>0, there exists an index n such that 7„с7.
Hence InnAndnAncInA, and thus, by c), we have the first ine-
/ a, \
quality of A5.21). Furthermore, by b) we have BnnA = Bnnl [j Aj\ = 0,
whence В„ с T*\A, whence 7„ n В„ с 7 n В„ с 7 n (Т*\Л), and thus, by
c), we also have the second inequality of A5.21).
Now we shall complete the proof by showing that the characteristic
function a0 of the set A has the required properties. This function is
obviously bounded and measurable and %0(t) = 0 for /e[0,1]^T.
Finally, since for every closed interval 7 с [0,1] with 1пТ*фф the
oscillation of a0 on 7 n T* is equal to 1 and cannot be diminished by any
change of the values of a0 on a set of measure 0 (by A5.21)), ao|r, is not
equivalent to any function belonging to the first Baire class1 on T*,
which completes the proof.
Lemma 15.6. The space 7_^([0,1]) (whence also every Banach
space Eo isomorphic to 7^([0, lj)) has property (p).
1 Indeed, it is well known (see e.g. [175], Ch. XV, § 3, theorem 3) that if у is
of the first Baire class on T*, then for any closed set Q <= T* the function y\Q
must have at least one point of continuity.

440
11. Special Classes of Bases in Banach Spaces
Proof. Let В be a subspace of1 ^([0,1]), such that for every
фоеП°([0,l~\) there exists a sequence {/?„} с В satisfying A5.10) and
A5.11) (with ФеЬх([0,1])*) and assume that В is separable. Choose
Гс [0,1] as in lemma 15.2 and for this T choose a0 as in lemma 15.5.
For фо = аоеП°([0,1]) let {/}„} <= В be a sequence satisfying A5.10) and
A5.11). Since by lemma 15.2 every fteB contains a function a with
a|reC(T), for each teT we can define a functional ф,еВ* by
ф,(Р) = ф) (fie В), A5.22)
where ae/?, a\TeC(T); this functional is well defined, since the relations
al,a.2eP, a^j-, %2\TeC(T) imply a1|r = a2|r, and it belongs to B*
since, T being perfect,
\ф,(Р)\= \a{t)\ < max|a(OI < ess sup|a(t')l= W\\ W^B).
ГеТ felO.l]
Consequently, by the Hahn-Banach theorem, for each teT there
exists a functional Ф,е1/°([0,1])* such that
A5.23)
where ae/?, a|reC(T). Then, by A5.11), there exist the limits
(feT), A5.24)
where а„е/?„, а„|геС(Т) (и= 1,2,...); furthermore, again by A5.11), we
have
sup ||^ || < oo. A5.25)
Now, let xel^fp), 1]) be an arbitrary element such that ?@ = 0
(re[0,l]\T, ?ex). Then, by A5.10), A5.24), A5.25) and the Lebesgue
theorem on integration of sequences of functions, we have
= lim/?„(*) = lim
whence, since c,\T is an arbitrary summable function on T, it follows
that ао|г is equivalent to the function lim а„(.)|г belonging to the first
Baire class on T, in contradiction with our choice of a0. This completes
the proof of lemma 15.6.
1 We identify (canonically) the spaces L°°([0,l]) and L'([O,l])*.
15. Some separable Banach spaces having no unconditional basis
441
Consequently, as we observed above, a separable Banach space E
containing a subspace Eo isomorphic to 1^([0,1]) has no unconditional
basis1. In particular, for the space ? = L'([0,1]) itself one can give
another simpler proof, as shown by
Theorem 15.3. Let E be a weakly complete separable Banach space
which is not isomorphic to any conjugate Banach space (e.g.2, E = L1 ([0,1]))
Then E has no unconditional basis.
Proof. Assume that E has an unconditional basis {х„} with the a. s. с f.
{/„} с Е*. We claim that {xn} must be boundedly complete. Indeed,
assume the contrary, i.e., that there exists a sequence of scalars {aj such
that sup
< oo and that ? aixi does not converge. Then,
since {х„} is an unconditional basis, for any ?,= +1 we have, by § 17,
theorem 17.1, sup
>= 1
С sup
l=Sn<oo
< oo, whence
by corollary 15.1, the series ? aixi is weakly unconditionally Cauchy.
i= 1
However, this series does not converge weakly to an element yeE, since
then one would have, by biorthogonality, fk(y) = lim fk У д,х, = ак
(fc=l,2,...), whence j = E fk(y)xk = E й*хь 'n contradiction to the
00 «C=l «C=l
assumption that E flfcxt does not converge. Consequently, E is not
k=1 f " ]
weakly complete (the weak Cauchy sequence < E ^-хЛ does not con-
u=i J
verge weakly to any element yeE), which contradicts our hypothesis.
This proves that {х„} is boundedly complete, and hence, taking into
account Ch. I, § 3, proposition 3.2, it follows that Eo is isomorphic to the
Banach space of sequences of scalars < {а„} с К
endowed with the norm |[ {а„} || = sup
sup
< oo
. However, by Ch. I,
§ 12, theorem 12.5 c), this latter space, whence also E, is isomorphic to the
conjugate space [/„]*, which contradicts our hypothesis. This completes
the proof of theorem 15.3.
1 Let us observe that the same assertion remains valid if we replace L'([0,1])
by any space L}(Q, v), where Q is a compact metric space and v a non-purely
atomic measure defined on the Borel sets of Q.
2 See [73], p. 265 and [46].

442
II. Special Classes of Bases in Banach Spaces
A Banach space E is said to have property (u) if for every weak Cauchy
sequence {zn} <= Е there exists a sequence {jn} с Е such that
00
a) the series ? yi is weakly unconditionally Cauchy;
b) the sequence < zn — ? yt > converges weakly to 0.
Our next aim is to prove
Theorem 15.4. Let E be a separable Banach space containing a sub-
space Eo which does not have the property (u) (e.g., by corollary 15.4
below, a subspace Eo isomorphic to the space J o/§ 4, example 4.1). Then E
has no unconditional basis.
For this purpose we shall need some auxiliary results. Let us prove
first a result on selection of basic sequences.
Proposition 15.1. Let {х„} be a basis of a Banach space E, with the
a.s.c.f. {/„} <= E*. If a sequence {yn} <= Е satisfies the conditions
inf ||jn||=?>0,
1 ^ n < oo
lim/jOg =0 (/=1,2,...),
A5.26)
A5.27)
then [yn] has a subsequence {yPn+l} which is a basic sequence, equivalent
to a block basic sequence with respect to {xn}.
Proof'. Let C=vlXn}= sup \\sn\\. By A5.27) we can construct two
1 $П< 00
increasing sequences of positive integers {/>„}, {qn} such that
AC
?
AC
I
1
(/1=1,2,...),
Indeed, take P, = l- Since
2" + 2
1
"^ лн+ 2 ^ ,,•••/¦
fi(yi)xi=yi> there exists a
A5.28)
A5.29)
such
that
I Я
AC-23
. Assume that we have already con-
constructed Pi,...,pn, ql,...,qn. Then, by A5.27), there exists арл+1 such
that \f(yPn+1)\<
(i=\,...,qn), whence we get
4C-2"
I l
i= 1
15. Some separable Banach spaces having no unconditional basis 443
4C-2"+2
i.e., A5.29). Since ? fi(yPn+l)Xi converges (to jPn+1), there also exists
i= 1
a qn+l such that we have A5.28) for n+l instead of и, which proves
our assertion.
Put
Z*= I /iO^.)*! («=1,2,...).
A5.30)
Then, by A5.26), A5.28), A5.29) and by C^ 1,
whence
4C-2
?
<2 +
. Tn+2
I fiiy^JXi
у («=1,2,...).
A5.31)
Consequently, for the sequence {hn} a [zn]* of coefficient func-
tionals associated to the block basic sequence {zn} we have, by Ch. I,
§ 3, formula C.8) and Ch. I, § 7, corollary 7.4,
(/1=1,2,...).
A5.32)
On the other hand, again by A5.28) and A5.29), we have
<
4C-2" + 2 4C-2"+3 4C-2"H
whence, taking into account A5.32), we obtain

444
II. Special Classes of Bases in Banach Spaces
Consequently, by Ch. I, § 10, theorem 10.1, {ул„ + 1} is a basic sequence,
equivalent to the block basic sequence {zn}, which completes the proof
of proposition 15.1.
Corollary 15.2. a) Let {х„} be a basis of a Banach space E, with the
a.s.c.f. {/„}<=?*. If [zn\ с E is a sequence with sup llzJIoo,
satisfying
Hm/l(z11) = 0 (/=1,2,...), A5.33)
and if zn-/->0 as n —> oo, for the weak topology a(E, E*\ then {zn} has
an infinite subsequence which is a basic sequence of type al+, equivalent
to a block basic sequence with respect to {xn}.
b) If {xn} is an unconditional basis of E, with the a.s.c.f. {/„} с Е*,
and if {zn} с Е is as in a), then {zn} has a subsequence which is a basic
sequence equivalent to the unit vector basis of I1, and which is equivalent
to a block basic sequence with respect to {xn}.
Proof, a) By our hypothesis, there exist an feE* and an infinite
subsequence {znj of {zn} such that1
inf |/(znJ|>0. A5.34)
Then
inf
1
inf |/(zJ|>0 and, by A5.33),
i«k«," "k" || f
II J и
lim/;(znJ = 0 (/=1,2,...), whence, by proposition 15.1, {znj has a
subsequence {znk } which is a basic sequence, equivalent to a block basic
sequence with respect to {xn}. By A5.34) and § 10, theorem 10.3 (implica-
(implication 2° => 1°), the basic sequence {znkj is of type al+.
b) If {xn} is an unconditional basis of E, the basic sequence {znkj
found in part a) above is unconditional, by § 17, corollary 17.2. Conse-
Consequently, by §17, corollary 17.1, {znkj is equivalent to the unit vector
basis of /', which completes the proof.
Corollary 15.3. Let {xn} be an unconditional basis of a Banach space E,
with the a.s.c.f. { /„} <= E*. If {zn} с ? is a weak Cauchy sequence in E,
satisfying A5.33), then zniE>0.
Proof. If zn -/*0 as n —> oc, for the weak topology a(E, E*), then,
by corollary 15.2, {zn} has a subsequence, say {znp}, which is a basic
sequence equivalent to the unit vector basis of I1. However, the unit
vector basis in I1 is not a weak Cauchy sequence (to see this, take e.g.
the functional defined by {1,0,1,0, ...}ет = A1)*), whence {znp} is not a
1 If the scalars are real, we can also find {znj с {zn} and feE* with
inf f(z )> 0, whence every basic subsequence {znk } of {znk} will be of type /+.
^ П < GO m
15. Some separable Banach spaces having no unconditional basis
445
tf([znj> IX,,]*)-Cauchy sequence and therefore (by the Hahn-Banach
theorem) not a <r(?, ?*)-Cauchy sequence, in contradiction with the
hypothesis that the whole sequence {zn} с ? is a weak Cauchy sequence.
This completes the proof.
The hypothesis that {xn} is an unconditional basis in corollaries
15.2b) and 15.3 is essential, as shown by
Example 15.1. Let E = c0 and let
(«=1,2,...)
A5.35)
where \en} is the unit vector basis of c0. Then, by § 14, example 14.1,
{х„} is a conditional basis of c0. Furthermore, ||zn|| = l («=1,2,...),
\zn) satisfies A5.33) (by biorthogonality) and zn/»0 as и->оо, for the
weak topology <r(c0, eg), since for the functional f(x)=^1 (х={?„}ес0)
we have f(zn)=l (n= 1,2,...). However, {zn} has no basic subsequence
equivalent to the unit vector basis of I1 (since c0 has no subspace iso-
morphic to I1). Finally, {zn} is a weak Cauchy sequence, since for
f(x) =
(x={Qec0), where
we have lim f{zn)
Proposition 15.2. Every Banach space E with an unconditional basis
{х„} has property (u).
Proof. We may assume, without loss of generality, that ||х„|| = 1
(n=l,2,...). Let {/„}c?* be the a.s.c.f. to {xn} and let {zn} с Е be
a weak Cauchy sequence. Then the limits
exist. Put
a, = lim f(zk) (/=1,2,...)
п = <ХпХ„=\ Mm. fn(zk)\ xn («=1,2,...).
Lit
A5.36)
Since {xn} is an unconditional basis of E, for every feE* the series
E f(xi)fi 1S <r(?*, ?)-unconditionally convergent to / (by Ch. I, § 14,
formula A4.8)). Consequently, by corollary 15.1, for every feE* there
exists a constant Mf > 0 such that
I Pif(Xi)fi
Mf («=1,2,...;

446
whence
i= 1
II. Special Classes of Bases in Banach Spaces
= 11ЖК1
lim ? [signf{xi)ai~]f{xi)fi(zk)
i= 1
My sup
sup II zfc||
<oo (feE*, «=1,2,...),
(since {zB} is a weak Cauchy sequence), and thus ? |/(y,-)l< °° for all
feE*, i.e. ? j; is weakly unconditionally Cauchy. Furthermore, we
have
- i y)=/)(z») - ? Liim /<(
= fj(zn) - lim /,.(zt) ^ 0 as n - a» (/= U, ¦ ¦ ¦),
and ta — ? j.-f is a weak Cauchy sequence (indeed, {zn} is a weak
I i=i J г , i
Cauchy sequence by our hypothesis and < ? j;> is a weak Cauchy
sequence since we have shown above that ? yt is even weakly uncondi-
i= 1
tionally Cauchy). Consequently, by corollary 15.3, the sequence
¦„ — Yj Ух \ converges weakly to 0, which completes the proof of
proposition 15.2.
Lemma 15.7. Every subspace Eo of a Banach space E with property (u)
has property (u).
Proof. Let {znj с Eo be a <r(?0,?J)-Cauchy sequence. Then, since
/|Eoe?* for all feE*, {zn} is also a <r(?,?*)-Cauchy sequence, whence,
since E has property (м), there exists a sequence \yn] a E such that the
series ? j, is a(E, ?*)-unconditionally Cauchy and that the sequence
i=l
{(;„} is cr(?,?*)-convergent to 0, where
A5.37)
15. Some separable Banach spaces having no unconditional basis 447
Then one can define inductively an increasing sequence of integers
0=po<Pi<p2< •• and a sequence of scalars {An} with Я„^0,
i.= l («=1,2,...), such that
1
where
"n= I a,-^ («=1,2,...).
A5.38)
A5.39)
Indeed, since г;„->0 for c(E,E*\ by a well known theorem of
S. Mazur1 there exists a sequence of finite convex combinations of the
elements vn which converges to 0 in the norm topology, whence there
pi
exist a positive integer^ and scalars lb ..., Xpi 5=0, ? A;= 1, such that
i=1
Я,-!),
1
—; the elements u1,...,un being constructed, м„+1
= Y, h l\ 's obtained in a similar way from the fact that the sequence
{iy, + fc}?=i converges to 0 for a(E,E*). Put
(«=1,2,...),
(«=1,2,...).
A5.40)
A5.41)
Then, since {zn} с ?0, we have {wn} с ?0, whence {j°} с ?0. We
ас-
shall show that the series ? j/? is <r(?0,?;!;(-unconditionally Cauchy
and that the sequence <zn — ? _v?> is <т(?0,?*)-сопуегёеп*
will complete the proof.
We have, by A5.41), A5.40), A5.37) and A5.39),
>'п=н'„+1-н'„ =
Pn+l /I
= Z ч
> = р„ + 1 V = 1
> = Pn - 1 + 1 \J = 1
1 See e.g. [50], p. 422, corollary 14.

448
where
II. Special Classes of Bases in Banach Spaces
Pn + 1 1 Pn '
= V A. V v. - У А,- У у-А
La * La " J La *¦ La J}
La * J ' J rt + 1 "!
Ai - Z я;=1-1=0 for
*i - E A = l - Z ^ for
X Я( for Pn+l
and thus /^ = 0 (Uj^.-Д 0<ju"<1 (р„.1+1</<^+1). Hence,
oo
taking also into account A5.38) and that J] J; is<r(?,?*)-unconditionally
Cauchy, we obtain, for every feE*,
ftl = Z
n=l
\j=l
Z Z
L
Zi/K+
n= 1 11=1 "=1
00
Consequently, by the Hahn-Banach theorem, ^ \ф(у°)\ < °о for all
oo
i$, i.e., ^ >>° is <r(E0,?g)-unconditionally Cauchy.
Furthermore, since {zn} is a <T(?0,?g)-Cauchy sequence, for every
феЕ% there exists the limit Vimф{г„) = Ф{ф). Let ?>0 be arbitrary,
and choose N = М(е,ф) so that Ф(ф) — к^ф{гп)^Ф(ф) + е for
Then
i = р„ - i + 1
15. Some separable Banach spaces having no unconditional basis
449
whence Мтф(н>п) = Ф(ф) (феЕ$) and thus
-w»:, =г„ —
con"
i = 0
verges to 0 for o(E0,E%), which completes the proof of lemma 15.8.
The main assertion of theorem 15.4 follows now from proposition
15.2 and lemma 15.7. In order to prove that the space E0 = J of §4,
example 4.1, does not have property (u), we need
Lemma 15.8. // a Banach space E contains no subspace isomorphic
to c0, then every weakly unconditionally Cauchy series in E is uncondition-
unconditionally convergent in the norm topology.
GO
Proof. Assume that there exists a series ? y. in E which is weakly
i=l
unconditionally Cauchy but not strongly unconditionally convergent.
Then for some permutation a of the set of all positive integers the series
CO
? ya(i) is not convergent, whence there exists an increasing sequence
i=l
O=qo<g1<q2< '" of integers such that
Put
inf
.= Z У Hi) ("=1,2,...).
A5.42)
A5.43)
Then, since ]T yi is weakly unconditionally Cauchy, the series
oo i= 1
Yjzn 's <r(?0,?g)-unconditionally Cauchy, where ?0 = [zJ, whence
n=l
{zn} converges to 0 for a{E0,E%); furthermore, by A5.43) and A5.42),
inf ||zJ|>0. Since E0 is separable, it can be embedded into a Banach
1 ^П< 00
space with a basis {xn} (e.g., into C([0,1]), by the Banach-Mazur theo-
theorem1). Consequently, by proposition 15.1, {zn} has a subsequence {zPn}
which is a basic sequence. We shall show that {zpn} is equivalent to the
unit vector basis of c0, whence [zpn] is a subspace of E, isomorphic to
c0, which will complete the proof. oo
If {<*„] is a sequence of scalars such that ? anzPn is convergent,
n=l
then limanzpii = 0, whence, by inf ||zJ>0, we have liman = 0. Con-
П-»00 1^П<00 П-»00
versely, assume that {<*„} is a sequence of scalars such that liman = 0.
1 See e.g. [10], p. 185, theorem 9.
29 Singer, Bases in Banach Spaces I

450
II. Special Classes of Bases in Banach Spaces
Since ? zpn is <r(?0,^-unconditionally Cauchy, by corollary 15.1
we have
whence
N + k
n = N
sup ? \ф{гРп)\ = С<со,
= sup
JV + fc
,n = JV
SUP 2. \Ф(гРп
sup
С sup |а„|->0 as N->oo
NiniN+k
c=l,2,...),
and thus, since Eo is a closed subspace of the complete space E, ? anzpn
is convergent to an element of Eo, which completes the proof. " = 1
Corollary 15.4. The space J does not have property (u).
Proof. Since J** is separable, J contains no subspace isomorphic
to c0, whence, by lemma 15.8, every weakly unconditionally Cauchy
series in J is strongly unconditionally convergent. Assume now that J
has property (к) and let {zn} be an arbitrary weak Cauchy sequence in
oo
J and {yn} a sequence in J such that ? yt is weakly unconditionally
Cauchy and \zn— ? Jjf converges weakly to 0. Then, by the above
00 ^ I = 1 J
remark, ? yt is strongly unconditionally convergent to an element
'¦=1 л
yeJ. Hence, taking also into account that zn— ? j,—>0 for o(E,E*),
we have, for each feE*, i = 1
/(;„)-/
i= 1
+ 11/
i= 1
>0 as и
>oo,
i.e., }zn} converges weakly to y. Since {zn} has been an arbitrary weak
Cauchy sequence in J, it follows that J is weakly complete, which is
impossible (e. g., since1 J* is separable and J non-reflexive). This completes
the proof of corollary 15.4.
Consequently, as we have observed above, a separable Banach space
containing a subspace Eo isomorphic to J has no unconditional basis.
Again, for the space E = J itself one can give other simpler proofs, e.g.,
from § 23, remark 23.1 and § 17, corollary 17.3 it follows that if a non-
reflexive Banach space E has no subspace isomorphic to I1 or c0 (we
See e.g. [43], Ch. Ill, § 4, remark G).
15. Some separable Banach spaces having no unconditional basis
451
have observed in the preceding that the space J satisfies these conditions),
then E has no unconditional basis.
Remark 15.1. Each of theorems 15.1, 15.2 and 15.4 obviously implies
that a separable Banach space E containing a subspace Eo which is iso-
morphically universal for all separable Banach spaces, has no unconditional
basis.
The examples of Banach spaces having no unconditional basis, given
above, have the property that they cannot be embedded into any Banach
space with an unconditional basis. It is natural to ask whether there
exists a Banach space having no unconditional basis, which can be
embedded into a space with an unconditional basis. We shall now show
that the answer is affirmative, namely, that the space I1 (which has an
unconditional basis) contains a subspace having no unconditional basis.
To this end, we need
Lemma 15.9. Let E,F be Banach spaces such that E contains no
subspace isomorphic to F and that there exists a bounded linear operator
и of E onto F, and let {Fx}xsA be a set of subspaces of F, directed by
inclusion, such that F= {JFX. Assume that for every a there exists a
meA
bounded linear operator vx:Fx-^E such that
u[vx(y)~]=y (jeFJ, A5.44)
IklKx, A5.45)
for some A>0 independent of a. Then the space Eo = u~1@) (with the
norm induced by E) is not isomorphic to any conjugate Banach space.
Proof. It is sufficient to prove that Eo is not complemented in any
conjugate Banach space, since if Eo is isomorphic to a conjugate space,
then there exists1 a bounded linear projection ?§*—>?0, whence Eo is
complemented in the conjugate space (?g)*.
Assume now, a contrario, that Eo is complemented in a conjugate
Banach space B*. We shall prove that in this case there exists an iso-
isomorphism v of F into E, in contradiction with our hypothesis, which
will complete the proof. For this purpose it will be sufficient to prove
that there exists a bounded linear mapping v. F—>•? such that
u\y{y)~\=y (yeF), A5.46)
since then v is an isomorphism (indeed, v is one to one, because v(y) = 0
implies у = иv(y) = м@) = 0 and v~l is continuous on v(F), since
vb>n)-*v(y) implies yn = u\y(yn)\->-u\y(y)\=y).
See e.g. [33], p. 122, exercise 16a).
29.

452
II. Special Classes of Bases in Banach Spaces
Since и maps E onto F, there exists a mapping ф-.F-^E (not neces-
necessarily linear or continuous) such that
ФШ=У
A5.47)
\\Ф(у)ЫчЫ (yeF), A5.48)
for a suitable t]>0 independent of у (indeed, by the open mapping
theorem there exists an i)>0 such that for every yeF one can find an
xeE with u(x)=y, \\x\\^ri\\y\\ and hence it is sufficient to take ф{у) = апу
such x).
Furthermore, since u^(y)-vx(y)~]=y-y = 0 for all yeFx (by A5.47)
and A5.44)), we have
A5.49)
Let us consider the product space
yeF
(where SB, = the unit cell of B*, endowed with the weak* topology
a(B*, B)). By the theorem of Tychonov, П is compact.
Let us assign to every aeA the point лхеП defined by
0
if
if y$Fx.
A5.50)
(This nx belongs indeed to Я, since by A5.49) we have nx{y)=vx(y)
-ф(у)еЕосВ* for yeFx and since by A5.45) and A5.48) we have
\К(У)\\ = \К(У)-Ф(У)\\^\КЬ>)\\ + \\ФЬ>)\\^ + П)\\У\\ for yeF.).
Let y^eF^, y2eFar Then, since {Fx}xeA is directed by inclusion,
there exists an aeA such that Fxl,FX2czFx, whence y1,y2,y1+y2eFa.
Therefore, by A5.50), we have
(/=1,2),
)=vab>i +У2)-ФЬ>1 +У2),
whence, by the linearity of va, we obtain
-ф(у1)-ф(у2)+Ф(У1+}'2)= -
A5.51)
Now, since П is compact, the net {na}aieA has a cluster point л.
Consider the neighborhood V(n) of n defined by
V(n)={n'en
|[яХУ,-)]М-|Ж>](*)|<? ('=1,2),
\[nl(y1+y2i](x)-[n{yi+y2)-]{x)\<e},
15. Some separable Banach spaces having no unconditional basis 453
where xeE and e>0 are arbitrary. Then, since я is a cluster point of
{лх}хеА, for every aeA there exists an a'eA with cc'^cc, such that
nx,eV(n), whence, by A5.51) (which also holds for every cc'^cc), we get
- [Ф(Уг +У2Шх)\ = |[я(у1)
\ < 3e.
Consequently, since xeE and e>0 were arbitrary,
)- A5.52)
Now, since Eo is complemented in B*, there exists a bounded
linear projection/? of B* onto Eo. Hence, by A5.52), the linearity of vx
and A5.49), we get
Consequently, for the mapping v0: [j FX-^E defined by
we have
voiyi +У2) =р[л(У1 +У2
A5.53)
i.e., v0 is linear. Furthermore, by p[n{y)\eE0 = u x@) we have
и(р\к(у)]) = 0 (ye \J FX\ whence, taking also into account A5.47),
=J (ye\jFa).
Finally, by A5.48) and the inequality Hj)K(/i-H)IMI (yeF)
(because пеП), we have
+ \\Ф(У)\\^\\Р\\\\П(У)\\+П\\У\\
and hence u0 can be extended by continuity to a bounded linear map-
mapping v.F^E satisfying A5.46), which completes the proof of lemma 15.9.

454
II. Special Classes of Bases in Banach Spaces
Now we can prove
Theorem 15.5. The space I1 has a subspace Eo (e.g., Eo = u~l@),
where и is any continuous linear mapping of I1 onto 1^([0,1])) which is
not isomorphic to any conjugate Banach space. Hence this space Eo has
no unconditional basis.
Proof. Let F = Ll{[0,l~]), let A be the set of all partitions of [0,1]
into a finite number of disjoint measurable sets, and for each aeA let
Fa be the subspace of ^([0,1]) spanned by the equivalence classes of
the characteristic functions of the sets in the decomposition. Then the
set {Fx}xeA is directed by inclusion and we have F= \JFX. Now let и
be an arbitrary continuous linear mapping of E = ll onto1 F = L1([0,l]).
Then by the open mapping theorem there exists an j/>0 such that for
every yeL1 ([0,1]) one can find an xel1 with u(x)=y, \\х\\^п\\у\\. Conse-
Consequently, for every a = {e{?\ ..., e(?}eA we can define a continuous linear
mapping vx:Fa^E = ll by
1
(/=1,...,#0
A5.54)
where v denotes the Lebesgue measure, xeo denotes the characteristic
function of the set eY\ and where x\x)eE — ll are such that
1
v(e\
М\
A5.55)
Then for every
Pi .
E 7r
i=l v(ei
we obviously have
E М"
n I \P\
i= 1
= П
Pi
i= 1
i.e., A5.44) and A5.45) with к—ц. Finally, E = ll contains2 no subspace
isomorphic to F = Lx([0,1]), and thus all conditions of lemma 15.9
1 Such mappings exist (see [12], theorem e) or [133], p. 283, theorem 1).
2 See e.g. [10], Ch. XII, theorem 1.
15. Some separable Banach spaces having no unconditional basis
455
are satisfied. Consequently, by this lemma, the space E0 = u l@) is not
isomorphic to any conjugate Banach space.
Since Eo is weakly complete (being a subspace of/1), by theorem 15.3
it follows that Eo has no unconditional basis, which completes the proof
of theorem 15.5.
One can also construct explicitly a subspace Eo of Z1 as in theorem
15.5. Indeed, let {xn} be the unit vector basis of Z1 and put
«(*2" + *-i) = 2"Zrj_ *+!] (fc = 0,l,...,2"-l; n=l,2,...), A5.56)
1.2- 2-J
where, as before, %e denotes the characteristic function of the set e <= [0,1].
Then we can extend и by linearity and continuity to a mapping
u: I1 ->Ll([0,1]) with ||m|| = 1, since for any finite sequence of scalars
^i,...,in we have
I \i<\ \\u(x,)\\ = E \ii\ =
We claim that и maps I1 onto Lx([0,1]). Indeed, let jeSti([0>1]) be
arbitrary1. Then there exists a sequence {у„} cL^O, 1]) of the form
2"-l
Уп = Е акЛ)Х[к k+il (и=1,2,...) such that lim \\y-yj =0 in
k = o L^'^^J
L}([0,1]); we may assume (omitting, if necessary, a finite number of
2--1 j
the Л) that \\yj^2. Then for wn = E -<$)х2Я+к.1е11 {"=1,2,...)
к = 0 l
we have \\u(wn)\\= \\у„\\^2 (и=1,2,...) and u(wn)=yn-+y as n^ со,
whence ye2u(Sti). Thus, 2m(S,i)=> Sli([0,X]p whence it follows2 that и
maps/1 onto ^([0,1]).
Now we shall prove that the subspace ?0 = м~1@) of I1 has the
following sequence {zn} as a monotone basis:
("= 1,2, ...)¦
A5.57)
Indeed, let us first observe that zneE0 (n = 1,2,...) and thus [zn] с Ео,
since
2^2-1
2+l
=0-
2n+1 2"+l
A5.58)
1 We recall that SB denotes the unit cell {yeB | \\y\\ sj 1} of the Banach space B.
2 See e.g. [263], p. 198, lemma 1 or [91], Ch. I, § 14, lemma.

456 II. Special Classes of Bases in Banach Spaces
Let us prove now the inclusion Eo с [zn]. Assume that for an
x = E Zixiell we nave м(х) = 0, and let ?>0 be arbitrary. Then
there exists an index N = N(e,x) such that
2N+l_2
X-
(=1
A5.59)
whence, by м(х) = 0 and ||м|| = 1,
2N+i_2
«I E «,*«
и(х-
A5.60)
By the definition A5.57) of {zn} one can find, by induction, scalars
a1,...,a2w_2, /?2N-n •••)/?2N + i-2 such that
У ?,х,- У atzt= У
i — 1 i = 1
Indeed, taking a1 = <^1, we have
1— 2
A5.61)
2N+1-2
i= 1
where /.3 = ^3-1^!, A4 = ?4-i?i and Я, = ^ for the other i>2. Fur-
Furthermore, if we have already found scalars tn1, ...,<xk, fik+i,..., /12~ + '-2>
where k^2N-3, such that
2N+i_2 k 2N+1-2
E ?ixi - E aizi = E №хм
then, putting afc+1 = /it+1, and taking into account that 2/c + 4
<2BJV-3) + 4 = 2JV+1-2, we obtain
2N+1-2 t+1 2N+1-2
E ^¦•X<-- E aizi= E toXi-xk+izk+1
2N +1 — 2
i=k+l
~2x2k + 3 ~
ViXi>
i=k+2
which proves A5.61). Hence, taking into account A5.58) and the defi-
definition A5.56) of u, we obtain
15. Some separable Banach spaces having no unconditional basis 457
2w+i-2
u[ E
i=l
2N+l-2 2N-2
x«~ E
i=l
i=l
У fljN + k-l
0
2N-1
l] @
w 2N
= 2№ Е ^v
« E A'
;=2N-i
1
= E l^+t-il =
2W+1-2
V 6 x
i=2N-l
=
2iV+ l _
E
i = 1
2
?;X
2N — 2
- E aiz.
i= 1
whence, by A5.60) and A5.59),
x-2N+E"
2n_2
x- E ^
i= 1
У 6х;- У aiZi
<?,
which proves that we have ?oci[zn] and thus ?0 = [zn].
Finally, let us prove that {zn} is a monotone basic sequence and
hence a monotone basis of ?0. For any scalars o^,..., <xH we have
&i{Xi~ 2X2i+l~ 2X2i+2J
i= 1
E
2n+2
= El
with suitable scalars fik. Hence, if <х„+1 is an arbitrary scalar,
n+ 1
=
=
=
which completes
П
i= 1
2n + 2
k=l
n
2 Xln + 3 2 *2" + 4
2n+2
E 1^*1 ~\Pn+ ll +li^n+ 1 +Kn
k= 1
2n + 2
E IW =
*=i
n
i= 1
the proof of our assertion.
2n + 2
л. V ft v
к+J K+il
+ il+ 2 2

458
II. Special Classes of Bases in Banach Spaces
§ 16. Some characterizations of unconditional bases among
^-complete or total biorthogonal systems and among bases.
Some characterizations by properties of the associated
cone. Multipliers
Let us introduce the following notations, which we shall also use
in the subsequent sections:
.#¦={1,2,3,...},
(9 = {{/„} rzjf | ix<i2<-},
oo
},
П = the set of all permutations of the set Ж
A6.1)
A6.2)
A6.3)
A6.4)
The set 3) defined by A6.3) is countable; we shall sometimes also
consider it as a directed set (by inclusion). Let us observe that in A6.3)
it is not assumed il < i2 < ¦ ¦ ¦ < in.
We recall some well known characterizations of unconditional con-
convergence, which will be used in the sequel.
Lemma 16.1. Let E be a Banach space and {xn} a sequence in E. The
following statements are equivalent:
1°.
xt is unconditionally convergent (i.e., ^ xl(i| is convergent
for every аеП).
2°. The limit lim У х-, exists (i.e., for every e>0 there exists a
d=d(?)e<2> such that
isd'
<e for all d'efj with d'^d).
3°. For every {ijcfi the series ? xt converges.
4°. For every {sn}czK with е„=+1 (и=1,2,...) the series
converges.
5°. For every {Pn}<=K with |/?H|<1 (n=l,2,...) the series
converges.
6°. The series ]Г |/
'=1
l (i.e., lim sup
converge uniformly with respect to feE*,
Proof. Assume that we have Г, but not 2°, and let x= ^ x{. Then
i= 1
there exists an eo>0 such that for every de<2> one can find a
16. Some characterizations of unconditional bases
459
with
and
?o (since otherwise we would have
lim Y,Xi = x). Since x= ? xf, there exists a positive integer N = N(e0)
e ied i= 1
such that
x —
Not let d1 = {l,...,N} and let d[e2 with d[=>dx be such that
llx— Y Х(||^?о. Let d2 = \\,..., max/} and let d'2e3 be such that
d'2=>d2, x—Y Х;||^?о- Construct in the same way d3,d'3,dA,d'A,...
and define then a rearrangement {<т(и)} of Jf, enumerating the elements
of the sets
dx,d\\dx,d2\d'x,d'2\d2,...
OO
Then the series Y xa(n) is not convergent, since
n= 1
L Xi -\{X- LXi)-[X- LXi
-I-
= — (и= 1,2,...),
2 2 V '
and hence Y xi 1S not unconditionally convergent. Thus, 1°=>2°.
Assume now that we have 2°, and let x = lim У xt. Let e>0 be
arbitrary and let d=d(e) be such that x— Yxt < ~ f°r a"
fed' 4
with d'-^d. Put N = N(e)= max/ and let n^N,l^\ and /e?* with
||/К 1 be arbitrary. Put
Then
f |Re/(xf)|=
|Re/(x;)|= S Re/

460
п. Special Classes of Bases in Banach Spaces
By a similar argument we also have
n + l
Consequently,
n + f
E U
i = n + 1
i=n+1
whence we infer lim sup
|/(x,)| = 0. Thus, 2°=>6°.
Н/И1
Assume now that we have 6° and let {/JJcK, |/?,|^1 (/=1,2,...),
. Then there exists an /nie?* with ||/.,|| = 1 such that
n + l
/»,«( E /U) =
n + f
E A
x,-
n + f
E A*,
i = n+ 1
n + f
n,f( E A:
, whence, by |/?(|<1 and 6°,
E !/n,f(*0
i = n + 1
and therefore the series E Ptxt converges. Thus, 6°=>5°.
i= 1
The implication 5°=>4° is obvious.
Assume now that we have 4° and let {ij}e&, /z^l, /^1. Then
ZY. — — I > p v _L- > p' v I whprp m — / iyi А-к — /' p — p' — 1
/=n+l -^ \i = m i = m /
for j = n + l, ..., n + l and ?;=1, ?j=—1 for ie{m, ..., m + k}
\{'„+1,*п + 2> ••-,'„+(}• Hence, by 4°,
m + fc
E x,
J = n + 1
i = m + 1
E ei*i
<?
and therefore the series E xi- converges. Thus, 4°=>3°.
Assume, finally, that we do not have 1°, i.e., that there exists a per-
00
mutation аеП for which the series E xa(t) 1S not convergent. Then
there exist an eo>0 and a sequence {mn}e& such that
E
A1= 1,2,...).
A6.5)
Choose an infinite subsequence {т„} с {mn} such that
min c@> max ^@ (/=1,2,...),
+l^i^ + l^i^
16. Some characterizations of unconditional bases
461
and rearrange the integers a(i) (mnj+ 1 ^i^mnj+1 ,j= 1,2,...) into an
increasing sequence, say {ij}. Then, by A6.5), the series E xtj ls not
convergent. Thus, 3°=>1', which completes the proof of lemma 16.1.
If ? is a Banach space and (х„,/„)({*„} <= E, {/„} с ?*) a biorthogo-
nal system, let us introduce the operators
), A6.6)
i= 1
and the sets
\imsd(x) = x\,
A6.7)
A6.8)
A6.9)
A6.10)
A6.11)
A6.12)
Theorem 16.1. Let E be a Banach space and (*„,/„) ({х„} с E, {/„} <= E*)
an E-complete biorthogonal system. Then
a) The following statements are equivalent:
1°. {xn} is an unconditional basis of the space E.
2°. For every xeE the limit lim Е/((х)х, exists.
aUx=\xeE Xmisd(x) exists[,
<cc},
r3 = \xeE lim||sd(x)\\ exists and <oo
lim ||5d(x)|| does not existi.
del
3°. For every {in}e(9 and xeE the series E fij(x)xi] converges.
4°. For every {en}cK with ?„=±1 (n=l,2,...) and xeE the
00
series E Eifi(x)xi converges.
5°. For every {fin}cK with |J?nKl (n=l,2,...) and xeE the
00
series E PifAx)xi converges.
i= 1

462 II. Special Classes of Bases in Banach Spaces
6°. We have
sup \\san(x)\\<cc (хеЕ,аеП).
1 ^П< GO
7°. W? /шие
sup ||s{j iinl(x)j|<co (xeE,{in}e(9).
1 ^ 0< GO
8°. We have
sup ||5(? w(x)||<co (xe?, e1,82,...= +
9°. We have
sup ||5|/JiiM(x)||<co (xe?, I^U^I,...s;
1 ^!<oo
10°-13°. We have
sup sup ||sff,n(x)||<oo (xe?),
, respectively, similar conditions for {in},' {е„} a«d {/?„}.
14°-17°. We have
sup ||5ет,„|| < oo (сте Л),
1 ^П< OO
, respectively, similar conditions for {/„}, {е„} аи^ {/?„}.
18°-21°. We have
sup sup
(ГеЯ 1 ^n< o
I < oo
?, respectively, similar conditions for {in},2 {?„} and {/?„}.
22°. For every xeE and feE* the numerical series
I l/.-WI 1Ж01
A6.13)
A6.14)
A6.15)
A6.16)
A6.17)
A6.18)
A6.19)
A6.20)
converges.
23°. For every xeE the series A6.20) converge uniformly with
respect to feE*, ||/||<1.
1 The condition for {/„} can be also written in the following forms:
11'. sup||^(x)||<co (xeE).
11". Ф2 = ?.
2 The condition for {[„} can be also written in the following form:
19'. sup Us,,|| <со.
f9
16. Some characterizations of unconditional bases
24°. For every xeE the series
? I/,(*)!*,
i= 1
converges, and there exists a constant С ^ 1 such that
(xeE).
463
25°. We have
i= 1
A6.21)
A6.22)
b) We have1
GO
sup||5d(x)||=sup sup ||sff,B(x)K sup ^\fi(x)\\f{Xi)\
= sup sup ||.s{/M,,(x)||<2 sup sup ||5{e
{Pn}c:K l^Koo {Snl^K l^Koo
\Pi\H ?(=±1
A6.23)
аи^/, /f {х„} is an unconditional basis of E, each of these numbers defines
a norm on> E, equivalent to the initial norm on E.
Proof, a) The equivalences 1°<»2О<»3О<»4О«.5О<*23О are conse-
consequences of lemma 16.1.
The implications 1°=>6°, 3°=>7°, 4°=>8° and 5°=>9° are obvious.
9°=>130.2 For any fixed xeE put
и«(Ш)=
,/=1,2,...).
Then each щ is a linear mapping of m into E, which is also contin-
continuous, since
{jS,}|| ({Р„}ет, /=1,2,...),
|
The same proof shows that in the particular case when the scalars are real,
sup sup \\s{. }l(x)\\ = sup sup ||5(?n),(x)|| (xeE).
2 It is also easy to prove directly that non 13°=>non 9°. Indeed, if we have
non 13°, then there exist sequences {lk}e& and {ft(t)} <= K, Ift^'K 1, such that
I I
whence, putting ft = ft№ (/t _ j + 1 sj К lk; к = 1,2,...), we obtain
i=l
i.e., non-9°.

464
II. Special Classes of Bases in Banach Spaces
where C,= ? |/f(x)l llx.ll (/=1,2,...). Furthermore, by 9° we have
i= 1
sup Ы{Р„})\\<со ({/?„} em).
ISKco
Consequently, by the principle of uniform boundedness, we have
sup ||m,||= sup sup ||5{pnM(x)||<oo.
l«!<eo IPniem 1«|<и
l|{/*n!ll«l
The proof of the implications 7°=>11° and 8°=>12° is similar. The
implication 6°=>10° will be proved via 6°=>7o=>llo=>10°.
6°=>7°. Assume that we have 6° and non-7°. Then by non-7°, there
exist an xeE and a sequence {in}e& such that
sup ||s{jl>i2 ,-п,(х)|| = оо, A6.24)
1 ^ П < GO
and by 6° we have sup \\fn(x)xn\\=M<cc. Furthermore, by A6.24)
1 ^ n< oo
there exists a sequence {т„\е(9 such that
Let {lj}e(9 be the complementary set to {/„} in Ж and define a
permutation аеП by
ij-n+i for ffl,-, + Kj<ran-l («=1,2,...; mo = 0),
/„ for j=mn (ii=l,2,...).
Then
.мл
У ft (x)x,
mn — n
Ef. (y\x
Ik
k=l
which contradicts 6°.
1Г=>10°. Let стеЯ be arbitrary. For any fixed пеЛ let iui2,...,in
denote the numbers стA), аB),..., a(n) arranged in increasing order and
let {ij}JLn+1 be an arbitrary sequence in Ж such that {ij}f=le(9. Then
s{ii,i2,...jn№) = so,n(.x) (xeE). Consequently, we have
sup sup |к„,„(хЖ sup sup lk(I-lji2j...iJn)(x)|| (xeE), A6.25)
whence we obtain the implication 11° => 10°. Thus also 6° => 10°.
16. Some characterizations of unconditional bases
465
The implications 10° => 18°, 1Г=>19°, 12° ^20° and 13° ^21° are
consequences of the principle of uniform boundedness1, and the impli-
implications 18°=>14°=>1°, 19°^15o=>3o,20o^16o^4° and 21°^17°^5°
are obvious. Furthermore, the implication 23°=>22° is obvious and
the implication 22°=>9° is a consequence of § 15, corollary 15.1.
5°n21o^24°. Putting2 pn = signfn(x) (n=l,2,...), from 5° it
GO ОС
follows that for every xeE the series ? \fi(x)\xt = ? Pifi(x)xi con-
converges. Furthermore, putting
E \
i= 1
and taking into account that (by 5°) {xn} is a basis of E and 21°, we
obtain
x\\ =
GO
E fi(x)xt
i= 1
< sup
1 $ 1 < OO
1
=
OO
E P
i= 1
V В f(z)x
i
i\fi(x)\Xi
<C\\z\\
=
= C
where C= sup sup lk{«nlill-
24°=>23°. Let e>0, xeE and feE* with Ц/К1 be arbitrary.
GO
Since by 24° the series ? \fi(x)\xt converges, there exists a positive
i= 1
integer N(e,x) such that
E i
— (n>N(s,x); /c = l,2,...),
where С is as in A6.21). Consequently, putting
/^sign/M/W if f{x,)*0, j5,-= 1 if/(*,) = 0 (i=n+\,...,n + k),
у = E
i=n+l
= E /iW^i.
1 See e.g. [50], p. 66, corollary 21.
2 For the definition of sign a (a complex) see Ch. I, § 12, footnote to example 12.2.
30 Singer, Bases in Banach Spaces I

466 II. Special Classes of Bases in Banach Spaces
and taking into account A6.21), we obtain
Z 1/iMI !/(*.•)! =
n + k
/ Z ft/ito*,
= n +
n + k
Z 1/iOOl*,
= с
n + k
< ?
for n>N(e,x) and /c=l,2,... whence, since e>0, xe? and feE*
with ||/IK 1 were arbitrary, we infer 23°.
11°=>25°. If we have 11°, then, obviously, % = % = E.
25°=>11°. Assume that we have non-11°. Then for every positive
integer и the set \xeE sup\\sd{x)\\^n\ is nowhere dense (since аИгФЕ)
and closed, whence the set
xeE
sup\\sd(x)\\^n
Ф2 =
is of the first category. On the other hand, for every de& the set
%={xeE\ \\sd.(x)\\7?\\x\\ + l forall d'e2,d'^d]
is closed and nowhere dense (since ^0 is dense in E), whence, by
E\% с (J %,
<ШЪ is of the second category. Thus <Ш2Ф1ШЪ, which completes the proof
of the equivalences I°o...o25°.
b) Let {in}e& be arbitrary. For any fixed пеЖ, take а аеП such
that
i, for
Then 5(r,nW =
sup sup
(xe?). Consequently, we have
<sup sup
<теП Кл<
(xeE),
which, together with the inequality A6.25) proved above, yields the first
equality of A6.23).
The inequality
sup sup ||s{il i2 . .jjx)\\ < sup sup \\s{fnUl(x)\\ (xeE)
is obvious.
16. Some characterizations of unconditional bases
467
Now let xeE, {fin} <= К with
Take д = д1Рп11хеЕ* with ||<
Z Pif{x)Xi . Then
i= 1
1
Eft/
(/= 1,2,...) and / be arbitrary,
such that gfl 2, Piftix)
: Z Iftl 1/iWI l»(*i)
i= 1
and hence
sup sup
sup
Conversely, let xeE and feE*, ||/|| = 1, be arbitrary. Put
ft = sign/,(*)/(*,) (i=l,2,...).
Then |ft|<l (/ =*1,2,...) and
/
whence
= II%.mWII (/=1,2,...),
sup X 1/iWI I/WK sup sup \\s{MA(x)\\ (xeE),
feE* (=1 |f,|cl! KKoo
which, together with A6.26), proves the second equality in A6.23).
Furthermore, assuming that the scalars are complex, let {/?„} с К,
|/?„|<1 (n= 1,2,...) be arbitrary. Take an xe? and a fixed ne,,K Then,
for1 Pk = yk + iSk, where yk = Reft., Ek = Imft. (fc=l,2, ...,n), we have
|y*l<l,|Et|<l(fc=l,2,...,n) and
Since {yt}" is in the unit cell Smn of the real Banach space2 mn,
there exist, by the theorem of Caratheodory, n+ 1 extremal points of
1 Here z=l/^T.
2 We recall that (the complex or real) mn is the space of all (complex, respec-
respectively real) и-tuples y= {r}k}\ endowed with the norm
ILK|| = max \щ
30»

468
II. Special Classes of Bases in Banach Spaces
Smn, i.e.1 n+1 points {s[j)}nk=1emn (j= 1,2, ...,n+1) with ?</>=±l
(/с =1,2, ...,n;j=\,2, ...,n+l), andrz+1 numbers A = ^0(/= 1,2,..., и+1)
with ? A;=l, such that
№=1,2,...,«).
Hence
п
Z;
n n+1
z z-
k=l j=l
max
Z
h Z
Consequently, we have2
k=l
SUp SUp
A6.27)
Since we also have a similar relation for {<5fc}", it follows that
n
Z Pkfk(x)xk < 2 sup sup \\slSn}J(x)\\,
k=l 1'"!с|; 1«!<оо
whence we obtain the penultimate inequality of A6.23).
The last inequality of A6.23) follows from
i= 1
where u^ = {ге,/К
Z /«(*)*,
ied.
WJti
, ?,= + 1}, ^2 = {/el/(/' ]
, ?,= -l}.
1 See e.g. [32], p. 82, example 2).
2 One can also give the following proof of A6.27), which makes no use of
the theorem of Caratheodory: Take a ge(?@)* (see Ch. I, § 1) such that
/ " \ I!
9\ Z 1кк(х)хЛ=
\ ]
| = 1,
У*/*(*)** and put ?4 = sign [/*(*) g(*4)] if fk(x)g(xk)^0
and ?4=1 if fk(x)g(xk) = 0. Then ?4=±1 (fc=l,...,n) and
n n
= Z r*/*W0(**) < Z \fk(x)g(xk)\
1*
k= 1
whence we infer A6.27).
Г " 1
= il Z 4fk(x)xk
L*=i J
4fk(x)xk ,
16. Some characterizations of unconditional bases
469
Finally, if {xn} is an unconditional basis of E, then, by the implica-
implication l°=>10° of part a) proved above, the numbers A6.23) are finite
for all xeE. Since they are obviously norms on E and since
x =
Z fi(x)xt
(xeE),
from the inversion theorem of Banach1 it follows that they are equiv-
equivalent norms to the initial norm on E. This completes the proof of theorem
16.1.
Let us observe that the equivalences 1 ° <=>... <=> 5° in theorem 16.1a)
also hold if the hypothesis of incompleteness is replaced by the assump-
assumption that {/„} is total on E. If E contains no subspace isomorphic to c0,
then, by § 15, lemma 15.8, a similar remark is also valid for the equiv-
equivalences Г<=>...o24°. In this latter case the assumption that ? contains
no subspace isomorphic to c0 is essential, as shown by the example
E = c, xn = the n-th unit vector, and /„ = the n-th coordinate func-
functional (then E =з c0 isometrically and (xn, /„) is a total biorthogonal
system satisfying 2°,3°, ...,24° but not Г, since [х„] = c0^?).
The assumption A6.21) in 24° of theorem 16.1 a) is necessary for the
validity of the implication 24° => 1°, as shown by
Example 16.1. Let E = ll and let {hn} be the conditional basis of Z1
considered in § 14, example 14.2, i.e.,
*!=/!, K=fn-X-fn (« = 2,3,...), A6.28)
where {/„} is the unit vector basis of I1. Then the convergence of Z a;^i
oo f=1
still implies the convergence of Z la;l^i-
i — 1
Indeed, as observed in § 14, example 14.2, ? a;/i; converges if and
i= 1
only if
-a,-|<oo.
Since
a,-4.1 -
|ai+1— a;|, it follows
that the convergence of Z af^i implies the convergence of ? Mht.
i= 1 i= 1
Comparing the equivalences 1°<=>1Г<=>25° of theorem 16.1 with
Ch. I, §5, theorem 5.1, it is natural to ask whether the condition 4lx=aU2
is also equivalent to 1°. The answer is negative. In fact, although 1°
obviously implies <%1 = %2( = E), the converse implication is not valid,
as shown by
Example 16.2. Let E = lx and let {xn} be any conditional basis of/1
(e.g., the basis considered in example 16.1 above). Then 6Ux=aU1.
1 See e.g. [10], p. 41, theorem 5.

470 II. Special Classes of Bases in Banach Spaces
Indeed, the inclusion °U^ с ^2 is obvious. Conversely, let xeaU2 be
arbitrary. Then, by § 15, corollary 15.1 (implication 3°=>Г) Y fi(x)xt
i= 1
is a weakly unconditionally Cauchy series. Hence, since I1 is (sequentially)
weakly complete and since in I1 weak convergence of sequence implies
GO
norm convergence, it follows that Y /j(*)*i is unconditionally con-
;= l
vergent1 and therefore, by lemma 16.1 (implication 1°=>2°), xsaU^.
Thus, аи2сЩи whence, finally ЩХ=Щ2,
We shall give now a characterization of unconditional bases among
incomplete biorthogonal systems in terms of a stability property.
Definition 16.1. A sequence {ynj in a Banach space ? is said to be
unconditionally co-linearly independent, if for any unconditionally con-
vergent series ? <xiy[ with Y a;j, = 0 we have <x( = 0 (i = 1,2,...).
;= l i= l
Obviously, every co-linearly independent sequence \yn\ is also
unconditionally co-linearly independent.
Definition 16.2. Let ? be a Banach space. An ?-complete biorthogonal
system (xn,fn) ({*„} <=. ?, {/„} с ?*) is said to be U-stable if every
unconditionally co-linearly independent sequence {_>>„} с ? for which the
GC
series Y || jc,- — jj /j converges, is complete in E.
i= 1
Theorem 16.2. Let E be a Banach space and (xH,/„)({*„} <= E, {/„} c?*)
аи E-complete biorthogonal system. {xn} is an unconditional basis of E
if and only if (xn, /„) is U-stable. Moreover, in this case every uncondition-
unconditionally co-linearly independent sequence {у„} с Е for which the series
on
Y Wxi ~У\ II .п converges, is an unconditional basis of E, equivalent to {*„}.
i= 1
Proof. Assume that {xn\ is an unconditional basis of E. Then, ob-
obviously, {/„} is total on E. Furthermore, let {у„} be an unconditionally
co-linearly independent sequence in E, such that the series Y 11*;— }';ll/;
i- 1
converges. Then, since {xn} is an unconditional basis of E, {/„} is an un-
conditional basic sequence (see § 17, theorem 17.7), whence ? ||*j— y'iWf]
i= 1
converges unconditionally. Consequently, by Ch. I, § 10, remark 10.4,
there exists a positive integer n0 such that {xn}*u x {yn}n0, whence
1 Actually, by § 15, lemma 15.8, the same conclusion holds in any space E
containing no subspace isomorphic to c0.
16. Some characterizations of unconditional bases
471
{Уп}% is an unconditional basis of [у„]вд- Since {_>>„} is unconditionally
«-linearly independent, it follows that \у„) is also co-linearly independent
and therefore, by Ch. I, §10, remark 10.4, \xn] ~ {yn}, whence {yn} is
an unconditional basis of ?.
Let us observe that this part of theorem 16.2 also follows from an
extended stability theorem mentioned at the end of § 11. Indeed, assume
again that {х„} is an unconditional basis of ? and {yn} an unconditionally
00
co-linearly independent sequence in ? such that the series Y \\xi~yi\\fi
00
converges and let {<х„} be any sequence of scalars such that Y aixt
OO '
= xoe?, Y а1У1 = ®- Then, since {xn} is an unconditional basis, both
oo I= 1 oo
Yaixt and Y Wxi—yi\\fi are unconditionally convergent. Let /?, be
j=l i=l
any scalars with |/?,|<1 (/=1,2,...). Then, by §17, theorem 17.7, and
§17, theorem 17.1 applied to the unconditional basic sequence {/„},
there exists a constant С ^ 1 such that
n+ p
У 8,<X,(jC,-V,)
= sup
Il/N« i
П + p
Y Pifi(xo)
n + p
n + p
sup
feE*
11/11 «1
Hence Y Pi°Li(xi-yi) converges and thus, by lemma 16.1,
00 '— 1 GO OO GO
Xai(*i—M whence also ^ a{y,:= ^ aixi + Z cctb;i~xil is uncon-
i=l i=i i=l i=l
GO
ditionally convergent. Therefore, by ^ aiJi = 0 and by our hypothesis
i= 1
on {>>„}, it follows that af = 0 (/=1,2,...). This proves that {yn} is
(?,{xn})-linearly independent. Consequently, since {yn} is ([/„],{/„})-
near to the ((?, (xn})-Besselian) basis {х„}, from the extended theorem
11.4 it follows that \yn} is a basis of ?, equivalent to {xn}, whence an
unconditional basis.
Conversely, assume now that (*„,/„) is [/-stable, but {xn} is not an
unconditional basis of ?. Then there exists an xoeE such that either
oo oo
Yfi(xo)xi is not convergent or Yft(xo)xi converges, but not un-
unconditionally. In both cases there exists an index n such that /n(x0)#0.

472
Put
II. Special Classes of Bases in Banach Spaces
, for i Ф n,
/,(*,)
1
A6.29)
xn for i = n.
L(x0)
Then, by an argument similar to that used in Ch. I, § 10, the proof
00
of theorem 10.6, it follows that ? ll*;-J>ll/i converges and {yn} is
1 ~~ 1 oo
unconditionally ю-linearly independent (observing that if ]T atyi con-
i=l
00 00
verges unconditionally, then so does ? а,х,, by A6.29), but
does not converge unconditionally). Hence, by our hypothesis, {yn}
must be complete in E, but this contradicts the relations
/„M = ./„(*,) - 77^/"(xo) = 0 (i= 1,2,...),
completing the proof of theorem 16.2.
Now we shall give some characterizations of unconditional bases
among E-complete total biorthogonal systems by properties of the
"associated cone". We recall (see § 10) that by "cone" we understand
"closed convex cone having the origin as extreme point", i.e., a closed
set Ж such that Ж + Ж с Ж, л Ж с Ж (Я ^ 0) and Жп(- Ж)={0}.
Definition 16.3. Let ? be a real1 Banach space and (х„,/„) ({х„} сЕ,
{/„}<=?*) a total biorthogonal system (i.e., a biorthogonal system
such that {/„} is total on E). The set
Ж(ХпМ = {xeE \ fn(x)^Q (n=l,2,...)} A6.30)
is called the cone associated to the biorthogonal system (х„, /„).
It is easy to see that Ж(Хп</п) is a cone. Indeed, Ж(Хп fn) is obviously
a closed set, satisfying Ж(ХпМ + Ж(Хп,/п) с Ж(Хп Л) and ХЖ(Хп fn) <= Ж(Хп fn)
(aJsO). Furthermore, if хеЛГ(ХпМп(-Ж(ХпМ), then, 'by A6.30),
/„(x) = 0 (и =1,2,...), whence, since {/„} is total on E, x = 0 and thus
In the particular case when {х„} is a basis of ? and {/„} the a.s.c.f.
to {х„}, we obviously have2 Ж(Хп/п) = Ж{Хп). Let us observe that some
properties proved in § 10 for Ж{Хп] (e.g., proposition 10.1 a),b) and first
half of c)) remain valid, with the same proof, for Ж,х r „ where (х„, fn)
is any total biorthogonal system.
1 See § 10, footnote to definition 10.2.
2 See § 10, definition 10.2.
16. Some characterizations of unconditional bases
473
A cone Ж is called minihedral if for every1 х,уеЖ there exists
zo = sup(x,j), i. e., the element zo^x,y with the property z^x,y^>z^z0
(we recall that x^y if and only if х—уеЖ). This condition is equiv-
equivalent to the following: for every х,уеЖ there exists гоеЖ such that
(х+Ж)п(у + Ж) = го + Ж. For the definitions of normal cones and
generating cones see § 10.
Theorem 16.3. Let E be a (real) Banach space and (х„,/„)({х„}с?,
{/n} с ?*) an E-complete total biorthogonal system. The following state-
statements are equivalent:
1°. {xn} is an unconditional basis of E.
2°. Ж(Хп/п) is normal and generating.
3°. JfL ,, is normal and for every xeE there exists an element
zeE such that
/„(*)= |/.(*)| ("=1,2,...).
4°. Ж(Хп/п) is generating and for every хеЖ{ХпМ the set2
A6.31)
A6.32)
is bounded (in the norm).
5°. Ж(ХпМ is generating and minihedral and for every хеЖ(ХпМ the
set 3?x above is linearly homeomorphic either to a finite dimensional cube
or to the fundamental parallelotope of Hilbert (hence compact).
Proof. 1°=>3°. Assume that we have 1° and let x,yeE be arbitrary
elements such that O^y^x. Then 0^/((у)^/((х) (г=1,2,...), whence
fi(y) = Pifi(x) with 0^K^1 (/=1,2,...). Consequently, by the impli-
implication 1°=>17° of theorem 16.1,
| = lim
== lim
Z fiifil
i= 1
where C= sup ||s{/j.} J. This proves that Ж(ХпМ is normal.
Furthermore, the existence of a zeE satisfying A6.31) is obvious
from the implication 1°=>24° of theorem 16.1.
3°=>2°. Let xeE be arbitrary and let zeE satisfy A6.31). Put
z — x
Jl =
У 2 =
A6.33)
1 Some authors modify this definition replacing the condition х,уеЖ by
x,yeE. It is readily seen that these latter cones Ж are nothing else than those
which are minihedral and generating (in the terminology used here).
2 For any cone Ж and any xeJf we have ЯГn(x-Jf)={yeE\O^y^x}.
Indeed, уех-ЛГ if and only if x-yeJf and thus х- Ж={уеЕ \у^х}.

474 II. Special Classes of Bases in Banach Spaces
Then x=y1—y2 and
,, л 1/„(*I+/„(*)„ r, ч l/»(*)l-/»(*>
i.e., Ji,J26-^(xn,/n)> which proves that JfiXtitfn) is generating.
The implication 2°=>4° is obvious.
4°=>1°. Assume that we have 4° and let хб^Хп/п) be arbitrary.
Since for sn(x) = E ft(x)xi we have, by biorthogonality,
i= 1
i [>»(*)] =
for i=l,..., и
for г = n + 1, и + 2,...
it follows that
whence, by the boundedness of the set 0>x defined by A6.32),
sup \\sn(x)\\=Mx<co.
A6.34)
A6.35)
A6.36)
Now let xeE be arbitrary. Then, since 3f(XnM is generating, we
have x=y — z, where y,zeJf(XnJn), whence, by A6.36) applied to у and z,
sup
sup ||5„0>)||+ sup \\sn(z)\\^My
Consequently, by [*„] = ? and Ch. I, §4, theorem 4.1, {х„} is a basis
of E.
Now let (*„(„,,/„(„,) be an arbitrary permutation of (*„,/„). Then
[х<т(л)] = ?> {/<т(п)} is total on ? and •^х„,„),/,(„)) = -^(х„,/„) is normal and
generating, whence, by the above, {xa(n)} is a basis of ?. Consequently,
{xn} is an unconditional basis of E.
1°=>5°. Assume that we have 1° and let х,уе,Ж(ХпГп) be arbitrary.
GO
Then the series ? [/¦(*)+./iO')]Jci is unconditionally convergent,
1 ~ 1 OO
whence, by lemma 16.1, so is the series E max[fi(x\ fi(y)lxi and the
i 1
sum of this latter is obviously sup(x,j). This proves that Ж(Хп fn) is
minihedral. By the implication 1°=>4° proved above, Jf(XnJn) is also
generating.
Now let xeX(XnSn) be arbitrary, such that the set
{п]} = {пе,Ж\их)>0} A6.37)
is infinite; in the case when the set {n,} is finite, the argument is simpler,
with obvious modifications.
16. Some characterizations of unconditional bases
475
Define a mapping Ф of the set 3?x (see A6.32)) into the fundamental
parallelotope of Hilbert
by
*iy) = \rr
-(j=\,2,-)\, A6.38)
A6.39)
Then the relations Уиу2е^х, Ф(у1)=Ф{у2) imply Ji=J2 (since for
n?rij we have /„(x) = 0, whence, by 0^yk^x, fn(yk) = O), i.e., Ф is one
to one. Furthermore, yuy2, alyi+a2y2e^x imply ФОх^ +а2у2)
), i.e., Ф is linear.
GO
Let {?j}eQ0 be arbitrary. Then, since Y, fm(x)xnl is unconditionally
i= 1
convergent and since O^/^l, the series ^ '^/п,(х)хт is uncondi-
i= 1
tionally convergent (by lemma 16.1) to an element yeE. Since 0^fn.{y)
x) and 0=fn(y) = fm(x) for пфп-р we have ye0>x and
which proves that Ф maps 0>x onto Qo.
Now, since Qo is compact, in order to prove that Ф is a homeo-
morphism it will be sufficient to prove that Ф is continuous. Let
yoe0>x and let l/= U{yo)={ye0>x \ \\y-yo\\ <e} be an arbitrary neigh-
neighborhood of j0 in 0>x. We shall find a neighborhood V=V(z0) of
{j
in Qo, such that
theorem 16.1 there exists an N = N(s,x) such that for all {?„}
|]3„|^ 1 we have
U. By 1° and
with
i = N+ I
= lim
с
Б

476
where C=
II. Special Classes of Bases in Banach Spaces
sup sup \\s,e .J. Consequently, for any z={f.}
0'=l, ...,
we have, by
GO GO
? /п;Ь>)Хт ~ ? fm(y0)Xni
i= 1
^/„,(*)*„,•
,y@)
i = N+ 1
i.e., jel/O'o), and thus we may take K(zo)={zeQo | |(z — zo|| <<5}.
Finally, the implication 5°=>4° is obvious, which completes the
proof of theorem 16.3.
Remark 16.1. One can also prove the implication 4°=>1° using
directly sd(x) {deS>) instead of sn(x) (obviously, A6.35) also holds for
sd(x)) and applying theorem 16.1. However, we have given the above
proof in order to show what is the difficulty in the problem of charac-
characterizing (not necessarily unconditional) bases among E-complete total
biorthogonal systems (*„,/„) by properties of the associated cone
X~(Xnjny namely, these should be such properties of Jf(Xntfn) which
depend on the order of {xn}.
In connection with this problem let us mention that the conditions
occurring in theorem 16.3 are not necessary in order that {х„} be a
basis of E, as shown by the following two examples:
Example 16.3. Let {xn} be the conditional basis
х„={1,...,1,0,0,...} (и=1,2,...) A6.40)
п
of E = c0, considered in § 14, example 14.1. Then the a. s. с f. is
whence
Ж = (x = !с }e<
Since the relations 0^x= {?„}^У={г]„}
imply ||x|j = sup
sup |»/B| = |
the cone Jfi
iXnJn)
is normal, whence, by theorem
16.3,
is not generating and does not satisfy A6.31).
16. Some characterizations of unconditional bases
Example 16.4. Let E = c0 and let
477
= ?(-l)" + 4- (n=l,2,...), A6.41)
where {en} denotes the unit vector basis of E = c0. Then ( —1)"х„
n
= ? (-1)'е,-(и = 1,2,...), whence, by Ch.I,§4, proposition4.3, {(-1)"х„}
is a basis of E and therefore {х„} is a basis of ?. Moreover, as in § 14,
example 14.1, we see that {( —1)"х„}, whence also {х„}, is a con-
conditional basis. The a. s. с f. to {х„} is
whence
The cone ^х„,/„) is generating, since every х={?„}ес0 can be
written as x = y — z— {r]n} — {?„}, with rjn^Q, ?„^0 (n= 1,2,...) and
then, by A6.42), 'y,zeJf(XnJn). Hence, by theorem 16.3, Ж{ХпМ is not
normal and there exists an xeJfiXnJn) with 0>x unbounded, whence
non-compact and therefore not linearly homeomorphic to a finite
dimensional cube or to the fundamental parallelotope of Hilbert.
Furthermore, Ж(Хп.м does not satisfy the second condition in 3°.
Indeed, if for every х={?„}ес0 there existed an element z={?n}ec0
satisfying A6.31), then we would have
f;
k=l
k=l
whence
k=\
would converge for every x={c,n}ec0.
However, this is not satisfied e.g. for x= {?„}ес0 defined by
since for each и =1,2,... we have
00
.
n+V
whence
|=-00.

478
II. Special Classes of Bases in Banach Spaces
Finally, Ж(Хп/п) is not minihedral. Indeed, assume that Ж(Хп/п) is
minihedral and let xeE be arbitrary. Then, since Ж(Хп4п) is generating,
we have x=y-z, with у,геЖ(ХпМ. Since Ж(ХпМ is minihedral,
sup(y,z) exists and hence there also exist the elements
x + = sup(x,O) = supCF-z,O)=sup(y,z)-z, x_ =sup(-x,0).
Since1 /„(х + ) = тах(/„(х),0),/„(х_) = тах(-./„(х),0), it follows that
for the element z = x++x_ we have A6.31), in contradiction with the
above. This proves our assertion.
The hypothesis [х„] = ? in theorem 16.3 is essential, as shown by
Example 16.5. Let E=c and let
х„={0,... ,0,1,0,...} (и=1,2,...), A6-43)
:?п (х={^}ес;и = 1,2,...). A6.44)
Then [хя~] = софЕ (and thus {х„} is not a basis of E), but {./„} is
total on E and Ж{ХпМ is both normal and generating.
Indeed, obviously,
Now, if Oscx= {?„}==:j= {>?„}, then
and thus Ж(
{}{} (Хп4п)
is normal. Furthermore, let х={?„}е? be arbitrary. Then x=x+-x^
= K«}-{C}, where ?„+=тах(?„,0), С=тах(-^,0); here x+,x_ec,
since if lim <*„ = <!;# 0, then sign ?„ = sign ? for all sufficiently large n.
Thus, ^Xn,/n) is generating.
However, such an example is no longer possible if E is reflexive,
i. e., in this case one can omit the hypothesis [х„] = E in the equivalence
Г«»2° of theorem 16.3. Indeed, we have
Proposition 16.1. Let E be a reflexive Banach space and let (х„,/„)
be a total biorthogonal system such that Ж{ХпМ is normal and generating.
Then {xn} is an unconditional basis of E.
Proof. Let xe.yfiXnJn) be arbitrary. Then by A6.34) we have
/,[>„(*)]
1 Indeed, x+^x,0 implies
imply z = anxn+ ^Мх + )х^х,
Mx) (i,n= 1,2,...),
U + ) 3s/„(*),() and the relations
whence z^x + , whence а„ = /
which proves that /„(*+) = max(/n(x), 0).
16. Some characterizations of unconditional bases
479
whence the limits lim/; [$„(*)] exist. Since E is reflexive and {/„} is
total on E, {/„} is also complete in ?*, and thus lim /[sn(x)] exists
for all / in a dense subset of E*. Since Ж(Хп/п) is normal, by A6.35)
we also have \\sn(x)\\ ^L||x|| (n=l,2,...). Consequently, {sn(xj} is a
weak Cauchy sequence. Since E is reflexive, whence weakly complete,
it follows that sn(x) converges weakly to an element yeE. By bi-
orthogonality and since {/„} is total on E, we have y = x, i.e. {^„(x)}
converges weakly to x.
Now, let xeE be arbitrary. Then, since Ж(Хп,!п) is generating, we
have x=y — z, where y,zeJf(XnJn), whence, by the above, sn(x)
= sn(y) — sn{z) converges weakly to y — z = x. Consequently1, хе[х„],
and thus [х„]=Е and the desired result follows by theorem 16.3. This
completes the proof.
Let us recall that a cone Ж is called regular if the relations
Ji^J2^'^z imply that {yn} is norm-convergent. One may assume
here that 0^yl^y2^---^z (by considering, if necessary, the sequence
{jn-Ji})- Replacing the normality of Ж(ХпМ by the stronger2 con-
condition of regularity of Ж(ХпМ, one can again omit the hypothesis
[х„] = ? in the equivalence \°o2° of theorem 16.3. Indeed, we have
Proposition 16.2. Let E be a Banach space and (х„,/„) ({х„} с Е,
{/„} с Е*) a total biorthogonal system. The sequence {х„} is an un-
unconditional basis of E if and only if Ж{ХпМ is regular and generating.
Proof. Assume that {xn} is an unconditional basis of E. Then, by the
implication 1°=>2° of theorem 16.3, Ж(Хп/п) is normal and generating.
We shall now prove that whenever {х„} is a basis and ^Xn/n) is
normal, then Ж(Хп fn) is also regular.
Let O^y^y^-^z. Then 0^(уп)^(уп+1)^№ (i,n= 1,2,-¦¦),
whence the limits lim /;(у„) = а((г= 1,2,...) exist and wehaveO^a;^/;(z)
П-» GO
(i= 1,2,...). Hence, since Ж(Хп fn) is normal,
i = n + 1
and therefore Y, ocixi=yeE. We shall show that lim ||j-jJ=0,
i= 1
which will prove that Ж(ХпГп) is regular.
1 See e.g. [10], p. 134, theorem 2.
2 One can show (see e.g. M. A. Krasnoselskii [137], Ch. I, section 1.2.2), that
every regular cone is normal, but not conversely. However, we shall not use this
result here.

480
II. Special Classes of Bases in Banach Spaces
Let e > 0 be arbitrary and take N = N(e, z) such that
< —. Then, by the normality of Ж{Хп/п),
J J—i
E .
i = N+ 1
E
i = N+ 1
E fiiyJXi
i=N+ 1
E .
i=N+ 1
3
On the other hand, since lim fi[yn) = ai=fi[y) (i= 1,2,...\ there
П-> GO
exists an M = M(e) such that
Consequently,
Wy-yJ =
E
e
<3
N
E
!= 1
E /00*
i = JV + 1
E
<?
which proves that lim ||j— jj =0. Thus, Ж(ХпГп) is regular.
«-•GO
Conversely, assume that ^х„,/„) 's regular and generating and let
хеЖ(Хп1-п) be arbitrary. Then, by A6.34) and since ¦%\Xnjn) is regular,
lim sn(x)=yeE. Hence, by biorthogonality and since {/„} is total
on E, it follows that y = x, and thus lim sn(x) = x.
«—¦GO
Now let xeE be arbitrary. Then, since ^(Xntfn) is generating, we
have x = y — z, where y,zeJfiXnJ-n}. Hence, by the above,
limsn(x)= lim^CF)- lim sn(z) = y-z = x,
and thus {xn} is a basis of E. Considering an arbitrary permutation
(xc(np /<7(n>) °f (xn>/n) ^ follows, as in the proof of the implication 4°=>lc
of theorem 16.3, that {xn} is an unconditional basis of E, which completes
the proof.
In the particular case when dim? = n<co and {x,}" is a basis of E,
by theorem 16.1 the associated cone ^|Xj]j is minihedral. Furthermore,
n
¦^{xX is obviously solid, since any x= E aixi witn af>0 (г=1,...,и)
i = i
is an interior point of -^{XjJi- The following converse is also true:
16. Some characterizations of unconditional bases
481
Theorem 16.4. Let E be a (real) Banach space with dim E = и < со
аи^ /е/ Jf cE be a solid minihedral cone. Then there exists a basis
{XjYi of E such that Ж1х.,л = Ж.
For the proof we shall make use of the following "decomposition
lemma" •.
Lemma 16.2. If Ж is a minihedral cone in a linear space E, then1 for
every pair of finite sequences {jj-=i, {zt}[=1c/, with r^2, s^-2,
satisfying
Еу,-=Е^, A6.45)
i=1 k=l
there exist rs elements wike.)f (i= \,...,r; k= 1,..., s) such that
yi=j^wik (i=l,...,r), z*= X>* (fc=l,...,s). A6.46)
к = 1 i = 1
Proof Assume first that r = s = 2. Let yi,y2,zl,z2eЖ be such that
Put
Obviously, wn,wl2,w2i^O. Furthermore,
Since
A6.47)
we also have
whence w2l ^y2 and thus for the element w22 defined by
w22=}'2-w2l = z2-wl2 A6.48)
we have w22$s0. Since by A6.47) and A6.48) we have A6.46) with
r = s = 2, the lemma is proved for r = s = 2.
Assume now that the lemma is true for all pairs of finite sequences
{yi}f=i,{zk}l=1cJf with 2^p^r,2^q^s and let yu...,yr, zu...,zs+1
r s+ 1
еЖ be arbitrary, satisfying E^>= E z*- Then ^ the induction
i=l k=1
hypothesis applied to the pair of finite sequences yl,,..,yr, z = z1+z2,
1 One can also prove the converse statement and thus this decomposition
lemma actually characterizes the minihedral cones.
31 Singer, Bases in Banach Spaces I

482
II. Special Classes of Bases in Banach Spaces
z3,...,zs+leJf, there exist r-s elements п>!кеЖ (i= 1,
such that
s
..,r; k=l,...,s)
z = ;
i= 1
i= 1
Furthermore, by the induction hypothesis applied to the pair of
sequences wtl,...,wrl, zi,z2eJf, there exist 2r elements щкеЖ
(i=l,...,r;/c=l,2) such that
Consequently, we have
Уг= E uu+ E ^
J=l k = 2
i= 1
i= 1
i.e., a decomposition of type A6.46) for the pair yu...,yr,zu...,zs+1eJf,
which, since the roles of j, and zk are symmetric, completes the proof of
lemma 16.2.
Now we can give the
Proof of theorem 16.4. Since dim? = n<co, by the classical theorem
of Minkowski-Weyl Ж coincides with the convex hull of its extremal
rays1, whence, since Ж is solid and dim? = n, it follows that Ж has at
least n distinct extremal rays. We shall show that Ж has exactly n distinct
extremal rays [0,Xi[,...,[0,х„[ which will complete the proof (since
then {Xj}1 will be a basis of ? such that Ж{х.,п = Ж).
Assume, a contrario, that Ж has at least n+l extremal rays, say
[0,Xj [,...,[0,х„+1[. Then, since dim? = n, x1,...,xn+l are linearly
n+l
dependent, i. e., there exist real numbers al,...,an + l with ?|а;|#0,
such that Y, atxi = 0- Put
}, S={k\ak<0}
A6.49)
1 For the definition of extremal rays of Ж see § 10.
16. Some characterizations of unconditional bases
483
and let r,s denote the number of elements of R and S, respectively. Then
and r + s = n+ 1 ^2. Put
yi = a.ixi (ieR), zk = (-ak)xk (keS). A6.50)
Then {y,}isR, {гк}ке5аЖ and J] j,= ^ zk. Let us prove that
ieR ksS
s^2. If we had r = 0, then 0= ^ z^zk^0(keS), whence
ak = 0(keS), which, together with R = 0, would contradict the assump-
tion ? |a,-|#0. Assume now that r—l, say R={i,}. Then, if а;, = 0,
i= 1
we arrive, as in the case r=0, to a contradiction with the assumption
? |af|#0. If a.^0, then 0/jil= ^ zk = \{2zkl) + \{l ^ zX where
i=l ksS teS\{ti) 7
2ztl^[0,j;il[ (since the rays [0,ztl[=[0,Jctl[ and [0,^,[=[0,хA[ are
distinct by our hypothesis), which contradicts the assumption that
[0,jfl[= [0,x;i[ is an extremal ray of Ж. Thus, r^2 and, similarly,
Consequently, by lemma 16.2, there exist r-s elements юл
(ieR, keS) such that
A6.51)
ksS
isR
Then for any wioko^Q the relations A6.51) for yio,zko imply, taking
into account that [0,jio[= [0,xio[ and [0,zto[= [0,xto[ are extremal rays,
where /o,yio>0. However, this contradicts the assumption that the
extremal rays [0,jIO[=[0,xio[ and [O,zto[=[0,jcto[ are distinct and
thus the proof of theorem 16.4 is complete.
Theorem 16.4 cannot be extended to infinite dimensional Banach
spaces, as shown by
Example 16.6. Let Ж be the natural positive cone in the space
? = C([0,l]) (i.e., хеЖ if and only if x(t)^0 for all fe[0,l]). Then Ж
is solid and minihedral, but there exists no E-complete total biorthogonal
system (х„,/„) such that Ж(Хп4п) = Ж.
Indeed, otherwise, since Ж is obviously normal and generating,
from theorem 16.3 it would follow that {xn} is an unconditional basis
of ?=C([0,1]), in contradiction with § 15, theorem 15.1.
We shall now characterize unconditional bases among total bior-
biorthogonal systems in terms of properties of the set of multipliers1.
1 For the definition of the sets of multipliers M(x,{xn,fn)) and М(Е,(х„,/„))
and for characterizations of bases by properties of the set M(E,(xn,fJ), see Ch. I, §5.

484
II. Special Classes of Bases in Banach Spaces
Theorem 16.5. Let E be a Banach space, (х„,/„) ({х„}<=Е, {/„}<=?*)
a total biorthogonal system and xeE. Then
a) If E is separable, the following statements are equivalent:
GO
1°. Y ft(x)xi is unconditionally convergent.
Т. М(х,(х„,/„)) = т( = П.
If E contains no subspace isomorphic to c0, statement 1° is equivalent
to the following:
3°. М(х,(х„,/„))^с0.
b) If E is separable, the following statements are equivalent:
1°. {х„} is an unconditional basis of E.
2°. М(?,(х„,/„))эт. Moreover, in this case М(Е,(х„, fn)) = m
and the identity mapping is an isomorphism of M(E,(xn, fn)) onto m.
If E contains no subspace isomorphic to c0, statement 1° is equivalent
to the following:
3°. M(E,(xn,fn))^c0.
c) If [х„] = ?, the statements b) 1°, b) 2°, and b) 3° are equivalent.
Proof, a) The implication 1°=>2° is a consequence of lemma 16.1
(implication 1°=>5°).
Conversely, assume that E is separable and that М(х,(х„,/„))от.
Then an argument similar to that used in Ch. I, § 5, the proof of theorem
5.2 (implication 2°=>Г) shows that the mapping ux: {у„}-*х{Уп] of m
into E is continuous. Now, for any {yn}em and any n the sequence
= {{yi' ••>}'„,0Д...}}*=1 is a Cauchy sequence for the
¦¦ l
weak topology <r(c0, eg), whence also for o(m,m*). Since every continuous
linear mapping of m into a separable Banach space transforms weak
Cauchy sequences into norm convergent sequences1, it follows that the
sequence < Y У;/;МХ;(" = \uA Y 7iei )r is convergent, whence, by
lemma 16.1 (implication 5°=>1°), the series Y fi(x)xi *s uncondition-
unconditionally convergent.Thus, 2°=> 1°. i=1
The implication 2°=>3° is obvious.
Conversely, assume now that E contains no subspace isomorphic to
c0 and that М(х,(х„,/„))эс0. Then, by the above, the mapping ux. {}•„}
—>xGn! of c0 into E is continuous. Hence, since the series Y ei ш со
is weakly unconditionally Cauchy, so is the series ? /;(х)х,= ? wx(ef)
i = 1 i = 1
1 See e.g. [88], p. 168 or [91], Ch. V, §4, exercise 12 and [50], p. 494, theo-
theorem 4.
16. Some characterizations of unconditional bases
485
in E. Consequently, by § 15, lemma 15.8, ? fi(x)xi is unconditionally
convergent in the norm topology. i ='
b) The implication 1°=>2° is a consequence of part a) above (im-
(implication 1°=>2°) and of Ch. I, § 5, proposition 5.4c) and the inversion
theorem of Banach. Alternatively, instead of the inversion theorem of
Banach one can also observe directly that in this case for any |у„}ет we
have, by theorem 16.1,
\Ы\\
м1ЕЛх„М)
sup sup \\s{.
. SKx xeE
sup \yn\,
A6.52)
where C= sup sup ILs1,» ., . The other implications are im-
|Pl|.|/Bl,...,«l 1«1<X
mediate consequences of part a), taking into account that {/„} is total
on E.
c) Assume now that [х„] = ? and that we have 3°. Let a be an
arbitrary permutation of Ж= {1,2,3,...}. Then, since
M(E,(xaM,/„<„,)) = {{ya{n)} | {Уп}еМ(Е,(хп,Щ,
from 3° it follows that M(?,(xtT(n),/„<„,)) contains every non-increasing
sequence tending to zero, whence, by Ch. I, § 5, theorem 5.2 (implica-
(implication 3°=>Г) {x,,,,,,} is a basis of E. Consequently, {х„} is an uncondi-
unconditional basis of E, which completes the proof of theorem 16.5.
The assumptions on E made in theorem 16.5 are essential, as shown
by the following two examples-.
Example 16.7. Let E = m, xn = the и-th unit vector, and /„ = the n-th
coordinate functional (и=1,2,...). Then (х„,/„) is a total biorthogonal
system, ? is non-separable, [х„] = со#? and for any xem\c0 we have
a) 2°, but not а) Г. Hence we have bJ°, but not b) Г.
Example 16.8. Let E = c, х„ = the и-th unit vector, and /„ = the n-th
coordinate functional (n=l,2,...). Then (х„,/„) is a total biorthogonal
system, ?эс0 isometrically, [х„] = со/?, and for any xec\c0 we
have a) 3°, but not а) Г. Hence we have b) 3°, but not b) Г.
Problem 16.11. Let E be separable, (х„,/„)((х„} с ?,{/„} с ?*) a
total biorthogonal system and xeE. a) If М(х,(х„,/„)) contains all
GO
sequences {е„} with ?„= + 1 (n= 1,2,...), is the series Y fAx)xi uncon-
1 Recently problem 16.1b) has been solved in the affirmative (see the Notes
and remarks).

486
II. Special Classes of Bases in Banach Spaces
ditionally convergent? b) If M (?,(*„,/„)) contains all sequences {е„}
with е„=±1 (и=1,2,...), is \xn} an unconditional basis of El What
happens if [х„] = ??
We shall now show that in the particular case when the linear sub-
space Vo of E* spanned by {/„} is of characteristic1 r(Vo)>0, the
answer to problem 16.1a), whence also the answer to problem 16.1b),
is affirmative (theorem 16.6 below).
Lemma 16.3. Let E be a separable normed linear space and V a linear
subspace of E* with r(V)>0. Then there exists a norm |||x||| on E, equiv-
equivalent to the initial norm \\x\\ on E and having the following two prop-
properties :
(K2)
If2 xn
//
(П
x0, then
х0 and lim |||xj| = |||xo|||, then lim |||х„-хо||| = 0.
Proof. Since E is separable, the unit cell SE»={feE* | ||/||^ 1} is
compact and metrizable3 for the topology a(E*,E). Let p be a metric
on S?,, inducing a topology equivalent to a(E*,E). Put
(O0(x)=\\x\\v= sup \f(x)\ (xeE),
feVr,SE*
A6.53)
юк(х)= sup \f(x)-g(x)\ (xe?,fc=l,2,...), A6.54)
11*111 = У i«
(xeE).
A6.55)
We shall prove that |||x||| has the required properties. Indeed, |||x|||
is a norm on E, equivalent to the initial norm ||xj| on E, since by Ch. I,
§ 12, formula A2.13) and by A6.53), A6.54), A6.55) we have
(xeE). A6.56)
Furthermore, let х„-^->х0 and let s>0 and /c^O (integer) be
arbitrary. Then there exist functionals j\g'eVnSE, with p(f',g')^-r
1 See Ch. I, §12.
2 We recall (see Ch. 1, § 13, the proof of proposition 13.1) that we write х„
or (V)-\imxn = x if /(*„)-»/(x) for all feV.
3 See e.g. [50], p. 424, theorem 2 and p. 426, theorem 1.
16. Some characterizations of unconditional bases
487
and \f'(xo)-g'(xo)\^wk(xo)-- (respectively, \f'(xo)\^mo(xo)-- if
/c = 0) and a positive integer N = N(e,k) such that \f'(x0)— f'(xn)\ <—,
\g'(xo) — g'(xn)\<— for all n>N. Hence
шк(х0) ^ \f'(xo)-g'(xo)\ + | ^ \f'(xo)-f'(xn)\ + \f'(xn)-g'(xn)\
\9'{xH)-g'{x0)\ < \f'(xn)-g'(xn)
(n>N)
(respectively, юо(хо) ^ \f'{xo)\ + у ^ \f'(xo)-f'(xn)\ + \f'(xn)\ + y
<|/'(х„)| + е for n>N), and therefore
cok{xn)>cok(x0)~e (n>N(E,k);k = 0,1,2,...).
Consequently, we have
Шсок(хп)^(ок(х0) (/с = 0,1,2,...) A6.57)
and hence it follows that |||x||| satisfies (K^. Indeed, by the remark made
below on the functions A6.59), for every e>0 there exists a positive
integer p=p(e) such that
N = N(s,p(s)) be such that
— for all k>p. By A6.57), let
o}k(xn)>cok(x0)--
Then
k=0
x 1 °° 1 e p e
k = 0 k= v +1 fe=:0
and consequently we have (Ki).
Finally, let х„-^>х0 and lim |||х„||| = |||jco|||. Then
П-*00 ^ = Q 2 П-*00
whence, by A6.57), it follows that
lim cok(xn) = wk(x0) (k = 0,1,2,...)
= 0,
A6.58)
(indeed, if A6.58) is not satisfied, then by A6.57) there exist an index
k0, an infinite subsequence [xn } of {xn} and a 5 > 0 such that

488
II. Special Classes of Bases in Banach Spaces
(»ko(xnp)-(oko{x0)^S {p=l,2,...), whence ? jkwk(xnp)^ ? ^ю*(х0)
CO 1
+ 2E (/?= 1,2,...), contradicting lim ? —^ [о>к(х„) — a>t(xo)] = 0).
Since for each xeE the function f->f(x) (/eS?.) is continuous on
S?, for the topology a(E*,E), whence also uniformly continuous for
this topology, the function
фx(f) = f(x) (JeVnSE,) A6.59)
is uniformly continuous on V n S?» for the topology induced by
a(E*,E). Hence, in particular, for anv e>0 there exists an integer
/co = /co(e)>0 such that wko(x0)<—. By A6.58) there exists an integer
N = N(e,ko)>0 such that \и>ко(хп) — соко(х0)\ <— for all n>N, whence
e
°hjyxn) < Oh0(xo) + -z < ? {n>N)
and thus, by A6.54), the sequence of functions {фХп} is equicontinuous
on VnSE*. Furthermore, by A6.56) and lim |||х„||| = |||xo||| we have
sup sup \фХп(/)\= sup
«n<co /e VnSE* l«n<c
sup |||xj|<cc,
and thus the sequence {фХп} is uniformly bounded on Vn SE,. Since
by xn-^-*x0 we have фХп{/)->фХ0(/) (JeVnSEt), from the theorem
of Arzela it follows that ]|х„ —xo|]K= sup \фх (/) — фхп(/)\ ->0 as
feVnSr.»
п-кх), whence, by A6.56), lim |||х„ —xo||| = O. Thus, |||x||| satisfies (K2),
П-* CO
which completes the proof of lemma 16.3.
Lemma 16.4. Let Ebea normed linear space and let {/„} be a sequence
in E* such that the linear subspace V of E* spanned by {/„} satisfies con-
condition (Ki) (with the initial norm on E). Then for every finite dimen-
dimensional linear subspace G of E and every e>0 there exists a positive
integer N = N(G,e) such that
|[v + x|| ^A — e)||j|| (yeG, хе[/1,...,/лг]1). A6.60)
Proof Let G be an arbitrary finite dimensional linear subspace of E.
We claim that for every jeG and e>0 there exists a positive integer
= N\y,-\ such that
(xe[fu...JN]±).
A6.61)
16. Some characterizations of unconditional bases
489
Indeed, assume the contrary, i.e., that there exist an element
and an eo>0 for which there is no such N. Then for each N=1,2,...
there exists an element xNeE such that
A6.62)
A6.63)
However, by A6.62) and since К is the linear subspace spanned by
{/„}, we have x^-^O, whence yo + xN-^-*yo and hence, by (Kj),
lim l|jo + x)vll > lljoll, in contradiction withA6.63). This proves the claim
JV^co
A6.61).
Since dimG<co, the unit sphere aG={yeG\ \\y\\ = 1} is compact
and hence it has a finite e-net, say y1,...,ym. Put
A6.64)
N(G,e)= max N[yh-
1 ^i^ \ 2
Let yeaG be arbitrary and let yt be such that ||j-Jil| < -. Then
for any xeE with/„(*) = 0 (и =l,...,N(G,e)) we have, by A6.61) applied
>(l --Jlb.-ll -2=1-?'
Consequently, for any yeG with j#0 and xeE with Jn(x) = U
(n=l,...,N(G,E)) we have
whence we get A6.60) for y?=0. Since A6.60) is obviously valid for
j = 0, the proof of lemma 16.4 is complete.
Proposition 16.3. Let E be a separable Banach space and (*„,/„)
{{х„} с Е, {/„} с ?*) a biorthogonal system such that for the linear sub-
subspace Vo of E* spanned by {/„} we have r(Fo)>0. Then there exist two
sequences of integers {т„}, {к„} with
= mo<k1<mi<k2<m2<''
A6.65)
such that for every xeE satisfying /;(x) = 0 for i = kn+l,...,mn,n=l,2,...
(or, respectively,fi{x) = Q for i = mn-l + l,...,kn; n=\,2,.), we have

490
II. Special Classes of Bases in Banach Spaces
со / mn
A6.66)
(respectively, x= ? ( ? Mx)xi)) ¦
n=l \i = kn+ 1
Proof. By lemma 16.3 there exists an equivalent norm |||jc||| on E,
with properties (КД (K2). Let l>?0>?1> •¦•, Угпе„ = 0 and let
/?„)=[х1,...,х„](и=1,2,...). Put
kn = N(P(mn_ , + !,,?„_!),
(n=l,2,...), A6.67)
where the integers N(G,e) are chosen according to lemma 16.4. Since
for ?<1 and each и =1,2,... we have N(PW, e)>n—l (because for
y = x = xn we have f1(x) = - = fn_l{x) = 0, but 0= ||j-x|| <(l-?)||y||),
it follows that {т„}, {/с„} satisfy A6.65).
Now let xeE be an arbitrary element satisfying
fi(x)=0
Put
A6.68)
A6.69)
Then, since by A6.69) and A6.68) fj(yr) =fj(x) (/= l,...,kr; r= 1,2,...),
we have yr^^x, whence, by (KJ,
Jimllbvlll^llWII-
A6.70)
On the other hand, since by A6.69) and A6.68) yrePikr)<= P(K+1)
and fj(x— yr) = 0 (j= l,...,mr) and since mr = N(iJtr+1), er), from lemma
16.4 it follows that
whence
which, together with A6.70), gives lim |||jj| = |||x|||. Since yr^x,
from (K2) we obtain
lim yr = x
in the norm |||x|||, whence also in the initial norm ||x|| on E.
The argument for the case when/;(x) = 0 (г = т„_1 + 1,...,/с„;и= 1,2,...)
is similar, which completes the proof of proposition 16.3.
16. Some characterizations of unconditional bases
491
Now we are ready to prove
Theorem 16.6. Let Ebea separable Banach space and (xn, /„) ({х„} с Е,
{/„}<=?*) a biorthogonal system such that the linear subspace Vo of E*
spanned by {/„} is of characteristic r(K0)>0. Then
a) For an element xeE the set M(x,(xn,Q) contains all sequences
GO
{?„} with ?„=±1 (и =1,2,...) if and only if the series ? fAx)xi is
unconditionally convergent. i=1
b) {х„} is an unconditional basis of E if and only if M(E,(xn,fn))
contains all sequences {?„} with ?„ = ±1 (n = 1,2,...).
Proof, a) If ? fiix)xi is unconditionally convergent, then, by theorem
Conversely, assume that for an element xeE the set M(x,(xn,/„))
contains all sequences {?„} with ?„= +1 (n= 1,2,...). Let {тп},{к„}
be as in proposition 16.3 and let {?„} be an arbitrary sequence with
?„=±1(и=1,2,...). Put
1 for i = kn+l,...,mn; n= 1,2,...
?j for i = mn-i + 1,...,/с„; п= 1,2,
.-1 for г = /с„+1,...,т„;и=1,2,...
Then, since by our assumption {е„}, {41'}, {e{n2)}eM(E,(xn,fn)), there
exist elements x(?ri}, jc{eu)}, х{Еи)}еЕ such that
Put
x' = ^(X{Al)} + x{Eu,}). A6.74)
Then, by A6.73), A6.71) and A6.72),
ffii/i(jc) for г = т„_1 + 1,...,/с„;и=1,2,...
0 for i = kn+l,...,mn;n=l,2,...
whence for x" = x|?n| —x' we have
0 for г = т„_1 + 1,...,/с„;п=1,2,...
/i(x) for i=kn+l,...,mn-,n=l,2,...
Consequently, by proposition 16.3,
к„

492 II. Special Classes of Bases in Banach Spaces
and hence, putting 12п_1=к„,12п = т„(п=\,2,...), we obtain
In
X,c ,=x' + x" = Hm Y ?;/;(*) X; (fi ¦= ± 1 ;J= 1,2,...). A6.75)
Assume now that there exists a sequence {en} with г„= + l(n= 1,2,...)
GC
such that the series Z ?,/;(х)х; does not converge. Then there exist
;= l
a d>0 and positive integers pl <p2 <••• such that
'Z е.-/д
A6.76)
Let {l'n} be a subsequence of {/„} with the property that for each n
there exists ар'„е{рп} such that Гп + 1^р'п + 1^р'п + 1^Гп + 1 and let e'n = ±1
In + 1
Z Шх
(и =1,2,...) be such that
(и=1,2,...). Then, by A6.76),
= max
?i= ±1
Z
?ifi(x)Xi- Z Zifi(x)xi~ Z
In^ 1
Z ?ifi(x)xi
which contradicts A6.75). This proves that all series Z ?iMx)xi with
i=l
?„= +1 (и = 1,2,...)converge, whence, by lemma 16.1 (implication 4° =>1°),
CO
Z fi(x)Xi converges unconditionally.
b) is an immediate consequence of a), taking into account that {/„}
is total on E (by r(Vo)>0), which completes the proof of theorem 16.6.
The following theorem on unconditional bases corresponds to Ch. I,
§ 5, theorem 5.3 (on general bases)-.
Theorem 16.7. Let Ebea Banach space and(xn,fn)({xn} a E, {/„} с Е*)
an E-complete biorthogonal system. The following statements are equiv-
equivalent:
1°. {х„} is an unconditional basis of E.
16. Some characterizations of unconditional bases
493
2°. There exists a continuous linear mapping {yn}^>v{ynj of m into
L(E,E), such that
en(x) = fn(x)xn
A6.77)
where en is the n-th unit vector {дп1}™=1 in m.
3°. There exists a continuous linear mapping {yn}^v{yn} of c0 into
L(E,E), satisfying {16.17).
Moreover, if {х„} is an unconditional basis of E, then the mapping
{y«}~*v{yn) 's a) an isomorphism of m onto a complemented subspace of
L(E,E) and Ъ) an isomorphism of с0 onto the subspace [uej o/1
and the subspace [uej is complemented in
Proof. Assume that {xn} is an unconditional basis of E. Then, by
theorem 16.5 (implication b)l°=>bJ°), we have M(E,(xn,fH)) = m and
the identity mapping m->M(?, (*„,/„)) is an isomorphism. Since by Ch. I,
§5, proposition 5.4b), the mapping {yn}->v{yn}, where v{yn](x) = x{yn}
(xeE), is a linear isometry of M(E,(xn,fn)) into L(E,E), it follows
that this mapping is an isomorphism of m into L(E,E) (and hence2
onto a complemented subspace of L(E,E)), satisfying A6.77).
The implication 2°=>3° is obvious.
Assume now that we have 3°. Then there exists a constant
such that we have, for any finite sequence of scalars }'i,•••,}'„
sup
xsE
I 7,e.
sjC sup |y;|
whence3, by theorem 16.1, {х„} is an unconditional basis of E.
Furthermore, assume again that {xn) is an unconditional basis of
f
E. Then, by the above, the mapping {yn}
v{yn)
is an isomorphism of
{}{yn)
c0 onto the subspace [oej of L(E,E). Let С be the norm of this iso-
isomorphism and let {у„}ес0 be arbitrary. Then
"{>>„}- Z
^(v.| —l
1 + 1
sup
¦ 0 as /->oo,
1 We recall (see Ch. 1, § 18) that we denote by «"(?,?) the subspace of L(?,?)
consisting of all compact linear mappings of E onto E.
2 See e.g. [43], Ch. V, §4, remark (9)(a) or theorem 3.
3 If ? contains no subspace isomorphic to c0 and (xn,fn) is assumed to be total
(but not necessarily ?-complete), the same conclusion holds (by § 15, lemma 15.8).

494
II. Special Classes of Bases in Banach Spaces
whence, since Y 7i"e, is °f finite rank, we infer that vtynj is compact.
Thus, [yen~] <= ^{E, E) and it remains to prove that [ve ] is complemented
in #(?,?).
We may assume, without loss of generality, that inf ||xJ>0
(whence sup ||/„|| < oo). We claim that i««<oo
1 5П< 00
{/»["(*¦.)]}«„ (иеЩЕ,Е)). A6.78)
Indeed, let {/„} be an arbitrary sequence of positive integers. Then,
since и is compact, {u{xin)} has a subsequence, say {u(xin)}, con-
converging to an element yeE. Consequently,
§ j < oo
as
was an arbitrary subsequence of {/„[«(*„)]},
whence, since {fin [u(x
we infer A6.78).
Thus, since by the above {yn}^>v{Vn] is an isomorphism of c0 onto
the subspace [uej of 4>(E,E), the mapping
00
n(u) = Y fi["(-*;)] vet (ue<?(E,E)) A6.79)
of ^{Е,Е) into [yen~] is well defined and continuous (actually, we have
\\n(u)\\^C sup Их,!! ||/;|| ||u|| for all иеЩЕ,Е)). Since by A6.77)
/' = 1,2,
it follows that тг is a continuous linear projection of <?{E,E) onto [ие„],
which completes the proof of theorem 16.7.
Finally, let us give some characterizations of unconditional bases
among bases.
Theorem 16.8. Let {xn} be a basis of a Banach space E. The follow-
following statements are equivalent:
1°. {xn} is an unconditional basis of the space E.
2°. For every increasing sequence of indices {in} the subspaces [x;j and
[Xj]JF, x(iri) are complementary to each other (i. e., E = [x( J © [xj]jS^\{in})-
3 . For every sequence of scalars {sn} with ?„= ±1 (и =1,2,...), {xn}
is equivalent to the basis {е„х„} of E.
4°. For every sequence of scalars {/?„} with 0<a < |j8J ^P< oo
(и =1,2,...), {х„} is equivalent to the basis {finxn} of E.
Proof. 1°=>2°. Assume that {xn} is an unconditional basis of E and
let, {/„} с E* be the a. s. с f. to {xn} and {*„} an arbitrary increasing
16. Some characterizations of unconditional bases
495
sequence of indices. Then every x= Y ft(x)xi can ^e written in the
form i=1
A6.80)
j= 1
where both series in the right hand side are convergent (by lemma 16.1,
implication 1°=>3°), the first to an element ^е[х,п] and the second to an
element ze[_x^]je^X{iri] (because these subspaces are closed). Since obvi-
obviously [*iJn[*;]ye>-4l-n)={0}, it follows that1 E = [xiJ®[xj~]jeJrX{ittj.
2°=>Г. Assume that {*„} is a basis of E, satisfying 2° and let
{/„} с E* be the a. s. с f. to \xn) and {/„} an arbitrary increasing se-
sequence of indices. Then, by our hypothesis, every xeE can be written, in
a unique way, in the form x = y + z, where уе[х(п] and ze\_x^\jsJV-Viin].
Hence
z) = ftj(y) (xeEJ= 1,2,...),
J=l
and therefore, by Ch. I, § 4, proposition 4.1, the series
= ? fi-(y)Xi.=y converges. Consequently, by lemma 16.1 (implication
i= i oo
3°=>1°), Yj fi(x)xi is unconditionally convergent for every xeE, i. e.,
i= 1
the basis {xn} is an unconditional basis of E.
1°=>4°. Assume that {xn} is an unconditional basis of E and let
0<a^|j8nKj8<c». Then {i3nxn} is a basis of E. Furthermore, if
QO
? <XjJC; converges, by lemma 16.1 (implication 1°^>5°) the series
i = l
00 1 CO
^ — Pi<XiXh whence also the series ^ ^,ос;Х;, converges, Conversely,
i=l P i=l
00
if Yj Pi^iXi converges, then again by lemma 16.1 (implication Г=>5°)
oo oo „ ro
the series ? a.aixi= ^ —^а^ь whence also the series J] aixi' con"
verges. i = 1 i = l^ i = 1
The implication 4°=>3° is obvious.
00
3°=>1°. Assume that we have 3°. Then ? a;X; is convergent if and
only if
is convergent for all {е„} with г;„=+1 (и=1,2,...),
1 Moreover, from theorem 16.1 (implication Г=>19°) it follows that the pro-
jections x-* Y fij(x)xij {xeE), where {in}e&, are uniformly bounded.

496
II. Special Classes of Bases in Banach Spaces
QO
which happens, by lemma 16.1 (equivalence I°o4°), if and only if ? а;х,
i = l
is unconditionally convergent. Thus {х„} is an unconditional basis of E,
which completes the proof of theorem 16.8.
Remark 16.3. The condition 2° of theorem 16.8 cannot be replaced
by the weaker condition that for every increasing sequence of indices
{/„} the subspace [x;j be complemented in E, since e. g. in the Hilbert
space I2 every subspace is complemented (admitting even a projection
of norm 1), but there exist conditional bases of I2 (see § 14, examples
14.4 and 14.5). Next, it is natural to ask whether every basis {х„} of
any Banach space E satisfies this weakened condition (i. e., that for
every {in}e& the subspace [x;j be complemented in E). We shall see
that the answer is negative, in Vol. II, Ch. IV, where we shall also study
other relations between bases and complemented subspaces.
It is also natural to ask the "unconditional analogue" of Ch. I, § 4,
problem 4.1, i.e., the question, whether in a Banach space E with an
unconditional basis every unconditional basic sequence can be ex-
extended to an unconditional basis of the whole space E. The answer
is negative, since e.g. in E = ll the natural basis {zn} of the subspace
B2 = (/2 x l\ x •••),! (see §8, example 8.1) is a bounded unconditional
basic sequence which is not equivalent to the unit vector basis of E = ll
and hence {zn} cannot be extended to an unconditional basis of E (since
by § 18, theorem 18.2, every bounded unconditional basis of E = ll is
equivalent to the unit vector basis of I1, which is obviously equivalent
to each of its subsequences).
However, the "unconditional analogue" of Ch. I, § 4, proposition 4.2,
is true, namely, we have
Proposition 16.4. Let G, F be two Banach spaces with unconditional
bases {у„} and {zn}, respectively. Then the sequence {х„} с Gx F defined
by
х2„-1 = {У„0}, х2я={0,гя} (и=1,2,...), A6.81)
whence also every permutation {xa{n)} of this sequence, is an unconditional
basis of Gx F.
Proof. Let ?„= +1 be arbitrary. Then, since {>>„}, {zn} are uncon-
unconditional bases of G and F respectively, by virtue of theorem 16.8 (im-
(implication 1°=>3°) J?2n_j,} is a basis of G, equivalent to {>>„}, and
{s2nzn} is a basis of F, equivalent to {zn}.
Now, if ? afXj converges, then, as we have observed in Ch. I, § 4,
' = 1 00 00
the proof of proposition 4.2, ? x2i_lyi and ]T a2izl converge, whence,
i = 1 i = 1
16. Some characterizations of unconditional bases
497
by the preceding remark, ? ?21-1 a2;-iJV; and Л ?2ia2.zi converge
and hence the series ]Г ?;(х;х, converges. Consequently, by lemma 16.1
1=1
(implication 4°=>1°), ? <х,Х; is unconditionally convergent, and thus
{х„} is an unconditional basis of GxF. Hence, by § 17, theorem 17.1
(equivalence I°o2°), every permutation {xa{n)} of {х„} is also an un-
unconditional basis of GxF, which completes the proof of proposi-
proposition 16.4.
In other words, proposition 16.4 says that the cartesian product of
two unconditional bases is an unconditional basis. For tensor products
of bases the situation is different, as shown by the following example
(for more complete results on tensor products, see § 17):
Example 16.9. The tensor square1 {.v,®x) of the unit vector basis
of I2 is not an unconditional basis of I2 ®y/ , nor of l2 ®я I2.
Indeed, to prove the first assertion we shall exhibit a functional
GC
Ae(/2®y/2)* and an element ? ао-х(®л;уб/2®у/2 such that
00 *' J ~ 1
? |/г(ао-х((х)Х;)| =сс. Let {А„} be the sequence of 2"x2" matrices
defined by
1/2
A —
n2
/1
2
1
2"
1
2"
1
\2
1
2"
1
~2"
1
I
1
1
2"
1
1
~~2
1
~2~
I\
1
~ 2"
1
~T
1
\ (A A \
, . . . , JLf, 1 1 5 • • • V •*J^i^
let u: 12->12 be the continuous linear mapping defined by the cartesian
product of these matrices (on I2 = (l\ x 1\г х--- х 1\„ x---)l2), i.e.,
1 See Ch. I, § 18, the remark made after the proof of theorem 18.1.
32 Singer, Bases in Banach Spaces 1

II. Special Classes of Bases in Banach Spaces
u(x) =
A6.83)
and let he{l2 ®yl2)* be the image of и by the canonical linear isometry
L(/2,/2)s(/2®y/2)*, i.e.,
h(x®y) = {x,u(y)) {x el2,у el2).
A6.84)
Furthermore, let
1/2" ]/Y" \/T
and let
(и=1,2,...), A6.85)
A6.86)
Then, since the supports of yhyj are disjoint for i^j, we have
Xi 1 "° 1
(v,,v,) = 0(;V/), whence ||z||2= У — ||Vn||2= У ^<oo, i.e., ze/2,
ОС ОС ОС
whence z® z= ? С(х;® ^ Cj^-= Z CiCjXi®Xjel2 ®yl2. On the
¦•=i j=i «,j=i
other hand, since C,- = C/ = , |(*;,и(х,-))| = (;j' = 2"-l,2",...
и l/2" l/2"
2"+1-2;и=1,2,...), we obtain
z
^ z Bf -
= z
= х.
which completes the proof of the first assertion of example 16.9. Now,
if J.Yj ® Xj] were an unconditional basis of I2 ®; /2, then its assoc-
associated sequence of coefficient functionals, which is again [xt ® Xj},
would be an unconditional basic sequence in (I2 ®; I2)* = l2 ®J2 (by
§ 17, theorem 17.7), which is impossible by the above.
17. Intrinsic characterizations of unconditional bases
499
§ 17. Intrinsic characterizations of unconditional bases. Some
more separable Banach spaces having no unconditional basis.
Properties of strong duality. Unconditional bases and
sequence spaces
Let ? be a Banach space, {xn} a sequence in E and d
' § 16, formula A6.3)). We shall use the notations
(see § 16,
\\x\\ = l} =
= {'i UeS
A7.1)
A7.2)
A7.3)
A7.4)
i=l
i=l
Y,%xh (m<n,im^k,im+i>k) A7.5)
!«„*„ (
and the notation F of Ch. I, § 6, formula F.18).
Theorem 17.1. Let E be a Banach space and {xn} a complete sequence
in E such that xn # 0 (n = 1,2,...). The following statements are equivalent:
1°. {xn} is an unconditional basis of the space E.
2°. Every permutation {xa(n)} (аеП) of the sequence {х„} is a basis
of the space E.
3°. There exists a countable directed set of endomorphisms {ud}
= {м>;, i ,} с L{E,E), where d={il,...,in} runs over the countable
directed set 9) (§ 16. formula A6.3)), satisfying
ud(x) =
(xeP(d),de2!),
(xeP(d\de2),
A7.6)
A7.7)
A7.8)
/и this case, the set \ud)deS is uniquely determined and coincides with
the set of partial sums {sd}de3 defined in § 16, formula A6.6).
32*

500 II. Special Classes of Bases in Banach Spaces
4G. There exists a constant M2 with 1 ^M2 < x, such that1
ZtX, X:
lj h
Y Of; X: + У (X, JC,
A7.9)
for any n,me..V and any scalars ail,...,ain,all,...,almeK whose indices
satisfy {«!,...,/„}п{/1,...,/т} = 0.
5°. There exists a constant M3 w/Y/i 1 ^ M3 < oc, such that
У С-О.-Х
L, WJ-i^-i
i= 1
;m,
Z «.*.
A7.10)
for any пе.Л" and any e,u...,cn, a.l,...,a.neK with e;=+l (<=1,...,и).
6°. There exist two constants m4,M4 with 0<m4^ 1
Z l«,i*,
i= 1
и
Z aixi
i= 1
«SM4
n
Z ««I*,
i= 1
A7.11)
/or дату яе ,/V and any scalars a.l,...,a.neK.
l'J. There exists a constant M5 with l^M5<x, such that
Z Pi*i*i
i- 1
z ««*«
A7.12)
/or anj иб,/Г аил? Р1,...,Р„, аи...,а„еК with |j8,-|^l (г= 1,...,и).
8 . We have
M6 = inf dist(cr(d), P(d>) > 0. A7.13)
9C. We have
infdist((j(d), o-<
A7.14)
10е. Г/геге exwfs a constant M7, l^M7<x, vw7A the following
property: for every de3> and poePid) there exists an feE* such that
f(Po)=\\Poh (П.15)
/(y) = 0 (yeP<d>), A7.16)
;M7. A7.17)
11". We have
M8 = sup sup \\Sd{p)\\<x.
dES pep
M
A7.18)
1 One can also write condition 4' in the following form:
for any dud2e& with d1czd1 and any {ai}ied2
17. Intrinsic characterizations of unconditional bases
501
Furthermore, for the above constants we have1
1 s= M! = inf M2 = = inf M7 = M8 sC inf M3 «S inf M5 ^ 2 inf M3
M6
^4infM2^x, A7.19)
where < x AoWs if and only if {xn} is an unconditional basis of E.
Proof. 1"=>2°. If {xn} is an unconditional basis of E with the a.s.c.f.
GO
{/„} and if <xe/7, then ^ /I(i)(x)xJ(i) is convergent, obviously to x
(since /Д ^ /,(j)(i)i,ffl =/jW, ./=1,2,...) and thus, by Ch. I, § 4,
\i=l /
theorem 4.1 (implication 2"=>1°), {x^)} is a basis of E.
2°=>Г. If every permutation {х„(п)}((теП) of the sequence {xn} is a
X)
basis of ?, then for every x= ^ /(х)х(е? (where [/„}cz?* is the
i= 1
a.s.c.f. to {х„}) we have x= ^ /ff(i)(x)xtr(j) (since the a.s.c.f. to {*ст(я)}
1 = 1 ОС
is obviously {/ff(n)}), and therefore ^ ,/i(x)x,- is unconditionally
i= 1
convergent. Thus, |х„} is an unconditional basis of E.
1 >=>3°.lf |х„} is an unconditional basis of E, then, by the implication
1°=>19° of § 16, theorem 16.1, the countable directed set of partial sum
operators {sd}ieS (defined by formula A6.6)) satisfies A7.6), A7.7) and
A7.8), and obviously it is the only set having these properties (since ud
is the projection of E onto P(d) along P(d)).
3"=>4°. By A7.6), A7.7) and A7.8) we have
У «i x.
Z «о^+ Z «/^ij
whenever {il,...,in}n{ll,...,lm} = 0.
4°=>5C. If we have 4C, then, for any c,-= ±1 (/=!,...,«),
Z ?iaix.-
i= 1
Z a.-
iAi
I—i l '
i= 1
where dl = {ie,Ar\ 1 ^/^и, г,= + 1}, й?2 = {г'е1/Г
•¦,et=-l}.
1 The same proof shows that in the particular case when the scalars are real,
we have infM5 = infM3.

502
II. Special Classes of Bases in Banach Spaces
5"=>7°. If we have 5°, then for any scalars al,...,an and Дь...,/?„
with \pk\ ^ 1 (fc= 1,...,и) we have, by the argument used in the proof of
formula A6.27) of §16,
Z Pi«iXi
sup
(respectively,
Z PitiXi
Z ?i*ixi
Z aiXi
7°=>6°. If we have 7°, then
i= 1
aiXi
i= 1
if the scalars are real).
n
z
i= 1
a.-IXj
=
1
n
i= 1
= tt Z E1ёпа;)аЛ
(sign «,)!«;I x,-
6°=>5°. If we have 6°, then for any ?;= +1
i= 1
n
Z tol*«
Z *««*«
i= 1
Z \^i\Xi
i= 1
= M4
V la.lx.
i= 1
M.
«4
Z af;
7°=>2°. Assume that we have 7° and let аеП, n,me.Y' and
affA))...,aff(ri+m)eK be arbitrary. Then, putting /= max a(i), a=0
(/e{l /}\{ff(l),...,ff(/n-m)»,j8e<I-)=l(/=l,...,/i) and /3_,. = 0(/е{1 /}
\{<хA),...,<х(и)}), we obtain
Z ^-«л-
Z «j*j
and hence, by Ch. I, § 7, theorem 7.1 (implication 4°=>Г), {xff(n)} is a
basis of E. „
4°=>8°. If we have 4°, then for any Z ai xi G(T(ii in) and any
m j= 1
У a, x, eP'1' ' we have
LOf; X; — У И, X,
1
Z «oxo
1
The equivalence 8C<*9C is an immediate consequence of Ch. I, § 6,
lemma 6.1 a) applied to F = Pid),G = P(d).
17. Intrinsic characterizations of unconditional bases
503
8°=> 10°. Assume that we have 8° and let poeP(d). If po = ®, there
exists an feE* satisfying A7.15), A7.16) and ||/|| =arbitrary. If />o#0,
we have, by 8°, dist (——, F<d) I > M6 > 0, whence there exists1 an
V ILpoII /
feE* satisfying A7.15), A7.16) and
1 1
H/ll = 7—- ^M/
\\Po\\
10°=>1Г. By A7.15), A7.16) and A7.17), for every p=
and {^...^'Je® we have, putting po = S{iu inj(p), ;
»! = Ml = 1/Ы1 = \f(p)\ <* Mn\
H°=>2°. If we have 1Г, and аеП, then
sup sup
p
sup ||S»||<oo,
pEP
Ml si
whence, by Ch. I, §7, theorem 7.1 (implication 10° =>1°), {xa{n)} is a
basis of E. Thus, 1 <=> ¦ ¦ ¦ о 11 .
Furthermore, from the above proofs of the implications 3'J=>4°=>8°
=>10°=>1Г it follows that
Ml > infM2 >ТГ^ infM7 > M8 $s 1.
Let us prove that M8 = Ml. If {xn} is not an unconditional basis
of E, then, by the above, we have M8 = M1 = oc. On the other hand,
if \xn} is an unconditional basis of E, then, by the above proof of the
implication Г=>3°, we have M1 = sup||id||< go, where {sd} are the
deS
partial sum operators defined by A6.6). Hence, by A7.18) and sd{p) = Sd(p)
eS)), M8 = M1<oo.
Furthermore, the inequality infM3>infM2 follows from
Z *ijxij
where /= max ij, eij=e'ij='i- for j=\,...,n and
/ce{l,...,/}\fC.,U.
1 /
l(
2 V
1
lek
k=l
4Xk
+
1
Z4
л=i
akXk
\
/
n
V Of X
k=l
= eL = — 1 for
fcfc— bk
1 See e.g. [10], p. 57, lemma.

504
II. Special Classes of Bases in Banach Spaces
Finally, from the above proofs of the implications 4°=>5°=>7° it
follows that we have the inequalities 4infM2 >2infM3 ^infM5, which
completes the proof of theorem 17.1'.
Definition 17.1. Let {xn} be a complete sequence in a Banach space E,
such that х„Ф0 (п= 1,2,...). Then the number vW, = sup||sj is called
dsi
the unconditional norm of the sequence {х„}. The number yfjnj = ~~
'*"'
is called the unconditional index of the sequence {xn}.
We have l^Vj^^v^^co; the sequence {*„} is an unconditional
basis of E if and only if v{$nj<co. Moreover, by theorem 17.1 (equiv-
(equivalence Г<=>2°), for an unconditional basis {х„} we have
supv.
A7.20)
Let us give now some corollaries of theorem 17.1.
Corollary 17.1. a) Every unconditional basis of type aP is equivalent
to the unit vector basis of c0.
b) Every unconditional basis of type al+ (whence also every uncon-
unconditional basis of type aP*) is equivalent to the unit vector basis of I1.
Proof. By § 16, theorem 16.8 (implication Г=>4°), it is sufficient to
prove the statements a) and b) for unconditional bases of types P and
/+, respectively.
a) Let {xn} be an unconditional basis of type P and let of1,...,otn
be arbitrary scalars with sup |a|^0. Put
sup a-
Then |j8j-|s$l (j=l,...,n), whence, by theorem 17.1 (implication
Г=>7°),
1 Alternatively, instead of the last part of the above proof one can also prove
directly that
infM3 = sup sup ||5{?|,||, infM5 = sup sup |k»,,|j
and then apply § 16, formula A6.23). Let us also mention that by the above proofs
of the implications 7°=>6°=>5° we have infM5 >max(infM4, inf—) and
inf—- > infMv
ГП4
17. Intrinsic characterizations of unconditional bases
505
E aixi
n
E Pi sup |ay|
M5 sup
HKoo
n
E
; = i
sup
xi
Sup Ofj
A7.21)
On the other hand, by theorem 17.1 (implication Г=>4°) we have
11 *4*11IIII /* 4 \
Z aix
к
whence
? ai
M2
inf
n
E aixi
1=1
>m4
n
у
i=l
From this inequality and A7.21) it follows, by Ch. I, § 8, theorem
8.1 d) (implication 2°=>6°) that {х„} is equivalent to the unit vector
basis of c0.
b) Let {xn} be an unconditional basis of type /+ and let аь...,а„
be arbitrary scalars. Then, by theorem 17.1 (implication 1°=>6°), we have
sup \\x,\\
where ?/>0 is the constant occurring in the definition of bases of
type /+ (§ 10, definition 10.1). Consequently, by Ch. I, § 8, theorem 8.1d)
(implication 2°=>6°), {х„} is equivalent to the unit vector basis of I1.
The same assertion for unconditional bases of type P* follows from
the fact that every basis of type F* is also of type /+ (§ 10, theorem 10.1),
which completes the proof of corollary 17.1.
Corollary 17.2. Let {*„} be an unconditional basis of a Banach space
E, let {mn} be an increasing sequence of positive integers, mo = 0, and let
Уп=
(«=1,2,...).
Then {>>„} is an unconditional (block) basic sequence1, with vf^sj v({^n).
Proof. By the implication Г=>4° of theorem 17.1, we have
for any dud2e® with dl<=d2 and any
. Hence, by A7.22),
I.e., {у„} is an unconditional basis of [>„]•

506
II. Special Classes of Bases in Banach Spaces
Z
У v- У ах-
jed'i i =mj- i + 1
Z Z y,
Z Z
Z
for any d[,d'2€& with d[ <= d'2 and any {у;}^^ <= JC. Consequently,
by the implication 4°=>1° of theorem 17.1, {>>„} is an unconditional basic
sequence, with v^, < v^,, which completes the proof.
The next natural question is whether we have a result on the ex-
extension of every block basic sequence with respect to an unconditional
basis {х„} to an unconditional basis of the whole space E, similar to
Ch. I, § 7, theorem 7.2, or at least whether in the case of an uncon-
unconditional basis {х„} every block subspace with respect to {х„} (i. e., every
subspace spanned by a block basic sequence) is complemented in E.
The answer to this latter question, whence also to the first question,
is negative, as shown by
Example 17.1. Let l<p<2, let {xn} be an unconditional basis of
E = L"{[0,1]), e.g., the Haar system (§ 14, theorem 14.1), and let r be
an arbitrary number such that p<r<2. Then there exists1 a sequence
{у„} с Е equivalent to the unit vector basis of Г. Since obviously {yn}
converges weakly to 0 and inf |[yn||>0, there exists, by § 15, pro-
1 ^ П < 00
position 15.1, a block basic sequence (with respect to {*„})
= Z
A7.23)
which is equivalent to some subsequence {yin} of {>>„}. Since by § 18,
proposition 18.1a) {yin} is equivalent to {у„}, it follows that (zn}~ {>'„},
whence [zn] is isomorphic to /r. Consequently, since for \<p<r<2
the space E = LP([0,1]) has2 no complemented subspace isomorphic
to f, it follows that [zn] is not complemented in E.
Another useful result on block basic sequences with respect to un-
unconditional bases is
Corollary 17.3. a) An unconditional basis {х„} is shrinking if and only
if it has no block basic sequence equivalent to the unit vector basis of I1.
b) An unconditional basis {х„} is boundedly complete if and only if
it has no block basic sequence equivalent to the unit vector basis of c0.
See [119] or [150], corollary 1 to theorem 7.2 and [10], p. 206, theorem 9.
See [119], P. 98.
17. Intrinsic characterizations of unconditional bases
507
Proof, a) By § 12, theorem 12.2 a), {х„} is shrinking if and only if
it has no block basic sequence of type /+, which, by corollaries 17.2
and 17.1b), happens if and only if {xn} has no block basic sequence
equivalent to the unit vector basis of I1.
The proof of part b) is similar, using § 12, theorem 12.2b) and cor-
corollaries 17.2 and 17.1a). This completes the proof of corollary 17.3.
We have the following "unconditional analogue" of Ch. I, § 8, pro-
proposition 8.3:
Corollary 17.4. Let {?„} be a sequence of Banach spaces and for
each n let En have a normalized unconditional basis {xl}, such that
sup v{"V,i: < oo. Then the sequence {ek} defined by
1 in< со к
for j=
for j=
A7.24)
is a normalized unconditional basis of the space E = (El x E2 x • • -),2, with
Proof. By Ch. I, § 8, proposition 8.3, {ek} is a basis of E. The proof
of the fact that {ek} satisfies A7.9) with M2 = sup v',"',,,,, is analogous
1 in oo
to the second part of the proof of Ch. I, § 8, proposition 8.3.
As an application of theorem 17.1 we shall now give some more
separable Banach spaces having no unconditional basis (among
which are some tensor products of Banach spaces with unconditional
bases, spaces of compact linear operators, and a reflexive Banach
space). For this purpose it will be convenient to work with spaces of
matrices.
Definition 17.2. Let M denote the linear space of all scalar valued
infinite matrices а = (а,-Д"/=1 such that a;j = 0 for "almost all" i,j
(i.e., such that the set {{iJ}\atj^0} is finite). A non-negative func-
function a on M is called a matrix norm if it satisfies the following con-
conditions:
(i) а(а) = 0оа = 0; a(ta)=\t\a(a)(aeM,teK); <x(a+b)^<x(a) + a.(b)
{a,beM).
(ii) а(м„т)=1 (n,m=l,2,...), where unm denotes the matrix in M de-
defined by
1
0
for i = n,j = m
for the other i,j= 1,2,...
A7.25)

508
II. Special Classes of Bases in Banach Spaces
(iii) а(Р„ >m(a))^a(a) (aeM, n,m = 1,2,...), where Pnm denotes the
linear mapping of M into M defined by
...alm О О...'
о о...
о ... о о о...
¦\
A7.26)
A matrix norm a is called symmetric, if
(iv) a(sI-^affl(i)jff20-)) = a(a) (aeM, av <x2eП, \st\ = |^| = 1 for /,y
= 1,2,...).
For any matrix norm oc we shall denote by Mx the completion of
the linear space M with respect to the norm a. The norm in the Banach
space Mx will be also denoted by a.
Proposition 17.1. // of is a matrix norm, then the double sequence
{unm}, arranged into a single sequence {zk} = {ui{khJ{k)} by the numbering
m
for k = n2
и + 1 for k = n2
п+\ for k = n2
и+1— т for k = n2
is a basis of Mx.
Proof. Put
Sn(a) = ]Г аЦк) j(k)uHk) як) {аеМ,п= 1,2,...).
Then, by A7.26), for each n= 1,2,... we have
A7.27)
A7.28)
A7.29)
Sr> D iD f / 1 it D
2 i i < ii ^= i i j , | — i , j _ i —|— _fV . I ( — I ..... ft , 1M r
whence, by condition (iii) of definition 17.2,
sup sup of(Sn(a)) ^ 3.
A7.30)
Consequently, since M is dense in Mx, from Ch. I, § 7, theorem 7.1
(implication 10°=>l°) it follows that {ui(khm} is a basis of Mx, which
completes the proof of proposition 17.1.
17. Intrinsic characterizations of unconditional bases
509
We shall show below that for some important concrete matrix
norms of the spaces Mx, and even their superspaces, have no uncon-
unconditional basis.
Let us observe that for every matrix norm of the space Mx can be
identified in a natural way with a space of matrices (by the correspon-
correspondence a->(hij(a))™j = l, where {hi(khj{k)} is the a. s. с f. to the basis
i,, ) nf м \ '
We return now, for a moment, to the linear space M.
Definition 17.3. For each и = 1,2,... the linear mapping Т„: М^М
defined by
aln 0 0 ...\
Tn(a)=
a-,, a,, ... a,. , 0 0 0 ...
(aeM)
A7.31)
is called the rc-th main triangle projection. If of is a matrix norm, we shall
?и(<х) = || 7j|„ = sup ofG^,(a)) (и=1,2,...). A7.32)
aeM
For each и=1,2,... we also define a projection Dn:M^>M by
m
0 0 ...\
о о...
о о ...
0
0
«32
0
a2m~ 1,2
0
«13
«23
«33
0
«2m-l,3
0
0
0
0
0
... a2m_
0
г2,2т-1
,2m- 1
в2т-2.2т-1 0 0...
«2m-l,2m-2 «2m-l,2m-l " 0 ...
0
{aeM),
D2m(a)= ? ? а;ум0 (aeM).
/c= 1 max(i,j)= 2k
If of is a matrix norm, we shall put
0 0 ....
A7.33)
A7.34)
A7.35)
p
aeM

510 II. Special Classes of Bases in Banach Spaces
Lemma 17.1. If n is a symmetric matrix norm, we have
*„(«) = </„(«) (и=1,2,...).
A7.36)
Proof. We shall consider only the case when n = 2m — 1, since for
n = 2m the proof is similar. Define а„еП by
n-i + 2
2
n+\ + i
for i= 1,3,5,. ..,n — 2,n
for г = 2,4,6,...,и-3,и-1
for i = n+l,n + 2,...
A7.37)
and a linear mapping Uan:M->M by
Then we have
since by A7.38), A7.31) and A7.33) both sides are equal to
аП-2,„-4---ап-2,1 «„-2,2 в»-2,4---в»-2,»-3
A7.38)
A7.39)
0 0..Д
«1» «l,n-2
«2„ «2,»-2
«1
«2
.1-4
.п-4
•••«11
... 0
0
0
0 ..
0 ..
0
. 0
0
0
0
0
0...
0...
0 ... 0 0 0 ... 0 0 0 0...
0 ... 0 0 0 ... 0 0 0 0...
0 ... 0 0 0 ... 0 0 0 0.../
\ О О
Hence, since UOn is an isometry (because a is a symmetric matrix norm),
*„(«)= sup а(Г„[С/^(а)])= sup «([/„„[Д,(а)])
M М
= sup а(Д,(а)) = </„(«),
asM
а(я) S 1
which completes the proof of lemma 17.1.
17. Intrinsic characterizations of unconditional bases
511
We return now to the spaces Ma.
Lemma 17.2. Let E be a Banach space, let {/„} <= E* and let a be a
symmetric matrix norm. If there exists an isomorphic embedding Uo:
Ma-^E, then there exists another isomorphic embedding U: Ma-^E
satisfying
)] = 0 (/,и=1,2,...), A7.40)
A7.41)
lim/„[ЩМ;у)] = 0
Proof. Since for each fixed pair of indices (n,r) the sequence
{/„[{УОКЯ)]}Г=1 is bounded (because а(и„)=1), one can find, by
the standard diagonal procedure, an increasing sequence of indices
{s})f=l such that the limits lim fn[_U0{ur s.)] (и,г=1,2,...) exist. Then,
with a similar argument, one can find an increasing sequence of indices
{г;}Г=1 such that the limits Нт/я[170(м s.)] (nj = 1,2,...) exist. Put
A7.42)
Observe now that, since a is a symmetric matrix norm, for any
no mo
Yj Z bijU^eM and any pair of increasing sequences of indices
l jl
i%u {4]Ij=y with
, we have
a( Z Z bP,.vuij) =a( Z Z Km,eMul
\i=l j=l / \i=l j=l
f к i
= « Z Ifty"
Vi = 1 j = 1
A7.43)
where <jl,<j2en are permutations such that o^(/)=/?,- (г= l,...,/c),
ff2(/) = 9j(/'=l,...,0 andforany {a}"°=i, {^}"=i we have
(«О то \ /"о то
Z !*««„.„ =« Z I
o o
Applying A7.43) to the matrix b=V(a)= J] Т.Ьчич (where
br»..2J = brit_l,.2J_l=-br2i.,2J_l=-br2l_i.llJ = aiJ and 6y = 0 for the
other i,j) and to Pi = r2i, qj = s2j, we obtain

512
II. Special Classes of Bases in Banach Spaces
f. ЬцщА = а(У{а)) (aeM). A7.45)
/
On the other hand, applying A7.44) to b=a, we get
A7.46)
Thus, by A7.45) and A7.46), V is an isomorphism of M into M
and hence it can be extended, by continuity, to an isomorphism Ц of
Ma into Mx. Put
t/=t/0K- (П.47)
Then U is an isomorphic embedding of Ma into ?, satisfying
fn[U(uij)]=fnlUoV1(uij)']=fn[Uo(ur2t,S2j)-Uo(ur2US2j_l)]
+/n[t/oK,_1,S2j_1)-^oK,_1,S2j)]^0 as у-юо
(«,/=1,2,...)
and, similarly, /и[Щиу)]->0 as /->oo (nj= 1,2,...), which completes
the proof of lemma 17.2.
Now we can prove
Theorem 17.2. Let E be a Banach space with a basis and let a be a
symmetric matrix norm. If there exists an isomorphic embedding U: MX->E,
then for every basis {xn} of E there exists an equivalent norm on E such
that in this norm
v\Z}^ sup tn(a). A7.48)
1 г?П< CO
Consequently, if E contains a subspace isomorphic to Ma, where a is a
symmetric matrix norm such that sup fn(a)=oo, then E has no un-
unconditional basis. i =s n < со
Proof. Let e>0 and let 5 be a positive integer. Then, by lemma 17.1,
there exists a matrix aeM with a(a)^l, such that
a(Ds(a))>ts(a)-E. A7.49)
We shall prove that for any basis {xn} of E we have, in a suitable
equivalent norm on E,
a))-e, A7.50)
17. Intrinsic characterizations of unconditional bases
513
whence, by A7.49), vfjn}^ts(a) — 2e, which, since e>0 and seJT
are arbitrary, will complete the proof.
Let {xn} be an arbitrary basis of E. By lemma 17.2 we may assume,
without loss of generality, that U satisfies A7.40) and A7.41), where
{/„} <= E* is the a.s.c.f. to the basis {xn}. Furthermore, by § 13, lemma
13.6, we may assume, replacing, if necessary, the initial norm of E with
a suitable equivalent norm, that U is a linearly isometric embedding.
We shall consider only the case when s = 2 n — 1, since for s = 2 n the
proof is similar.
By A7.40) and A7.41) one can construct, inductively, three increasing
finite sequences of indices {mk}skt\, {/>*}*= i> and {(/iJiUi such that for
the matrices
the following inequalities hold:
Z |/г[^(^)]| H*rll <y^. (k=l,---,s) A7.52)
r= 1 25
m' g
У fr\U(bS\xr <—^ (m' = mk + i + l,mk + ,+2,...;k=l,...,s).
A7.53)
Indeed, put m! =/?! = 1 and, by A7.41), choose qx such that we have
i.e., A7.52) for /c=l. Then, since ^ = Z fr(t>i)xr (because {xn} is a
r=l
basis of ?), there exists an m2 > тг such that we have
e
'2?
i. e., A7.53) for k=l. Next, by A7.40) we can choose p2 such that
m2 g
la2ll /_i \fr\J^^UPi,qi)\\ \\XA < ^
r= 1
and then, by A7.41), g2 such that
VII -452
r=l ' r=l
33 Singer, Bases in Banach Spaces I

514 II. Special Classes of Bases in Banach Spaces
whence we obtain
E \fr[U(b2)~\\ \\xr\\ =
r=l
2s2
i. e., A7.52) for k = 2. Since b2= ? fr(b2)xr, we can choose тъ>т2
r=\
so as to have A7.53) for /c = 2. Continuing in this way, in a finite
number of steps we achieve the construction of {mk}lt\, {pk}sk = l and
ми-
Conditions A7.52) and A7.53) imply
E fr[U(bk)]xr-U(bk)
*,+ I
• _ (k—\ ч\ П7 541
s
and
r=mk+l
mk+ I
r= 1
¦¦ 2 Ё 1Л
r= 1
r = mi + i +1
; 2 sup
m( + i ^ m' < go
mj(
r = 1
8
5
whence
E /r№/)]*r
?
s2
Now, put
x= t U(bk).
A7.55)
A7.56)
Then, since U is a linear isometry and since the symmetric matrix
norm a satisfies A7.44), we have
1 If fe = /+l, the term
will not occur.
17. Intrinsic characterizations of unconditional bases
515
U(bk)
: s
A7.57)
and, similarly, recalling that s = 2n—l,
.1=1
= a( E *21-1
1 = 1
= «(E E «у"*.,;
\1 1 i 2
E
i=i
= « E E ациу =«(D,(a)). A7.58)
\1 = 1 max(i,j)=21-l /
Consequently, by A7.57), A7.58), A7.56), A7.54) and A7.55) we
obtain
.ilk 11^
n
E
-1
n
E
1=1 r-
Щ*2!-1)-
M
= i V
U(*2,-l
E
к =
1
m2i
E
J
1+1
E
»"-m2! - l +
)-
/
+ 1
Ш21
E
= m21-
\{x)xr
fr(x)xr
1
n
E <л^-1)
1=1
^<x(Ds(a))
/rEt/^j,.!)]^
1+ 1
\)]xr
)
/
It #21- 1
i.e., A7.50), which completes the proof of theorem 17.2.
Now we shall give some concrete symmetric matrix norms a having
the property sup tn(a) = oo.
Let 1
oo. Put
p2(a)= SUP 1ЕЕ«.7^^1 (аеМ),
A7.59)
зз»

516
II. Special Classes of Bases in Banach Spaces
where 1 =1 (/c=l,2;— =0) and where in the case qk=co the
Pk Як °o
sum ?|?;|Чк (°r EM*") is replaced by sup|^,-| (respectively, supl^-l).
Then, obviously, AP1,P2 is a symmetric matrix norm.
Lemma 17.3. // l^:p1^p3^cc and I^p2^p4.^cc, then
A7.60)
JL J
where —+ —= 1 (k= 1,2,3,4).
Pk Як
/1 " V
Proof. Since — У \?i\q у is1 an increasing function of q and
, we have
SUp
I ? «1^.- ^
= sup
I I
sup
I I «y^f^
l l
whence A7.60), which completes the proof of lemma 17.3.
Let us recall that the Hilbert matrices hn (и= 1,2,...) are defined by
1
-— for i+j=?n+l,i,j=l,...,n
for the other i,j
A7.61)
1 See e.g. [263], p. 8, problem 2.
17. Intrinsic characterizations of unconditional bases
517
and that for each p with 1</><оо there exists1 a constant Bp>0
such that
lpJhn)^Bp (и =1,2,...), A7.62)
1 1
where —|— = 1.
P q
1 1
Lemma 17.4. Let I^p1,p2<co, 1 ^ 1. Then there exists a
constant CPl P2 5ис/г г/ra?
Pi Pi
Consequently, lim ?n(lpi, P2) = со .
Proo/ By A7.61), A7.31) and A7.59) we have
i+j=2
= sup
A7.63)
1 1
И*1 И«=
A7.64)
!_J._J.
«Чпи,
with a suitable constant C>0 (independent on n,pup2). We shall
consider three cases:
а) 1<^<оо. Then, since Р„,„(Л„) = Л„ and since the hypothesis
— ^1 = — implies p2^qi, from A7.60) with ръ—Р\, p^ = q
Pi Pi 4i
A7.62) we obtain
1 See e.g. [51], Ch.XI, §7 or [79], Ch. Ill, § 10.

518 II. Special Classes of Bases in Banach Spaces
whence, by A7.64), l 1
_ С
в
" Pu *-PuP2(hn) в пугъ Вр.
b) Pi = 1 and 1<^2<оо. Then, by A7.60) with pl = l, p3 = 1i,
p4,=p2 and A7.62) we obtain
_L _L
lx p2(hn) = At Р2(Р„,„(Л„)) < Л,2>Рг(Л„)иР2 ^ BqinP2,
whence, by A7.64) with p1 = 1,
К p (Tn(hn)) Си'"«1пи С
t.U. .)> UPlK " '> ; = -—1ПИ.
c) pl=p2 = l- Then, from A7.60) with p3=p^ = 2 and A7.62) we
obtain
Я1Д(Л„) = Кл(РпЖ)) < ^-2,2(Л„)и < В2и,
whence, by A7.64) with рг = 1, g2 = co,
Я, ,(Т„(Л„)) Си1пи С ,
which completes the proof of lemma 17.4.
Remark 17.1. The assumptions pup2<<n in lemma 17.4 are essen-
essential. Indeed, we have
л„.Р2(в)= sup Z(Zaij4j)ti\ = suP SUP |1«и
whence
A7.65)
and. similarly, ApiJa)=sup(Y\aij\pyi(aeM,l^pl^co), whence
Гп(ЯР1Ж)=1 A</?1^оо;и=1,2,...). A7.66)
If a is a matrix norm, the conjugate norm a* is defined by
a*(a)= sup УУ.анЬ,! (aeM). A7.67)
17. Intrinsic characterizations of unconditional bases
519
Since for aeM fixed, ga(b) = Y^Y^aij^ji таУ ^е regarded as a
linear functional on a suitable finite dimensional space containing a,
and since a*(Pnm(b))^a*(b), we obtain
a**(a)= sup
ЬеМ
i J
= a(a) (aeM),
A7.68)
whence
tn(a*)= sup <x*(Tn(b))= sup sup
bsM ЬеМ яеМ
a*(b)< 1 a*(b)< 1 я(а)^1
Я+1
= sup a**(TB(a)) = sup а(Т„(а)) = г„(я) (и=1,2,...). A7.69)
Consequently, from lemma 17.4 it follows
Corollary 17.5. Let 1
constant Cpi P2 such that
, p2<oo, 1 ^ 1. Then there exists a
Pi Pi
A7.70)
From theorem 17.2, lemma 17.4, and corollary 17.5 we obtain im-
immediately
Theorem 17.3. Let 1 ^рг, р2<са, —| > 1 and let E be a Banach
Pi Pi
space with a basis, containing a subspace isomorphic to MXp . P2 or M^ _ p^.
Then E has no unconditional basis.
Remark 17.2. In particular, like the space./ (see§ 15, theorem 15.4), the
space M)ul is an example of a space with a basis which is isomorphic to a
conjugate Banach space and which cannot be embedded into any
space with an unconditional basis. Indeed, since for any
Я1Л [ ? ? ачич
\ i = 1 i = 1
infinite matrix a = {ai}) the relations sup
= sup
sup
I I <*i
< oo imply aeMXut, the basis
{ui(k)j(k)} of M/bl (see proposition 17.1) is boundedly complete, whence,
from § 6, theorem 6.2 (implication 1°=>3°) it follows that MXlil is iso-
isomorphic to the conjugate space [g^-]*, where {gi(k)j{k)} c Mft>t is the
a. s. с f. to the basis {ui{k) J(fc)}. Moreover, one can also show1 that in
MXui weak and strong convergence of sequences coincide.
1 See [145], example 2.1.

520
II. Special Classes of Bases in Banach Spaces
Let us observe now that if E and F are Banach spaces with norm-
normalized monotone bases {xn} and {>>„}, respectively, and if a is a uniform
crossnorm on their algebraic tensor product E ® F, then one can
define in a natural way a matrix norm a = a(E,F, {xn},{yn},a) by
A7.71)
Indeed, а(мо-) = а(х,-®у,.)= ||Х;|| Щ = I (i,j= 1,2,¦¦¦) and, if si, si de-
denote the partial sum operators associated to the bases {х„} and {yn},
respectively,
a(PB,m(fl)) =« Z Z flyx,- ®у. = а
V i /
X aijyj)] = a [(si
= a
x,
(яеМ).
Consequently, by A7.71), proposition 17.1 and Ch. I, § 18, theorem
18.1, E®aF is linearly isometric to MS.
In particular, to ? = /"', F = V2 (l^Pl,p2^ou and /°° stands for c0),
{*«}> {У„} = the unit vector bases of E and F, respectively, and1 a = l
there corresponds the matrix norm
k(a) =
= sup
= sup
V |f/-|42^ 1
where —|—=1 (fc = l,2) and where in the case fft=oo the sum
Pk 4k
Т\Ц"к (ОГ Т\Ц:
/") is replaced by sup|^| sup|^,|). Hence lp'®JP2
is linearly isometric to MXp and consequently lPl ®ylP2 is linearly
isometric to MA» , where —I—= 1 (/c = 1,2) and where by /°° we
mean c0. Therefore, theorem 17.3 can be also reformulated as follows:
1 As in Ch. I, § 18, we denote by A the least crossnorm whose associate is a
crossnorm and by у the greatest crossnorm.
17. Intrinsic characterizations of unconditional bases
521
Theorem 17.4. Let 1 ^n,, p2<ao, —I ^1 and let E be a Banach
Pi Pi
space with a basis, containing a subspace isomorphic to one of the spacesx
lp43jP2=Wlqi,lP2) or /ei®,,/«2, where — + — =1 (/c = 1,2) and where
Pk 4k
/°° stands for c0. Then E has no unconditional basis.
In order to give a similar result for tensor products of the spaces
Lp([0,l]) and C([0,l]), we need
Proposition 17.2. Let l<p<co, and let rn (n= 1,2,...) be the Rade-
macher functions on [0,1]. Then
a) {rn} is a basic sequence in Lp([0,1]), equivalent to the unit vector
basis of I2 and hence the subspace [>„] of Lp([0,1]) is isomorphic to I2.
b) The mapping
A7.72)
u(x)=fi(]x{t)rl(t)dt)ri (xeLp([0,l]))
\ /
is a continuous linear projection of Lp([0,1]) onto [г„] and hence the
subspace [г„] is complemented in Lp([0,1]).
Proof, a) By § 14, formula A4.2), {rn} is a block basic sequence with
respect to the Haar basis {yn}. By the Khinchin inequality A4.56),
{rn} is equivalent to the unit vector basis of I2.
b) Let p^2 and let xeLp([0,l]). Then xeL2([0,l]) and
Z K*,r,)|2 f
\x(t)\2dt I2 < oo and thus, by the Khinchin inequality A4.56),
Z {х,ФЛ
Consequently, for every xeLp([0,1]) the series ? (*>rdri converges
in Lp([0,1]) and hence A7.72) is a continuous linear mapping of Lp([0,1])
into [г„]. Finally, by the orthogonality of the Rademacher functions
(§ 14, corollary 14.1), we have
= Z
and hence и is a continuous linear projection of Lp([0,1]) onto [rB].
See Ch. I, § 18, footnote to corollary 18.5.

522
II. Special Classes of Bases in Banach Spaces
Finally, assume that l<p^2 and let —|—= 1. Then
P 4
whence, by the above, for any /eLp([0,1])* = L«([O,1]) the series
X)
?(г;,/)г( (where the rt are considered as elements of L4([0,1])) is
i= 1
convergent in L4([0,1]) and by a) this convergence is unconditional.
Hence
oo
for any xeLp([0,l]), /eLp([0,1])* = L«([O,1]), where we have put
?i = sign/((x,/',)ri). Thus the series ? (x,rdri m LP([O,lJ} *s weakly
i- 1
unconditionally Cauchy, whence, by § 15, lemma 15.8, it is convergent in
the norm topology of Lp([0,1]). Consequently, again the mapping A7.72)
is a continuous linear projection of Lp([0,1]) onto [rn], which completes
the proof of proposition 17.2.
Theorem 17.5. Let 1 ^pl7p2 ^ oo and let E be a Banach space with a
basis, containing a subspace isomorphic to one of the spaces LPl([0,1])
®ALP2([O,1]) or LPl([0,l])®yLP2([0,l]), where L=°([O,1]) stands for
C([0,1]). Then E has no unconditional basis.
Proof. If I<p1,p2<oo, then, by proposition 17.2, the space /2®A/2,
respectively /2®T/2, can be embedded1 isomorphically into LP1([0,1])
®ALP2([0,1]), respectively into LP1([O,l])®yLP2([0,1]), whence the
desired conclusion follows by theorem 17.4 applied to p1=p2 = 2. The
remaining tensor products (with Ll([0,1]) or C([0,1])) contain a sub-
space isomorphic to ?/([0,1]), respectively C([0,1]), whence the desired
conclusion follows by § 15, theorems 15.2 and 15.1. This completes the
proof of theorem 17.5.
Let us consider now the symmetric matrix norms <3p defined by2
(aeM), A7.73)
for
[Я2>2(а) for p = cc
where a^ = aj{ (i,j= 1,2,...) and where trZ> = ?
1 See [89], p. 93 and p. 40, corollary 1, or [223], Ch. II, lemma 2.12 and Ch. III.
corollary 3.4.
2 For a detailed study of the spaces Ms see e.g. [164] or [51], Ch. XI, § 9 or
[79], Ch. III.
17. Intrinsic characterizations of unconditional bases
523
Lemma 17.5. We have
A7.74)
with a suitable constant С > 0.
Proo/ Since lim <Зр(а)=<300(а) (яеМ), we have, by A7.62) and
A7.64), p~">
Нт6р(Л„)=<300(Л„) = 12,2(
lim Sp(Tn(/O)=;.2,2
whence
C
(«=1,2,...).
1 1
Since it is well known1 that <3* = <34, where —|— =1, we also
have, by A7.69) and the above, P a
С
limfB(S )=hm?n(S*)= lim fn(<3 ) Ss —Inn (n=l,2,...),
P->1 P-»1 q-*<n й2
which completes the proof of lemma 17.5.
Let us mention that by A7.69) and A7.63) we also have
fB(®1)=USf)=^(S»)=^2,2)^Clnn (и=1,2,...). (П.75)
Theorem 17.6. Let {pk} be a sequence of positive numbers such that
l<pk< oo (fc= 1,2,...) and that either lim/>fc=oo or \impk=l and
/e? E be a Banach space containing for each /c=l,2,... a subspace iso-
isomorphic to the space Me , such that these isomorphisms and their in-
inverses are uniformly bounded. Then E has no unconditional basis.
Consequently, in particular the space E0 = {MSpi x M3p2 x •••),2 /5
a reflexive Banach space with a basis, which cannot be embedded into any
space with an unconditional basis.
Proof. For every basis {.*„] of E we have, by our hypothesis and by
the proof of theorem 17.2,
vY>n]>A sup tn(&Pk) (fc=l,2,...),
1 <П<0О
where A>0 depends only on {xn}, whence, by lemma 17.5,
v\"Jn]>A sup sup ?n(<SPk)=co,
l<fe<00 1<Л<00
and thus E has no unconditional basis.
See e.g. [79], Ch. Ill, § 1 or [164], p. 265, §4.

524
II. Special Classes of Bases in Banach Spaces
Furthermore, by proposition 17.1 each MePk has a basis, whence,
by Ch. I, § 8, proposition 8.3, the product space ?0 has a basis and thus,
by the above, ?0 has no unconditional basis. Since each MSj,t is reflexive1,
their /2-product space ?0 is reflexive, which completes the proof of
theorem 17.6.
Problem 17.1. Let 1<р<со,рф2. Does the space MSp have an
unconditional basis ?
Let us give now some properties of strong duality for unconditional
bases.
Theorem 17.7. Let {xn} be an unconditional basis of a Banach space
E and let {/„} <=?* be the a.s.c.f. Then {/„} is an unconditional basic
sequence and we have
1 < '¦([/„]) yfxl, < V!/ni < Vflv <17-76)
Hence, in particular, if /¦([/„])= I,2 then
,,(") —,,(") П7 771
Conversely, if {х„} is a basis of E, such that the a.s.c.f. [fn}c?*
/5 an unconditional basic sequence, then {xn} is an unconditional basis
of E.
Proof. If {х„} is an unconditional basis of ?, then every permutation
{•"^(п)} is a basis of ?, whence, by Ch. I, § 12, theorem 12.1, every permu-
permutation {/„(„,} is a basic sequence in ?*, and thus {/„} is an unconditional
basic sequence in ?*.
The first inequality in A7.76) follows from v(Xnj ^ v',"^, and Ch. I,
§ 12, theorem 12.2 a). The proof of the other inequalities in A7.76) is
similar to that of the corresponding inequalities for v(Xn!,v(yn) (Ch. I,
§ 12, theorem 12.3) considering instead of sn the operators sd.
Finally, assume that {х„} is a basis of ?, such that the a.s.c.f. {/„}<=?*
is an unconditional basic sequence. Then, by the first statement proved
above, the a.s.c.f. {ф{х„)}<=[/„]* to the basis {/„} of [/„] is an un-
unconditional basic sequence in [/„]* and by Ch. I, § 12, theorem 12.2,
the canonical mapping ф of ? into [/„]* is an isomorphism, whence {*„}
is an unconditional basis of ?. This completes the proof of theorem 17.7.
The inequalities A7.76) are, in a certain sense, the best possible,
since for orthogonal bases (i.e., for which v(!x)n!=l) they all become
equalities. However, they are, in general, strict inequalities, as shown
by example 12.3 of Ch. I, §12, since in that example the basis {х„} is
1 See e.g. [79], Ch. Ill, § 1 or [164], p. 265, §4.
2 This happens e.g. when the basis {х„} is monotone, in particular when
,)= 1 (such bases are called orthogonal, see §20), or when {xn} is shrinking.
17. Intrinsic characterizations of unconditional bases
525
does not
unconditional, г([/„]) = л, v((xJi!^v(Xn) = 2 + -^- and v^,
A
depend on A (by Ch. I, § 12, formula A2.33)). Furthermore, as shown by
Ch. I, § 12, example 12.4, there exists a biorthogonal system (xn,fn)
with [*„] = ? and {/„} total on ?, such that {/„} is an unconditional
basic sequence, but {xn} is not even a basis of ?.
Let us also mention that if {xn} is an unconditional basis of ? and ф
the canonical mapping of ? into [/„]*, then, similarly to Ch. I, § 12,
theorem 12.4 and corollary 12.3, we obtain
^„,, = v({:L, A7.78)
1 < КШ) v(,l, ^ v((;»(XnI^ V&,. A7.79)
We also observe that the "unconditional analogue" of Ch. I, § 12,
problem 12.1, i.e., the problem "if ?* has an unconditional basis, does ?
have an unconditional basis?" has a negative answer, as shown by the
example E = C(coa) considered in § 5, example 5.3. Indeed, as observed
there, ?* = I1 and so ?* has an unconditional basis, but we shall see in
Vol. II that ? = C(wm) has no unconditional basis.
The following property of strong duality will be used in § 21:
Proposition 17.3. Let {xn} be an unconditional basis of a Banach space
E, with the a.s.c.f. {/„} <= ?*, let {хПк} be a subsequence of {xn} and let
?o = IXJ, </>nj=/njlEoe?o (/=1,2,...). Then {</>„.} /5 a basic sequence,
equivalent to the basic sequence {/„,}.
Proof. By Ch. I, § 12, theorem 12.1 and Ch. I, § 4, proposition 4.1a),
{/„.} and {</>„} are basic sequences. Since {х„} is an unconditional basis,
by § 16, theorem 16.8 the mapping
GO
"(*)= Z Lj{x)xnj (xeE)
j"=i
is a well defined bounded linear projection of ? onto ?0 and thus for
any scalars a1,...,xk we have
И It И \ к
z ш
к
?></>„,.[>(*)]
к
Z ъФ*,
i= 1
=
/ *
\i=l
1
sup
xeE
xll«l
к
i= 1
?o
Z «l/-.
i = 1
Г ос 
4
Z
-.7=1
= sup
xeE
11*11 «1
= sup
xeE
11*11 «1
whence {фП]}~ {/„.}, which completes the proof.

526
II. Special Classes of Bases in Banach Spaces
In the above proposition the assumption that {*„} is unconditional
is essential, as shown by
Example 17.2. Let {en} be the natural basis of E = c0 and let
ха=Уе, («=1,2,...) A7.80)
be the conditional basis of ? considered in § 14, example 14.1. Then every
subsequence {xn.} of {х„} is equivalent to the basis {xn}, but {/2j} is
equivalent to the unit vector basis of I1, whence unconditional, whence
not equivalent to {ф2]}-
Indeed, we have seen in § 14, example 14.1, that for any increasing
sequence of indices {n^} and any scalars ah...,a[+k we have, putting
I Pj*j
= sup
j=p
= sup
l+k
Consequently, ]Г а,х„. converges if and only if ? af converges,
i 1
and thus {xn.} is equivalent to {х„}.
On the other hand, we have
whence
p
I aj2l
(x={Qeco,j=l,2,...)
= sup
fSnJeco
m + p
= 2
Consequently, ]Г а;/2; converges if and only if ]T |a,| converges,
i = 1 i = 1
and thus {f2j} is equivalent to the unit vector basis of/1, which completes
the proof of the assertions of example 17.2.
Finally, we shall give a relation between unconditional bases and
sequence spaces, similar to Ch. I, § 12, theorem 12.6 (on general bases
and sequences spaces).
We recall that the a-dual of a sequence space S is the sequence space
S" defined by
Sa = \{Pn}czK ? I/Jjtx,-1 < oo forall {an}eS>. A7.81)
For every sequence space S we have, obviously, S<=Sm. A sequence
space S is said to be a-perfect if SM = S.
17. Intrinsic characterizations of unconditional bases
527
It is also obvious that for any sequence space S we have
where Sy is the y-dual of S defined in Ch. I, § 12, formula A2.58). We shall
consider now a class of sequence spaces for which we have the equality
Sy = S".
A sequence space S is called normal if for every {an}eS and every
}( = /°°) we have {yBa,}eS.
Lemma 17.6. If S is a normal sequence space, then Sy = S* and hence
Syy = S™. A7.82)
Proof. If {fin}eSy and {txn}eS, then, since S is normal,
{[sign(/?nan)]an}eS, whence
< 00,
i.e., {j3n}eS*. Thus, STcS* and hence, by the opposite inclusion S*cSy
observed above, S7 = S".
Since for any sequence space S the a-dual Sa is obviously normal,
it follows that for any sequence space S we have S*y = S™. Consequently,
if S is normal, CTT CC,T C[M
which completes the proof of lemma 17.6.
Theorem 17.8. A sequence space S is associated to an unconditional
basis of a Banach space if and only ifS contains all unit vectors en and there
exists an a-perfect В K-space T such that [en]r = S. In this case, T=Sm.
Proof. Assume that S is associated to an unconditional basis {х„}
of a Banach space E, i.e., S = <{y.n}<^K ]T aixi convergesV. Then,
by Ch. I, § 12, theorem 12.6, S contains all unit vectors en and there
exists a y-perfect В K-space Tsuch that [en]r = S and T=Syy.
Now, since {xn\ is an unconditional basis of E, from § 16, theorem
16.1 (implication 1°=>5°) it follows that S is normal. Hence, by lemma
17.6, Saa = Syy=T. Since T=(Saf is normal and since Tis y-perfect, by
lemma 17.6 we also have T*x=Tyy=T, i.e., Tis a-perfect.
Conversely, assume now that S is a sequence space containing all
unit vectors en and that there exists an a-perfect В .K-space T such that
[en]r = S. We shall prove that {en} is an unconditional basic sequence
in T, whence S is associated to the unconditional basis {xn} = {en} of
the Banach space E = S.
Since T=(T*y is normal, by lemma 17.6 we have Tyy=Tm=T,
i.e., Tis also y-perfect. Hence, by Ch. I, § 12, theorem 12.6, {en} is a basis of

528 II. Special Classes of Bases in Banach Spaces
Now let {yj}em be arbitrary. Define a linear mapping u{ ,: T->Tby
viyjMxj})= {yj<*j} (ЫеТУ, A7.83)
since T=(T"f is normal, we have indeed {у^}еТ. Let {af*}, {aJeT
be such that
lim {txf} = {a,}, lim и. ,({а'-п)})= lim {y,af>} = {5=}.
n~-*oo n-*oo J n-*co
Then, since Tis a BX-space, we have
lim «y> = a,., lim yyaf = ^- (/=1,2,...),
whence
i.e., {57-} = f{yi}({aj})- Thus, v{yj] is closed, whence continuous and there-
therefore (since S=[en] and v{yj]{en) = ynen for all n=l,2,...)
v{yj]{S)<=S ({yj}em),
i.e., S is normal. Consequently, by § 16, theorem 16.1 (implication
5°=>1°), {е„} is an unconditional basis of S, which completes the proof
of theorem 17.8.
One can also prove the sufficiency part in theorem 17.8 with an
argument similar to the proof of Ch. I, § 12, theorem 12.6, by introducing
on V the norm
||{/?;}||= sup sup ?lftail= sup Z lfta;l i{Pj}eTX)
1|{«7>1и1 1""<°° '=1 ll/«7ilf<i '=1 A7.84)
(we have || {/?,•} || < oo by the principle of uniform boundedness applied to
n
the sequence of continuous non-linear functionals Fn({a7})= Z lftaiL
where {ajeT, n=l,2,...) and on j=Taa the equivalent norm
GO
|||{a,.}|||= sup Zl^^l ((аЛеТ) <17-85)
and observing that for any scalars a1,...,an,y1,...,yn with
(/= 1,...,и) we have
Z
SUP
18. Equivalence and permutative equivalence of unconditional bases 529
§ 18. Equivalence and permutative equivalence of unconditional
bases. Universal unconditional bases
In finite dimensional Banach spaces all bases are unconditional
and equivalent, hence also permutatively equivalent. It is natural to
ask whether in an infinite dimensional Banach space with an uncon-
unconditional basis there exist two non-permutatively equivalent bounded1
unconditional bases, or at least two non-equivalent bounded uncondi-
unconditional bases. We shall give now the answer in some concrete Banach
spaces with unconditional bases. In § 24 we shall determine the Banach
spaces which have, up to equivalence, a unique bounded unconditional
basis.
Theorem 18.1. In the space I2 all bounded unconditional bases are
equivalent.
Proof. By § 14, proposition 14.1, every bounded unconditional basis
of I2 is both Besselian and Hilbertian and hence, by § 11, corollary 11.3,
equivalent to the unit vector basis of I2, which completes the proof.
Now we shall show that a result similar to theorem 18.1 holds also
in the spaces I1 and c0. We shall first prove some lemmas, culminating
in lemma 18.3.
Let S"~1 = {xeE"\ \\x\\ = 1} denote the (и — l)-dimensional sphere
in the n-dimensional real euclidean space E". Let \i be the rotation
invariant Borel measure on S" normalized so that /x(S"~1)=l. Let
n
(x,y)= Z iifli (*={?¦}"= i, y={tli}1=ie^") denote the usual inner pro-
product. >=i
Lemma 18.1. Let x,yeS"~l. Then
=l в(х,у),
A8.1)
s,,-i
where в = 9(x,y) is the unique number satisfying cos в — (x,y) and 0<9<iz
(i.e., 0 = arccos(x,y)J.
Proof. Applying, if necessary, a rotation, we can choose the basis in E"
in such a way that x={l,O,O,...,O} and y= {cos$,sin0,O,...,O} (i.e.,
that x is the first basis element, у is in the two-dimensional plane spanned
by the first two basis elements and (x,y)= l-cos0 + O-sin0 + O = cosfl).
This being done, we shall use polar coordinates ф={ф1,...,ф„-1} to
write the points z={A,...,tn}eS"~1 as г{ф)= {d(</>),•••,С»Щ, where
1 We have already observed in a footnote in Ch. I, § 8, that a boundedness
condition is necessary in problems of this type.
34 Singer, Bases in Banach Spaces 1

530
II. Special Classes of Bases in Banach Spaces
n- 1
CiW>)=
i= 1
A8.2)
..,и-1), A8.3)
A8.4)
For any bounded measurable function g on S" we have1 then
j g(z)dn(z) = —Ц- j д{2(ф)K(фЩф), A8.5)
s*-i in-1
where
,...,и-1)}, A8.6)
A8.7)
~Чф(. A8.8)
|s-m =
= 2яП
i = 2
Let h{z) = {x,z) (y,z). Then, by our choice of the basis in E" we have
), whence, by A8.2) and A8.3),
n-l \2
= I ]~[ sinф; 1 sinф1 (sinфгcos0 + cost/)!sin0).
Consequently, for the function
<5((z) = sign(x,z)sign(j,z) = sign/r(z) (zeS"'1) A8.9)
we obtain
Since 0<9<n, we have
/(ф!, 0) = sign [sin ф j sin (ф j + 0)] =
(indeed, e.g. for ^^(О,^ —0) we have ф! + Ое@,я)с:(О,л), whence
sin<?i>0, sin(^1 + 0)>0, etc). Consequently, by A8.5)-A8.8) we get
A8.10)
-1 if фге{п-в,
1 See e.g. [59], p. 401-402.
18. Equivalence and pcrmutative equivalence of unconditional bases 531
g(z)dpL(z) =
IS"*11
ДфивK{ф)<Цф)
2n
Vl' Г •
i I I I (sinm
1=2 J
—,
я
= 2n] ЯФи1
о
which completes the proof of lemma 18.1.
if f=1 te a real-valued NxN matrix and let
Lemma 18.2. Let
M > 0 be such that
N N
;/,/=!,....N). A8.11)
Then for arbitrary elements xhyt (i,j= \,...,N) in a real inner product
space Ж we have
N N
]T ^T aij{xhyj) ^CMsup ||Xj|| sup||>*j||, A8.12)
where С is a constant independent of N and xhy-, satisfying
С ^ sinh - =
2
A8.13)
Proof. Let us first observe that if a matrix (а,7) satisfies A8.11) then
for arbitrary real numbers с\,с'- (/,_/= 1,...,7V), the matrix (a^), where
ai, = ciayc; (/,/=l,...,A0, A8.14)
also satisfies A8.11), with the constant M' = Msup|c||sup|cj|. Indeed,
¦ j
this is obvious if sup|c;|sup|c-j'| = 0; if sup|c||sup|Cj'|#O, then by
A8.11) we have ' j J
; /,y=l,...,7V),
¦ = i A sup \c'kI sup|c,"|
Ik I
к I
whence the assertion follows.
Let us proceed now to the proof of lemma 18.2. Since any 2N ele-
elements in Ж belong to some 2 JV-dimensional subspace of Ж which is iso-
isometric to E2N, we may assume without loss of generality that xhyjeE2N

532
II. Special Classes of Basis in Banach Spaces
(i,j=l,...,N). Furthermore, we may also assume that ||х;|| = \\yj\\ = 1
(i,j= l,...,N). Indeed, if lemma 18.2 is true in this case, then for arbitrary
elements хьухеЖ we have, by the above remark applied with c\= ||х;||,
whence, by our assumption applied to the matrix (ay||x;||
and to the elements x\ =
Xi
У)
if
jj||)fj=i
and х\,у)
= arbitrary elements of norm 1 of E2N for the other i,j, we obtain
E
E E aij(Xi,
j
which proves the assertion.
Thus, let xhyjeE2N, \\Xi\\ = \\y}\\ = 1 (iJ=l,...,N). Define
A8.15)
Then, by A8.11),
A8.16)
whence, integrating over S2iV~! with respect to the normalized rotation
invariant measure /л we get, by lemma 18.1,
A8.17)
for any matrix (a;j) satisfying A8.11).
We claim that the matrix (a|j'), where
= ач\ \ - 0(х„У]
A8.18)
nM
also satisfies A8.11) with M replaced by . Indeed, if |^| ^ 1, |^.| ^ 1
(i,j= l,...,N), then by the remark made at the beginning of this proof,
the matrix (a(;y>Sj>), where afy'Sj)= a^s,- (i,J= l,...,N), also satisfies
A8.11), whence, applying A8.17) for this matrix instead of (ai}),
18. Equivalence and permutative equivalence of unconditional bases 533
4f l f
which proves our assertion. Therefore, repeating the argument of
integration over S2iy~\ we obtain
L L ч\ j У ""j>
;=i j=i Lz
[л 1
Putting a(;2> = a(;jM Ofx,-,^) and continuing in the same
manner, we obtain inductively
E Iflyk-
lz
j) M (и=1.2.-)- A8.19)
Applying this for 2 n +1 instead of n and multiplying by
(-1)"
-, we get
t П2"
and hence
n
~2
2» + l M
2) Bn + :
M
Bи+1)!
со N N
N N со
= E E «у Е
i = 1 j = 1 n = 0
(-1)" y-0(Xi,J,.)
E ciijSm - - 6»(х;,^.) =^ ^ aycosfl^.jj-)
1l t jj^
¦=ij=i z
which completes the proof of lemma 18.2.

534
II. Special Classes of Bases in Banach Spaces
Corollary 18.1. Let (аг]) be a real valued NxN matrix satisfying
A8.11). Then for arbitrary elements yj(j=l,...,N) in a real inner product
space Ж we have
E
E аиУ]
A8.20)
Proof. For each i=\,...,N choose an element x.eJf such that
? ву^.**) =
Then, by using these elements xi,yi in A8.12), we obtain A8.20),
which completes the proof.
Lemma 18.3. Every continuous linear mapping u: Iх —>/2 is "absolutely
00
summing", i.e., carries each unconditionally convergent series Y z, in
oo '~ 1
Iх into an absolutely convergent series Y u(zt) in I2.
Proof. It is clearly sufficient to prove the statement for real spaces.
-jo
Let Y zi be an unconditionally convergent series in I1. Then, by
§ 15, corollary 15.1, there exists a constant M >0 such that
OG
(/?(/!)* = /°°). A8.21)
Let {х„} be the unit vector basis of Iх and {/„} cf/1)* =./¦*= the
a. s. с f. Then
('=1,2,...),
A8.22)
where ey=/}(zi)(i,./=U,...)-
Let t:,S: be arbitrary scalars with \tt\, \s,\^ 1 («',/'= 1,2,...,N). Then
for f= Y, sjfje(li)* = ^ж we have /(z,)= J] atjSj{i=l,...,N),
||/||= max |.^|^1, whence
E «№
JV
<E
i= 1
IM
JV
<E
E
18. Equivalence and permutative equivalence of unconditional bases 535
Hence, by corollary 18.1 applied for yj = u{Xj)el2 (/=1,...,N), and
by ||x,-1| = 10'= 1,2,...) we infer
I
E aiAxj)
E aij"(xj)
sup ||м(х_,.)|| ^ CM\\u\\
whence, for N-»oo, ^] ||m(z,-)|| ^ CM\\u\\ (n= 1,2,...). Consequently,
oo '=1
^ ||m(z;)|| < oo, which completes the proof of lemma 18.3.
Now we can prove
Theorem 18.2. In each of the spaces Iх and c0 all bounded uncon-
unconditional bases are equivalent.
Proof. Let {xn} be an arbitrary bounded unconditional basis of
? = /', with the a. s. с f. {/„}. Then, by § 14, proposition 14.1, {х„} is
a Besselian basis, and hence, by §11, theorem 11.1a) (implication Г=>3°),
there exists a continuous linear mapping u: Iх^>I2 such that
м(х„) = е„ (и=1,2,...), A8.23)
where {en} is the unit vector basis of I2. By lemma 18.3, и is absolutely
summing, whence, since ||м(х;)|| = ||е;|[,2= 1 (/= 1,2,...), we obtain
00
E
00
l = E
00
ii«(*i)li = E H
<18-24)
Conversely, since {х„} is a bounded basis, by Ch. I, § 3, lemma 3.1b)
we have the implication
2, |a;|< 00=> 2. '
i=l i=l
converges,
A8.25)
which, together with A8.24), proves that {х„} is equivalent to the unit
vector basis of I1. This proves the assertion of theorem 18.2 for the
space E = lx.
Now let {х„} be an arbitrary bounded unconditional basis of E = c0,
with the a. s. с f. {/„}. Then, by § 17, corollary 17.3a), {х„} is a shrink-
shrinking basis and hence, by § 17, theorem 17.7 and Ch. I, § 3, corollary 3.1,
{/„} is a bounded unconditional basis of E* = lx. Therefore, by the
above, {/„} is equivalent to the unit vector basis of I1. Consequently,
by Ch. I, § 12, proposition 12.1, {х„} is equivalent to the unit vector
basis of c0, which completes the proof of theorem 18.2.
Our next aim is to show that the fact that in /2 and L2([0,1]) all
bounded unconditional bases are equivalent (theorem 18.1) character-
characterizes the space /2 among the spaces V (the space L2([0,1]) among the

536
II. Special Classes of Bases in Banach Spaces
spaces Lp([0,1])) for 1 <p< oo; moreover, in each of the spaces lp and
Lp([0,1]), where \<p< oo, рф2, there exist two non-permutatively
equivalent bounded unconditional bases. For this purpose we need
some propositions and lemmas. By "projection" we shall understand
"continuous linear projection."
Proposition 18.1. In E = lp A ^p < oo) let {х„} be the unit vector basis
and let {zn} be a block basic sequence with respect to {xn}, i. e.,
A8.26)
i=mn- i + l
where О = то<т1< ¦¦¦ Then
a) There exists a linear isometry и of lp onto [zn], such that
A8.27)
b) There exists a projection v of norm 1 of lp onto [zn].
Proof, a) For any scalars ^,...,^n we have
mj
I
V=l
/ "
= ( L
\ J_
/
\ l
II
n
J= 1
P m
^J
¦7 1
p\i
IP
A8.28)
whence the assertion follows.
b) Put ?„=МГ=т„-, + 1 (и =1,2,...). Since zneEn, there exists a
functional фпеЕ* such that ф„(гп)=\, \\ф„\\=~-(п=\,2,...). Put
ll^ll
This series is convergent and ||d|| = 1, since by A8.28) and
Л =-,—7 we have
1*Л Е «
VI = Fllj - 1 + 1
j=l
•Л Е
1
cc m.j
E E
,j=l i=mj-
i=mj-i+l
P M-
= X .
18. Equivalence and permutative equivalence of unconditional bases 537
Finally, since ф„(гп)=1,
whence г;(х) = х for all хе[г„], which completes the proof.
Proposition 18.2. Le7 {г„} йе а йа^г'с sequence in a Banach space E
with the a.s.c.f. {hn} a [zB]* and let {yn} be a sequence in E. If there
exists a projection v of E onto [zn] such that
Ml E II^IMlJn-zn||
A8.30)
then {yn} is a basic sequence equivalent to {zn} and there exists a projection
wofE onto [>„].
Proof. By A8.30) we have
whence, by Ch. I, § 10, theorem 10.1, {yn} is a basic sequence equivalent
to {zn}. Put
i= 1
(the series in A8.31) is convergent since by и(х)е[г„] the series
CO
Л ht \y{x)~\ Zj is convergent and since {yn} is equivalent to {г„}). Then,
byA8-3O),
||/?-m||= sup ||x-m(x)||= sup
v(x)-
i= 1
= sup
i= 1
whence и is an isomorphism of E onto ?. Furthermore, by v(zn)=zn
and h^z^dij we have
CO CO
u(zn) = zn~v(zn)+ X /ф(г„)Ь= Е 4zn)yi=yn (и=1,2,...),
whence м([г„])=[у„]. Put
A8.32)

538
II. Special Classes of Bases in Banach Spaces
Then
and
whence w is a projection of E onto [>>„], which completes the proof.
Lemma 18.4. Let E be an infinite dimensional subspace oflp A ^p < oo).
Then E contains a subspace F which is complemented in lp and isomorphic
to I".
Proof. Let {х„} be the unit vector basis of lp. Since dim?=oo,
we can construct a sequence {yn} a E and a sequence of indices 0 = m0
<m1<-- such that
yn= t «! (»=U,-), A8.33)
I «i"»**
(и=1,2,...), A8.34)
(и=1,2,...). A8.35)
with lljjll = 1 and take
—. Assume that we have constructed
4
E ос|"+1>х.
Indeed, take an arbitrary yx =
mx such that
ух,...,у„ and т,,...,т„. Then, since dim?=oo and codim[xmn+k]f=1
= mn < oo, there exists an element yn+l= ? я!"+
i = m,,+ 1
with ||jn+1ll = l, whence also an index т„+1 with
^ , which proves our assertion.
^n + 2 x
PUt
zB = X «i"'** (и=1,2,...)
¦ = т„ - i + 1
Then, by A8.33), A8.36) and A8.35) we have
A8.36)
A8.37)
whence, taking into account A8.34),
A8.38)
18. Equivalence and permutalive equivalence of unconditional bases 539
and thus {zn} is a block basic sequence with respect to xn. Let {hn} a [zj*
be its a.s.c.f. Then for the sequence of coefficient functionals {AJ|zJ|}
associated to the basis
of [zn] we have, by proposition 18.1 a),
= l(n = l,2,...), whence, by A8.38),
1 1
I|AJ=-
1 -
1
A8.39)
Furthermore, by proposition 18.1b) there exists a projection v of
norm 1 of /" onto [zB] and we have, by A8.39) and A8.37),
00 00 1 1
IN I IIA.II 11^-z.K I Г'^г<1.
1 -¦
Consequently, by proposition 18.2, the subspace F=[jvl is com-
complemented in lp and isomorphic to [zn], whence also to lp (by proposi-
proposition 18.1a)), which completes the proof of lemma 18.4.
Let \^p<co and let {?„} be a sequence of Banach spaces. We
denote by (Ex x E2 x ¦ • -)iP the space of all sequences {х„} with xneEn
(n=l,2,...), for which {||;cj}e/p, endowed with the norm
A8.40)
It is easy to verify that (Er x E2 x ¦ ¦ -\P is a Banach space. The finite
product (?,x-x ?m)(P is defined similarly. By ?,x-x?, (where
m<oo) we shall denote the product space endowed with an arbitrary
norm equivalent to the norm of (?x x ••• x Em)lP.
We shall write ?^F, respectively, E = F, if the spaces ? and F
are isomorphic, respectively equivalent (i. e., linearly isometric).
Lemma 18.5. Let l^p<co. Then
a) // ?, F are Banach spaces such that ? = F, then
(?x?x---),p = (FxFx---),p. A8.41)
b) We have
(lpxlpx---\Pslp. A8.42)
c) For any two Banach spaces E, F we have
((? x F)lP x (E x F\P x •••),„ = ((? x ? x ••¦),, x(FxFx ¦••)„)„ A8.43)
and hence
((? x F\P x (? x F)lP x •••)!.> = (? x ? x ¦¦¦\P x(FxFx ¦¦¦),„. A8.44)

540
II. Special Classes of Bases in Banach Spaces
d) For any Banach space E,
(E x (? x E x •••)„)„ s (? x E x •••),„
and hence
? x (? x ? x •••),„ ^ (? x E x •••)„.
A8.45)
A8.46)
Proof, a) If m' is an isomorphism of ? onto F, the required iso-
isomorphism may be given by the formula
u({xn})={u'(xn)} (xneE,n=l,2,...)-
A8.47)
Indeed, и is one to one, linear and onto, and since Cx \\x\\ ^ ||м'(х)
|x|| (xe?) for suitable constants Cl,C2>0, we have
c, \\{xH}\\ = c,
({xn}e(ExEx -)lP
b) Let {en} be the unit vector basis of F and let и be an arbitrary
one to one mapping of the set of elements
onto the set {en}, extended by linearity to the set of all finite linear
combinations. Then we have, for any scalars atj (i=\,...,n;j=\,...,m),
M( E E
,i= 1 j = 1
E E ay"(zy)
= I EK-lt
= I
«ye,
V
E aiiei'---> E «;те;,0,0,...
i= 1
E E «yzy
whence the assertion follows, since {ztj} is linearly independent and
[zy] = (/'x/'x •¦•),,, [е„] = /".
с) The required isometry may be given by the formula
«({KjJ^II^-W} (xne?jnGF,n=l,2,...). A8.49)
18. Equivalence and permutalive equivalence of unconditional bases 541
Indeed, и is one to one, linear and onto, and we have
= E ll*.ll'+ E «л»
\i=l i=l
Hence, we also have the isomorphism A8.44).
d) The required isometry may be given by the formula
u({x,{xn}})={x,xl,x2,...} (x,xneE,n=l,2,...). A8.50)
Indeed, и is one to one, linear and onto, and we have
CO
.IIP I V II V I
¦11 + Li WXil
i= 1
\\p\p -
— \\ix Ix \\\\
Hence, we also have the isomorphism A8.46), which completes the
proof of lemma 18.5.
Lemma 18.6. IfE is an infinite dimensional subspace oflp, complemented
in lp, then E is isomorphic to lp A ^p < oo).
Proof. By lemma 18.4, ? contains a subspace F which is comple-
complemented in F, whence also in ?, and which is isomorphic to F. Thus there
exist1 subspaces El of F and F, Fx of ? such that
F^ExEu A8.51)
E^FxFu A8.52)
F^F. A8.53)
Hence, taking into account that the operation of product of Banach
spaces is associative and commutative and applying lemma 18.5, we
obtain
S(/p x /" x ¦ • •)„> x (Fi x Ex) =*((? x Ex) x (E x Ex) x ¦ ¦ ¦),, x (Ft x Et)
s(?x?x •••)jPx(?1x?1x •••),Px(?1xF1)
1 We use the well known fact (see Ch. I, § 4, lemma 4.1) that if F has a comple-
complement Fj in E, then ?SF x Ft.

542
II. Special Classes of Bases in Banach Spaces
Ex ¦••)(* x(El x?,x •••),p x Ft
EJxiExEJx ¦¦¦),PxF1={lpxlpx ¦¦¦Ip*F1
which completes the proof of lemma 18.6.
We shall write E s F, if there exists an isomorphism и of E onto F
satisfying
(xeE). A8.54)
Lemma 18.7. Let l^p<ao. Then
a) // {?„} аия? {Fn} are sequences of Banach spaces such that there
с
A8.55)
exists a constant C^ 1 for which En~ Fn(n= 1,2,...), then
с
(E1xE2x ¦¦ -),P ^{F1xF2x ¦¦ -),p•
b) // {?„} is a sequence of Banach spaces and Fn is a subspace of En
such that there exists a projection vn of En onto Fn (n= 1,2,...) and
sup ||i>J|<oo, then there exists a projection v of (ElxE2x •••);p
1 < П < CO
onto (F1 x F2 x ¦¦¦)(?¦
Proof, a) For each и= 1,2,... let м„ be an isomorphism of En onto
Fn satisfying
|| (xeEn). A8.56)
Then the required isomorphism can be given by the formula
«({*„})= {««(*¦,)} ({х,}б(?,х?2х...IР). A8.57)
Indeed, и is one to one, linear and onto (since so is each м„) and for
any {xn}e(E1 xE2 x ¦¦¦%„ we have, by A8.56),
= 11{«„(х„)}!| = цм({х„})||
b) The required projection can be given by the formula
v({xn})={vn(xn)} ({xn}e(ErxE2x-)lP). A8.58)
18. Equivalence and pcrmutative equivalence of unconditional bases 543
Indeed, v is linear and continuous, since
<*> \i / =°
L H^a*;/" 1^12. II ^H
sup \\vj I X 1|х,1Г)Р=( sup Kll)ll{
and f is a projection onto (F,xF2X"-),P since for {х„}6(Ft xF2 x• • •),„
we have vn{xn) = xn{n=\,2,...\ whence
v({xn})={vn(xn)} = {xn}.
This completes the proof of lemma 18.7.
Lemma 18.8. Let l<p<co and let rn(n= 1,2,...) be the Rademacher
functions on [0,1]. Then for each n the mapping
. ri -i
t)n(x)= ? jjc(f)r,.(O^ r; (xeLp([0,l])) A8.59)
i= 1 Lo J
w a continuous linear projection of Lp([0,1]) onto [rj,...,rn] a«J г/геге
exists a constant Bp depending only on p, such that
\\vn\\^Bp (n=l,2,...). A8.60)
i
Proof. Since the functional f(x)= J х(г)г;(г)Л are continuous
о
on Lp([0,1]), each vn is continuous. Furthermore, each vn is a projection
n
of Lp([0,1]) onto [гь...,ги], since for x= ^ a^r^e^!,...,^] we have,
by the orthogonality of the Rademacher functions (§ 14, corollary 14.1),
z«*
^= E ^-ji=x.
Finally, by §17, proposition 17.2b), for every xeLp([0,1]) the
limit u(x)= lim г;„(х) exists, whence, by the principle of uniform bound-
n—¦ со
edness, we obtain A8.60), which completes the proof.
Theorem 18.3. Let 1<р<оэ,рф2. Then the space lp has two bounded
unconditional bases which are not permutatively equivalent.

544
II. Special Classes of Bases in Banach Spaces
Proof. We shall prove that lp has a bounded unconditional basis
which is not equivalent to the unit vector basis of V; since every per-
permutation of this latter is equivalent to the unit vector basis of lp, it follows
that our basis is not permutatively equivalent to the unit vector basis of /p.
Let Fp be the space of all sequences of scalars {?„} such that
\\\{Q\\\ =
k= 1
k(k+ 1)
2
I
(fc-l)k
<oo.
A8.61)
It is easy to verify that Fp endowed with the norm A8.61) is a Banach
space1. Furthermore, the unit vectors yn={0,...,0,1,0,0,...} (и =1,2,...)
n- 1
form a normalized unconditional basis of Fp (because [yn] = Fp and by
§ 17, theorem 17.1, {yn} is an unconditional basic sequence in Fp, with
Since for each p with l<p<2 there exists a sequence {^n}eFp
CO
such that E l?ilP:
i=l
..., ,и=1,2,... and <^-=0 for the other j) and for each p> 2
1/
e.g. take Ы/- for j =
there exists a sequence
such that |||{?„}||| = оо but
i= 1
it follows that the basis {yn} of Fp is not equivalent (and hence, as observed
above, not permutatively equivalent) to the unit vector basis {xn} of /p.
Therefore, in order to complete the proof it is sufficient to prove that
the spaces Fp and lp are isomorphic. For this purpose it will be sufficient,
by lemma 18.6, to prove that there exists a subspace Ep of lp isomorphic
to Fp and complemented in lp.
Let \_ru...,rn~] be the subspace of Lp([0,1]) spanned by the
equivalence classes of the Rademacher functions rx,...,rn and let
G2n be the subspace of Lp([0,1]) spanned by the equivalence classes of
the characteristic functions Хгк^х k-i (/c=l,...,2"; n= 1,2,...). Since
G2n is equivalent to /?.„, the subspace [rl,..., /¦„] of Gln is equivalent to an
и-dimensional subspace Rn of l\n (n= 1,2,...).
Define Ep to be the image of the subspace (RxxR2x
{lp2 x l\i x • • )(P in lp by the canonical linear isometry2 (/f. x /f.2 x
We shall show that Ep has the required properties.
1 ActuaUy,F, = (/»x^x...IP,bythemapping{U-{51,{52,{3},{54,«s.
2 This isometry is given by the formula {{?ь?2},{^ъ,и,ЪЛв),•••}
of
6},}
"¦ {?„}¦
18. Equivalence and pcrmutaiive equivalence of unconditional bases
545
By the Khinchin inequality (§ 14, formula A4.56)) there exists a
constant Cp>0 such that Rn *k l\ {n= 1,2,...). Consequently, by lemma
18.7 a), the spaces Ep and (/2x/f x ••¦),, are isomorphic. On the other
hand, it is obvious that Fp = (l\ xl\x ¦¦¦\P. Thus ?p = Fp-
Since G2n=> [г1(...,ги], from lemma 18.8 it follows that there
exists a projection1 и'„ of G2n onto \_r1,...,rn], of norm ||m;||^Bp
(n= 1,2,...), whence, by the definition of Rn, there exists a projection un
of /?„ onto Rn of norm \\un\\^Bp (n=l,2,...). Consequently, by lemma
18.7b), (RjxRjX-jjp is complemented in (/f x/f2x •••),P and thus ?p
is complemented in /p, which completes the proof of theorem 18.3.
In order to prove that a result similar to theorem 18.3 also holds in
the spaces Lp([0,1]), where 1<р<со, рф2, we shall need some pro-
propositions and lemmas.
Proposition 18.3. // {zn} is a sequence in Lp([0,1]) (K/xco),
such that zn^0 (n=l,2,...) and that the sets An= {te[Q,l]\zn{t)^Q}
(и =1,2,...) are mutually disjoint, then {zn} is a basic sequence and the
subspace [zn] is isometrically isomorphic to lp, by the mapping
-^ > xn {n= 1,2,...), where {*„} is the unit vector basis oflp.
Proof. For any scalars
sets An,
^„ we have, by our hypothesis on the
z-
dt=
\Zj(t)\f
\\ЦР
dt
= I \Zj\'
and hence the mapping м:х„->-™- (и =1,2,...) can be extended, by
ll^nll
linearity and continuity, to an isometric isomorphism of /p onto [zj.
Lemma 18.9. Let {Ak} be an infinite sequence of distinct sets Akc[0,1]
(/c=l,2,...), such that if kx<k2 and Лк1пАк2Ф0, then Akl=>Aky Then
there exists a subsequence {AK} of {Ak\ such that either
А, глАк =0 whenever
or
A8.62)
A8.63)
1 Namely, u'n = vn\O2n(n=h2,...), where vn is defined by A8.59).
35 Singer, Bases in Banach Spaces I

546
II. Special Classes of Bases in Banach Spaces
Proof. Assume that no subsequence of {Ak} satisfies A8.63). Then
for every index к there exists an index ф(к) such that Аф(к)с Ak and
that Аф(к) contains no other А^Аф(ку Since {Ak} is infinite, the set
Х={ф(\),ф{2),...} is infinite. From the hypothesis on [Ak] and from
the definition of Z it follows that if /c^,/cveZ are such that /cu#/cv,
then Ak nzlkv = 0. Hence the subsequence {Akv}kveZ of {Ak} satisfies
A8.62), which completes the proof of lemma 18.9.
Proposition 18.4. Let {yn} be the Haar basis of LP ([0,1]) A ^p< oo).
Then for every sequence of indices пх<пг<-- the space [уПк] contains
a subspace isomorphic to F and hence [у„к] is not isomorphic to I2 if'рф 2.
Proof. Put
U] 134@/0}
A8.64)
then the sequence \у„
I
> where t,eAk2Jv=l,2,...),
where we assume, for convenience, that each Haar function ynk is 0
in its dyadic dividion points (i.e., in the endpoints and midpoint of Ak).
Then the sequence {Ak} satisfies the assumption of lemma 18.9 and
hence it must have a subsequence {Akv} satisfying either A8.62) or
A8.63). Now, if {Akv} satisfies A8.62), then the sequence {у„К} satisfies
the assumption of proposition 18.3, whence the subspace [у„к ] is iso-
metrically isomorphic to /p. On the other hand, if {Akv} satisfies A8.63),
- """'^г^У
IJnk2v('v)l
satisfies the assumption of proposition 18.3, whence it spans a sub-
subspace isometrically isomorphic to /p, which completes the proof of
proposition 18.4.
Lemma 18.10. Let \<p<co. Then the product space Lp([0,l])x/2
is isomorphic to the space Lp([0,1]).
Proof. By § 17, proposition 17.2 b), there exists a Banach space E
such that Lp([0,l]) = ?x/2. Since /2x/2 = /2, we obtain
LP ([0,1]) x /2 ^ (E x I2) x I2 ^ E x (I2 x I2) ^ E x /2 s LP ([0,1]),
which completes the proof.
Now we can prove
Theorem 18.4. Let 1<р<оэ, рф2. Then the space L"([0,1]) has
two bounded unconditional bases which are not permutatively equivalent.
Proof. Let {у„} be the Haar basis of Lp([0,l]) and let {xn} be the
unit vector basis of I2. Then, by § 14, theorem 14.1, and § 16, proposi-
proposition 16.4, the sequence Jz,}cLp([0,l])x/2 defined by
A8.65)
18. Equivalence and pcrmutative equivalence of unconditional bases
547
is a bounded unconditional basis of Lp([0,l])x I2. Since by lemma 18.10
the space Lp([0, l])x/2 is isomorphic to Lp([0,1]), it follows that the
image {z'n} of {zn} in Lp([0,1]) under this isomorphism is a bounded
unconditional basis of Lp([0,1]) containing a subsequence equivalent
to the unit vector basis of I2. On the other hand, for the bounded un-
f yn ) f у }
conditional basis < -^— > of Lp( [0,1 ]) no permutation < ——У- > has such a
subsequence (by proposition 18.4), and therefore {/„} and
are not
permutatively equivalent, which completes the proof of theorem 18.4.
The spaces c0,11 and /2 are not the only ones (up to an isomorphism)
in which all normalized unconditional bases are permutatively equiv-
equivalent. Indeed, one can prove1 that e. g. the product space I1 x /2 also
has this property. However, if we replace "permutative equivalence" by
the more restrictive "equivalence," then the situation is already different.
Namely, we shall see in § 24 that the only spaces (up to an isomorphism)
in which all normalized unconditional bases are equivalent, are c0, I1
and/2.
Finally, we shall use permutative equivalence to prove the unique-
uniqueness, up to an isomorphism, of Banach spaces having an unconditional
basis which is universal (in the sense of § 13) for the family @u of all
bounded unconditional bases. To prove first the existence of such bases,
we shall need the following "unconditional analogue" of Ch. I, § 5, pro-
proposition 5.3 a), b):
Proposition 18.5. Let E be a Banach space, (xn,fn)({xn}cE, {/„}<=?*)
a biorthogonal system and let
Ux=lxeE
CO "I
Y,ft(x)xi is unconditionally convergent >, A8.66)
J
>,
J
endowed with the usual vector operations and with the norm
|||x|||= sup sup \\s{ISnU{x)\\.
l/3i|,|/32l,...«l l«l<»
Then
a) Ux is a Banach space.
b) {xn} is an unconditional basis of U^.
Proof, a) Consider the linear space of sequences of scalars
<>=
A8.67)
а(Х; is unconditionally convergent >, A8.68)
1 See [53].

548 II. Special Classes of Bases in Banach Spaces
endowed with the norm
||{а„}|| = sup sup
A8.69)
Then, by an argument similar to that used in the proof of Ch. I,
§ 3, proposition 3.1, A[u) is a Banach space. Since the mapping
W-Ьл A8.70)
is a linear isometry of A("] onto Uu it follows that Ux is a Banach space.
Then, by A8.67),
(.X6l/,,n=l,2,...).
1
A8.71)
x||| (xeUun=l,2,...)
and hence фпе11? (и= 1,2,...). Since {xn,fn) is a biorthogonal system,
it follows that (xn, ф„) is a biorthogonal system. Furthermore, by A8.67),
A8.71), A6.26), A8.66) and § 16, lemma 16.1 (implication 1°=>6°) we have
x -
= sup sup
= sup sup
sup
E PjfjWxj
whence, by Ch. I, § 4, theorem 4.1, {х„} is a basis of Ul. Since by the
same argument1 every permutation {х„(п)} of {xn} is a basis of Uu it
follows that {xn} is an unconditional basis of 1715 which completes the
proof of proposition 18.5.
From the above proof it follows that the unit vectors en={5nj}f=1
(и =1,2,...) constitute an unconditional basis of the Banach space A{"]
defined by A8.68), A8.69). However, in contrast to Ch. I, § 8, proposi-
proposition 8.1b), {х„}, considered as a sequence in E, need not be equivalent
to {е„}, as shown e.g. by any conditional basis {xn} of E.
Let us also observe that by A8.67) we obviously have
(n=l,2,...). A8.72)
XJ = Х„
1 Alternatively, one can observe that ^ „,~,,
i= 1
finite sequence of scalars al,..., а„, д1,..., &„ with \5
§ 17, theorem 17.1.
for every
III ' = 1
i.|^l ((=!,...,n) and apply
18. Equivalence and permutative equivalence of unconditional bases
549
Theorem 18.5. The family
contains a universal element.
% of all bounded unconditional bases
Proof. Let {xn} be a bounded basis of a Banach space E, which is
universal for the family of all bounded bases (by § 13, corollary 13.3,
there exist a continuum of mutually non-equivalent such bases) and
let {/„}c?* be the a. s. с f. to {xn}. For this biorthogonal system
(*„,/„) define the Banach space Ul as in proposition 18.5 above. We
shall prove that the bounded unconditional basis {xn} of Vx is universal
for the family <MU of all bounded unconditional bases.
Let {yn} be an arbitrary bounded unconditional basis (of a Banach
space, say F). Then, since {xn} is universal for the family of all bounded
bases, there exists a subsequence {xir} of {х„} such that {yn}~{xir}.
Hence, since {yn} is an unconditional basis, from Ch. I, § 8, theorem
8.1d) (implication 6"=>1°) it follows that {xin} is an unconditional
basis of the subspace [x,J of E. Therefore, [x,J с (у, and by § 16,
theorem 16.1b), the norm on [x,J induced by the norm A8.67) of U^
is equivalent to the norm on [x,-n] induced by the initial norm of E.
Consequently, {yn} is equivalent to the subsequence {x,J of the
bounded unconditional basis {х„} of l^, which completes the proof
of theorem 18.5.
Let us observe that by § 16, theorem 16.8 (implication 1°=>2°) every
universal unconditional basis is complementably universal and hence
theorem 18.5 also shows the existence of a complementably universal
element in the family $8U of all bounded unconditional bases. Moreover,
the "unconditional analogue" of the uniqueness theorem for com-
complementably universal bases (§ 13, theorem 13.2) is also valid, yielding
the uniqueness of universal unconditional bases (in contrast to § 13,
corollary 13.3 on the non-uniqueness of universal bases):
Theorem 18.6. Any two bounded unconditional bases, which are uni-
universal for the family 8$u of all bounded unconditional bases, are per-
mutatively equivalent. Hence they span isomorphic Banach spaces and
therefore the Banach space U1 in the above proof of theorem 18.5 is
unique up to an isomorphism.
The proof is analogous to that of § 13, theorem 13.2, taking into
account § 16, proposition 16.4, § 17, corollary 17.4 and the above re-
remark that every universal unconditional basis is complementably
universal.
Corollary 18.2. There exists a Banach space E with an unconditional
basis such that every Banach space with an unconditional basis is iso-
isomorphic to a complemented subspace of E.

550
II. Special Classes of Bases in Banach Spaces
We shall denote by Eu the Banach space, unique up to an iso-
isomorphism, which has a bounded unconditional basis universal for the
family of all bounded unconditional bases. The following corollary
shows that the property described in corollary 18.2 also characterizes
Eu up to an isomorphism:
Corollary 18.3. If E is a Banach space with an unconditional basis,
such that every Banach space with an unconditional basis is isomorphic
to a complemented subspace of E, then E is isomorphic to Eu.
The proof is analogous to that of § 13, corollary 13.2.
Problem 18.1. Are all normalized unconditional bases of the space Eu
permutatively equivalent?
Problem 18.2. Let a Banach space E with an unconditional basis
contain a subspace isomorphic to Eu. Is E isomorphic to ?„?
Concerning the problem of existence of universal bases for other
classes of unconditional bases we mention that by § 13, the proof of
theorem 13.5, we actually have the following sharpening of that theo-
theorem: There exists no shrinking basis which is universal for the family
of all normalized unconditional shrinking bases. An affirmative answer
to § 17, problem 17.1 would again imply, by § 17, theorem 17.6, that the
family of all normalized unconditional shrinking bases has no universal
element.
§ 19. Best approximation in Banach spaces with
unconditional bases
In the present section we shall consider for unconditional bases
some notions and problems analogous to those studied in Ch. I, § 19
for arbitrary bases. We shall see in theorem 19.1 and example 19.1
below that the relations between the "unconditional analogues" of
T-norms and K-norms are different from those between T-norms and
K-norms.
Definition 19.1. The norm in a Banach space E with an unconditional
basis {х„} is called an NT-norm (with respect to the basis {х„}), if
a) for every xeE and d={iu...,in}e@ there exists a unique poly-
polynomial Уо = лР d)(x)eP(d)=\_xh,...,xitJ of best approximation of x;
b) this polynomial coincides with the af-th partial sum of the ex-
expansion of the element x with respect to the bais {xn}, i.e.
t ={iu...,QeS). A9.1)
We shall denote the NT-norms (( ((„.
19. Best approximation in Banach spaces with unconditional bases
551
Definition 19.2. The norm in a Banach space E with an unconditional
basis {х„} is called an NK-norm (with respect to the basis {х„}), if
a) for every xeE and d={iu...,in}e& there exists a unique poly-
polynomial complement уо = лРи>(х)е P(d) =[Xj]Jejr. d of best approxima-
approximation of x;
b) this polynomial complement coincides with the af-th rest of the
expansion of the element x with respect to the basis {х„}, i. e.
npW(x)=rd(x) = x-sd(x) (xeE,d={h,...,in}e&). A9.2)
We shall denote the NX-norms by )) ))„.
Definition 193. The norm in a Banach space E with an unconditional
basis {х„} is called an NTK-norm (with respect to the basis {х„}) if it
is simultaneously an NT-norm and an NX-norm with respect to this
basis.
We shall denote the NTX-norms by (( ))„.
For instance, the natural norm in a separable Hilbert space H is
an NTX-norm with respect to any orthogonal basis {х„} of the space H.
Let us first give a useful characterization for each of these norms.
Proposition 19.1. Let E be a Banach space with an unconditional basis
{х„}. Then
a) The norm in E is an NT-norm if and only if
A9.3)
for every pair dl,d2e@ with d1ad2 and every sequence of scalars
a;| #0, for which the series in A9.3) are convergent.
A9.4)
iGd2\di
b) The norm in E is an NK-norm if and only if
for every pair dx,d2&3) with d1ad2 and every finite sequence of scalar s
Ыш2 with t I^MO.1
ied2\di
Proof, a) Assume that the norm in E is an NT-norm. Let ? aixi
i?,\'\d2
be convergent, where d2 e 3), and let dx <= d2 be arbitrary. Then
E «;Х( has a unique element of best approximation in P(d2) = \_xi]ied2>
ieA'\di
namely
ОС;*;,
ied2\di
1 The bases {х„} satisfying A9.4) (i.e. such that the norm in E is an NK-norm
with respect to {х„}) are called strictly orthogonal (see § 20).

552
II. Special Classes of Bases in Banach Spaces
whence, since 0ePA,2),
X «1*1 = X 0.^-np
X <*i*i
i. e. we have A9.3).
Conversely, assume now that we have A9.3). Then for every
GC
x= X ^iXjeE, de3s and p= X /J^
A9.3) with d2 = d,dt = $)
with рФ$й(х), we have (by
X «i*
iej d
and thus the norm in E is an NT-norm.
b) Assume that the norm in E is an NX-norm. Let {oc;}iEd2 be a
finite sequence of scalars and let d1<=d2 be such that X lail^0-
ied2 di
Then X ixixi has a unique element of best approximation in
^" = [3,/^ namely
] <*¦*¦)= X aixi>
z aixi - ^p*' (z aixi)i < z aixi
V ' id
whence, since OeP(dl),
ied,
i. e. we have A9.4). oo
Conversely, assume now that we have A9.4) and let x = X а;Х;,
i = l
rfe® and /?= Z PiXieP(d) with P^rd(x)— Z aixi ^e arbitrary.
Then there exists in Ж\й a smallest index, say i0, such that /?in^ai().
Hence, applying A9.4) successively, we obtain
isd
- X (ft-«<)*<
is.»" d
ied
(ft-«0х,-
>,.*,- X (ft-««)*,• j
I ied ie*« d
and thus the norm in E is an N K-norm, which completes the proof of
proposition 19.1.
19. Best approximation in Banach spaces with unconditional bases 553
Let us consider now the relations between N T-norms and NK-norms.
Theorem 19.1. Let E be a Banach space with an unconditional basis
{х„}. Then every NT-norm with respect to {х„} is an N K-norm (whence
also an N TK-norm) with respect to {х„}.
Proof. We shall prove that the relations A9.3) imply the relations
A9.4). The idea is to complete the a,-'s in A9.4) with zeros until we obtain
suitable sums of the form X ai-vn to which we can apply A9.3).
ieJi \d
Let d1,d2e@, dY с d2 and {a,}ied2 <
ft =
К be such that
a; for i ed2
0 for /e./K'W,,
Then
@, d[<=d'2, X lftl= Z
X
ed2 d,
A9.5)
A9.6)
and the series
X ft*;= X aixt is convergent. Hence, since the norm is an N T-norm
ieJ cl\ ied2
with respect to {х„}, by A9.3) we have
x aixi!
ied.
X
= I Z ftxi I <
ie \"\d'2
Z ftxi
ieJ/"\d\
i.e. A9.4), which completes the proof of theorem 19.1
The converse implication is not true, i.e. there exist ./VK-norms which
are not N T-norms, as shown by
Example 19.1. Let E be the space c0 endowed with the equivalent
norm ))x)) given in Ch. I, § 19, example 19.2, formula A9.10). Then
))x))u = ))x)) is an N K-norm, but not an N T-norm, with respect to the
unit vector basis {х„} of c0.
Indeed, proposition 19.1 and Ch. I, § 19, formulas A9.13), A9.14)
show that ))x))u is an N K-norm, but not an NT-norm, with respect to {х„}.
Theorem 19.2. Let E be a Banach space with an unconditional basis
{xn\ and let {/„} a E* be the a.s.c.f. Then one can introduce on E an
N TK-norm equivalent to the initial norm on E, by the formula
(W)u=X «\\fi(
sup
<ii 'n!e
У fi(x)xi
A9.7)
Proof It is obvious that ((x))u is a norm on E. This norm is equivalent
to the initial norm on E, since for every x e E we have

554
II. Special Classes of Bases in Banach Spaces
IWK((x))u< max ||/((х)х(||
ПК 00
: max
1 ^i< oo
Z fM)Xj
sup
i- 1
Z fh(x)xtj
Z fj(x)xj
+ sup
lU-ll.
Finally, in order to prove that ((*))„ is an N TK-norm it will be suffi-
sufficient, by theorem 19.1, to prove that it is an NT-norm. Let dud2eS> with
dd{i}ieJrW2whh ? |af|#0 be such that ? a, x^ converges.
We have then Jr\d1 = (, I \d2) и (rf2Wi), whence
1
z «,*, = z
У - Ha x II
Zv Ti I' ' >l
SUP
{i, i,)eS
SUP
Z ««л
j= 1
n
and thus, by proposition 19.1a), ((x))u is an NT-norm, which completes
the proof.
Let us mention that one can also prove, with a similar argument, that
in the conditions of theorem 19.2 it is possible to introduce another
equivalent N TK-norm on E, by the formula
1
-Ii
sup
1 ^n< oo
sup Zl
A9.8)
Remark 19.1. Theorem 19.2 admits, in a certain sense, a converse.
Namely, one can define weak NT-norms ( = weak NK-norms= weak
N TK-norms) with respect to an arbitrary sequence {х„} с Е with
x^O (n=l,2,...) and [х„] = Е, similarly to remark 19.1 of Ch. I, § 19.
If one can introduce on E such a norm, equivalent to the initial norm on
E, then by § 17, theorem 17.1, {х„} is an unconditional basis of ? endowed
with this new norm, and therefore also of E with its initial norm.
20. Orthogonal bases. Strictly orthogonal bases. Hyperorthogonal bases 555
§ 20. Orthogonal bases. Strictly orthogonal bases.
Hyperorthogonal and strictly hyperorthogonal bases
In this section and the subsequent ones we shall study some special
classes of unconditional bases. In the present section we shall con-
consider some "unconditional analogues" of monotone bases (defined by
the condition that one of the constants of § 17, theorem 17.1, be equal to
1) and of strictly monotone bases.
Definition 20.1. A basis {xn} of a Banach space E is said to be ortho-
orthogonal, if we have
Za,- X:
Z *i,xlj + Z
B0.1)
for any п,т€.Ж and any scalars
indices satisfy {/1,...,/n}n{/1,...,/m} =
strictly orthogonal, if we have
У «i x,.
аA,...,а!п, аA,...,а,тбК whose
. The basis {xn} is said to be
B0.2)
for any
aijAij "r Zv °%- О
and any scalars ah,...,ain, au,...,almeK with
{/1,...,/m}n{/1,...,/m} = 0, such that f
j= i
For instance, the unit vector basis of lp (/?> 1) is strictly orthogonal,
while the unit vector basis of c0 is orthogonal, but not strictly orthogonal.
Similarly to the geometric interpretations of monotone and strictly
monotone bases, given in § 1, we see that a basis {х„} of E is orthogonal
(respectively, strictly orthogonal) if and only if we have [хA,...,х(п]
±[xh,...,xlm\ {respectively, [xh,...,xin~] 1 l\_xh,...,xlm\) for any indices
such that {il,...,in}n {I1,...,lm} = $. From this remark and from §1,
lemma 1.1, it results that for a basis {xn} of a Hubert space E the following
statements are equivalent: 1°. {х„} is orthogonal. 2°. {xn} is strictly ortho-
orthogonal. 3°. {xn} is orthogonal in the usual Hilbert space sense (i.e., (х;,х;) = 0
for all 1ф}).
One may also express these conditions in the following form:
for any di,d2eS> with dt<=d2, respectively
<II.S(
for any di,d2^^ with dl<=d2, such that J]

556
II. Special Classes of Bases in Banach Spaces
Let us also observe that for every orthogonal basis {xn\ with the
a.s.c.f {/„} we have
Ы\\и = 1 (n=l,2,...), B0.3)
and hence, in particular, every normalized orthogonal basis {xn} is normal.
Indeed, by B0.1),
L[ Z aixi
\i= 1
1
1
a- =-
Z
Y а.гх(еЕ, n= 1,2,... I,
whence B0.3) follows.
By § 17, theorem 17.1, a basis {х„} of a Banach space E is orthogonal
if and only if it is of unconditional norm v'^' , = sup Ц^Ц = 1 (whence, in
particular, unconditional). From this remark it follows that the problem
of existence of orthogonal and strictly orthogonal bases has a negative
answer in the class of all infinite dimensional Banach spaces with bases.
Furthermore, from this remark and § 17, theorem 17.1, formula A7.19),
there follow at once several intrinsic characterizations of orthogonal
bases. Some other intrinsic characterizations of such bases are given in
Theorem 20.1. Let E be a Banach space and {х„} a complete sequence
in E such that х„ # 0 (n = 1,2,...). The following statements are equivalent:
1°. {xn} is an orthogonal basis of E.
2°. Every permutation \xain)} (аеП) of {х„} is a monotone basis of E.
3°. Every subsequence {xin} of {xn} is a monotone basic sequence.
00
Moreover, in this case the relations Y а(х(еЕ and
(n=l,2,...) imply
and
i= 1
00
i= 1
cc
Z aixt
i= 1
B0.4)
Proof 1°=>2°. If {xn} is an orthogonal basis and аеП, then by
B0.1) we have, for any n,me.Jf and any scalars астA), а(ТB),...,аG(п + т),
Z ««о*
L, a<r(i)
{xa(n)}
20. Orthogonal bases. Strictly orthogonal bases. Hyperorthogonal bases 557
2°=>1°. Assume that we have 2" and let {il,...,in}n{ll,...,lm\
= 0, a.-p...,^, a.h,...,a.lmeK be arbitrary. Take any аеП such that
Then, since {xaM} is a monotone basis,
Z %xh
У а
"(j)
L, a"U)x"U)
Z «1^0+ Z
and thus {х„} is an orthogonal basis.
The proof of the equivalence 1°<=>3° is similar.
Finally, for any ке.Л ' and y1,...,yk_l, yk+l,...,yneK the continuous
real function
ф(А) =
k- 1
(-oo<A<oo) B0.5)
is convex (obviously, ф(а2.1+A— а)Д2)<а0(Я1) + A— а)ф(л2) for any
0<а<1 and any — оо<Я1,Я2<оо) and, by 1°, ф(Х) has a minimum at
Я = 0, whence for any onk,yk with 0^yk^ak(k=l,...,n)
k-\ n
Z iixi
Z lixi
Z
n
У У-Х-
i= 1
n
У a-x-
i= l
Applying this successively for k=l,...,n it follows that for any
ne,/F and 0^yk^ak {k= l,...,n)
B0.6)
whence, taking и->со, we infer B0.4), which completes the proof of
theorem 20.1.
Remark 20.1. With a similar argument to the above proof of B0.4),
it follows that for a strictly orthogonal basis {х„} of E the relations {а„},
GO
{у„}еК, Y <xixieE,0^yn^txn(n=l,2,...) andyno<txno for some index n0
GO
imply Yj 7ixieE ап{$, for any ne.V,
B0.7)
n
Z 7Л
i= 1
<
n
i= 1

558
II. Special Classes of Bases in Banach Spaces
n
У В а х
i= 1
У а х
i= 1
conversely, if {х„} has this property, then it is obviously strictly ortho-
orthogonal. However, B0.7) need not hold for n = oo, as shown by § 19, example
19.1.
Definition 20.2. A basis {х„} of a Banach space E is said to be hyper-
hyperorthogonal if we have
n n
B0.8)
for every finite sequence of scalars аь...,а„, Р1,...,р„ with |/?(|<1
(/= \,...,n). The basis {xn} is said to be strictly hyperorthogonal if we have
B0.9)
for every finite sequence of scalars а1,...,а„, /?!,...,/?„ with |j3;|^l
(i=l,...,n) such that |/?io|<l, aio#0 for some index i0.
For instance, the unit vector basis of lp (p^l) is strictly hyper-
hyperorthogonal, while the unit vector basis of c0 is hyperorthogonal but not
strictly hyperorthogonal.
Some intrinsic characterizations of hyperorthogonal bases are col-
collected in
Theorem 20.2. Let E be a Banach space and {xn} a complete sequence
in E such that х„#0 (n= 1,2,...). The following statements are equivalent:
1°. {xn} is a hyperorthogonal basis of E. oo
2°. The relations {а„}, {у„} с К, |у„|<|а„| (п=1,2,...), ? а^еЕ
n
V R
i= 1
<
n
i= 1
imply
and
GO
TbXi
i= 1
GO
i= 1
3°. We have
GO
Е«л
i= 1
=
00
E
i= 1
E «i
B0.10)
B0.11)
4°. For every finite sequence of scalars аь...,а„, ?ь...,е„ wi?/i |?;| =
(/= l,...,n) we
B0.12)
П
У ?-а х
i= 1
П
E «,*,
i= 1
1 Naturally, a corresponding condition could be also added in § 16, theorem
16.1 and § 17, theorem 17.1, but there it would be an obvious consequence of the
equivalences 4°<t>5° and 5°<t>7° respectively.
20. Orthogonal bases. Strictly orthogonal bases. Hyperorthogonal bases 559
// E is a real Banach space, these statements are equivalent to the
following:
5°. {х„} admits a total sequence {/„} a E* such that /;(х;) = с>0- and the
space E, endowed with the natural order induced by Ж(Хп^п) (i.e., x>0
if and only if fn(x)>0 for n=l,2,...) is а КB-lineal (that is1, a normed
vector lattice in which the relations x,yeE, |x|<|j| imply \\x\\ ^ \\y\\).
Furthermore, in this case all constants occurring in § 17, theorem
17.1, are equal to 1.
Proof 1°=>2°. If |у„|<|а„| (n=l,2,...), then у„ = Д„а„ for suitable ]?„
with |/?„|<1 (n=l,2,...) and hence, by definition 20.1,
П
У y-x-
i = 1
=
n
V R
i—j ' i i ^
i= 1
n
i= 1
whence, taking n->co, we obtain 2°.
The implication 2° =>3° is obvious, since \at|< ||af11< |a;|.
3°=>4°. If we have 3°, then for any |?j|=l (г=1,...,п)
n
У Е0СХ-
I—j i i i
i= 1
=
n
E
i= 1
?,(Х;|Х;
=
П
E la;lx-
i= 1
=
n
i= 1
4°=> Г. If we have 4° and the scalars are complex, then, since every
complex number /}, with |/?;|<1 can be written in the form
eU> , ?B)
, where \e\l)\ = |e|2)| = 1, we have
E
E
E
If we have 4° and the scalars are real, then, by the proof of § 16,
inequality A6.27), we again obtain B0.8). Thus, 1°«>---«>4O.
Assume now that ? is a real Banach space.
Гп2°=>5°. If we have Г, then, by § 16, theorem 16.3, ? is a normed
vector lattice. By virtue of Г and 2°, the relations x,yeE, \x\^\y\
imply ||xj|<||j||, and thus ? is a KB-lineal.
The implication 5° => Г is obvious, by § 17, theorem 17.1.
Finally, the assertion concerning the constants of § 17, formula
A7.19) is an obvious consequence of that formula. This completes the
proof of theorem 20.2.
If {хл} is a hyperorthogonal basis, then, obviously, every sequence
{?„*„}, where |?„|=1 (n=l,2,...), is an orthogonal basis (in particular,
every hyperorthogonal basis is orthogonal). The converse is not true, as
shown by
1 See e.g. [129], p. 211.

560
II. Special Classes of Bases in Banach Spaces
Example 20.1. Let E be the two-dimensional Banach space of all
pairs of scalars x = {?i,?2} endowed with the norm
B0.13)
and let x1 = {l,0}, x2 = {0,1}. Then for any scalars a1,a2,?1,e2 with
?l|= \e2\ — 1 we have
whence every {EiXl,e2x2} (where |e!l=|?2| = l) is an orthogonal basis
of E, but
j|x1+x2]|=max(l,l,2) = 2, ||x1-x2||
and hence \xl,x2} is not hyperorthogonal (by theorem 20.2, implication
Proposition 20.1. A basis {х„} of a Banach space E is strictly hyper-
hyperorthogonal if and only if {х„} is both strictly orthogonal and hyper-
hyperorthogonal.
Proof. Clearly, every strictly hyperorthogonal basis is both strictly
orthogonal and hyperorthogonal.
Conversely, assume that {х„} is both strictly orthogonal and hyper-
hyperorthogonal. Let аь...,а„, /Jb...,/}„ be arbitrary scalars with |/?;|<1
(i=l,...,n) and |0J<1, aio^0. Then |0loaJ<|aJ and hence, by re-
remark 20.1 and theorem 20.1, implication 1°=>3°, we have
Z ft*,*,
which completes the proof.
We have the following theorem of "orthogonalization":
Theorem 20.3. Let E be a Banach space with a basis {х„}. The fol-
following statements are equivalent:
1°. There exists on E a norm \\x\\u equivalent to the initial norm
on E, such that in the norm ЦхЦ, the basis {xn} is strictly hyperorthogonal
(hence also hyperorthogonal, strictly orthogonal, orthogonal).
2°. There exists a Banach space F with a strictly hyperorthogonal
basis \yn\, such that {х„}~{у„}.
3°. {х„} is unconditional.
Z №
= 1
<
п
[XI
i = 1
=
и
У а х-
i = 1
20. Orthogonal bases. Strictly orthogonal bases. Hyperorthogonal bases 561
Proof. 1°=>3°. If we have 1°, then, as observed in the above, {х„} is
unconditional in the norm \\x\\u whence also in the initial norm of E.
3°=>2°. Assume now that we have 3° and let1
Z aixi
= Z ;*ll«i*.ll +.SUP.
sup
i—j ' i ^ i
i= 1
z
B0.14)
Then, by § 17, theorem 17.1, \\x\\l<oo {xeE) and
/ GC
\x\\ ^ \\x\\i ^ max ||cxfjcf|| +M5 ||x|| ^ 2M5 ||x|| i=y».
and hence the norm ||jc|| t on E is equivalent to the initial norm of E.
Furthermore, in the norm ||x||, the basis {х„} is obviously strictly hyper-
hyperorthogonal. Hence, for F = E endowed with the norm ||jc|| t and
Ш = Ы(ЬП we have 2°.
2° => Г. If we have 2°, then, by Ch. I, § 8, theorem 8.1 d) (implication
6°=* 1°), there exists an isomorphism и of E onto F, such that и(х„) = уп
(n = 1,2,...). Then, defining
X ,= MX
(xeE),
we obtain a norm on E, equivalent to the initial norm and such
that {х„} is strictly hyperorthogonal in the norm Цх^ (since {«(*„)]
= {у„} is a strictly hyperorthogonal basis of F), which completes
the proof.
Finally, let us give some relations between positive bases, the
disjoint support condition (§ 3, remark 3.2) and various types of
orthogonal bases.
Proposition 20.2. Let {х„} be a positive strictly hyperorthogonal
basis of a Banach space E, satisfying B0.7) for n=co. Then (E, {x,,})
satisfies the disjoint support condition, i. e., for any linear isometry
Т.Е-уЕ with T{xj)= Z aijxi(j=\,2,...) we have
i = l
a4 -aim = 0 (i,j, т=\,2,...;]фт).
B0.15)
Proof. Assume the contrary, i.e., that for some indices io,jo,mo
we have aiojoainmo^0. Then \aiojo\ + \auimo\ > \aiojo\- \aiomo\, whence
1 Let us also mention that by § 19, theorem 19.2, one can introduce on E an
N TK-norm equivalent to the initial norm and, by § 19, already in every NK-norm
(with respect to {х„}) the basis {х„} is strictly orthogonal.
36 Singer, Bases in Banach Spaces I

562
II. Special Classes of Bases in Banach Spaces
for the linear isometry T+ with T+(xj)= ? \аг1\х{ (/=1,2,...) we have,
by B0.7) for n=oo, i=i
i= 1
= ||T+(x,-xJ||,
which is impossible, since by the hyperorthogonality of {х„} we have
||xj + jcm|| = \\Xj — xm\\. This completes the proof.
Let us observe that in proposition 20.2 one cannot replace "strictly
hyperorthogonal" by "hyperorthogonal", since e.g. the natural basis {xn}
of c0 is positive and hyperorthogonal, but (со,{х„}) does not satisfy
the disjoint support condition (by §3, lemma 3.1).
Proposition 20.3. Let {х„} be a hyperorthogonal basis of a Banach
space E, such that (E, {xn}) satisfies the disjoint support condition. Then
{xn} is a positive basis.
00
Proof. Let T:E->E be a linear isometry, with T(xj)= Y, aijxi
i= 1
(/=1,2,...), and let аь...,а„ be arbitrary scalars. Then, by the disjoint
support condition, for each / there is, among the n numbers an,...,ain,
at most one =?0, whence
ZKI»j
У а а
whence, taking into
account the hyperorthogonality of {х„} (the implication 1°=>4° of theo-
theorem 20.2), we get
Z
Z
and hence, by §3, proposition 3.4, {xn} is positive, which completes
the proof.
From propositions 20.2 and 20.3 we infer
Proposition 20.4. A strictly hyperorthogonal basis {xn\ of a Banach
space E, satisfying B0.7) for n=co, is positive if and only if (E, {xn})
satisfies the disjoint support condition.
Note that if {xn} is a hyperorthogonal basis of a Banach space E,
such that (E, {х„}) satisfies the disjoint support condition, {д;„} need
not be strictly hyperorthogonal, as shown by the unit vector basis
of /„».
Let us observe, finally, that a positive basis need not be orthogonal,
since we have seen in §3 that there exist finite dimensional Banach
spaces in which all bases are positive.
21. Subsymmetric bases
563
§ 21. Subsymmetric bases
Definition 21.1. A basis {xn} of a Banach space E is said to be sub-
symmetric, if it is an unconditional basis and for every increasing se-
sequence of positive integers {nj the basis {xn.} of the space [х„.] is equiv-
equivalent to the basis {*„}.
For instance, the natural bases of c0 and F (/?> 1) are subsymmetric.
The normalized Haar basis {zn} in Lp([0,1]) (l</?#2) is not sub-
subsymmetric, since the support of the function z2kti_l is contained1 in
--ГГГ, l--r) (fc=l,2,...), whence by §18, proposition 18.3,
2 2 /
[22k+i-i]J°=i is isometrically isomorphic to lp, which is not isomorphic
to Lp([0,l]) for p #2.
Moreover, we shall now show that the space Lp([0,1]) {\<рф2)
has no subsymmetric basis at all.
For 1</7<оо and e>0 put2
B1.1)
/i \I
where v is the Lebesgue measure and where ||x||p= I \\x{t)\pdt у.
Lemma 21.1. The classes MP have the following properties:
a) // 2^/7<co,?>0, then
\\x\\P> \\A\i
B1.2)
b) // 2</?<oo, e>0, and {х„} is a sequence in MP such that the
GO °°
series ? х„ is unconditionally convergent in Lp([0,1]), then ? ||xj
xjp<oo.
n=1 n 1
c) // xeLp([0, l])\{0} (K/?<oo) does not belong to MP, then there
exists a set A a [0,1] such that
v(A)<e,
x{t)
B1.3)
Proof, a) If xeLp([0,l]), where 2</?<oo, then by the Holder ine-
inequality we have
j\x(t)\2dt < (j(\x(t)\pdtjp (jV^
= (j\x(t)\pdt
1 We assume now that z2kt,_1(;)=0 for t=\ - ^f+т, t=\ - -j; this does
not change the class z2k+,-l.
2 For simplicity, we shall denote both x and x by x; this will lead to no confusion.
36»

564 II. Special Classes of Bases in Banach Spaces
Consequently, whenever
and thus, in particular, we have the first inequality in B1.2).
Put
Sfp(x)={?e[0,l]| \x(t)\^e\\x\\p}.
Then, if xeMf, we have v[Sp(x)]>e, whence
B1.4)
B1.5)
j
SP(.v)
which proves the second inequality in B1.2).
b) Assume that 2s?/?<oo,e>0 and let {х„} be a sequence in Mp
CO
such that the series ? xn is unconditionally convergent in Lp([0,1]).
n= 1
Then, by B1.4), Z x» 's a^so unconditionally convergent in L2([0,1]),
л = 1 XJ
whence, by §14, lemma 14.10, Xl|xJ|<co. Hence, since xneMP
n= 1
(n = 1,2,...), by part a) proved above we get
00 00
Z Wlp^e1 Z ll*J22<co-
n=1
c) Assume that xeLp([0,l])\{0}, хфМрЕ and let A be the set Sp(x)
defined by B1.5). Then, since хфМр, we have v{A)<s. Furthermore,
since |x@lp<?pIWiP for te[O,l~\\A, we have
\\x\
p=\\x(t)\pdt + j \x(t)\pdt^l\x(t)\pdt +
[0,1] A
whence the second inequality in B1.3), which completes the proof of
lemma 21.1.
Proposition 21.1. Let 2</?<co and let {х„} be an unconditional
basis of Lp([0,1]) for which there exists an ?>0 such that
xneMP
B1.6)
Then I—— > is equivalent to the unit vector basis of I2.
21. Subsymmetric bases 565
Proof. Let {а„} be an arbitrary sequence of scalars such that
x. Г Xn I
Za;7~T converges. Then, since <——> is an unconditional basis,
? xi
Lai],—17 is unconditionally convergent. Furthermore, by B1.6) and
the definition B1.1) of MP we have а„—~eMp (и =1,2,...). Conse-
Consequently, by lemma 21.1 b), "x""
ZKI2=Z
X,-
< GO.
Conversely, let {а„} be an arbitrary sequence of scalars such that
Г x )
Z laj2 < go. Then, since by § 14, proposition 14.1, <^—— > is a Hilbertian
(II^IIJ
x-
basis, Y, ai тт~7 converges, which completes the proof of proposition
i = l llXill
21.1.
Proposition 21.2. Let ]^p<co and let {х„} be a sequence in Lp([0,1 ])
with the property that for every ?>0 there exists an index nc such that
хПсфМР. Then there exists a subsequence {x'n} = {xkn} of {х„}, such that
x' )
is a basic sequence equivalent to the unit vector basis of lp.
Proof. We first show that one can construct successively a subse-
subsequence {x^} of {х„} and a sequence of sets {An} in [0,1] such that
B1.7)
i)p
z
i= 1
X[(t)
dt<
1 )p
B1.8)
Indeed, take ? = —. Then by our assumption there exists an index
/c1=/c1(fi) such that х[=хк1фМ^2. Hence, by lemma 21.1c) there exists
a set A1 с [0,1] such that
x[(t)

566
II. Special Classes of Bases in Banach Spaces
Assume that we have already constructed x[,...,x'n and Al,...,An.
Then, since the set function ф(А) =
x't{t)
x
dt is absolutely con-
continuous (because each summand is absolutely continuous), there exists
an e>0 such that ф(А) < Aln+Un whenever v(A)<e. We may assume,
without loss of generality, that
1
^у- Then, by our assumption,
there exists an index kn+l = /cn+1(e) such that x'n+l = хкп+1фМР. Hence,
by lemma 21.1 c), there exists a set An+1 с [0,1] such that
1
and by the first inequality we also have
\\x,\\
< —(
(n+l)
, which proves our assertion.
Now we shall prove that the sequence
properties. Put
А'„=А„\ у А,
has the required
(n=l,2,...), B1.9)
for
("=1,2,
0 for t$A'n
B1.10)
Уп =
B1.11)
Obviously, А'ппА'т = Ф for n#m and {?e[0,1]
(п=1,2,...), whence, by § 18, proposition 18.3, \yn} is a basic sequence
and the subspace \y^\ is isometrically isomorphic to F, by the mapping
jn—>е„, where {е„} is the unit vector basis of /p; hence for the a.s.c.f.
{вп}^\Уп~\* to {yn\ we have ||д„|| = 1 (n=l,2,...). Furthermore, by
B1.7) —B1.10) we have, for every n= 1,2,...
21. Subsymmetric bases
567
p
[0
1
г
J
l]\A'n
"I
x'n(t)
IK II
Х'п({)
\\х'„\\
dt+
1 1
i = n+ 1
P
- Z
f
00
= n + 1
X
II
f
til ~r 1
J
II n II
p
dt
X'n(t)
IK II
B1
dt
1 " 1 1
1 У — > 1 .
4<n+i)p i-i ^ip 4ПР
From B1.11) —B1.13) we infer
llx;
1
~~ 4" +
z"
n (
!
Z" J
lk.ll-i)
4"
B1.13)
which, together with the relation \\gn\\ = 1 (n= 1,2,...) above, implies
Z
V 1
»= i 4"
Consequently, by Ch. I, § 10, theorem 10.1,
is a basic se-
sequence equivalent to {yn}, whence also to the unit vector basis of /",
which completes the proof of proposition 21.2.
Remark 21.1. Assume, in particular, that {*„} is the Haar system in
Lp([0,1]) A</7<оо). Then for any e>0 there is only a finite number
of indices n such that xneMP (because even the number of all indices и
such that v({?e[0,1] | х„(г)#0})^е is finite for any ?>0), whence, by
proposition 21.2, for any n1<n2<-- there exists a subsequence {х'„} of
f x' }
{х„к} such that <—^-> is a basic sequence equivalent to the unit vector
basis of lp. This result implies again proposition 18.4 of §18.

568
II. Special Classes of Bases in Banach Spaces
We shall also need the following result on duality of subsymmetric
bases:
Proposition 21.3. // {х„} is a subsymmetric basis of a Banach space E,
then the a.s.c.f. {/„} с ?* is a subsymmetric basis of [/„].
Proof. By § 17, theorem 17.7, {/„} is an unconditional basis of [/„].
Let {nk} be an arbitrary increasing sequence of positive integers. Then,
since {xn} is subsymmetric, {х„} is equivalent to the basis {xnj of [xnj,
whence, by Ch. I, §12, proposition 12.1, {/„}~{0nJ, where {фПк}
с [х„к]* is the a.s.c.f. to the basis {х„к} of \_х„к\ Since {х„} is uncondi-
unconditional, {0nJ = {/nJ [,„,]}~UJ' ьУ virtue of §17, proposition 17.3.
Thus {/„} ~ {ф„к} ~ {/„J, which completes the proof.
This being said, we can prove now
Theorem 21.1. The space Lp([0,1]) A<рф2) has no subsymmetric
basis.
Proof. Assume first that 2<p< со and that {х„} is a subsymmetric
basis of Z/([0,1]). Then there are two cases: a) If there exists an ?0>0
Г Хп 1
such that х„еМ?0 (n=l,2,...), then, by proposition 21.1, <—— > is
t II*n II J
equivalent to the unit vector basis of I2, whence we obtain that Lp([0,1])
is isomorphic to I2, contradicting the assumption рф2. b) If there
exists no such eo>0, i.e., if for every e>0 there exists an index nE
such that х„ фМ%, then, by proposition 21.2, there exists a subsequence
( x'n )
{x'n} of {xn} such that ¦<—^> is equivalent to the unit vector basis
of lp, whence we obtain that Lp([0,1]) is isomorphic to /p, contradicting1
again the assumption рф2. Thus theorem 21.1 is true for the spaces
Lp([0,l]) with 2<p<cc.
Assume now that l<p<2 and that {*„} is a subsymmetric basis
of E = Z/([0,1]). Then, since every basis of a reflexive space is shrinking
(see §4, example 4.3), by proposition 21.3 the a.s.c.f. {/„} с E* is a
subsymmetric basis of [/„] = ?* = L4([0,1]), where —|—= 1, which,
P 4
since 2<q<co, contradicts the fact proved above, that theorem 21.1
is true for the spaces Lp([0,1]) with 2<p<co. This completes the
proof of theorem 21.1.
Let us observe that in definition 21.1 of a subsymmetric basis it is
essential to assume separately that {х„} is unconditional, as shown by
§ 17, example 17.2.
1 See e.g. S. Banach [10], p. 206, theorem 9.
21, Subsymmetric bases
569
Proposition 21.4. Every subsymmetric basis {xn} is bounded.
Proof. Assume that for a basis {х„} there exists an infinite sequence
of positive integers {mk} such that
||xmj>/c
B1.14)
B1.15)
j>l (n=l,2,...), B1.16)
the series ? fi{x)xmu will be divergent, whence {xn} is not subsym-
Then, taking an x = ? /|(х)х;е? such that
i= 1
./„(x)/0 (n=l,2,...)
and an infinite subsequence {mkn} of {mk} such that
i= 1
metric. We have thus proved that for every subsymmetric basis {х„} we
have sup ||xn||<co.
Assume now that there exists an infinite sequence of indices {mk}
such that
B1.17)
Then, taking an infinite subsequence {mkn} of {mk} such that
(n=l,2,...), B1.18)
the series Y xm will be convergent, while the series У x.
II Y II k" ^ II Y II
n=l Нлл11 n=l Нлл11
is, of course, divergent, whence {xn} is not subsymmetric. Thus for
every subsymmetric basis {xn} we have inf ||jcb || > 0, which completes
the proof of proposition 21.4. i«n<°o
As before, we shall use the following notation:
(9 = the set of all increasing sequences of positive integers. B1.19)
If {xn) is a subsymmetric basis of E, then for every xeE and
00
({nij}, {rij})e(9 x(9 the series Y^ fm.(x)xnt converges. Indeed, since {xn}
00 '= 1
is unconditional, Y fmi{x)xm. converges, whence, since {xmj}~{xn}
oo i= 1
~ {xn.}, ? fmi{x)xm converges. Therefore the linear operators
i= 1

570
II. Special Classes of Bases in Banach Spaces
are well defined on E, and below we shall see that they are uniformly
bounded.
Some characterizations of subsymmetric bases are given in
Theorem 21.2. Let {xn} be a basis of a Banach space E, with the a.s.c.f.
{/„}. The following statements are equivalent:
1°. {xn} is a subsymmetric basis of E.
2°. We have
3°.
4°.
We have
sup
We have
Urn
sup
к
i= 1
y},{»y}ll<Q0-
< oo (xeE).
B1.21)
B1.22)
sup
k
Z /тД*)*„,
<oo
). B1.23)
5°. For every xeE and {{mJ),{nj))e(9y.(9 the series ? fm,(x)xni й
convergent.
6°. We have
and
sup sup
sup sup
Z /»¦(*)*.-
i= 1
i=l
< oo (xe?) B1.24)
< oo {xeE). B1.25)
7°. Wfe have
and
sup
sup
z.
i= 1
к
z.
;= l
< oo {xeE, {mje0) B1.26)
< oo (xe?, (nje^). B1.27)
8°. For every xeE and ({raj,{п}})е(9 х &
00
? /(х)х„. are convergent.
i= 1
i= 1
Proof. Assume that 2° is not valid. Then, by the principle of uniform
boundedness, there exists an xeE such that
sup \\A{mh{ }
({my), {яу))е<9 x 0
x)\\ = sup
({my), {n,))e« x
Z /»<(*)*-,
= oo. B1.28)
21. Subsymmetric bases 571
Then there exist sequences {nxa}, {n.-Jefi? and a positive integer /t
such that
ii
z.
i= 1
B1.29)
Let us denote
jrp={p,p+l,p + 2,...} (p=l,2,...), B1.30)
) = the set of all increasing sequences of elements of s/, B1.31)
where л/ а Ж Observe that
sup
({my), {nj))ee(jrp)xe(jrp)
V f {
= oo (/7=1,2,...), B1.32)
since otherwise for some/» one would have, by proposition 21.4,
sup
({my),{ny))e<Px«i
Z Lt(
(p-l) jup \\xj ^sup ||/„|| ||x
GO
I/,
+ sup
({my}, {4j))eeKJrp)xeHJrp)
< 00,
in contradiction with B1.28).
Putting kt= max (тп,пй) and applying B1.32) for p = k1 + \, it
1 < i^l l
follows that there exist sequences {mi2}, {ni2}?&(^kl + l) and a positive
integer /2 such that
12
у /¦
i= 1
1.
B1.33)
Putting k2 = max (mi2,ni2) and continuing in this manner, we obtain
sequences {lj}e(9, {kj}e(9 and sequences {mij}liJ=le(9(,Arkj^1 + l\A/kj + l),
{п}^еФ{Ж\^ )( 12;/c 0) such that
[/=l,2,...;fco = 0), such that
(/=1,2,...).
Define now the following sequences {m,}, {n,}
i= 1 ¦
B1.34)
B1.35)

572
II. Special Classes of Bases in Banach Spaces
For these sequences, by B1.34) the series ? fmi{x)xn, is divergent,
i = 1
whence {х„} is not subsymmetric. Thus 1°=>2°.
Assume now that we have 2°. Since {*„} is a basis, there exists a
constant C^l such that
sup
i= 1
= C\\A{mjUW(x)
whence, by 2°, we obtain B1.22). Thus, 2°^3°.
The implication 3°=>4° is obvious.
Assume now that we have 4°. Then, by the principle of uniform
к
boundedness applied to the operators и{т.]Лп.]к(х) = ?/„,(*)*„,
(ле?, ({mj},{nj})e(9x(9, /с =1,2,...), we have '' '='
sup ||и{1Я,,.{1у,>к||<оо (({/и,.},{/1;})е0х0),
B1.36)
whence, since for every finite linear combination p = Y_, %jxj and every
C we have MM|nijli|n.l,t(p)= ^«m.xB., where am. = 0
for wf^/+ 1 and since [*„] = ?, it follows that we have 5°. Thus, 4°=>5°.
Assume now that we have 5°. Then, in particular, for every xeE
oo
and {и;}еб> the series ?/„,(*)*„, converges and hence, by §16, theo-
i = 1 oo
rem 16.1, {*„} is an unconditional basis. Furthermore, if ? <xfx,- con-
oo '=1
verges, then, by 5°, the series ? а,-хИг converges for {nt}e(9. Conversely,
00 '=1
if ? а(хИ1 converges, say to xeE, then а( = /и.(х), whence, by 5°,
i = 1
ОС 00
? <xfxf = $]/„,(*)*,¦ a'so converges, which proves that {xn} is equiv-
i = 1 i= 1
alentto {*„.}. Thus, 5°^1°.
The implications 3°=>6°^>7° are obvious, and the proof of the impli-
implication 7°=>8° is similar to the above proof of 4°=>5°.
Assume, finally, that we have 8° and let xeE, {щ}е& be arbitrary.
00
Then by 8° Y.f"i(x)xi converges, say to yeE, whence ft(y) = fn.(x)
21. Subsymmetric bases
573
(/=1,2,...), whence, again by 8°, Ytfni(x)xni=YtfMxn, converges.
i = 1 i = 1
Consequently, by §16, theorem 16.1, {xn} is an unconditional basis.
Now, as the above proof of the implication 5°=>Г, we see that {х„} is
equivalent to every subsequence [хщ}, and therefore {xn} is subsym-
subsymmetric. Thus, 8°=>Г, which completes the proof of theorem 21.2.
Let us observe that the conditional basis {xn} of c0 given in §17,
oo
example 17.2, has the property that ? j\(x)xn. converges for every
i ~ 1 oo
xeE and {«JeC (and conversely, the convergence of ? я(хи. implies
oo i= 1 oo
that of Y, aixi), since this amounts to the convergence of ?/,•(*),
i = 1 i = 1
00
but Y,fmi(x)Xi is not necessarily convergent for every xeE and
'=1 oo
{тг)е& (since ? /„.(i) need not be convergent, or, alternatively, by
theorem 21.2, implication 8°=>1° and the conditionality of {*„})¦ Dually,
the conditional basis {hn} of/1 given in § 14, example 14.2, has the prop-
аДе/1 and {mJeC (since
1 ~ 1 oo
:ami-ami+1l<°o), but 5]а;А„. is not
ОС
erty that ? ат./г; converges for every
00 1= 1 00
5] К- — а,+ 1|<оо implies
necessarily convergent for every J] щк^е1х and {«JeC However, we
i=l
don't know any example of an unconditional basis {хи} such that
00 00
fi{x)xn, converges for every xeE and {nt}e(9 (or such that
il
converges for every xe? and {mf}e6>) but {х„} is not subsymmetric.
In the usual concrete Banach spaces with a subsymmetric basis all
subsymmetric bases are equivalent. In fact, by the results of §18, in
Co,/1 and /2 all bounded unconditional bases are equivalent, while for
the other spaces P we have
Proposition 21.5. In the space E = lp (l<p<cc,p^=2) all subsym-
subsymmetric bases are equivalent.
Proof. Let {yn} be an arbitrary subsymmetric basis of E. Then, by
proposition 21.4, {у„} is bounded, and since E is reflexive, by §7, the
remark made before theorem 7.2, jn->0 weakly. Consequently, by §15,
proposition 15.1, one can extract a subsequence {ynk} which is a basic
sequence, equivalent to a block basic sequence {zn} with respect to the
unit vector basis \xn} of E = lp. Since \yn} is subsymmetric, {у„}~{у„к}

574
II. Special Classes of Bases in Banach Spaces
and, on the other hand, by § 16, theorem 16.8 and § 18, proposition 18.1,
'{*„}. Thus {у„}~ {у„к} ~{г„}~{х„}, which completes
the proof of proposition 21.5.
Problem 21.1. In a Banach space ? with a subsymmetric basis are
all subsymmetric bases of ? equivalent?
§ 22. Symmetric bases. Symmetric spaces
Definition 22.1. A basis {х„} of a Banach space E is called symmetric if
sup sup
аеП |/?i|Sl
l^S/Koo
E Pifi
i= 1
< со
(xeE),
B2.1)
where {/„} <=?* is the a.s.c.f. to {х„}, and where, as before, П denotes
the set of all permutations of the set Jf= {1,2,3,...}.
For instance, the natural bases of c0 and lp (p^l) are symmetric.
The normalized Haar basis {zn} in Lp([0,1]) A <рф2) is not symmetric,
and moreover, the space Lp([0,1]) {\<рф2) has no symmetric basis at
all, since it has no subsymmetric basis (by §21, theorem 21.1) and since
we shall see below that every symmetric basis is subsymmetric.
Taking in B2.1) the identical permutation a=i it follows (by §16,
theorem 16.1) that every symmetric basis {xn} is unconditional. Further-
Furthermore, if {хи} is a symmetric basis of E, then for every xeE and (p,<r)
00
еПхП the series E /p(o(x)x<r<o converges. Indeed, by B2.1) we have
IIXL -
E Я
i= 1
<oo (xe?, <теЯ),
B2.2)
whence, by the principle uniform boundedness applied to the operators
n
и„Лх)= ЕЛ
i=l
B2.3)
r(i) (xeE, aeU, n= 1,2,...), we have
HxL^CJ|x|| (xeE, аеП),
where the constant Ca depends only on cr, and thus
I/,
^ sup
Ю<оо
n+p
V f I
i = n
к
E
t i
Гп + р
fi\ E^
<C
a
n + p
22. Symmetric bases
575
which shows that ?/;(*)*„(;> converges for every хеЕ,аеП. Ар-
plying this to the permutation р^оеП, we infer that ? fAx)xP-^a)
i= 1
converges for every xe?, (р,<т)еЯх П, whence, since {х„} is un-
00 00
conditional, it follows that E/p(>)(x)x^<o= X/iW^p-Mo converges
i = 1 i = 1
for every xe?, (р,<т)еЯх П. Therefore the linear operators
ap,Ax)=
B2.4)
i= 1
are well defined on ? and we shall see below that they are uniformly
bounded isomorphisms of ? onto ?.
Some characterizations of symmetric bases are given in
Theorem 22.1. Let {хи} be a basis of a Banach space E, with the
a.s.c.f. {/„}. The following statements are equivalent:
1°. {хи} is a symmetric basis of E.
2°. We have
sup \\A || < oo. B2.5)
3°. We have
sup sup
4°. We have
fp(i)\X)Xa(i)
i= 1
< со (xe?).
B2.6)
sup
k
I /„(!>(*) *,*)
< со (хе?, (р,а)еПхП). B2.7)
5°. For every xeE and (р,а)еП х П the series
converges.
6°. Ж
7°. We have
sup sup
sup
E
xx.
< со (xe?). B2.8)
< со (xe?, erei7). B2.9)
8°. For every xeE and аеП the series E fAx)xa(i) converges.
i= 1

576 П. Special Classes of Bases in Banach Spaces
9°. We have
sup sup
E fp(i)(x)xi
< со {xeE). B2.10)
10°. We have
sup
E fp(iM)xi
< oo (xe?, реЯ). B2.11)
1Г. For every xeE and реП the series ? /р(,)(х)х> converges.
{х„} к a basis of the
12°. ?rery permutation {х„(и)} o/ rt
space E, equivalent to the basis {*„}.
Proof. Assume that we have Г. Then, since {х„} is unconditional,
we have
sup
= sup
(р,<т)еП х П
= sup
(р,гг)еП хП
sup sup
1 5= k< oo
i= 1
Jp(i)\X)Xa(i)
I ^i/iW^O
i= 1
< со
whence, by the principle of uniform boundedness, we infer B2.5). Thus,
1°^2°.
Assume now that we have 2°. Since {хи} is a basis, there exists a
constant C^l such that
sup
1 <ik< да
Jp(i)\X)Xa(i)
i= 1
с
/_, Jpt.i)(x)xrr(i)
whence, by 2°,
sup sup
)ПП Щк
= C\\ApJx)\\^C\\ApJ\\x\\ (xeE),
Jp(i)\x)xa(i)
С sup \\AP
| < oo {xeE),
and thus 2°=>3°.
The implications 3°=>4°^7° and 3°=>6°^7° are obvious. The
implication 7°=>8° has been already proved in the above (before the
statement of theorem 22.1).
22. Symmetric bases
577
Assume now that we have 8°. Then the series ]T fa(i){x)xa(i)
oo Г да П i=l
= E/f E f№)xs-4j) хт@ also converges for every xe?, <теЯ,
i = 1 L j = 1 J x
i.e., {х„} is an unconditional basis, and therefore E/p(o(x)x<t(o
oo i=l
= E /i(x)xp-^<o converges for every xe?, (р,<т)еЯхЯ. Thus,
The implications 3°=>9o=>10° are obvious.
Assume now that we have 10°. Then, by the principle of uniform
к
boundedness applied to the operators upJi(x) = ]T /Ра)(х)х( (xe?, реЯ,
/c= 1,2,...), we have
i= 1
sup ||Mp>k||<oo (реЯ),
B2.12)
whence, since for every finite linear combination p = ? «/Xj and every
реЯ we have lim ир t(/?) = E ap(i)x^ where ap(f) = 0 for p
and since [х„] = ?, it follows that we have 11°. Thus, 10° => 11°.
Assume now that we have 11°. Then, as in the proof of Ch. I, § 12,
theorem 12.1, it follows that for every /e[/J and реЯ the series
00
E f(xi)fpd) converges, whence, by the implication 8°=>5° proved
i= 1
above, {/„} is an unconditional basis of [/„] and hence, by the last
statement of § 17, theorem 17.2, {хи} is an unconditional basis. Hence,
да oo
again by 1Г, it follows that the series ? fp{i)(x)xa(i) = E L-iPii){x)xi
i=l i=l
is convergent for every xe? and (р,<т)еЯхЯ, i.e., we have 5°. Thus,
1Г=>5°.
Assume now that we have 5°. Then, applying 5° for all р = аеП we
obtain that {хи} is an unconditional basis, whence every permutation
00 00
(x^(n)} is a basis of ?. If ? oefxf converges, then, by 5°, ]T ^iXa(i) also
i = 1 i = 1
00
converges. Conversely, if E aix,r(i) converges, then, since {хи} is un-
i= 1
conditional, we can write
^-i^x,-, whence, by 5°, the
series ? ааа-Ч1)х{ = J] afxf also converges. Thus, 5° => 12°.
37 Singer, Bases in Banach Spaces I

578
II. Special Classes of Bases in Banach Spaces
Assume, finally, that we have 12°. Let us first show that in this case
the basis {хи} is bounded. Assume that there exists an infinite sequence
{mk}e(9 such that
\\xmk\\>k (fc=l,2,...); B2.13)
we may also assume, without loss of generality, that the set Ж\{тк) is
00
infinite. Take an x = ? /j(x)xfe? such that
/„(*)/0 («=1,2,...)
and an infinite subsequence {mkn} of {mk} such that
*Я,Л>1 («=1,2,...).
B2.14)
B2.15)
Then, taking а аеП such that <j(nij) = mk. for y=l,2,...,
00
<j(ri)e<Ar\{mkj\ for пеЖ\{тк}, the series ? /i(x)xJ(j) will be diver-
i= 1
gent, contradicting the assumption 12°. Thus, 12° => sup ||х„||<со.
1 ^П< 00
On the other hand, assume that there exists an infinite sequence
{mk} e (9 such that
||xmj|<- (fc=l,2,...) B2.16)
and that the set Jf\ {mk} is infinite. Then, taking an infinite subsequence
{mkn} of {mk} such that
B2.17)
and a permutation аеП such that а{т^) = тк. for y=l,2,..., a(n)
eJf\{mkj} for иб./Г\{шк}, and putting
JC—
1
the series
series
«„^(n) = ?
for n = m}, 7=1,2,...
0 for the other n,
1
B2.18)
x will be convergent, while the
anxn = ? —~— x<"j 's divergent, contradicting the as-
llXll
nl jl l
sumption 12°. Thus 12°
||xJ|>0.
22. Symmetric bases
579
Now we can show that {хи} satisfies 1°. Indeed, assume the contrary,
i.e., that there exists an xe? such that
sup sup
1 5k< oo
We claim that in this case
к
sup sup
m+1^ k<
V R f(x\x
= со.
B2.19)
i = m+ 1
= со (w=l,2,...), B2.20)
where Я(,/(^, + 1) denotes the set of all permutations of the set1 ^Vm+1.
Indeed, by the boundedness of the basis {хи}, proved in the above,
we have, for any fixed те.Ж,
sup sup
«П |/>i|«l
sup sup
||
к
i= 1
sup ||xj|<co,
i= 1
E'
i = m+ 1
B2.21)
B2.22)
с || sup ||/i|| sup ||х^||<со.
Now, let geYI be arbitrary. Then the cardinal numbers of the finite
are equal. Let ф be an arbitrary one to one mapping of M± onto ,Мг
and let xo(i) = a(i) for ie._AfmJr^\.Mx, т„(/) = <т[ф(/)] for ie,Mx. Then
and
Г
i=m+l
Therefore if B2.20) were not true, one would have, taking into
account that by 12° the basis {х„} is unconditional,
sup sup
к
E'
С sup sup
m+1ik<
к
E
< со,
1 For the definition of the sets Ж, see § 21, formula B1.30).

580
II. Special Classes of Bases in Banach Spaces
where C depends only on the basis {х„}, and this, together with B2.21)
and B2.22), would contradict B2.19). Thus, B2.19) => B2.20).
Using this remark one can inductively construct two infinite se-
sequences {nj},{nij}e(9, a sequence of permutations т^еЛ(.Л?, +1)
0=1,2,...; mo = 0) and a sequence of scalars {/?;} of modulus |
(;'= 1,2,...), with the following properties:
i:= max
J m, - i + 1 « i
E
0=1,2,.--),
(y=l,2,...),
(y'=l, 2,.-.).
B2.23)
B2.24)
B2.25)
Since by т,-еЯ(Л^._1 + 1), B2.23) and B2.24) the cardinal numbers
h fii { } {
of the finite sets {ie^V \ nj+
and {ie.Ar
mj-1
j j j
jjj ), ...,Tj(rij)} are equal, let фу be a one to one
mapping of the first of these sets onto the second (_/= 1,2,...). Let
Ы0 for ^
j(i) for nj
0=1,2,
0=1,2,
Then аеП, and by B2.25) the series Y^ Pifi{x)xa(i) is divergent.
i= 1
On the other hand, since by 12° the basis {хи} is unconditional and
00 00
]T f(x)xa(i) converges, the series Y^ pif{x)xrr(i) must be convergent,
i= 1 i= 1
a contradiction. Thus, 12° => Г, which completes the proof of theorem
22.1.
The following proposition is useful for applications-.
Proposition 22.1. Let {х„} be a symmetric basis of a Banach space E.
Then
a) There exists a constant C^l such that
I ft «!*„«>
I<v
B2.27)
for any аеП and any finite sequence of scalars a1,...,an, Р1,...,Р„
b) The numbers
\\x\\\ = sup sup
||
E ft/i
B2.28)
22. Symmetric bases
581
groe a «ew «orw on E, equivalent to the original norm of E. In this new
norm we have B2.27) with C=\ and
У , f .(
= |||x||| (хе?,(р,<т)еЛхЛ,|в(|=1), B2.29)
lll*illl = 111*2111=-= SUP llxnll- B2-3°)
Proof, a) Since {х„} is a symmetric basis, for the operators
к
/ \ v^ ft f (x\x
i= 1
we have, by the principle of uniform boundedness,
sup sup IliVj/jjj.fcl! =C< со
and hence ~~ °°
B2.31)
B2.32)
E ft«i-
•"@
M^,(/fi!."( E KiXi
i= 1
b) Obviously, |||x||| is a norm on ?. Since {х„} is a symmetric basis,
by part a) proved above we have
II vll <lllrlll < f"ll rll (yi=F\ (?? 33^
and thus the norm |||x||| is equivalent to the original norm of ?.
Furthermore, for any аеП and any finite sequence of scalars
<x1,...,txn,pi,...,pn with \Pi\^l {i= 1, ¦-.,«) we have
E Р^
= sup sup
к
E
fid
>iaiXxa(i)
л
Eaixi
i= 1
B2.34)
i.e., B2.27) with C=l. Hence it follows that for any аеП and any
a1;..., а„, e1;..., е„ with |ef|= 1 (/= 1, ¦¦-,«) we also have
i= 1
E ?iaix^)
B2.35)
Indeed, since the cardinal numbers of the finite sets My = |l,2,...,
max <т(/)}\{<гA), ...,a(n)} and JB = (l,2,..., max a(/)]\{l,...,«} are
equal, let ф be an arbitrary one to one mapping of M^ onto Jt2 and let
for /g
for

582 II. Special Classes of Bases in Banach Spaces
where / = max a(i). Then, putting
1 Si^n
'-'(Л^-'О-) for ye {<r(l),..., <т(л)
0 for jeJt^,
from B2.34) we obtain
41)
E yjxj
E ?лх««)
i.e., B2.35).
Now, by theorem 22.1, E/р(о(х)х<т(О converges for every xe?,
{р,(т)еП хП. Since {хи} is unconditional, ]T ei/p(i)(x)xJ(f) also con-
i= 1
verges for |e;| = l (/=1,2,...) and we have, taking into account B2.34)
and B2.35),
L_, ?iJp(i)(X)Xtr(i)
i= 1
!/«(¦
XX;
= x
Finally, applying B2.28) to Xj= ? fi(xj)xi = E V* (/=1Д-),
we get i=1 <=i
||x.|||=sup sup
1= sup
(/=1,2,...),
number v$n} = sup sup \\uaAfiilk\\=sup sup
стеП |/S|«l П |/?|« 1
стеП
which completes the proof of proposition 22.1.
Let us mention that the converse statements are also true, i.e., each
of B2.27) and B2.29) above characterize symmetric bases, as shown by
the implications 8°^1° and 5°^1° of theorem 22.1.
Definition 22.2. Let {хи} be a basis of a Banach space ?, with the
a.s.c.f. {/„}. The least constant C^l for which B2.27) holds i.e., the
\\ sup
|| xeE
Ю<00 Ю<ОО ||Х||<1
is called the symmetric constant (or symmetric norm) of the basis {хи}.
The number vs(?) = inf v\s^ ,, where @)s denotes the set of all sym-
metric bases of the space E, is called the symmetric constant of the space
E. Any norm |||x||| on E for which we have B2.29) or, equivalently, B2.27)
with C=l, is called symmetric with respect to the basis {х„}. We shall
call symmetric space any couple (E,{xn}\ where ? is a Banach space
with a symmetric basis and {х„} a symmetric basis of ?, such that the
original norm of ? is symmetric with respect to {хи} (or, equivalently,
22. Symmetric bases
583
Proposition 22.2. Every symmetric basis {xn} is subsymmetric.
Proof. Let {nt}e(9 be arbitrary. Then by B2.33) and B2.29) we have
m + p
E aiXi
i = m
m + p
E aixm
i = m
m + p
E aiXi
i = m
m + p
E aiX»<
i = m
=
m + p
E aiXm
i = m
m + p
E aiXi
i = m
m + p
E XiXm
i = m
m + p
E a*x*
i = m
whence {xn} is equivalent to the basis {х„(} of [х„.], which completes
the proof.
The converse of proposition 22.2 is not valid, as shown by
Example 22.1. Let G be the space of all sequences of real numbers
x=ttn\ such that
00 II I
||x|| = sup У —-<oo. B2.36)
Then G is a Banach space, and the unit vectors {хи} are a subsym-
metric but non-symmetric basis of G.
Indeed, let us first prove that G is complete. Let ги={?(„т)}"=1
(w=l,2,...) be a Cauchy sequence in G and let e>0 be arbitrary.
Then there exists an N = N{e) such that
-zm+p\\= sup
B2.37)
Hence, taking i= 1, nx — 1,2,... it follows that
and therefore the limits lim С(„Р) = СИ («= 1,2, •••) exist Now, by B2.37)
we have
к i(-(m)_ r(m + p)|
{m>N,p=l,2,...;k=l,2,-;{nj}e(9),
whence, for /?->oo we obtain
, /c= 1,2,...;{^}
and hence, putting z={(n}, we get zm — zeG, \\zm — z||<e (m>N). Con-
Consequently, z = (zm —z) + zmeG and zm->z as m->oo, which proves the
completeness of G.

584 II. Special Classes of Bases in Banach Spaces
Let us prove now that
x- X с,*,-
as
>оэ (x={QeG).
B2.38)
Let xeG and e>0 be arbitrary. Then by B2.36) there exists a se-
sequence {n{}e6 such that
II II ^ \~* I^Hjl i, и ?
11*113* E ~i=> IWI -T'
;=i yi J
whence also a /ce./T such that
E -77 >
и
\x\\ --r-
B2.39)
Observe now that we must have ?„-*•() as n->co, since otherwise
there would exist an infinite subsequence {?„.} with inf |fm.|
J 1 ^ 7 < 00 J
whence ||x||
= со, which is impossible. Therefore
¦=i
there exists an N = N(s) such that
e
We claim that
if
Pn =
~ E ?ixi
{i>N). B2.40)
(и>Л0. B2.41)
Indeed, take a fixed n>N. Then, by B2.36), there exists a sequence
{lj}e<9{jrn+1) such that
P.>I^>P.-T- B2-42)
By virtue of B2.40) we have then
^/7
,= i 1/7 " 3Ц;
" 3
and we also have
у Ы<1.
f=f+i j/7" з
22. Symmetric bases
585
since otherwise for the sequence {т}}е(9 defined by mj = rij(j=\,...,k),
mj = lj(j=k+l,k + 2,...) one would get, by B2.36) and B2.39),
\\*\\>i ^= E ^+ E ^>l
which is impossible. Consequently,
00 1С I t IE I 00 IE
^ ICiJ v K;,l v I?
i=l yi i=\ У I i = k+l у
whence, by B2.42), we infer
Z 16.1 e 2e e
which proves B2.41) and thus also B2.38).
By B2.36) it is clear that for any finite sequence of scalars ? 1,...,
and any sequence of indices 1 ^ m, < • • • < mp ^ к we have
B2.43)
and thus, by § 17, theorem 17.1, {х„} is an unconditional basic sequence.
This, together with B2.38), implies that {хи} is an unconditional basis
of G. If {/„} с G* is the a.s.c.f. to {х„}, then by B2.36) it is also clear that
p
2w ъпнХпц
i= 1
E ti*i
i= 1
sup
<co
(xeG,{{mj},{nj})eC)xe)),
whence, by §21, the implication 3°^>Г of theorem 21.1, {х„} is a sub-
symmetric basis of G.
Finally, let us prove that {хи} is not symmetric, by showing that
condition 2° of theorem 22.1 is violated. Consider the sequence {yk} с G
defined by
Ук =
,...,1,0,0,..Л (fc=l,2,...).
B2.44)
We claim that this sequence is bounded in G. Indeed, for a fixed k,
there exists a finite set of positive integers 1 <«!<••• <ns^k such that
1Ы1=Е
Let
& \/i{k+l-nt)
Щ for I = 1, ..., 5 — 1,
к for i — s.
B2.45)

586 II. Special Classes of Bases in Banach Spaces
Then by B2.36) and B2.45),
S
-I-
i
i
Vs{k+l-na,
1 / 1
which, together with ns^k, implies that ns = k. Continuing with a
similar argument, we obtain ns.1=k-l, ns_2 = k-2,...,n1=k+l-s,
i.e., ni = k + i — s(i=l,...,s), whence, by B2.45),
= E
1
B2.46)
However, we have
E
s-J_
2 2 1 1
+ 2 У + for 5=3,5,7,...
"^ ~^i) 5+1
for 5 = 2,4,6,...
and hence, since the function
ш-
ш = E
i= 1
is non-increasing for
dt
J yt{s+i-t)
s+ 1
2
2 f Л
2
V~s
-2
dv
2 2 ^ 5-12
Л = h 2 arc sin 1 ^ 3 + 7Г,
5+1 T/7 5+1 5+1
which proves the assertion that {yk} is bounded.
22. Symmetric bases
Now for each ке.У define a permutation ркеП by
ffc+1-л for n=l,2,...,k
587
Pki") =
for n = k+l, k + 2,...
B2.47)
and let геП be the identical permutation i(n) = n (n= 1,2,...). Then
|/2 1Д
whence
* 1
X -
j=i J
as
which, together with the relation sup \\yk \\ < со proved above, shows
1 ik< oo
that the condition 2° of theorem 22.1 is violated, and this completes the
proof of the assertions of example 22.1.
In the usual concrete Banach spaces with a symmetric basis all
symmetric bases are equivalent. Indeed, we have made a similar remark
for subsymmetric bases in §21, and by proposition 22.2 above every
symmetric basis is subsymmetric. Therefore it is natural to ask
Problem 22.1. In a Banach space E with a symmetric basis are all
symmetric bases of E equivalent?
We have seen in § 17, example 17.1, that a block subspace with
respect to an unconditional basis {х„} of a Banach space E need not be
complemented in E, whence a block basic sequence with respect to an
unconditional basis {х„} of E cannot be extended, in general, to an
unconditional basis of E. It is natural to raise
Problem 22.2. Can every block basic sequence, with respect to a
symmetric basis {хи} of a Banach space ?, be extended to a symmetric
basis of the whole space El Or, at least, to an unconditional basis
of the whole space El Or, at least, is every block subspace with
respect to a symmetric basis {хи} of a Banach space E complemented
in El
We shall give now some partial results concerning these questions,
which will be useful in determining the Banach spaces which have, up
to equivalence, a unique normalized unconditional basis (§ 24). First
we shall show that in a symmetric space (?, {х„}) the subspace spanned
by a (finite or infinite) sequence of non-overlapping finite sums of the
basis elements х„, is complemented in E, admitting a projection of
norm 1.

588
II. Special Classes of Bases in Banach Spaces
Proposition 22.3. Let (?, {х„}) be a symmetric space, {An\ a (finite
or infinite) sequence of disjoint finite sets of positive integers and /„#0
[Iе0
E xj
, _,, r „(Лп| „^ „_„„_, jeAk -l*=l
1
«M.»( E «.¦* I = I Г I «; E X
E <
i=i
B2.48)
Г 1е0
и a projection of norm I of E onto E ХЛ
Lje/lk _U=1
Proo/. Let us first show that for every positive integer N and every
00
sequence of scalars {<х;} for which ? ai*i converges, we have
Е ЛЪъ
E a.xf
B2.49)
Let Г denote the set of all permutations с of \J Ak such that
k=l
N
(r(Ak) = Ak(k=l,...,N). The cardinality of Г is M = ]~[ (lk\) (since each
n = i
ael is an ЛГ-tuple {o1,...,ofl}, where ak is a permutation of At and
since the cardinality of the set П(Ак) of all permutations ak of Ak
is lk\).
Furthermore, we have
E *
B2-50)
i
since the coefficient of a,- in the left hand side is — ? xa(i) and since
™ .те!
for any pair i,je \J Ak with jeAko there are exactly D0-l)! П C*!)
*=» t=i
permutations сеГ such that o(i)=j if /e/lk and no such
M
lk0
permutation if i$Ako.
1 For the simplicity of notation, we shall unify the finite and infinite cases, by
writing always [ ]j*L, and ? .
22. Symmetric bases 589
Now, since (?,{х„}) is a symmetric space, for every ael we have
N
L ajx°
whence, by B2.50), we obtain
iX;
t
k=\ lk
i. e., B2.49).
V or x
i = l
jeAk
Applying B2.49) to
we get
E
/
аЛ( J] xj) converges, i.e., и{Лк, is well defined.
whence ]T
Furthermore, u{Ak} is obviously linear and, by B2.49) for N->co, we
have ||ии„)К1. f 100
On the other hand, obviously u{An} maps ? into ]T *,L = 1 and
jeAk
«M.i ( E x.) = f ( Il)( E xi) = E xj №=1^2,...), whence u{Ani is
a projection of ? onto ]T xJ . Since any projection has norm ^ 1,
by the above it follows that ||uMJ| = l, which completes the proof of
proposition 22.3.
Definition 22.3. Let (?,{*„}) be a symmetric space and let {An},
1пф0 be as in proposition 22.3. The projection u{An} above is called the
averaging projection with respect to \An\.
Now we shall show that in a symmetric space (?, {х„}) certain un-
unconditional basic sequences, consisting of weighted finite sums of the
basis elements xn, span complemented subspaces with complements
having unconditional bases and hence they can be extended to un-
unconditional bases of the whole space ?.
Proposition 22.4. Let {E,{xn}) be a symmetric space, let {nkJ}kxtj=1
be an enumeration of the set Jf = {1,2,3,...} as a double sequence and let
\Pkik= i cЛЛ{ 1 }• Furthermore, let
zb „ =
B2.51)

590 II. Special Classes of Bases in Banach Spaces
and for fc= 1,2,... let
1 1 ?
— zfc , =— > xn, for (=1
Ук,1 =
1
1
Pk-l
= ¦*„
zt , —x
к,2
Pk-
i
Pk
2,
i = 2,...,pk
B2.52)
«к. i \ к,т~т~ Zk,m+ l)
= х _if?x
/or i = (m— \)pk-\-
Finally, let E, = [*-,,]?= i, ^2 = l>*.i]"=i 3 I>*.J=i.. = 2
Г/геи а) ? = ?2ф?1ф?3; b) {yk.i}k=i,i = 2 " an unconditional basis
ofE3; hence с) {укл}?=1и {х„м}4ш=1и {jMK°=i,, = 2 й an unconditional
basis of E.
Proof a) Let us first observe that the algebraic sum E1+E2 + E3
contains all basis elements х„(п = 1,2,...) and hence is dense in E. Indeed,
for every fc= 1,2,... we have xnk ieE1, zk YeE2 and
1
1
¦2 л-i
1
A-l
z,
It, 2
- _L_ v Г _ ! / . l 1
- 1 L \Xnk,i AZk,l-Xn ) - —ZK 2
B2.53)
whence, for i=2,...,pk and k= 1,2,...
1 1
JC«k,.=J't.i + r(zktl--\k ,) + -z^e
Pk-1 ' Pk
Similarly, if we assume that {xm i}|=71)Pk(=?1+?2 + ?3 and
{zkiJ.}™=1<=?3 for a pair fcSsl, w^2, then, taking into account the
relations
22. Symmetric bases
Pk
— z = У x —(z+z )
- У \x --{z +z I= У v
we obtain, for i={m—l)pk+l,...,mpk
1
""¦' *'' Pk k'm k'm
591
B2.54)
which proves the assertion that х„е?1+?2 + ?3 (и =1,2,...); actually,
the above argument shows that we have even xn .e?3 for i=pk-\-\,
pk + 2,... and k= 1,2,....
We show next that there exists a bounded linear projection of ?
onto ?, which carries ?2 and ?3 into 0, whence E = E1 ф(?2ф?3).
Indeed, let {/„} с ?* be the a.s.c.f. to {хи} and put
n — f V f (Ь—\1 \ ПО SSI
Then, by the biorthogonality relations /;(^) = йу, we obtain
Since gk =
B2.56)
(fc=l,2,...)
^-[fnhl~- E /„,, , and -^-
and since (?, {xn}) is a symmetric space, we have, for every xe? and
every positive integer N,
k=l
E ./..,(
k=\
+ 2
N Pk I
E E -/,,,(*•,>
k= 1 ;=! /»fc
iV 1 / Pk \ / Pk
E - I/*» 1^..
k= 1 Pk \i= 1 / \i=l
where м(л(к0)] denotes the averaging projection with respect to the
sequence A[0) = {nKi}f^t (fc=l,2,...) of disjoint finite sets of positive
integers. Hence, by proposition 22.3,
E
<4||x|| (xe?, JV=1,2,...),
B2.57)
i.e., sup ||ид,||<оо, where uN{x)= Y, дк(х)х„к1 {хеЕ, N= 1,2,...).

592
II. Special Classes of Bases in Banach Spaces
Consequently, since for every finite linear combination p= E <*,*,
the limit lim uN(p) exists (by B2.55) and /;(х.) = <5;.) and since [х„] = ?,
N-> со
GO
it follows that E 0tCx)*nk , converges and thus
"(*) = Z 9k(x)xnui (xeE)
B2.58)
is a well defined bounded linear mapping of E into ?t. By B2.56), we
get that и is a projection of ? onto Ex such that м(?2+?з) = 0.
Finally, to complete the proof of a), we shall show that E2+E3
= E2@E3. Let и{д0)) be the averaging projection which was con-
considered above, i.e.,
V>M= Z ^
Then for every /г = 1,2,... we obtain
B2.60)
whence the restriction of u{Am} to E2 is the identity and и{Лт}(Е3) = 0,
which proves a).
b) Let «t be the averaging projection with respect to {A,m}it°m=i
where ^,m=K.(m_1)Pk+J-}j'i1, i.e.,
"i(*)= Z -( ? A.(
k,m=l Pk \j=l
j=l "
). B2.61)
Then for every Л= 1,2,... and г'^2 we obtain, by biorthogonality,
1 Ph 1
Уи,д= Z-4,mp +J= zh,m+i (i = (m-l)ph +
Ph j=l " Ph
Indeed, for 2^i^ph we have
/>fc-l ( = 2
B2.62)
2- ( L
k,m=\ Pk \j=l
GO
z
m=l
A1
/ Ph
z /;„,
22.
(m- 1)P)]
Symmetric bases
(x l
Л'"-" Рн-
Ph
v
1 1 = 2
593
p»
Ph
J=1
1 Г/л
Z -v
1 = 2 // j=l
I"
j=l
1 \
Zj "h,P||+l/ Zj "h,P|i+j
Ph 1=1 / j = 1
- -a
and for the other i the computation is similar.
Hence, taking into account proposition 22.3 and that (Е,{х„})
is a symmetric space, we have, for any scalars ak h
N Npk
Z Z и-клУкл
k=\ i=2
N Npk
Z I<
fc=li=2
B2.63)
mpj< + j I fc,m + 2
JV J Г Рк ЛГ-1 / pk
Z - Z «».izu+ Z Z «».
fe=l/'kLi = 2 m = l\j=l
Z — Z ^,.Zfc,l+ Z Z *k,mpk + j)Zk,m+l
k = \Pk\_i = 2 m = l\j=l / J
Z — Z ak,.-(zk,i—-v«fc,)
k=\Pk\_i = 2
JV-1 / Pk
z z «*..
m = l \J=1
V
—г Z «и^
N JV-1 J / Pk
+ Z Z Г"! Z afc,ш
fc = l m = lPk\j=l
38 Singer, Bases in Banach Spaces I

594
II. Special Classes of Bases in Banach Spaces
Consequently,
JV JVpk
Z Z <*k,ib>k,i-xnkj
k=l i = 2
z :
JV три / i
Z Z akA—)(zk,n
1
Pk'
m = 2 i = (m-l
Z - Z «KiZk,2
k=lP
N-l / Pk \ 
Z (Z ak,mpk + jJZk,m + 2
JV
z
1
\k = lPk—l 1 = 2
JV JV-1 л / pk
+ z z- z
ak,i(zk,l — Xnk ,)
Z
and thus
Z Z <*клУкл
JV JVpk
Z Z «м*пМ
JV JVpk
Z Z <*клУкл
k=l i=2
B2.64)
B2.65)
\k=l i = 2
On the other hand, again by proposition 22.3, we have
JV JVpk
Z Z «и*.*.,
k=l i = 2
N Npk
Mi Z Z afc,i-v.
t = l 1 = 2
N 1 | Pk
N-l / pk
Cl
, < , . < , ..,. ..,- ^ , . < , k,mph+j
\\k=lPk\_i-2 m = l\J=l
and thus, by using B2.63) we obtain, as in B2.64), that
; I Zk,m + 1
JV JVpk
Z I«
/t = i ; = 2
JV JVpk
Z Z <*клхпк,
k=i i=2
B2.66)
From the inequalities B2.65) and B2.66) it follows, by Ch. I, § 8,
theorem 8.Id) (implication 2°=>1°) that {jM}"=1 ii = 2 is strictly equiv-
equivalent to the unconditional basic sequence {х„к .}^L 1 i = 2, whence
{Укл}кЮ=1л = 2 1S an unconditional basis of E3.
c) Since {*„} is unconditional, the subsequence {х„к1}?°=1 is an
unconditional basis of Ex and, by applying § 17, theorem 17.1, it
follows that the sequence {jM}f=i is an unconditional basis of
E2 (same proof as for § 17, corollary 17.2). Hence, by a), b) and
§ 16, proposition 16.4 on cartesian products of unconditional bases,
22. Symmetric bases
595
{yk,i}k=i^{xnKi}k=i^{yk,i}k'=i,i = 2 is an unconditional basis of E,
which completes the proof of proposition 22.4.
Proposition 22.5. // {*„} is a symmetric basis of a Banach space E,
then the a. s. c.f. {/„} с Е* is a symmetric basis of [/„] and
Hence, in particular, if (?,{*„}) is a symmetric space, then so is
(Unim)-
Proof. Consider the operators
= Z
/*= Z Pi
B2.68)
where ф denotes the canonical mapping of E into [/„]*. We have
/= max <j(i) and
1Д= 1,2,...), where we have put
0,
0 for je{l,...,l}\{<j(l),...Mk)}.
Consequently, taking into account that ||и|[Л1||<||и|| and ||и*|| = ||и||,
(; sup llf.r.jpa.tl^Ksup sup \\uaAfii),k\\ = v\s^},
p
which completes the proof.
Remark 22.1. In B2.67) the sign < may also occur. Indeed, the
bounded basis {*„} of I1 given in Ch. I, § 12, example 12.3 is uncon-
unconditional, whence, by § 18, theorem 18.2, equivalent to the unit vector
и - ii
basis of I1, and thus symmetric. By the expression of
(Ch. I, § 12, formula A2.33)), v\s}ni does not depend on /•([/„])"= Я, but
v'j^,^ vlXni = 2 + -—>oo as Я-+0, which proves our assertion.

596
II. Special Classes of Bases in Banach Spaces
Let us give now a relation between symmetric bases and sequence
spaces, similar to § 17, theorem 17.8 (on unconditional bases) and to
Ch. I, § 12, theorem 12.6 (on general bases).
We recall that the a-dual of a sequence space S is the sequence
space S" defined by
к
t\Piap{l)\<co forall («„JeS.peM. B2.68)
For every sequence space S we have, obviously, S a S"". A sequence
space S is said to be a-perfect if S™ = S.
It is also obvious that for any sequence space S we have S" с Sx,
where Sx is the a-dual of S defined in § 17, formula A7.81).
A sequence space S is called symmetric if for every {«„}eS and
every реП we have {ap(n)}eS.
Lemma 22.1. // S is a symmetric sequence space, then S" = S°' and
hence
S°°=S™. B2.69)
Proof. If {Pn}eS*, {an}eS and реП, then, since S is symmetric,
{aHn)}eS, whence
У |ftap(i,| < go,
i.e., {pn}eSa. Thus, S*<^S° and hence, by the opposite inclusion
S'cS1 observed above, Sa=S*.
Since for any sequence space S the u-dual S" is symmetric (because
00 OO
{j3n}eSff implies ? |j3t(Oap(i)| = ? |ftat-ip(i)|<oo forall {<xn}eS and
i=l 1=1
р,хеП), it follows that for any sequence space S we have S"' = S'"'.
Consequently, if S is symmetric,
which completes the proof of lemma 22.1.
Theorem 22.2. A sequence space S is associated to a symmetric basis
of a Banach space if and only if S contains all unit vectors en and there
exists a a-perfect ВK-space T such that [en]r = S. In this case, T = S'"'.
Proof. Assume that S is associated to a symmetric basis {х„} of
Г CO "I
a Banach space E, i. e., S = < {а„} czK ? <*Л converges)-. Then, since
{xn} is unconditional, by § 17, theorem 17.8, S contains all unit vectors
en and there exists an а-perfect BK-space T such that [en]r = S and
T = SX*. Since {xn} is symmetric, {ap(n)}eS for all {an}eS, реП (by
22. Symmetric bases
597
theorem 22.1, implication Г=>1Г) and thus S is symmetric. Hence, by
lemma 22.1, S'"'=SXIX=T. Since T = (S<T)'T is symmetric and since T
is а-perfect, by lemma 22.1 we also have т""=Тс'с'=Т, i.e., T is
cr-perfect.
Conversely, assume now that S is a sequence space containing all
unit vectors en and that there exists a u-perfect BK-space T such that
[en]r = S. We shall prove that {en} is a symmetric basic sequence in T,
whence S is associated to the symmetric basis {х„} = {<?„} of the Banach
space E = S.
Since T = {T")" is symmetric, by lemma 22.1 we have тая=Т'"'=Т,
i.e., Г is also а-perfect. Hence, by § 17, theorem 17.8, {en} is an uncon-
unconditional basis of S = [е„].
Now let реП be arbitrary. Define a linear mapping up: T->T by
ир({а„}) = {ар(п)} ({<х„}еГ); B2.70)
since Т = (Тау is symmetric, we have indeed {ар(п)}еТ. Then, since
T is a BX-space, the mapping up is closed, whence continuous and
therefore (since S=[en] and up(en) = ep{n) forall и=1,2,...),
up(S)
(реП),
i. e., S is symmetric. Consequently, by theorem 22.1 (implication 11°=>1°),
{en} is a symmetric basis of S, which completes the proof of theorem 22.2.
One can also prove the sufficiency part in theorem 22.2 with an
argument similar to the proof of Ch. I, § 12, theorem 12.6, by intro-
introducing on T" the norm
j8n}||= sup sup
({/JB}eT")
B2.71)
and on Y=Ta" the equivalent norm
||{а„}|||= sup sup ? |/Зт(Оа(| = sup sup
1Р„(еГ» i?l7 i = 1 {/S^fT" реП ;
ap(i)| ({<х„}еГ)
\LL.U)
and observing that for any |у;|< 1 (/= 1,2,...), <теЛ andn= 1,2,... we have
а„}||| ({an}eS).
B2.73)
Moreover, this remark makes also possible the following proof of
the implication 12°=>Г of theorem 22.1: If every permutation {xa(n)}

598
II. Special Classes of Bases in Banach Spaces
of a basis {*„} is a basis, equivalent to {xn} and if S = <{<xn} <=¦ К
E aixi
converges >, then {*„} is an unconditional basis and hence, by §17,
theorem 17.8, there exists an a-perfect BK-space Tsuch that [en]r = S
and T = Saa. Furthermore, by our assumption, for every {<xn}eS and реП
the series
a,-xp-i(i)
converges, whence, since {*„} is unconditional,
Y, &P(i)Xi= E aixp-Hi) converges, i.e., {ap(n|}eS and thus Sis symmetric
i = 1 i = 1
Therefore, by lemma 22.1, T=S™ = S'"' and thus Tis symmetric. Conse-
Consequently, by the sufficiency part of theorem 22.2, {en} is a symmetric basis of
S and hence {*„} is a symmetric basis of E (since {е„} ~ {*„}), which
completes the proof.
Finally, we shall consider finite-dimensional symmetric spaces. In
such spaces one can introduce analogues of the function systems of
Haar and Rademacher and prove a certain abstract analogue of the
Khinchin inequality (see § 14, formula A4.56)), which is useful for
applications.
Definition 22.5. Let (Е2„, {*,}) be a 2"-dimensional symmetric space,
where и<оо. We shall call Haar system (with respect to {*„}) the se-
sequence [jj}j 11 defined by
B2.74)
"-*-1, B2.75)
—2Jn"fe-1 and 2/'2""'Ui<2".
We shall call Rademacher system the sequence {rk}l=1 defined by
*= E »-' + / (k=l,...,n),
1=1
B2.76)
where {j,-} is the Haar system in (?2„, {*;}).
Proposition 22.6. Let (E2n,{xj}) be a 2"-dimensional symmetric
space. Then the Haar system {}'j}j=i is a monotone basis of Е2„. Conse-
Consequently, the Rademacher system {rk}l=1 is a monotone block basic
sequence with respect to the Haar system {yj}j= 1 •
22. Symmetric bases
599
Proof. Let m be an arbitrary integer such that l^m^2"—1, and
let a1;...,am+1 be arbitrary scalars. Then, since {*,} is a basis of Е2„,
there exists a sequence of scalars {?,};" i such that
B2.77)
E «л= Z T^i-
Let (k,l) be the couple of non-negative integers determined by the
following properties: l</<2fc, 2k + l = m + l. Then, by B2.77), B2.74),
and B2.75),
m+ 1
E «л
—
=
E
)
E
Since (E2n, {xj}) is a symmetric space, this number is equal to
i i i ^^ v*i m т l / i
E
2"
E
ViX,
i = 2I-2"
Adding these equalities and multiplying by j, we get
?—1 j у j
1n
У y-x-
У a-v.
ь-и J J J
which proves the first assertion of proposition 22.6. By B2.76) and
Ch. I, § 7, corollary 7.4, the second assertion is a consequence of the
first one, which completes the proof of proposition 22.6.
Proposition 22.7. Let (Е2„, {xj}) be a 2''-dimensional symmetric
space and let kn =
a.s.c.f. to {xn}. Then we have
Proof. We have, obviously,
2"
Шс?* is the
B2.78)
B2.79)

600
II. Special Classes of Bases in Banach Spaces
2" 2"
On the other hand, let /= ? f and let x= ?
that /(*) = /;„, |W| = 1. Then
be such
¦2- \/2- \ 2-
Z/J Z«;*j = Z«r B2-80)
\;=i / \j=i / j=1
Let мА be the averaging projection with respect to the sequence
consisting of the single set A = {1,...,2"} (see definition 22.3). Then, by
B2.80),
»a(x) = ^ ( Z «,) ( Z ^) = ^ ft. S ^. B2.81)
whence, since ||мА!| = 1 (by proposition 22.3),
1
2"
B2.82)
which, together with B2.79), gives B2.78). This completes the proof
of proposition 22.7.
Theorem 22.3. Let (E2n,{xj}jl1) be a 2"-dimensional symmetric
space and let {rk}"k = l be the Rademacher system in this space. Then
for any scalars a t,..., <xB we have
Z
B2.83)
Proof. Since {xj} is a basis of Е2„, there exists for each integer к with
К к <и a unique sequence of scalars {>¦*,¦}?= i such that
2"
rk=YJrkixi (k=l,...,n). B2.84)
Precisely, by B2.74), B2.75) and B2.76) we have
1 for Bl-2J"-k+l^i^Bl-lJ"-k
-1 for Bl-lJ"-k +
B2.85)
(,,,;,,)
Let rk(-) be the usual Rademacher functions on [0,1] (see § 14,
formula A4.2)). We claim that for any scalars al,...,an we have
Z
in i—i
B2.86)
22. Symmetric bases
601
Indeed, let us denote by (l\»,{e}}) the 2"-dimensional symmetric
space in which the norm is defined by
2"
I U
i= 1
|Cil and by
i= 1
/i-l /Л
/,-(•) the characteristic function of the interval (——, —I (/= 1, 2,..., 2").
/2" \ { 2"
Then the mapping и defined by у ( ? ?, ^j I = — X ^ e> ^s obviously
\i=i / 2" i=1
a linear isometry of the 2"-dimensional subspace [zi,---,Z2«] c ^([0,1])
2"
onto /|„. Since rk(t)= Y, гыХМ (?е[0,1],/c= 1,...,и), it follows that
n  n
/t=l J fc=l
whence since v is an isometry, we infer B2.86).
By B2.86) and the usual Khinchin inequality in ^([0,1]) (see § 14,
the remark after formula A4.56)) we have, for any scalars <х,,...,а„,
z
1
B2.87)
On the other hand, since (?2„, {х7}) is a symmetric space, we have,
by B2.84) and B2.85),
r, =
=¦¦¦= rJ =
= А„.
B2.88)
Now let аь...,а„ be arbitrary scalars and let ef = sign ]
(j=l,...,2"). Then, by proposition 22.5, (E%n,{fj}) is a symmetric
space (where {/,} с E\n is the a. s. с f. to {x;}), whence
If,
L—t i J i
. Consequently, taking into account B2.88),
proposition 22.7, B2.84) and B2.87), we get
n у
X я. ^
*=Л ||гк||
"A.
1
= 2"
n
fc= 1
i= 1
~ 2"
n
fc= 1
л
z
(^
1
1
iX- Z

602
II. Special Classes of Bases in Banach Spaces
2"
E
i= 1
2"
E
i= 1
* E «л,
E <v«
/t=i
2"
1
which completes the proof of theorem 22.3.
§23. Applications: Existence of non-equivalent normalized
bases and conditional bases in infinite dimensional
Banach spaces with bases
As a first application of symmetric bases we shall prove that the
equivalence of all normalized1 bases characterizes finite dimensional
spaces among Banach spaces with bases. The idea of the proof is to
reduce the problem first to symmetric spaces and then to one of the
spaces c0, I1 or I2, where we know the existence of both unconditional
and conditional bases (§ 14), whence also of non-equivalent normalized
bases.
Proposition 23.1. Let E be a Banach space with a basis, in which all
normalized bases are equivalent, and let {xn} be a normalized basis of
E. Then
a) {xn} is a symmetric basis.
b) If {yn} is a bounded block basic sequence with respect to {*„},
then2 {*„}- {>•„}.
c) {*„} is a Besselian basis.
d) E is reflexive.
Proof, a) By our hypothesis, for any е„=+1 (и =1,2,...), {х„} is
equivalent to the normalized basis {е„х„} of E, whence, by § 16,
theorem 16.8 (implication 3°^»Г), {*„} is a normalized unconditional
basis of E. Hence, by § 17, theorem 17.1 (implication 1°=>2°), every
permutation {xCT(n)} of {xn} is a normalized basis of E. Since by our
hypothesis the bases {*„} and {х„(п)} must be equivalent, it follows by
§22, theorem 22.1 (implication 12°=>1°) that {*„} is a symmetric basis
of E.
1 As we have observed in a footnote in Ch. I, § 8, some boundedness con-
condition is necessary, since otherwise it is easy to find non-equivalent bases (e.g.
and S"]r-"]rf) in any infinite dimensional Banach space with a basis {х„}.
Such bases {х„} are called perfectly homogeneous (see §24, definition 24.1).
23. Existence of non-equivalent normalized bases and conditional bases 603
b) Let
Jn = E
т„-i+l
(fi=l,2...;mo =
B3.1)
to a normalized
be an arbitrary bounded block basic sequence with respect to {*„}.
Then, by Ch. I, § 7, theorem 7.2, we can extend
basis {zn} of E. By our hypothesis we have {xn}~{zn} and by part a)
. Furthermore, since 0 < inf ||jj < S"P lljj<o°, by
IIjJ
Уп
§ 16, theorem 16.8 we have
~{jnb which proves b).
'{у„}. Thus, {xn}~{zn
c) By the assertion a) proved above, {*„} is a symmetric basis. We
may assume, without loss of generality, that (E, {х„}) is a symmetric
space (introducing, if necessary, the equivalent norm |||x||| defined in
§22, proposition 22.1b); the basis {*„} will remain normalized in this
new norm).
Assume now that {*„} is not Besselian, i. e., that there exists a se-
GO °°
quence of scalars {yn} for which ? y,*, is convergent, but ? |y;|2 = oo.
Then there exists an increasing sequence of positive integers {mn}
such that
Let
(n=l,2,...;mo = O).
<7o = 0, qn = E V A1=1,2,...),
B3.2)
B3.3)
B3.4)
of
and let ?2Р„ denote the 2""-dimensional subspace [Xj]qf=qn_l
E (и = 1,2,...). Furthermore, let {^}*=«n_, + i denote the Haar y
and {г;}Г="т , + i the Rademacher system in the symmetric space
(E2P.,{x!}qj"-nq~\ + 1). Then, by § 22, proposition 22.6, WJS=,B_1 + i is a
monotone basis of ?2Р„, whence, by Ch. I, § 7, corollary 7.3, the sequence
{yj}
j = qn- l+l
B3.5)
n=l
is a basis of the space E. Since by our hypothesis the normalized bases
{*„} and
\\Уп\
of ? are equivalent and since ? у;х; is convergent,
1
00 у.
the series ? yt —^ is convergent.
i=l II Jill

604
II. Special Classes of Bases in Banach Spaces
Put zi = v^j7 (/= 1,2,...). Since by §22, proposition 22.6, the se-
sequence {zj is a normalized block basic sequence with respect to the
У П I n .-. л Г1 УI
normalized basis
, of E and since Y
IjJj ,¦ = ,
converges, from
the assertion b) proved above it follows that the series ? ytzt is con-
convergent, whence т„ '-1
li X 7i*i = 0. B3.6)
lim
On the other hand, by § 22, theorem 22.3 and by B3.2) we have
which contradicts B3.6). This proves c).
d) Let us first show that {xn} is both shrinking and boundedly
complete. Indeed, assume that {*„} is non-shrinking. Since by the
assertion a) proved above, {*„} is unconditional, it follows from § 17,
corollary 17.3 a) that {xn} admits a block basic sequence \yn] equiv-
equivalent to the unit vector basis of I1. Now there are two cases:
1) If the subspace [у„] is complemented in E, then, denoting by F
an arbitrary complementary subspace of [у„], we have the isomorphisms
?x/1^(Fx/1)x/1^Fx(/1x/1)^Fx/1^?. Hence, taking a condi-
conditional basis of I1 (e. g. see § 14, example 14.2) we obtain, by Ch. I, § 4,
proposition 4.2, a conditional basis of E, whence also a normalized
conditional basis of E, which, since {*„} is unconditional, contradicts
the assumption that all normalized bases in E are equivalent.
2) If the subspace [>„] is not complemented1 in E, then extend
{>'«}. by Ch. I, § 7, theorem 7.2, to a basis {zn} of E. Since [у„] is non-
complemented in E, by virtue of § 16, theorem 16.8 (implication 1°=>2°),
{zn} is a conditional basis of E, which again contradicts the assumption
that all normalized bases in E are equivalent. Thus, \xn} must be
shrinking. The proof of the assertion that {*„} is boundedly complete,
is similar, using § 17, corollary 17.3 b), which leads to a block basic
sequence {yn} equivalent to the unit vector basis of c0.
Now the reflexivity of E follows easily. Indeed, since {*„} is
boundedly complete, by § 6, theorem 6.2 (implication 1°=>2°) we have
1 One can show (as we shall see in Vol. II, Ch. IV) that this second case cannot
occur, i. e., every subspace spanned by a block basic sequence of type /+, with
respect to an unconditional basis, is complemented. However, we shall not use
this remark here.
23. Existence of non-equivalent normalized bases and conditional bases 605
?** = я:(Е)ф[/„]±, where n denotes the canonical mapping of E into
?** and {/„} the a. s. с f. to {*„}. Since {*„} is shrinking, we have
[/„] = ?*, whence [/„]±={0} and hence Е** = к(Е), i. e., E is reflexive,
which completes the proof of proposition 23.1.
Remark 23.1. The final part of the above proof of proposition 23.1 d),
together with § 4, example 4.3 and § 6, example 6.3, shows that a Banach
space E with a basis {xn} is reflexive if and only if {х„} is both shrinking
and boundedly complete. We shall study the reflexivity of Banach spaces
with bases in more detail in Vol. II, Ch. IV.
Now we can prove
Theorem 23.1. In every infinite dimensional Banach space with a basis
there exist two non-equivalent normalized bases.
Proof. Let E be an infinite dimensional Banach space with a basis,
such that all normalized bases of E are equivalent. Then, by proposi-
proposition 23.1 d), E is reflexive, whence, by Ch. I, § 12, corollary 12.2, E* has
a basis. Let {#„}, {hn} be two normalized bases of E*, with the
a. s. с f. |Ф„}с?** = 7с(?) and {•?„}<=?** =n(E), respectively, where
)](/) = /(*) (xeEJeE*). Then, by Ch. I, § 12, corollary 12.1, the
1 l h h
sequences {yn} = {п~1{Фп)},
zn} =
zn} =
are bases of E, such that
{} {} {}
g.(y.) = dij, hi(zj) = 5tj. Since by Ch. I, § 3, theorem 3.1, we have 1 <\\y}\\,
\\zj\\^M<cc (/=1,2,...) and since by proposition 23.1a) {yn} and {zn}
are unconditional bases of E, by virtue of § 16, theorem 16.8 (implica-
tion Iе
are equivalent to
\\yn\
and
resPec"
tively. Since by our hypothesis
bJI
and
are equivalent, it
follows that {yn} and {zn} are equivalent, whence, by Ch. I, § 12, pro-
proposition 12.1, {gn} and {hn} are also equivalent. Thus all normalized
bases of E* are equivalent.
Now, let {*„} be a normalized basis of E and let {/„} a E* be the
a. s. с f. to {х„}. Then, by the above arguments, {/„} is a basis of ?*,
equivalent to the normalized basis <—— i . Since all normalized bases
of E* are equivalent, it follows, by proposition 23.1c) applied to E*
and
that
A
II/J
whence also {/„}, is Besselian. Since by
proposition 23.1c) {*„} is Besselian as well, from § 11, corollary 11.2
(implication 5°=>2°) we infer that E is isomorphic to I2. Consequently,
E has a normalized conditional basis (§ 14, examples 14.4 and 14.5),
in contradiction with the hypothesis that in E all normalized bases are
equivalent. This completes the proof of theorem 23.1.

606
II. Special Classes of Bases in Banach Spaces
We shall also give two other proofs of theorem 23.1 in § 24, remarks
24.1 and 24.2.
As a second application of symmetric bases we shall prove now
that the unconditionality of all bases characterizes finite-dimensional
spaces among Banach spaces with bases.
Proposition 23.2.Let {xn} be a normalized non-symmetric unconditional
basis of a Banach space E. Then there exists a block perturbation of a
suitable permutation of {*„}, which is a conditional basis of E.
Proof. We claim that there exists a permutation of the basic se-
sequence {x2j} which is not equivalent to the basic sequence {x2j_l}.
Indeed, assume that all permutations of {x2J} are equivalent to {x2j_l\.
Then, by § 22, theorem 22.1 (implication 12° =>Г), {x2J} is a symmetric,
whence also subsymmetric, basic sequence and thus {x2j} is equivalent
to its subsequences {х4,_2} and {x4j}. We shall show that the map-
mapping x2j_1->x4.J_2, x2j->x4.j defines an equivalence of the basis {*„}
with its subsequence {x2j}, which is a contradiction since {*„} is non-
00
symmetric. Indeed, since {xn} is unconditional, ^ atXi converges if
ОС OO
andonly if both ? %2i-ix2i-i and E a2ix2i converge. Since {x2J_1},
i = 1 i - 1
{x2j} are equivalent to {x4j_2} and {xAj}, respectively, this happens
CO GO
if and only if E a2.-i*4i-2 and E a2i*4i are convergent, i. e., (since
i=l i = l
00
{xn} is unconditional) if and only if E ixix2i 1S convergent.
Thus, let {xTBj)} be a permutation of {x2j} such that {x2j_1}
and {x42J)} are not equivalent. Let {х„(п)} be the permutation of {*„}
defined by
[x. for n = 2i--
xt(n) for n = 2j
(/¦=1,2,...)
B3.7)
and let {z'k}, {z'k} be the following two block perturbations1 of the
basis {xain)}:
1<т(к)
for к — 2 n — 1
t-n + *T(it) for k = 2n
(#i=l,2,...), B3.8)
<т(к)
for k = 2n-\
for k = 2n
B3.9)
1 See Ch. I, § 4, definition 4.7.
23. Existence of non-equivalent normalized bases and conditional bases 607
By Ch. I, § 4, proposition 4.4, {z'k} and {z'k'} are bases of the space E.
We shall complete the proof by showing that at least one of these bases
must be conditional.
Assume that both {z'k} and {z'k} are unconditional bases of E. Then,
since {z'k} is unconditional, by § 17, theorem 17.1 there exists a constant
C'^l such that we have, for any scalars au...,a.neK,
XaBi]
/_,
n
- У а((х„2; ! + xa2i)
n
i = l
n
1 = 1
Similarly, since {z^'} is unconditional, there exists a constant C"
such that we have, for any scalars а1г..,а,еК,
Hence, by Ch. I, § 8, theorem 8.Id) (implication 2°=>6°), the basic
sequences {xaBj_r)} and {xaBj)} are equivalent, which contradicts
the construction B3.7) of the permutation {х„(п)} and completes the
proof of proposition 23.2.
Theorem 23.2. In every infinite dimensional Banach space E with a
basis there exists a conditional basis.
Proof. Assume that all bases of E are unconditional and let {*„}
be a normalized basis of E. Then, by virtue of theorem 23.1, there
exists a normalized basis {y2j} of the subspace [x2j] which is not
equivalent to {x2j}. Since by § 16, theorem 16.8 (implication 1°=>2°),
[*2j-i]© [*2j] = ?> the sequence {zn}c? defined by
Z2._1=X2J_1, z2j = y2j (/=1.2,...) B3.10)
is a normalized basis of E (by Ch. I, § 4, proposition 4.2 and lemma 4.1)
and hence, by our hypothesis, {zn} is an unconditional basis of E. Con-
sequently (since either
{x2j-i
{X2j} or {x2j.1}
{x2j}
b
{y2j}),
i
either {xn} or {zn} is a normalized unconditional non-subsymmetric,
whence non-symmetric, basis of E and hence, by proposition 23.2, the
space E has a conditional basis, in contradiction with our hypothesis
that in E all bases are unconditional. This completes the proof of
theorem 23.2.

II. Special Classes of Bases in Banach Spaces
We shall give another proof of theorem 23.2 in § 24, remark 24.3.
Now we can also prove the existence of "many" mutually non-
equivalent normalized conditional bases.
Theorem 23.3. In every infinite dimensional Banach space E with a
basis there exist a continuum of mutually non-equivalent normalized
conditional bases.
Proof. By theorem 23.2, there exists in ? a normalized conditional
basis {*„}. Then, by § 16, theorem 16.1 (implication 8 =>Г), there
exist sequences of scalars {<х„}, {е„} with е„=+1 (и=1,2,...) such that
di-Xi is convergent but
sup
= 00.
B3.11)
Hence there exists an increasing sequence of positive integers {mn}
with the following properties:
B3.12)
B3.13)
(fi=l,2,...;mo =
Now, for each increasing infinite sequence of positive integers {/?,}
let us define a normalized conditional basis {y'f'} of E by
Xj for Мр^^ + КуЧИр, (и = 1,2,...),
for the other j.
B3.14)
We claim that for any increasing infinite {/?;} and {p'l} such that
the set ({/>;}\K}MK}\{/>!}) is infinite1, the bases {j]p;>} and {jf1}
are not equivalent. Indeed, assume that {р\}\{р'1} = {р\к} is infinite (the
treatment of the case when {p"}\{pj} is infinite is similar) and let
°<-j for mpl|j_1 + l<./
[0 for the other;.
Then, by B3.15) and B3.14) we have
(k= 1,2,...),
where
a,- for
0 for the other ;
B3.15)
B3.16)
B3.17)
1 Here the symbol {pt} denotes the set of all elements of the sequence
24. Perfectly homogeneous bases. Spaces with an unique unconditional basis 609
and, by B3.12), the series B3.16) is convergent. On the other hand, by
B3.15), B3.14) and the definition of \p'ik}, we have
where
g
j
for «pi.-i+l
for the other j,
B3.18)
whence, by B3.13) and since {p'ik} is infinite, the series B3.18) is divergent.
Thus the bases {y'fiY} and {jjp!l} are not equivalent. Since there
exists a continuum of increasing sequences of positive integers {/?;}
such that for {р-}ф{р'1} even both {р',}\{р-} and {p'-}\{p'i} are in-
infinite1, the corresponding collection of bases {j-Pil} of E has the prop-
properties required in theorem 23.3, which completes the proof.
§ 24. Perfectly homogeneous bases. Application: Banach spaces
with an unique normalized unconditional basis
Definition 24.1. A basis {*„} of a Banach space E is called perfectly
homogeneous if it is a bounded basis and every bounded block basic
sequence {>>„} (with respect to {*„}) is equivalent to {*„}.
For instance, by § 18, proposition 18.1 and § 16, theorem 16.8, the
unit vector basis in F A </?<oo) is perfectly homogeneous and, by a
similar argument, so is the unit vector basis in c0. Now we shall prove
that these are, up to an equivalence, the only perfectly homogeneous
bases.
Proposition 24.1. Every perfectly homogeneous basis {xn) of a Banach
space E is an unconditional basis.
Proof. Since for any sequence {е„} with е„= ± 1 (n= 1,2,...) the se-
sequence {snxn} is a bounded block basis with respect to {х„}, we have
h ilii °1°),
{е„х„}~{х„} and hence, by §16, theorem 16.8 (implication 3
{xn} is an unconditional basis of E.
1 Indeed, let ф be a one-to-one mapping of Jf= {1,2,3,...} onto the set of
all rational numbers. Take, for each real number a a sequence of rational num-
numbers {^a)} such that Iin4a) = a and let р^ = ф~\№) ("= 1,2,...). Then the
collection of all sequences {p{a)} (rearranged, if necessary, to form increasing
sequences) has the required properties.
39 Singer, Bases in Banach Spaces I

610
II. Special Classes of Bases in Banach Spaces
Proposition 24.2. Let {х„} be a perfectly homogeneous orthogonal
normalized basis of a Banach space E, let {{ydn}}d€A be the set of all
normalized block basic sequences with respect to \xn}, where A is the
suitable index set, and for each deA let ud be1 the isomorphism of E
onto [yf] satisfying
ud{xn) = /n (n=\,2,...). B4.1)
Then there exists a constant M ^ 1 such that
\\щ\\^М, \\щх\\^М (deA). B4.2)
Proof. Assume first that sup ||wd|| = oo. Then, by the principle of
deA x
uniform boundedness, there exists an element x= ? агхгеЕ with
||x|| = l, such that sup ||wd(jc)|| = oo and hence there exists a sequence
deA
id.} a A such that
= \\udn(x)\\^n+l (#i= 1,2,...)-
B4.3)
We shall construct inductively two sequences of positive inte-
integers {pk}, {qk}, as follows: Take px = \ and take q^px so that
I
¦¦I. Assume that we have already constructed pu...,pk,
such that
I «^
B4.4)
B4.5)
and that if Mj (respectively, Nj) is the least (respectively, the
largest) index of the xf's which appear in the representation of
/p'.j,ydlr/ + l,...,/q^, then
B4.6)
Put
! = тах(Л^ь^)+1. B4.7)
Since {xn} is a normalized orthogonal basis, we have |<х,-|= ||<х,-*,|1
aixt
= \\x\\ = 1 (j= 1,2,...), whence, since ||jjp«c+i|| = 1 (/=1,2,...),
+ 1 1
1 By Ch. I, § 8, theorem 8.1 d) (implication 6°=>1°).
24. Perfectly homogeneous bases. Spaces with an unique unconditional basis 611
and thus, taking into account B4.3),
I
I ^
= 2.
Hence we can choose
\ such that
i.e., that we have B4.4) for y=fc+l. Since by B4.7) pk+1>qk, we
also have B4.5) for j=k+\. Finally, since the representation of each
yj'K+i (j=\,...,pk + l — l) contains at least one xt, we have pk+i^Mk + 1
and by B4.7) we have Nk<pk+1, whence Nk<Mk + u i. e., we also have
B4.6) for j= k.
Now, by B4.6), the sequence [j {yf^ }fi Pk constitutes a normalized
к = 1 go qk
block basic sequence. By B4.4) the series ? ? a^f"- does not con-
oo <jk t = l i = pk
verge, while the series ? ? а;х; converges, since {*„} is an uncon-
k = 1 i = pt oo
ditional basis. Thus the normalized block basic sequence [j {*,-}?= Pt
/t=i
OO
is not equivalent to the normalized block basic sequence [j {j?Pt }?= Pk,
k=l
which contradicts the hypothesis that {xn} is a perfectly homogeneous
basis.
Assume now that sup HmJ || = oo. Then there exists a sequence
deA
{dn} a A such that Ци^Ц ^(и+1J"+1 (и= 1,2,...), whence also a se-
GG
quence zn= ? а(рх}еЕ (п= 1,2,...) such that
L, aj >j
B4.8)
We shall construct again sequences {pj}, {qj} such that we have
B4.5), B4.6) (for 7=1,2,... and ./ = 2,3,..., respectively) and
= 1
and
4k
j = Pk
take
•j\Pk
i
4i
>XJ
>\
! such
(k = l,
that
2,...).
«i
j = pi
B4.9)
Assume
Put
that we have already constructed Pi,--.,pk, Чъ----,Чк- Define pk + 1 by

612
II. Special Classes of Bases in Banach Spaces
B4.7). Since {xn} is an orthogonal basis, {yj"} is an orthogonal basis of
[/}"] (by § 17, corollary 17.2), whence, since it is also normalized, we have
: — < 1 (/, и = 1,2,...), and thus, since {xn}
is normalized,
Z «Ту
again into account B4.8),
j = Pk*l
+ 1 —1. Therefore we have, taking
Pk+ 1 -
z
J = l
= 2
and hence we can choose qk + l ^pk+i such that
L~t 3 J
Now, since {у*"*} is an orthogonal basis of
we have
z «rv
l = Pk
and thus the series
by B4.9), the series
L aj Уз
1
1 2Pk
[yj"*], for any /t with
(fc=l,2,...), B4.10)
k=1 j=Pk
k=l j =
Z ^к)у]"" converges. On the other hand,
ocf^Xj does not converge, and thus the
normalized block basic sequence (J {ydj"*})k= Pk is not equivalent to the
k= 1
oo
normalized block basic sequence [j {Xj}]k=Pk, in contradiction with the
k=l
hypothesis that {xn} is a perfectly homogeneous basis. This completes the
proof of proposition 24.2.
Theorem 24.1. A basis {xn} of a Banach space E is perfectly homo-
homogeneous if and only if it is equivalent to the unit vector basis ofc0 or lp for
some p with l^p<co.
Proof. The sufficiency part is obvious, since we have observed above
that the unit vector bases in c0 and V A</)<оо) are perfectly homo-
homogeneous.
Conversely, assume that {xn} is a perfectly homogeneous basis of a
Banach space E. Then, since {х„} is a bounded unconditional basis, we
may assume that it is hyperorthogonal and normalized (by considering, if
necessary, a suitable equivalent norm on E; indeed, by Ch. I, § 3, theorem
24. Perfectly homogeneous bases. Spaces with an unique unconditional basis 613
3.2, one can take an equivalent norm \\x\\' in which {xn} is normalized
and then the equivalent norm
Z «л-
= sup sup
Z ft**
i= 1
,B4.11)
in which |||xn|||= ||xj'= 1 and {х„} is hyperorthogonal). Hence, by
proposition 24.2, there exists an M ^ 1 such that for every normalized
block basic sequence {zn} and all scalars au ..., а„
M
Put
n
z ««*«
i= 1
Z
n
z
i
1
(A
= 1,2,.
n
;= l
¦•)•
B4.12)
B4.13)
Then, since {xn} is a normalized monotone basis,
^л^Лг^Лз^--- B4.14)
Taking in B4.12) the normalized block basic sequence Zj = x
and the scalars а^=1, it follows that for every increasing sequence
{Pj}"j= i of positive integers we have
1
M
Zxn.
м.
B4.15)
Taking in the right side of B4.12) the normalized block basic se-
quence zf =
obtain
M2
Z x(.--i
M2
~M
M2
M2
and using the left side of B4.15) we
n Zj {i— l)nk {+ j
2^ „k-i
i*(«-.
B4.16)

614 II. Special Classes of Bases in Banach Spaces
Hence, by induction,
M
'2 k
J = l
(n,k = 1,2, ¦¦¦);
B4.17)
indeed, for к = 1 this is trivial, and if M2k 2
J=l
then, multiplying by the respective sides of B4.16), we obtain B4.17).
On the other hand, by the left side of B4.12) and the right side of
B4.15), we have
z *,
z
J"=l
Z *<i-l)nl'-i
Xi
whence, again by induction,
n
I*.
i= 1
^M2fe
I*«
i= 1
By B4.13), we can write B4.17) and B4.18) in the form
Щк2к
M'
B4.18)
B4.19)
For any positive integers N,n and к let h = h(N,n,k) be the non-
negative integer for which
Nh^nk<Nh+1. B4.20)
Then, by B4.19), B4.20), B4.14) and again B4.19), we have
log и
Since by B4.20) h < к —— < h + 1, if follows that we have
logJV
logn
'logJV
whence, dividing by fclogn,
Г—-—1>
LlogJV fclognj
log и
logiV
21ogM 21ogM bgAn
logJV logn logn
24. Perfectly homogeneous bases. Spaces with an unique unconditional basis 615
and hence, for fc->oo,
<21ogM -
-) +
logJV " ° \logiV logn/ logn
Interchanging the roles of n and N, we obtain
logn " '"""\\ogN logn/ logN'
which, together with B4.21), yields
B4.21)
B4-22)
logln
logn logiV
1
logN logn
Consequently, the sequence < -> converges to a limit с Since
I lognj
1<1„<и, we have 0<logл„^logn, and thus
in B4.21), we obtain
l. Taking JV->oc
and similarly, from B4.22) for JV-»oo we obtain
whence
--^- < с logn,
M2
nc
M2
B4.23)
Consequently, if c = 0, we have л„<М2 (и =1,2,...), i.e., {xn}
is an unconditional basis of type P and hence, by § 17, corollary 17.1,
{х„} is equivalent to the unit vector basis of c0.
Assume now that 0<c<l and put p = —. Then we can write
B4.23) in the form c
\_
—.< YXi ^M2n" (n=l,2,...). B4.24)
M2 fei
kt
Let rl,...,rn be arbitrary positive rational numbers, say rt = —
(j=l,...,n), where fc,-,m are positive integers and let k= ^ kt. Then,
using that {*„} is a hyperorthogonal basis, the right side of B4.24) for

616
II. Special Classes of Bases in Banach Spaces
n = kt, B4.12) for the normalized block basic sequence
j— i m4"] m
Ух-,
/=i .Z,k-+J
and for the scalars a, =
, the left side
of B4.15) and the left side of B4.24) for n replaced by k= ? fc,-, we get
У r"x
L, '• xt
m
5s-
Z khi
i= 1
z
n k,
M2m"
n
z
i= 1
ki
j=i
У X
II^I
М6т"
Hence, taking into account the hyperorthogonality of the basis {xn},
it follows that for any scalars al,...,an we have
n
z
i= 1
=
n
z
i= 1
|a,|x,
Similar arguments yield
IP \p
B4.25)
n 1
У r^Xj
1
~ I
mp
n 1
Y.kfXi
i = l
M2
^ 1
mp
n
z
1 = 1
k,
мэ
M4
T
mP
n
z
t= 1
У л:
24. Perfectly homogeneous bases. Spaces with an unique unconditional basis 617
whence, by the hyperorthogonality of {xn}, for any scalars a1;...,an
we obtain
Z aixi
B4.26)
Thus, by B4.25) and B4.26), {xn} is equivalent to the unit vector
basis of I", which completes the proof of theorem 24.1.
Remark 24.1. With the aid of theorem 24.1, one can give the follow-
following proof of § 23, theorem 23.1: Let E be an infinite dimensional Banach
space with a basis, such that all normalized bases of E are equivalent.
Then, by §23, proposition 23.1b), every such basis is perfectly homo-
homogeneous. Consequently, by theorem 24.1 above, E is isomorphic to one
of the spaces c0 or I" (p^ 1), in contradiction with the hypothesis that
in E all normalized bases are equivalent. This completes the proof.
Now we shall give a sharpening of theorem 24.1, which will be
useful in determining the Banach spaces which have, up to equivalence,
a unique normalized unconditional basis. For this purpose, let us first
observe the following sharpening of proposition 24.2:
Proposition 24.3. Let {xn} be a subsymmetric orthogonal normalized
basis of a Banach space E, with the property that for every increasing
sequence of positive integers {mn} the basis {х„} is equivalent to the
normalized block basic sequence
Уп =
z
i = mH- i 4
z
i = mn- i 4
Xi
1
Xi
1
(и=1,2,...;шо =
B4.27)
let {{.Уп}}<*ел be the set of all normalized block basic sequences of the
form B4.27), where A is the suitable index set, and for each deA let ud
be the isomorphism of E onto \yd^\ satisfying B4.1). Then there exists a
constant M^\ such that we have B4.2).
Proof. Since {xn} ~ {yn} and {xn} is subsymmetric, it follows that
{у„} is subsymmetric and thus for any two increasing sequences of
positive integers {hJ, {mj we have {xn.}~{xn}~{yn}~{ym.}, whence
,}{У*,}
Now, assuming that sup ||Md|| = oo, it follows, as in the above proof
deA
of proposition 24.2, that there exists a normalized block basic sequence
QO
of the form \J {/ip*}TLPl< (where each {ydn} is of the form B4.27)), which
k=l
is not equivalent to [J {xt}ftpk, in contradiction with the above (since

618
II. Special Classes of Bases in Banach Spaces
U MP*}?=Plt can be extended to a sequence {yn} of the form B4.27)).
k=l
Similarly, assuming that
Ци^'Ц = оо, we obtain again a contra-
diction with the above, which completes the proof.
Now we can give
Theorem 24.2. Let {xn} be a subsymmetric basis of a Banach space E,
with the property that for every increasing sequence of integers {mn}f=0
with mo = O the basis {xn} is equivalent to the normalized, block basic
sequence B4.27). Then {xn} is equivalent to the unit vector basis of c0
or lp for some p with 1 ^p < со.
Proof. Observe that for any equivalent norm on E, the basis {xn} is
still equivalent to B4.27), by §16, theorem 16.8 (implication 1°=>4°)
and that in the above proof of theorem 24.1 only subsequences of {xn}
and normalized block basic sequences of the form B4.27) have been
used, for a suitable equivalent norm. Thus, applying, in that argument,
proposition 24.3 and §21, theorem 21.2 (implication l°=*-2°) instead
of proposition 24.2, the desired conclusion follows.
Proposition 24.3 and theorem 24.2 are sharpenings of proposition
24.2 and theorem 24.1, respectively, since every perfectly homogeneous
basis {х„} is subsymmetric (it is unconditional by proposition 24.1 and
equivalent to any subsequence because any subsequence is a bounded
block basic sequence) and equivalent to any normalized block basic
sequence of the form B4.27) and since a priori we don't know whether
the converse is also true; however, it turns out that this is indeed the
case, namely, every such basis is already perfectly homogeneous (by
theorem 24.2).
As an application of the above we shall prove now that up to an
isomorphism, the only Banach spaces with an unconditional basis in
which all normalized unconditional bases are equivalent are those of
§ 18. theorems 18.2 and 18.1, i.e., c0, I1 and I2. If the answer to the first
or second question of § 22, problem 22.2, were affirmative, then the
desired result would follow easily, with the same argument as that
used in remark 24.1 above, taking into account also § 18, theorem 18.3.
We shall obtain the same result by using theorem 24.2 above and § 22,
proposition 22.4.
Theorem 24.3. The only infinite dimensional Banach spaces with an
unconditional basis, in which all normalized unconditional bases are
equivalent, are (up to an isomorphism) c0,11 and I2.
Proof. Let E be an infinite dimensional Banach space with an un-
unconditional basis, in which all normalized unconditional bases are
24. Perfectly homogeneous bases. Spaces with an unique unconditional basis 619
equivalent, and let {xn} be an arbitrary normalized unconditional basis
of E. Then, by § 17, theorem 17.1 (implication 1°=>2°), every permuta-
permutation {*„<„)} of {xn} is a normalized unconditional basis of E. Since by
our hypothesis the bases {х„} and {xa{n)} must be equivalent, it follows
by §22, theorem 22.1 (implication 12°=>1°) that {*„} is a symmetric
basis of E. Consequently, by §22, proposition 22.1b), we may assume
that (?,{*„}) is a symmetric space. Let {т„}?=0 be an arbitrary in-
increasing sequence of integers with m0 = 0, and let
z
Уп =
Z
...; mo =
B4.28)
We shall show that {х„} is equivalent to the normalized block basic
sequence {yn}, which, by theorem 24.2 above and § 18, theorem 18.3,
will complete the proof.
Let
рй = тй-та-1 («=1,2,...). B4.29)
Observe that if р„= 1 for an index n, then у„ = хтп. Hence, if pn> 1
only for a finite number of indices n, say nu...,nk, then {У]}^ж\{щ,...,пк)
is a subsequence of {*„}, and thus, since {х„} is symmetric, {у^еЖ\1п1,...,„к}
is equivalent to {xj}f=k+u whence the basic sequence {yn} is equivalent
to {х„}.
Assume now that there are infinitely many indices «with pn>\,
say n1,n2, ¦¦¦ We claim that in this case we may assume
(и = 1,2,
Indeed, putting
J=l
4k
z
Z ¦
i=qk- l + 1
(/c=l,2,...)
:=1,2,...),
B4.30)
B4.31)
B4.32)
we obtain a normalized block basic sequence {zk} with qk-qk-i =Pnk> 1
(k= 1,2,...) and, since (E, {*„}) is a symmetric space, {zk} is equivalent
to the normalized block basic sequence {у„к}. Now, if {*„} is equivalent
to {zj and if the set Л\{пип2,...} has a finite number of elements,
say /, then, since {xn} is symmetric, {х}}^, + 1~{хп}~{гк}~{уЯк},

620
II. Special Classes of Bases in Banach Spaces
whence the basic sequence {yn} is equivalent to {xn}. On the other hand,
if {xn} is equivalent to {zk} and the set ,/V\{nl,n2,...} is infinite, then
{xn} ~ {zk} ~ {уПк} and, since {xn} is symmetric, it is also equivalent to
its infinite subsequence {jJ}je^\|ni>n2,...,, whence, again since {xn} is
symmetric, it follows1 that {х„\ is equivalent to {у„}. This proves the
claim that we may assume B4.30).
Now let {nkj}kj=l be an enumeration of the set .#' = {1,2,3,...}
as a double sequence. For this double sequence and for {pn} defined by
B4.29) and satisfying B4.30), construct yki as in §22, proposition 22.4.
Then, by that proposition, {укл}к=1 и {*„„,,}*°=i и{Л,.}?=1,, = 2 is an
unconditional basis of E. Since by our hypothesis all normalized un-
unconditional bases of E are equivalent, after normalization this basis
becomes equivalent to {xn}, whence, by the symmetry of {xn} we get
that {xn} is equivalent to <—k-^— > . Since by the symmetry of {xn}
we also have the equivalence
11л, 11
(observe that both
ykl and yk have pk = mk — mk-l summands), it follows that {х„}~ {>>„},
which completes the proof of theorem 24.3.
Remark 24.2. From theorem 24.3 there follows again § 23, theorem
23.1 on the existence of non-equivalent normalized bases in infinite
dimensional Banach spaces with bases. Indeed, if in E with dim E= oo
all normalized bases are equivalent, then, as we have seen in § 23, proposi-
proposition 23.1a), they are all unconditional, whence, by theorem 24.3 above,
E must be isomorphic to c0,11 or I2, but in each of these spaces we know
examples of conditional bases. This contradiction completes the proof.
Remark 24.3. The above proof of theorem 24.3 shows also that if
a Banach space E has an unconditional basis and every normalized un-
unconditional basis of E is symmetric, then E is isomorphic to c0, /' or I2
(since {xn
{xeittl}j°=l
\\Ук,1П)к=1
by the symmetry of {xn} and
1 Indeed, let ,Л\ = {/,} <=,/K ./^ = {rj c/K be any pair of infinite sequences
of indices such that .Аг1иЖ2 = ,У, ЛхъЖг = §. Then, {х„} being unconditional,
oo oo oo
Y aixi converges if and only if both У a, x, and У a, xr converge. Since
i = i i = i ;=i
oo oo
{xn} is symmetric, this happens if and only if ? a.uxx and ? arxi converge.
i=l ' i=l
GO GO
Since {х„} ~ {yt.}, {х„} ~ {yr}, this happens if and only if both ? а, .у,, and ? a,,}',,
il il
converge, which (since {у„} is an unconditional basic sequence) happens if and
oo
only if ? а,_к; converges.
i = l
Ук,1
25. Absolutely convergent bases. Uniform bases
Ук.1
621
respectively). In other words,
J.y*,ilJ* i (JJ
the only spaces with an unconditional basis, which do not satisfy the
hypothesis of § 23, proposition 23.2, are (up to an isomorphism) c0, I1
and I2. From this remark we obtain again § 23, theorem 23.2.
§ 25. Absolutely convergent bases. Uniform bases
One can obtain some new classes of bases by requiring the conver-
00
gence of all expansions ? А(х)х( in a stronger sense, e.g. this idea
i= 1
yields the very useful notion of unconditional bases. However, a further
00
strengthening of the mode of convergence of all expansions ? /j(*)Xj
i= 1
leads, in general, to very restrictive classes of bases. In the present
section we shall consider two such instances.
Definition 25.1. A basis {xn} of a Banach space E is said to be an
absolutely convergent basis if every convergent expansion ? а(х, is
i=1
absolutely convergent, i.e., ? ||а,л:,|| < со.
ii
Obviously, this amounts to
B5.1)
where {/„} <= E* is the a.s.c.f. to the basis {xn}.
For instance, every basis in a finite dimensional space E and the
unit vector basis of E = l\ are absolutely convergent bases. However,
the converse statement is also true, i.e., these are, up to a normalization
and an equivalence, the only absolutely convergent bases, as shown by
Theorem 25.1. Let {xn} be an absolutely convergent basis of an infinite
dimensional Banach space E. Then
basis of I1.
Proof. If
II*» II
is equivalent to the unit vector
at
^ = ?
-xt is convergent, then, since {xn} is
i=l 11-411 i=l WXiW ' oo oo
an absolutely convergent basis, we have ? |a;| = X
;=i i = i
a,-
-x,
< 00.

622 II. Special Classes of Bases in Banach Spaces
CO
Conversely, if ?|af|<oo, then from
X,
n + p v n + p
~ I l«<l
i =n
and from the completeness of E it follows that ? a
converges,
which completes the proof.
Definition 25.2. A basis {*„} of a Banach space ?, with the a.s.c.f.
{/„}, is said to be a uniform basis, if the series ? /j(x)Xj converge uni-
i= 1
formly in the unit cell SF= {xeE | |jx|| < 1}, i.e.,
sup
I /¦•(*)*.-
= sup
xeE
11*11 «1
i= 1
>0 as и-^оо. B5.2)
For instance, every basis in a finite dimensional space ? is a uniform
basis. However, the converse statement is also true, i.e., these are the
only uniform bases, as shown by
Theorem 25.2. Let E be a Banach space with a uniform basis {xn}.
Then dim E < oc.
Proof. By B5.2) we have
>0 as и->оо,
B5.3)
where IE is the identical mapping of E onto itself and sn is the n-th
partial sum operator associated to the basis {xn}. Since by dimsn(.E)
= dim[x1, ...,xn] = n<oo each sn is compact, from B5.3) it follows that
IE is compact, whence the unit cell SE = IE{SE) of E is compact. Con-
Consequently, by a classical theorem of F. Riesz1, dim?<oo, which com-
completes the proof.
Notes and remarks
Part I. §1. Bases satisfying A.1), respectively bases with vUn) = l, have
been considered, independently, by R. С James [114] and V. Ya. Kozlov
[135] and called by them "orthogonal bases"; this term is still used by
some authors, e.g. V.I. Gurarii. The term "monotone basis" was sug-
suggested by M. M. Day [43]. Strictly monotone bases were introduced in
[141], definition 2.1 and proposition 1.1 was given in the same paper
([141], proposition 2.1).
1 See e. g. [10], p. 84, theorem 8.
Notes and remarks
623
Theorem 1.1 and example 1.1 are due to M. Z. Solomiak [253]. As
we already observed in § 1, H. F. Bohnenblust [29] proved that for
every integer n ^ 3 there exists an и-dimensional Banach space En
satisfying A.6), namely, a suitable subspace of a space /^ with k> 2 (In — 3)
and/)# integer. The construction culminating in theorem 1.2 is due,
essentially, to V. I. Gurarii [92], [96] to whom we are indebted for
sending us a part of [92]; we slightly modified here the arguments of
[92], following L. S. Pontryagin [199], Ch. I. For the definition and
properties of the opening 0(Gl,G2) (formula A.15)) see M. G. Krein,
M. A. Krasnoselskii and D. P. Milman [141]. The modified "opening"
0{Gl,G2) (formula A.17)) was introduced by I. C. Gohberg and A. S.
Markus [80]. For lemma 1.10 see I. C. Gohberg and M. G. Krein [77],
theorem 6.2.
Lemma 1.12 and theorem 1.3 were given by V. I. Gurarii [95].
Lemmas 1.13—1.15 and theorem 1.4 are due to V.I. Gurarii [94],
[97] and so are lemma 1.16, proposition 1.2 (verbal communication)
and corollary 1.1 (see [97 a] for the first part and [94], [97] for the
second part of corollary 1.1).
Problem 1.1 was raised by M. M. Grinblium [82].
§2. Normal bases were already considered by S. Banach ([10],
p. 238), but the term "normal basis" has been introduced by S. Karlin
[131].
Lemma 2.1 is due to G. Ascoli [6].
In the particular case of finite dimensional Banach spaces, the equiv-
equivalence Го2° of theorem 2.1 was observed by M. M. Day [42], the
equivalence 1°<=>3° by A.E.Taylor [255] and the implication 1°=>4°
by A. Yu. Levin and Yu. I. Petunin [146], who have also observed the
equivalence 1°<=>2° in the infinite dimensional case.
Proposition 2.1 has been given, with a different proof, by S. Karlin
([131], theorem 7).
Theorem 2.2 for real Banach spaces was stated, without proof, in
the monograph of S. Banach ([10], p. 238), with the mention that it is
due to H. Auerbach; again for real Banach spaces, it has been redis-
rediscovered independently, with different proofs, by M. M. Day [42] and
A. E. Taylor ([255], theorem 2). The proof given here can be found in
A. F. Timan ([256], p. 407) with the mention that it is due to M. I. Kadec;
later, the same proof was also given by A. F. Ruston [218].
Theorem 2.3 is due to C. Bessaga (verbal communication). Cor-
Corresponding to §1, where it was observed (after proposition 1.3) that
every basis of a Banach space E can be "monotonized", one could ex-
express theorem 2.3 by saying that every bounded basis of a Banach
space E can be "normalized"; however, this might be misleading, since

624
II. Special Classes of Bases in Banach Spaces
the term "normalized basis" is used in a different sense (Ch. I, § 3, defini-
definition 3.2).
§ 3. The notion of positive basis was introduced by W. J. Davis [37].
Lemma 3.1, proposition 3.1, lemma 3.2, lemma 3.3, remark 3.2, propo-
proposition 3.2, lemma 3.4 and proposition 3.3 are given in the same paper
[37]. The example of an и-dimensional Banach space with only two
linear isometries, given before problem 3.1, was communicated to us
by С Kottman. Problem 3.2 was raised by W. J. Davis [37].
§ 4. Bases {*„} such that [/„] = ?* (where {/„} is the a.s.c.f. to {xn})
have been considered by M. M. Grinblium and L. A. Gurevich [87] and
S. Karlin [131] who have proved the result given in Ch. I, § 12, corol-
corollary 12.2. Bases with the property that lim||/||B = 0 for all feE*,
where ||/||„ is defined by D.3), were considered by R.C.James [113].
The term "shrinking basis" was suggested by M. M. Day [43]. For
fc>0, the fc-shrinking bases were introduced in [244].
Lemma 4.1 was given in [244], lemma 1. Proposition 4.1 was proved
in [237]; however, later V. F. Gaposhkin [66] observed that a more
general result had been given by S. M. Nikolskii ([178], theorem 1).
Proposition 4.2, except its last equality, and theorem 4.1, except its
equivalence Го5°, were given in [244], lemma 2 and theorem 2. The
equivalences Го2°о5°о6° of theorem 4.2 were proved by R. С James
([113], theorem 3) and the equivalences ГоЗ°«4° by A. Wilansky
([262], theorems 1 and 2). For monotone bases the implication 1°=>7°
of theorem 4.2 has been given by R. С James [113] and in the general
case in [239], corollary 3. The implication 1°=> the first part of 8° is
due to B.R. Gelbaum ([69], theorem 6). The equivalences Го7°о8°
were observed by J. R. Retherford [205].
Example 4.1, except its parts g) and h), was given by R. С James
[113]; part g) was observed in [241], example 2.3. Example 4.2 was
observed in [241], example 2.4. As we already mentioned, example 4.3
was found by M. M. Grinblium and L. A. Gurevich [87] (see also
S. Karlin [131], lemma 2).
§5. Retro-bases in conjugate Banach spaces were introduced by
B. R. Gelbaum [69].
The equivalences l0o2°o4° of proposition 5.1 were given by
B.R. Gelbaum ([69], theorems 17 and 5). Problem 5.1 was raised in
[241], problems 2.4, 2.3' and 2.4'.
The problem of the existence of normal non-retro-bases in conjugate
Banach spaces was raised by S. Karlin ([131], problem 1). Example 5.3
was given by A. Pelczynski (verbal communication) and, later, proposi-
Notes and remarks
625
tion 5.2 and example 5.2 by J. R. Retherford [207]. Lemma 5.1 is due
to J. P. Williams ([264], lemma).
§ 6. Boundedly complete bases have been considered by N. Dunford
and A.P.Morse [49], L. Alaoglu [la] and, later, by S. Karlin [131]
and R. С James [113]. The term "boundedly complete basis" was sug-
suggested by M. M. Day [43]. For fc>0, the fc-boundedly complete bases
were introduced in [244].
Theorem 6.1 and corollary 6.1 were given in [244], theorems 1 and 4.
Examples 6.1 and 6.2 were observed in [241], Ch. II, § 5. Example 6.3 is
due to R. С James ([113], theorem 1).
The implications Г=>4°=>2°=>3° of theorem 6.2 were given, in a
slightly weaker form and with different proofs, by S. Karlin ([131],
theorems 9 and 10) and the implication 3°=>Г in [235], theorem 5; see
also M. M. Day [43], Ch. IV, § 3, lemma 2 for the equivalences lDo2°o4°
in the case of monotone bases and [239], theorem 5). The implication
1°=>6° was stated, without proof, by A. R. Lovaglia ([154], p. 234); the
proof given here and, essentially, example 6.4, were communicated to
us by A. Pelczynski. The implication 1°=>7° of theorem 6.2 as well as
problem 6.2 were communicated to us by C. Bessaga, with the mention
that they are due to J. Lindenstrauss.
§7. Bases of type wco,swco,{wco)* and (swcof were introduced in
[62], p. 932. For bases of type wc0, A. Pelczynski and W. Szlenk [197]
have suggested the term "semi-shrinking basis"; however, W. Ruckle
[215 a] has used the term "semi-shrinking basis" for bases of type (wc0)*.
Theorem 7.1 was given in [62], theorem 1 and corollary 2. Problem
7.1, except the last question of part b), was raised in [62], p. 940, prob-
problem 2. Recently part of this problem has been solved in the affirmative,
namely, J.R. Holub [108] has proved that C([0,l]) has a basis of type
(swcof and ^([0,1]) has a basis of type swc0. Actually, the following
more general result holds ([108], proposition D.2)): Let E be a Banach
space with a basis {х„}, in which weak and norm convergence of sequen-
sequences are not equivalent. Then E has a basis {zn} of type swc0, such that
W = Ш' where Ш> (M E* are the a-s-c-f- t0 the bases W
and {zn} respectively. Consequently, {hn} is a basis of type (swc0)*
Indeed, by hypothesis there exists a sequence {yn} с Е such that
yn^0, inf |[yJ>0. Then, by §15, proposition 15.1, {yn} has a
l«n<oo
subsequence {yPn+1} which is a basic sequence, equivalent to a block
basic sequence {У„ +1} with respect to {*„}. Extend {y'Pn+l}, by Ch. I,
§7, theorem 7.2, to "a basis {zj of E, such that [Л„] = [/„]. Then, since
40 Singer, Bases in Banach Spaces I

626
II. Special Classes of Bases in Banach Spaces
/РI+1Д-0 (because
+ 1-»0), the basis {zn} is of type swc0. Hence,
by the "swc0-analogue" of Ch. II, §7, proposition 7.3, {hn} is a basis
of type (s w c0)* of [/„], which completes the proof.
Since e.g. the sequence {smlnnt} czl}{\Q,1]) is weakly convergent
to zero but not norm convergent, it follows that 1^([0,1]) has a basis
{zj of type swc0. Since for the a.s.c.f. {/„} to the Haar basis of
Ll{[0,l]) we have [/„] =s C([0,1]), (see Vol.11, Ch. V), -it follows that
C([0,1]) has a basis of type {swc0)*.
Let us also mention that if weak and norm convergence of sequences
in E are equivalent, then E clearly has no basis of type swc0 (this was
observed in §7 for E = ll).
Proposition 7.2 is due to A. Pelczynski ([192], p. 544, lemma).
Recently W. Ruckle ([215 a], lemma 1) has observed that a basis {х„}
of a Banach space is of type (wc0)* if and only if I1 <= S77 <= c0, where
5 = Л1({х„}) is the sequence space associated to the basis {х„}; this
amounts, essentially, to the equivalence ГоЗ° of proposition 7.3 (see
Ch. I, § 12, proof of theorem 12.6).
§8. Weakly closed and (weakly closed)* bases were introduced in
[62], p. 932.
Proposition 8.1c) was given in [62], lemma la). Theorem 8.1 was
given in [62], theorem 1 and corollaries 1 and 2. Remark 8.1 was made
in [62], p. 937. Problem 8.1 was raised in [62], p. 940, problem 2 b).
Since every basis of type P* (respectively P) is weakly closed (respec-
(respectively (weakly closed)*) (see § 12, theorem 12.1) and since ^([0,1]) has
bases of types P* and P (see the Notes and remarks to §9, problem 9.1)
it follows that the answer to problem 8.1 is affirmative (J. R. Holub
[108], remark 1).
Remarks 8.2 and 8.3 are due to A. Pelczynski (verbal communication).
§ 9. Bases of types P and P* were introduced in [237]. For bounded
biorthogonal systems conditions of this type occur in V. Ptak [201],
theorems 1 and 2. In aP and aP* the letter a stands for "almost". For
every family 38 of bases one can define in a natural way the families
a& and &*, as follows: {xjeaf if there exists a sequence {еп}сК
with |е„| = 1 (n=l,2,...) such that {snxn}e@, respectively {х„}е@* if
{fn}e&, where {/„}<=?* is the a.s.c.f. to {xn}. Then, obviously,
a(P*) = (aP)* and therefore we use for this family the notation aP*.
Actually, the families of bases of type (wco)*,(swco)* and (weakly
closed)* considered in § 7 and §8, as well as the bases of types (/+)* and
(a/+)* in §10, are families 08* for the corresponding families 3&. One
can also introduce the family s<ffl as follows: {xn}es38 if there exists a
subsequence {xin} of {xn} such that {xir}e3S.
Notes and remarks
627
Theorems 9.1 and 9.2 were proved in [237]. Let us also mention the
following characterization of bases of type P: A basis {xn} of a Banach
space E, with inf \\х„\\ > 0, is of type P if and only if {е„}>{х„\, where
1 in< oo
{en} is the unit vector basis of the space bvo = {xe{?n}ebv\ lim?n = 0}.
Indeed, if {xn} is of type P, then {en}>{xn} by the implication 1°=>3C
of theorem 9.1 and by an argument similar to that used in Ch. I, §5,
proof of theorem 5.2, implication 3°=>2°. Conversely, if {е„}>-{х„}, then,
by Ch. I, §8, theorem 8.1 d), implication 6°=>1°, there exists a continuous
linear mapping u:bvo^>E such that и{е„) = х„ (п = 1,2,...,), whence,
since sup
i= 1
it follows that sup
bv0
I-
< oo, which
completes the proof. Hence it follows, using Ch. I, § 12, proposition 12.1,
that a basis {*„}, with sup ||х„|| < oo, is of type P* if and only
if
>
where {en} is the unit vector basis of the space
with
11*11= SUP
Similar
? ?,¦ converges >,
i J
results, in terms of sequence spaces, have been given, with a different
proof, by W. Ruckle ([215], proposition 5.6).
If {xn}, {yn} are bases of Banach spaces E and F respectively, then,
obviously, {*„} x {yn} is of type P (or P*) if and only if both {xn} and
{у„} are; a related result for sequence spaces has been given by W. Ruckle
([215], theorem 5.7). H. Joiner has observed [116] that a similar result
also holds for the basis {х,®^} of ?®aF, where a is any uniform
crossnorm on E®F, such that l<a<y; the proof of the sufficiency
part is analogous to that of Ch. I, § 18, theorem 18.1, expressing the
finite sums ?*(®У/ with the aid of ^x; and ?_у^.
Theorem 9.3 was given in [62], theorem 2 and corollary 1. Re-
Remark 9.1 was made in [62], p. 940, remark 2. Corollary 9.1 is a part of
[62], theorem 2. Problem 9.1 was raised in [62], problem 2. Recently
J. R. Holub [108] has given the following affirmative solution of this
problem: Let {z*,1»} = |4^[ be the normalized Haar basis of Lx([0,l])
I HJnllJ
(Ch.II, §2, formula B.3)) and let {hn} <=?* = L°°([O,1]) be the a.s.c.f.
t {j1»} (Ch II §2 fl B4)) Pt
§, ()) {n}
to {zj,1»} (Ch. II, §2, formula B.4)). Put
for i#2m-l, m=3,4,...
2m-2
h2m_l- ? hj for /=2m-l, m = 3,4,...
40»

628
II. Special Classes of Bases in Banach Spaces
Then, since ||/г,|| = 1 (/= 1,2,...) and
Z hj
= 1 (m = 3,4,...),
j *¦
{/,} is basis of [Л„] (by virtue of Ch. I, §4, proposition 4.4 on block
perturbations of bases). Furthermore, H/J^l (г=1,2,...) and for any
m = 3,4,... we have
2""+l
i = 1
whence, since
3
If,
i= 1
=
+
3
I/.
;= l
3
I/l
i= 1
3
Z/i
i=3
+
+
P
Z ^2J- 1
J
= 3
2
m — 1
Z /i
i = 4
+
+
m
m+ 1
Z ^2J-1
m+ 1
Z h2l^
J = 3
= 1(^ = 3,4,
2m.
Z
i = 2
+
¦¦)
H
m
2m + (
i — 2m
for 1=1,...,
for /=2m-l
2m + i|| for / = 2m,
and
2m + l
Z ^i
i =
2m
= 1 (/=1,...
it follows that {/,} is a basis of type P of [Л„]. Now let {Ф„}, {•?„} с [/„]*
= [Л„]* be the a.s.c.f. to {/„} and {hn} respectively. Then {Ф„} is a basic
sequence of type P* and by the proof of Ch. I, §4, proposition 4.4, we
have [Ф„] = [•?„]¦ However, since the canonical mapping ф:1}([0,1~\)
->[Л„]* is an isomorphism (by Ch. I, §12, theorem 12.2e)), we have
[fn] = [4»(z<1»)] = 4)(L1([O,l]))^L1([O,l]), whence [Ф„] s ^([0,1]), and
consequently Ll([Q,1]) has a basis of type P*, whence also a basis of
type P, which completes the proof.
Example 9.2 was observed in [237].
§ 10. Bases of type /+ were introduced in [237] and bases of type
al+ in [62], lemma 1 and problem 1, but the term "basis of type a/+"
was used only in [163]. Bases satisfying conditions 2°, 3° of theorem
10.2 were also considered later by D. P. Milman and V. D. Milman
[168] (who were apparently unaware of the paper [237]) and bases
satisfying condition 4° of theorem 10.1 were studied later by R. C. James
([115], p. 116, condition 35).
The equivalences ГоЗ°о50о6° of theorem 10.1 were given, for
real Banach spaces, in [237].
The cartesian product {*„} x {yn} or the tensor product {xt®yj}
of two bases {xn}, \yn} is of type /+ if and only if both \xn} and {yn} are;
see the corresponding results for bases of types P, P* in the Notes and
remarks to § 9.
The cone associated to a basis of an infinite dimensional Banach
space has been studied by H. H. Schaefer [220], R. E. Fullerton [63]
Notes and remarks
629
and others. Proposition 10.1a) was given in [163], proposition 1 and
the rest of proposition 10.1 is due, essentially, to R. E. Fullerton [63].
The equivalences Го5°«>4° of theorem 10.2 were proved in [163],
theorem 3 and proposition 3. The proof of the implication 2°=>3C was
communicated to us by C. Foias..
Corollary 10.1 was given in [163], theorem 4. Proposition 10.2 was
observed in [163], corollary 1. The example of a compact base of a
cone, which is not a hyperbase, given after lemma 10.1, was exhibited in
[163]. Proposition 10.3 was given in [163], proposition 4a), c), d). The
equivalence l°<s>6° of theorem 10.3 was proved in [163], proposition 4b).
Theorem 10.4 is partially contained in [62], corollary 1. The fact
that the sequence (9.16), where the е„ are defined by (9.15), is a basis of
type /+ of C([0,1]), was pointed out in [62], theorem 2.
Example 10.1 was observed, essentially, in [237].
§11. Besselian and Hilbertian bases have been studied first in Hilbert
spaces (see e.g. N. K. Bari [15] and the references mentioned therein). In
general Banach spaces they have been considered, independently, by
several authors (see e.g. [196], p. 23, definition 6 and Z. A. Canturija [35],
В. Е. Veic [260]).
In the particular case when ? is a Hilbert space, the equivalences a),
b) 1°<=>3°<=>6° and the implications a), b) 1°=>4° of theorem 11.1
were given by N. K. Bari [15]. As we already mentioned, the equivalences
a), b) 2°<=>3°<=>4° in general Banach spaces are consequences of the
equivalences b) 6°<=> Го2° of Ch. I, § 8, theorem 8.1 (therefore, see the
Notes and remarks to Ch. I, § 8; see also В. Е. Veic [260], for the im-
implications a), b) 1°=>3° and 1°=>6°). Corollary 11.1 has been proved
in [251], corollary 1.
Problem 11.1 was raised by A. Pelczynski ([193], problem 13); see
also В. Е. Veic [260].
Example 11.1 was given by S. Karlin ([131], theorem 1.2).
Theorem 11.2 is due, essentially, to В. Е. Veic [260]. Some of the
implications of theorem 11.3 have been given by N. K. Bari [15], I. C.
Gohberg and A. S. Markus [81] and В. Е. Veic [260]. In the case when
? is a Hilbert space, the equivalence I°o2° of theorem 11.4 was given
by I. C. Gohberg and A. S. Markus [81]; for arbitrary Banach spaces
theorem 11.4 was proved, with a different argument, by В. Е. Veic [260].
Problem 11.2 was raised in [196], problem 4. Recently, J. R. Rether-
ford has communicated to us in a letter that he was able to answer both
questions in the affirmative, by combining the method indicated in [196],
remark 3, a perturbation lemma and Ch. I, § 7, theorem 7.2.
Example 11.2 is due to K. I. Babenko [8] (see also B. R. Gelbaum
[70], M. §. Altman [2 a] and V. F. Gaposhkin [64]; the proof given here

630
II. Special Classes of Bases in Banach Spaces
is based on V. F. Gaposhkin [64] and N. K. Bari [16]. There also exist
bounded bases of L2([ — n, n~\) which are simultaneously non-Besselian
and non-Hilbertian, e.g. the sequence {xn}, where
= ——\t\acosnt,
and where 0<a<| (M. S. Altman [2a]).
If {en} is a symmetric basis, with symmetric norm v({^n(=l, of its
closed linear span [en] (in the sense of §22, definitions 22.1 and 22.2),
then Ф({а„}) =
5>
is a "symmetric gauge function" [223] and
conversely; the class of Besselian bases with respect to such a function
was studied by I. C. Gohberg and A. S. Markus [81]. In the case when F
is an arbitrary Banach space with a basis {en}, the (F, {en})-Besselian and
{F, {en})-Hilbertian bases were introduced by Z. A. Canturija [35]. For
the extension of theorem 11.1 to such bases, see the Notes and remarks
to Ch. I, §8, theorem 8.1 (see also В. Е. Veic [260] for the case when
F = l", Kp< oo, {е„} = the unit vector basis of F and Z. A. Canturija
[35]; for the general case, see [251], theorem 4). The extensions of the
results of § 11 to (F, {en})-Besselian bases, mentioned at the end of § 11,
were given in [251] (see also the references of [251], for some particular
cases). Theorem 11.5 can be found in the paper of V. D. Milman [167],
with the mention that it is due to V. I. Gurarii.
§ 12. The part of theorem 12.1 concerning shrinking bases, boundedly
complete bases, bases of types P, P*, l+ and the five complementary
classes of bases was given in [237]. The implication al+ => weakly closed
was observed in [62], lemma 1 b) and the implication (weakly closed)*
=>non-(swc0)* in [62], corollary 2. We raised the problem of the exist-
existence of a non-shrinking basis of type wc0 in a letter to A. Pelczynski.
The affirmative answer to this problem, given in example 12.2, is due to
A. Pelczynski and W. Szlenk [197]; however, one can give a direct
proof, simpler than that of [197], of the fact that {/„} is a non-shrinking
basis of W*, of type wc0. The space d of example 12.1 is a particular
case of a space introduced by G. G. Lorentz [153] (see also [39], [78]).
The proof of formula A2.7) is based, essentially, on the argument of [78].
The fact that {xn} is a non-shrinking basis of type wc0 of the space d
has been proved by J. R. Retherford [208]. Recently, W. Ruckle has
communicated to us the following more general example (unpublished):
Let {xn} be a symmetric basis of a Banach space E (see § 22, definition
22.1), which is not equivalent to the unit vector basis of c0 or I1 and
let {/„} <= E* be the a.s.c.f. to {xn}. Then both {xn} and {/„} are bases
Notes and remarks
631
of type wc0 of ? and [/„] respectively, but, whenever E is non-reflexive,
either {xn} or {/„} is non-shrinking (by §23, remark 23.1 and §6, corol-
corollary 6.1); this is a corrected version of [215 a], theorem 3.
Example 12.4 was given in [237].
Recently J. Lindenstrauss and J. R. Retherford have informed us that
problem 12.1 (concerning part b) of theorem 12.1) has been solved in the
negative by J. R. Holub [108]. The original construction of J. R. Holub
involved perturbed members of the Schauder basis of C([0,1]), but in
the referee's report to that paper a simpler construction has been given,
which is equivalent, essentially, to the following: Let {an} be any dense
sequence in [—1, 1] and let {en} be the unit vector basis of c0. Then,
by Ch. II, § 14, example 14.1 and Ch. I, § 4, proposition 4.2 (on cartesian
products of bases) and lemma 4.1, the sequence {yn} с Е defined by
2n- 1
У*п-Ъ=
,-1=1
2n
i= 1
is a basis of c0. Hence, by Ch. I, § 4, proposition 4.4 (on block perturba-
perturbations of bases), the sequence {xn} с с0 defined by
2n- 1
x = Y e ¦ x
i= 1
2n
2n-l 2n-l In
i= 1 1=1 1=1
2n
4n= Z e2i-
is a basis of c0. We claim that this is a weakly closed basis of type non-al + .
Indeed, \\xj\\ = 1 (_/= 1,2,...) and for the functionals f0, goe(co)* defined
by
we have max(\fo{xn)\,\go{xn)\) = l (и = 1,2,...), whence х„фУиво.л{0)
(n= 1,2,...) and thus {xn} is weakly closed. Furthermore, if
is an arbitrary continuous linear functional on c0 (hence ||/||= Z I
< oo), then i=1
2n-l 2n-l
iim. I /(*4n-2)l = ii
а„ Z »/2i-i+ Z ^2;
i = 1 i = 1
2n 2n
I z ( — 1 « i^t * ^ t

632
II. Special Classes of Bases in Banach Spaces
and since the sequence {an} is dense in [— 1,1], one of these two numbers
is =0 (because either
n = 0 or min
i = 1
/ °°
/ ?
i= 1
ОС
\ .-=
>2i
a
1 =
00
I
\
i
1).
Thus, {х„} is of type non-a/+ (by § 10, theorem 10.3), which completes
the proof.
Let us observe that the construction of this example is similar to
that of Ch. II, § 12, example 12.4, (of a bounded basis {zn} of c0 such
that there exists an /e(c0)* with /(zn)^l (и =1,2,...) but there exists
no/e(c0)* with |/(zn)| = l(«=l,2,...)).
The implications а) 2°оЗ°=>Г and b) 2°=>Г of theorem 12.2
were given in [237].
§ 13. Universal and complementably universal bases were introduced
by A. Pelczynski [195]; block-universal bases also occur in the same
paper [195], without bearing any denomination. The construction
culminating in theorem 13.1 as well as remark 13.2, corollary 13.1,
problem 13.1 (and the comments to it), theorem 13.2, corollary 13.2 and
remark 13.3 are due to A. Pelczynski [195]. Lemma 13.6 was observed
by A. Pelczynski ([190], proposition 1). The particular case of theorem
13.3 when ? = C([0,l]) and {en} = the Schauder basis of C([0,l]),
occurs in [195]; in the general case, theorem 13.3 was communicated to
us by A. Pelczynski.
Remark 13.5 and problem 13.2 were given in [250].
The equivalence Г<=>3° of theorem 13.4 was given, with a different
proof, by A. Pelczynski [195], theorem 2; actually, that proof also
contains, implicitly, the equivalence 2°o3°. The proof presented here
was communicated to us by A. Pelczynski.
Remark 13.6 and problem 13.3 were given in [250].
The remark concerning subsequences of the Schauder basis of C([0,1]),
made before corollary 13.3, is due to A. Pelczynski ([195], proposition 4).
Corollary 13.3, theorem 13.5 and problems 13.4, 13.5 were given by
A. Pelczynski [195]. Problem 13.6 is also due to A. Pelczynski (verbal
communication).
Part II. § 14. Some authors, e.g. S. Karlin [131], B. R. Gelbaum [69],
R. E. Fullerton [63] have used the term "absolute basis" instead of
unconditional basis and "non-absolute basis" instead of conditional
Notes and remarks
633
basis, but to-day the terms "unconditional basis" and "conditional
basis" are adopted by the majority of specialists in the theory of bases.
The fact that for l<p<co the Haar system is an unconditional
basis of Lp([0,1]) (theorem 14.1) was proved by J. Marcinkiewicz [155].
The crucial lemma in this proof, namely, lemma 14.3, is due to R. E.
A. C. Paley [187]; for another proof of this lemma see S. Yano [269].
The proof of lemma 14.3, given here, is essentially the one given by R.
E. A. C. Paley [187], with some simplifications. The proof of lemma 14.8
has been communicated to us by G. Albinus. It would be desirable to
have a simpler proof of theorem 14.1. In Vol. II, Ch. V we shall consider
the problem of unconditionality of the Haar basis in Orlicz spaces.
Examples 14.1 and 14.2 were given in [237]; the first example of a
conditional basis of c0 was produced by B. R. Gelbaum [69].
Lemmas 14.9 and 14.10 are due to W. Orlicz [186]. For the space I2,
proposition 14.1 has been obtained, independently, by B. R. Gelbaum
([69], p. 193, theorem 14) and N. K. Bari [15]; see also В. Е. Veic [260].
Example 14.3 was given by S. Karlin ([131], theorem 12).
Example 14.4 is due to K. I. Babenko [8] (see also B. R. Gelbaum
[70], M. S. Altman [2 a]). V. F. Gaposhkin has proved the following
more general result ([64], theorem 2 and p. 370): Let z(-) be a measurable
junction on [ — л, л] for which the function y(-) = z2(-) has the properties
a) y( — n)=y(n),
b) y(r,t)^cn(r,t) @^r< 1, — n^t^n), where y{Q=y(r,t) denotes
the harmonic function defined by
1
2^
y(s)
l-r2
l+r2-2rcos(s-0
ds
l, ( = re"),
a«J ?/(Q = ??('¦, 0 its conjugate function and where с is a constant such that
с > 0 for
pn
с >
tg*
for
/и f/w case the equivalence classes \xn} and {x'n} of the sequences
xn(t) = —— z(t) eint (n = 0, ± 1, + 2,...)
*„(•) and {х'„{-)} defined by
1
2я z(t)
(и = 0, ±1, ±2,...)

634
II. Special Classes of Bases in Banach Spaces
form bounded bases of the space L2([ —я, я]). // there exist constants
m>0 and M^m such that m<|z(?)|<M a.e. (on [ — n,n~\), then {xn}
is simultaneously Besselian and Hilbertian; if there exists m>0 such
that m^ \z{t)\ a.e., but there does not exist M such that \z(t)\^M a.e.,
then the basis {xn} is Besselian and non-Hilbertian; if there exists M such
that \z(t)\^M a.e., but there exists no m>0 such that m^\z(t)\ a.e.,
then the basis {xn} is Hilbertian and поп-Besselian; if there exists no
m>0 such that m^\z(t)\ a.e. and no M such that \z(t)\^M a.e., then the
basis {xn} is non-Besselian and non-Hilbertian.
Taking in this theorem the function z(;)=|?|a, where 0<a<| (in
[64] it is shown that this function satisfies the conditions of the theorem),
one obtains again the non-Besselian bounded Hilbertian basis {xn} of
L2{\_ — n, n~\) constructed by Babenko.
For conditional bases in L2([ — n, я]) see also H. Helson and G.
Szego [105].
J. Lindenstrauss and A. Pelczynski have called to our attention that
from a recent paper of С A. McCarthy and J. Schwartz [165] follows
the explicit construction of a conditional basis of I2 of the form
GO
xn= ? У;я)е,- (и=1,2,...), where {е„} is the unit vector basis of I2.
i= 1
Namely, from [165] one can obtain, for each n, a basis x{"\ ..., x[n) of the
и-dimensional Hilbert space I2 such that v)x(n)!»=1<M<oo (и =1,2,...)
but v^U.^j-^oo as«->oo, whence, by Ch. I, §7, corollary 7.3, the
sequence
{0,х(!2),0, ...}
vB)
', о,...},...
is a conditional basis of (l\ x l\ x ••• x I2 x ¦¦¦)l2. Hence, by the natural
canonical equivalence of (l\ x l\ x ••• x I2 x •••),2 with I2, we obtain a
conditional basis of I2 of the form xn =
,. Example 14.5 gives
i= 1
somewhat more, namely, conditional bases of I2 of the "triangular forms"
A4.69) and A4.70). These bases of/2 were obtained in [147 a].
Problem 14.1a) was raised by A. Pelczynski (verbal communication).
Recently J. Lindenstrauss has communicated to us that the space c0
has a monotone conditional basis, namely, the following: Let {?„} be a
sequence of numbers such that
0<г„<1
linun=l,
Notes and remarks
635
(e.g., one can take г„=1 , и = 1,2,...) and let
и+ 1
where {en} is the unit vector basis of c0. Then, obviously, xnec0
(и =1,2,...). Since
*„-'-*„+1 = У '-> 0,...} = tnen (n= 1,2,...),
n-l
we have [х„] = с0. Furthermore, since 0<г„<1 (и =1,2,...), for any
scalars аь...,а„ we have
i= 1
= тахМа1г1|, \oLltlt2+a2t2\,...,
n-l n-l
= ma\(\a1\t1,\a1t1+a2\t2,...,
V
n-l n-l
k=l j=k
n-2 n-2
*=i i=k
, max
ПИ< CO
'n-l,
at П o+a"
max П',
n-l n-l
П ?;+a»
whence
i= 1
L= 1
for all a,,...,i,+ 1eK and thus \xn} is
a monotone basis of c0. Finally, for every n>m we have
/n-l n-l
= max
(
k=l j =
n n
n n
= max A,AB + Bv. X По)^ Tj П'r
\ k—1 j=k / k=n—mj—k
Keeping m fixed and letting и-юо, the right hand side tends to
m+1 (since Нт(„=1) and hence sup
= 00, i.e. {xn} is of
type non-P. Consequently, by § 18, theorem 18.2, {х„} is a conditional
basis of c0, which completes the proof.

636
II. Special Classes of Bases in Banach Spaces
From the computation made in example 14.2 it follows that the
conditional basis {/*„} of I1 given in that example is monotone (for the
subsequence {/*„}? this was also observed in § 1, after proposition 1.4).
However, we don't know whether the space /p, where 1 <p < oo, has a
monotone conditional basis.
Recently J.R. Holub and J. R. Retherford have observed ([109],
example 1) that the space c0 has a conditional shrinking basis of type P,
namely, the following:
where {е„} is the unit vector basis of c0.
Indeed, if {hn}c(c0)* is the a.s.c.f. to {en} (i.e. the sequence of
coordinate functionals on c0), then for the sequence {/„} с eg defined by
и-1
f1=h1+h2, fn = hn + hn + 1 (и = 2,3,...)
we have fi{Xj) = 8ij (i,j= 1,2,...) and
П E
- У to)x- =
i=l
4-1)"
I Zte,
И L j=2 J i=n+l
for each x= {4}ec0, whence {х„} is a basis of c0. Furthermore,
j=2 У"'
whence /ii?[/,-]. Consequently, Alle[//-](n = 2,3,...), whence (co)* = [An]
с [/.] с (с0)* and thus {х„} is shrinking. Since by the above the series
? —: fj converges in (c0)*, but the series ? — fj
J=2- / 1 1 \ J
= У h:-\—hi+1 clearly does not converge in (c0)*, it follows that
j = 2 V — 1 j J
{/„} is a conditional basic sequence and hence, by §17, theorem 17.7,
{х„} is a conditional basis of c0. Finally, a simple computation shows
that
;= l
= 1 (и= 1,2,...) and thus {xn} is of type P, which comple-
completes the proof.
From the above it follows immediately, by duality, that {/„} is a
conditional boundedly complete basis of type P* of the space I1 [109].
Let us also mention the following special class of (conditional) bases,
introduced by V. G. Vinokurov [261]: A conditional basis {х„} of a
Banach space E is called a basis with a discontinuity of the uncondition-
Notes and remarks
637
ality, if {xn} is the union of two disjoint subsequences {xin} and {xln}
which are unconditional bases of [x;j and [x,J respectively. The dis-
discontinuity of the unconditionality of the basis \xn} is called simple, if
every permutation of at least one of the sequences {xin}, {x,n} is a part1
of a permutation {xa{n)} of the basis {х„}, with the property that {xa(n)}
is also a basis of E. For instance, the unit vector basis of the space J
(§4, example 4.1) is a basis with a discontinuity of the unconditionality.
The conditional basis of lp (\<p^2), mentioned after example 14.3,
is a basis with a simple discontinuity of the unconditionality. Such
bases were introduced for the study of the problem of existence of con-
conditional bases in Banach spaces with bases; since in the meantime this
problem has been solved (see §23, theorem 23.2 and §24, remark 24.3),
we have omitted them.
§ 15. Lemma 15.1 and corollary 15.1 have been given, with different
proofs, by W. Orlicz [184] and I. M. Gelfand [73].
In the particular case when ? = C([0,1]), theorem 15.1 was proved
by S. Karlin [131]; the fact that a separable Banach space containing
a subspace isomorphic to C([0,1]) has no unconditional basis, was
proved by С Bessaga and A. Pelczynski [23] (see also A. Pelczynski
[189]).
Theorem 15.2 is due to A. Pelczynski [191] and so are the lemmas
15.2, 15.5 and 15.6. Since the proof of lemma 15.5, given in [191], was
not accurate (it claimed that a certain set is closed, but this was not the
case), the proof given here is partially based on some arguments com-
communicated to us by M. I. Kadec and A. Pelczynski. For some results in
certain quotient spaces of L1-spaces similar to those of [191] and with
analogous proofs, see also В. М. Byckov and V. M. Grober [34].
Theorem 15.3 has been proved, independently, in [230], [231] and
in the paper of A. Pelczynski [190]. In Vol.11, Ch.V, we shall also
consider the problem of existence of unconditional bases in Orlicz spaces.
The notion of a Banach space having property (м) was introduced by
A. Pelczynski [189]. Theorem 15.4, proposition 15.2, lemma 15.7 and
corollary 15.4 were stated, without proof, by A. Pelczynski [189] (see
also the paper of С Bessaga and A. Pelczynski [23], which contains a
proof of the fact that a separable Banach space containing a subspace
Eo isomorphic to J has no unconditional basis); the proof of lemma
15.7 was communicated to us by A. Pelczynski. Proposition 15.1 on
selection of basic sequences is due to C. Bessaga and A. Pelczynski [22]
and so are corollary 15.2 b) and corollary 15.3 [23]. In the particular
1 A permutation {*t(ln)} of the elements of the subsequence {xin} is said to
be a part of a permutation {*„<„)} of the elements of the sequence {х„} , if
т(«„)=G(д(и=1,2,...).

638
II. Special Classes of Bases in Banach Spaces
case when ? is weakly complete, lemma 15.8 has been proved by
W. Orlicz ([184], theorem 2) and in the general case by С Bessaga and
A. Pelczynski [22].
The problem of the existence of a Banach space ?0 having no un-
unconditional basis, which is a subspace of a Banach space with an un-
unconditional basis, was raised by C. Bessaga and A. Pelczynski ([23],
problem 5.3), who conjectured that the answer is affirmative [23].
Theorem 15.5, which substantiates this conjecture and lemma 15.9,
were proved by J. Lindenstrauss [149]. Recently, J. R. Holub and
J. R. Retherford ([109], example 2) have proved that the monotone
basic sequence {zn} с /' = (c0)* defined by A5.57) is the sequence of
coefficient functionals for a suitable basis of c0, whence {zn} is a
H>*-Schauder basis of I1.
§16. The equivalence 1°<=>3° of lemma 16.1 is due to W. Orlicz
[186]. The equivalences 1°<=>2°<=>6° of this lemma have been proved by
T. H. Hildebrandt [106] and the implication 6°^3° also by С W.
McArthur [160]. The equivalences 1°<=>4°<=>5° are also well known.
There exist many other characterizations of unconditional convergence,
but we did not need them here.
The equivalence I°o3° of theorem 16.1 was given by M.M. Grinblium
([82], theorems B) and Б)). The implication 1°=>7° can be found in the
book of M. M. Day ([43], Ch. IV, §4, theorem 1). The implication
12°=>20° was proved by I. M. Gelfand [74]. The equivalences Го22°
o23°<=>24° and example 16.1 were given by В. Е. Veic [258] (see also
[260a]). The equivalence Г<=>25° and example 16.2 are due to J. R.
Retherford [202].
Definition 16.1 was given by В. Е. Veic [258]. Theorem 16.2, with
a different proof, is due, essentially, to В. Е. Veic [258]; the observation
that the necessity part of this theorem also follows from a extended
stability theorem mentioned at the end of § 11, was made in [251].
The equivalences 1°<=>4°о5° of theorem 16.3 have been given, in a
slightly different form, by R. E. Fullerton [63]. The equivalence \co2'
of this theorem is due to H. H. Schaefer ([220], propositions E.3), E.4);
see also [221], p. 251, exercise 8) and has been rediscovered later by
L. A. Gurevich [102], who also gave the equivalence 1°<=>3° and, essen-
essentially, examples 16.3 and 16.4. Propositions 16.1 and 16.2 were proved
by L. A. Gurevich [102]. Some other characterizations of unconditional
bases among ?-complete total biorthogonal systems by properties of
the associated cone and partial order have been given recently by
Ya. M. Ceitlin [35a] and Nguen Van Khue [177]. The problem of
characterization of general (i.e. not necessarily unconditional) bases by
cone and order properties was raised by R. E. Fullerton [63].
Notes and remarks
639
Theorem 16.4 was proved by A. Yudin [271]. Lemma 16.2 and the
proof of theorem 16.4, given here, are due to B. Sz.-Nagy [172]. Another
proof of this theorem has been given by M. G. Krein and M. A. Rutman
[144].
The implications aJ°=>l° and a) 3°=>1° of theorem 16.5 were proved
by M. I. Kadec and A. Pelczynski ([128], theorems 5 and 6); under the
more restrictive assumption that {х„} is a basis of E, with the a.s.c.f.
{/„}, theorem 16.5c) was observed by S.Yamazaki [268]. This latter
result, together with the results of Ch. 1, § 5, suggests naturally the
question, whether there exist bases {*„} for which M (?,(*„,/„)) is
distinct from bv and m. By Ch. I, §12, theorem 12.8, this problem is
equivalent to that of the existence of y-perfect BK-algebras containing
all unit vectors en and the identity e= {1,1,...}, distinct from bv and m.
An affirmative answer has been given by R. J. McGivney and W. Ruckle
([166], theorem 6.2 and example). Recently, L. Sternbach has commu-
communicated to us the following simple way of obtaining bases {xn} for which
M(E,(xn,fn)) is distinct from bv and m: Let \xn} be a conditional basis
of a Banach space E, such that there exists a complemented infinite
subsequence {xin} of {xn} (in the sence of Ch. II, § 13, definition 13.2);
such a basis is e.g. any cartesian product {xn} = \yn} x {zn} of two bases
{yn}, {zn} (see Ch. I, §4, definition 4.6), one of which is conditional. Then
M(E,(xn,fnj) (where {/„} is the a.s.c.f. to {xn}) is distinct from bv and m,
since it contains the sequence {yj}$bv defined by у7=1 for y = г\,/2,...
and y-=0 for j^il,i2,... and since {xn} is a conditional basis of E.
There remains open the problem of classifying bases {xn} in terms of
their multiplier algebras M(?, (*„,/„)).
Problem 16.1 was raised by B. S. Mityagin [169] (see also A. Pelczynski
[194], problem 1). M. I. Kadec and A. Pelczynski [128] expressed the
opinion that "it is very probable" that the answer to this problem is
affirmative. Indeed, recently problem 16.1b) has been solved in the
affirmative [41]; we shall give the proof of this result in Vol. II, Ch. III.
Lemma 16.1 is due to M.I. Kadec and A. Pelczynski [128] (see also
M. I. Kadec [120], [121], [122] and [123] and V. Klee [132]). The
sufficiency parts of theorem 16.6 a), b) were given by M. I. Kadec [124];
for this theorem, lemma 16.4 and proposition 16.3 see also M. I. Kadec
and A. Pelczynski [128] (for a version of lemma 16.4 in the particular
case when r(V)>0, see also A. Pelczynski [191a], p. 371, lemma).
The final part of theorem 16.7 has been proved, independently, by
S.Yamazaki [266] and by A. Pelczynski (verbal communication, 1964,
unpublished).
The equivalence l°<s>2° of theorem 16.8 was observed by В. Е. Veic
[258] (although he stated it in a slightly different form, this is what he
actually proved; see also B. E. Veic [260a]); the implication 1°=>2° was

640
II. Special Classes of Bases in Banach Spaces
observed also by M. I. Kadec and A. Pelczynski [127]. The implication
3°=>1° occurs, implicitly, in [196], proof of proposition 4(a).
Proposition 16.4 was observed by A. Pelczynski ([195], lemma 11).
Example 16.9 is due to B. R. Gelbaum and J. Gil de Lamadrid [72].
Various generalizations of unconditional bases and of spaces with
unconditional bases, as well as corresponding extensions of the char-
characterizations of unconditional bases and of the properties of spaces
having unconditional bases, will be given in Vol. II.
§ 17. The equivalences I°o8°ol0° of theorem 17.1 were given, in a
weaker form, by M. M. Grinblium [82] and the equivalences I°o5°o7°,
in a weaker form, by L. A. Gurevich [98] (namely, both in [82] and [98]
it is assumed that \xn} is a basis of E). The equivalence 1°<=>4° has been
given by R.C.James [113] and the equivalence Г<=>7° in the general
case by C. Bessaga and A. Pelczynski [22]. The equivalence 8°o9° is
the "unconditional analogue" of a result of S. Yamazaki [265] (see the
Notes and remarks to Ch. I, § 7, theorem 7.1); this equivalence was also
given by J. R. Holub and J. R. Retherford in [110].
Corollary 17.1 was observed in [237]. Corollary 17.2 is due, essen-
essentially, to С Bessaga and A. Pelczynski [22].
Example 17.1 was given by A. Pelczynski [195].
The sufficiency parts of corollary 17.3 a), b) are contained, implicitly,
in a paper of R. С James ([113], the proofs of lemmas 1 and 2). Corollary
17.4 is due to A. Pelczynski ([195], lemma 12).
The material of § 17 on Banach spaces of matrices and other Banach
spaces having no unconditional basis (definitions 17.2, 17.3, proposition
17.1, lemmas 17.1-17.5, theorems 17.2-17.6, corollary 17.5 and prob-
problem 17.1) are due to S. Kwapien and A. Pelczynski [145]. Proposition
17.2 was given by A. Pelczynski ([190], proposition 5).
The relations A7.76) and A7.77) of theorem 17.7 were given in [249].
Proposition 17.3 is due to M. I. Kadec and A. Pelczynski [127]. An
example that in this proposition the unconditionality of {*„} is essen-
essential, different from example 17.2, has been given by M. I. Kadec and
A. Pelczynski [127]. The first part of example 17.2 was observed in
[238]. Recently, J. R. Holub and J. R. Retherford ([110], theorem
C.4)) have proved that the property occurring in proposition 17.3
actually characterizes unconditional bases, namely, if this property
holds for every subsequence {хПк} of {*„}, then \xn} is an un-
unconditional basis.
Lemma 17.6 and theorem 17.8 are due to W. Ruckle [214].
§18. Theorem 18.1 has been proved by N. K. Bari [15] and I. M.
Gelfand [74]. Other proofs of this theorem have been given by M. I. Kadec
Notes and remarks
641
and A. Pelczynski ([127], p. 168, remark 3) and J. Lindenstrauss ([148],
p. 246, remark); the short proof presented here was given in [248].
The problem, whether in the spaces c0 and I1 there exist two non-
equivalent bounded unconditional bases, was raised by A. Pelczynski
([190], p. 224]; the same problem, for Banach spaces non-isomorphic
to I2 and having an unconditional basis, was raised in [196], p. 22,
problem 3. Lemmas 18.1 and 18.2 are due to A. Grothendieck ([90],
p. 59 —64) and so are, essentially, corollary 18.1 and lemma 18.3; we have
reproduced them from the paper of J. Lindenstrauss and A. Pelczynski
[150]. Theorem 18.2, which solves the above problem, was given by
J. Lindenstrauss and A. Pelczynski ([150], corollary 1 to theorem 6.1).
Proposition 18.1 is due to A. Pelczynski ([190], lemma 1) and pro-
proposition 18.2 to С Bessaga and A. Pelczynski ([22], theorem 2). Lemmas
18.4-18.8 and theorem 18.3 were proved by A. Pelczynski ([190],
lemma 2, propositions 3, 4 and 5 and theorem 7). Propositions 18.3,
18.4, lemmas 18.9, 18.10 and theorem 18.4 were given in the same paper
of A. Pelczynski ([190], lemma 4, proposition 8 and theorem 7); let us
mention that, independently, V. F. Gaposhkin [65] has proved that for
1<р<2,рф2, the Haar system {yn} has a permutation {ya(n)} which
is not equivalent to {yn}.
Theorem 18.5 was proved by A. Pelczynski ([195], theorem 1); the
proof given here (via proposition 18.5), communicated to us by
A. Pelczynski, is due to M. Zippin.
Theorem 18.6 and corollaries 18.2, 18.3 have been given by
A. Pelczynski ([195], theorem 3 and corollaries 3, 4). Problems 18.1
and 18.2 were raised by A. Pelczynski ([195], problems 4 and 5).
§19. NT-norms, NX-norms and NTK-norms were introduced in
[236], but this terminology has been used in [243]; here N stands for
the Romanian word for unconditional.
The necessity part of proposition 19.1b) is contained, implicitly, in
[236], proof of theorem 3. The other statements of theorem 19.1 were
proved, with a slightly different argument, by J. R. Retherford and
R. С James ([210]. theorem B.4) (b) and (d)); for bases satisfying A9.3)
they have used the term "strictly co-orthogonal bases").
Theorem 19.1 was given by J. R. Retherford and R. С James ([210],
remark B.2) (c)). Example 19.1 is a slightly changed version of an example
due to J. R. Retherford and R. С James ([210], example B.5)); see the
Notes and remarks to Ch. 1, § 19, example 19.2.
Theorem 19.2 was proved, partially, in [236], theorem 2, namely,
there it was proved that A9.7) is an NX-norm on E, equivalent to the
initial norm and the problem was raised, whether there exists on E an
equivalent NT-norm. This latter question has been answered in the
41 Singer, Bases in Banach Spaces I

642
II. Special Classes of Bases in Banach Spaces
affirmative by J. R. Retherford [206], who has used, instead of A9.7),
the norm defined by the right hand side of A9.8); however, as shown
by theorem 19.2, the norm A9.7) given in [236], theorem 2, also yields
an affirmative answer to the problem.
§20. Bases with property B0.1) have been considered by S. Karlin
[131] and M. M. Day [43]; the term "orthogonal basis" for such bases
was introduced in [236]. Strictly orthogonal bases were introduced in
[236]. In Banach sequence spaces which are "symmetric" in the sense
of A. Sobczyk [252], the unit vectors form a hyperorthogonal basic
sequence.
A part of theorem 20.2 was given, without proof, in [241], theorem
2.13. A slightly weaker version of theorem 20.3 was given in [236],
theorems 1 and 3; let us mention that M. M. Day [43] has observed
that in every Banach space with an unconditional basis one can intro-
introduce an equivalent norm in which the basis {xn} is orthogonal.
Propositions 20.2-20.4 are due to W. J. Davis [37].
§21. Bases with the property described in definition 21.1 were con-
considered in [238], condition {SB2), but the term "subsymmetric basis"
was not used there (see the Notes and remarks to § 22).
The sets Mf (formula B1.1)) were introduced by M. I. Kadec [118].
Lemma 21.1, propositions 21.1, 21.2 and remark 21.1 are due to M. I.
Kadec and A. Pelczynski ([127], theorems 1, 2 and lemma 1). Theorem
21.1 was proved by M. I. Kadec and A. Pelczynski ([127], corollary 8).
The remark that the condition of unconditionality of \х„} in defini-
definition 21.1 is essential, was made in [238], remark 1.
Proposition 21.4 was proved in [238], lemma 1. Theorem 21.2 was
given, essentially, in [238], lemma 2 and corollary 1 (equivalences
2M6)G))
Proposition 21.5 and problem 21.1, with "symmetric basis" instead
of "subsymmetric basis", are due to I. Edelstein (verbal communication).
§22. Symmetric bases in Banach spaces were introduced in [232],
[233]. Independently, M. I. Kadec and A. Pelczynski [127] have con-
considered bases satisfying condition 12° of theorem 22.1 and called them
"permutatively homogeneous bases". In Banach sequence spaces which
are both "symmetric" and "permutable" in the sense of A. Sobczyk
[252], the unit vectors form a symmetric basic sequence, with symmetric
constant 1.
The implications 1°=>5° and 1°=>12° of theorem 22.1 were proved
in [233], theorem 1 and p. 161, remark 2°. The equivalences 1°<=>6°
<=>7°<=>12° of theorem 22.1 was proved in [238], theorem (equivalence
Notes and remarks
643
) and corollary 1 (equivalences (SB1)<=>(SB4MSB5)). The
equivalences I°o9°ol0° are due, essentially, to D. J. H. Garling [68].
Recently W. Ruckle has observed ([215 a], theorem 1) that if S is
the sequence space associated to a symmetric basis of a locally convex
F-space, then either S = 5 (the space of all sequences of scalars) or
P^S^Cq. However, for Banach spaces this merely says that every
symmetric basis is bounded (by Ch. I, §3, lemma 3.1), a fact which was
observed in the proof of theorem 22.1 (implication 12°=>Г). For similar
remarks on the sequence space associated to a symmetric basis see also
D. J. H. Garling [68].
Proposition 22.1 b) was given in [233], theorem 1. The "subsymmetric
analogue" of proposition 22.1a) appears in [196], p. 22, formula C7).
Definition 22.2, with a slightly different notion of symmetric constant,
was given in [196], p. 23 and p. 6, definition 1.
Proposition 22.2 was proved in [238], theorem (implication (SBJ
=>(SB2)); independently, M.I. Kadec and A. Pelczynski [127] proved
the equivalent result that every "permutatively homogeneous basis"
(see above) is subsymmetric. In [238] it was claimed that the converse
implication also holds, i.e., every subsymmetric basis is symmetric.
This claim has been disproved by D. J. H. Garling [68], who has given
as a counterexample the closed linear span of the unit vectors in the
Banach space G of example 22.1. The proof of the fact that this subspace
of G actually coincides with G, has been communicated to us by
V. I. Gurarii; later, a similar proof was given to us, independently, by
K. I. Oskolkov, who also observed that the same fact remains valid if
the sequence in B2.36) is replaced by any sequence a,^0 with
lA
CO
Y, a,-=oo satisfying the following additional condition: there exists a
p < 1 such that for every positive integer n one can find an index i0 = io{n)
with the property that а(A — p) — ai + n^0 (i^i0).
As we already mentioned (in the Notes and remarks to §21), prob-
problem 22.1 is due to I. Edelstein.
Problem 22.2 was raised in [250].
In the particular case when the sequence {An} consists of one single
set A and the space E is of dimension 2", proposition 22.3 is contained,
implicitly, in [196], proof of lemma 1. In the general case, proposition
22.3, definition 22.3 and proposition 22.4 were given by J. Lindenstrauss
and M. Zippin ([151], lemmas 4, 5 and 6).
Lemma 22.1 and theorem 22.2 were proved by W. Ruckle [214].
The proof of the implication 12°=* Г of theorem 22.1, given after theo-
theorem 22.2, is due, essentially, to W. Ruckle [214], [213].

644
II. Special Classes of Bases in Banach Spaces
Definition 22.5 of Haar system and Rademacher system with respect
to a symmetric basis of a finite dimensional symmetric space, proposi-
propositions 22.6, 22.7 and theorem 22.3 (which may be considered as an
"abstract analogue" of the classical Khinchin inequality) were given in
[196], proposition 1, corollary, lemma 1 and proposition 2.
§23. Proposition 23.1a), c), d) was proved in [196], proposition 4
and proposition 23.1b) in [250]. Theorem 23.1 was established in [196],
theorem. Proposition 23.2 and theorem 23.2 were given in [196], propo-
proposition 3 and theorem. Finally, theorem 23.3 was proved in [196], re-
remark 1.
§24. Perfectly homogeneous bases were introduced by C. Bessaga
and A. Pelczynski ([122], definition 2).
Propositions 24.1, 24.2 and theorem 24.1 are due to M. Zippin
[272] (see also H. F. Bohnenblust [28a]); for some previous results
on perfectly homogeneous bases, which are covered now by theorem
24.1, see [238], corollary 2 and remark 2 and W.J.Davis and D. W.
Dean [39].
The proof of §23, theorem 23.1, presented in remark 24.1, was given
in [250].
Proposition 24.2, theorems 24.2, 24.3 and remarks 24.2, 24.3 are due
to J. Lindenstrauss and M. Zippin ([151], lemma 2, theorem 1 and p. 124,
note A)).
§25. Absolutely convergent bases and uniform bases were intro-
introduced by S. Karlin [131].
Theorems 25.1 and 25.2 were proved by S. Karlin [131].
Finally, we mention a generalization of uniform bases, due to
С W. McArthur and J. R. Retherford [161]. Let ? be a Banach space
which is also a topological linear space for a topology T, where T is not
necessarily the norm-topology. A T-Schauder basis {xn} of ?, with the
n
a.s.c.f. {/„}, is called [161] T-uniform if x = lim У/Дх)х; for the
topology T, uniformly in the unit cell SE = {xeE\ |jx||^l}, i.e., if for
every T-neighbourhood К of 0 in ? there exists a positive integer
N = N(V) such that
n
x-Y.Mx)x,eV (xeSE,n>N).
i= 1
Theorem 25.2 says that an infinite dimensional Banach space ad-
admits no norm-uniform norm-Schauder basis. In contrast to this, C. W.
McArthur and J. R. Retherford have observed ([161], theorem 2) that
Notes and remarks
645
every w*-Schauder basis {/„} of a conjugate Banach space E* is w*-uni-
form. Indeed, by Ch. I, § 14, theorem 14.1, {/„} is the a.s.c.f. to a basis
\xn) of ?, whence, given a ^-neighbourhood V= Vyi^^ym.B{0) of 0 in
?*, there exists a positive integer N = N(V) such that
<8
and hence for any feSE*,n>N and j=l,...,m we have
/( I
i = n+ 1
<?,
which completes the proof. From this result it follows that every norm-
basis {х„} of E is <t(?,[/n~])-uniform, where {/„}<=?* is the a.s.c.f. to
{х„} (clearly, {xn} is also a <r(?, [/„])-Schauder basis); indeed, {ф{х„)}
is a H>*-Schauder basis of [/„]*, where ф is the canonical mapping of ?
into [/„]* and ф is an isomorphism (by Ch. I, § 12, theorem 12.2e)).
On the other hand, С W. McArthur and J. R. Retherford proved
([161], theorem 3) that a w-Schauder basis \х„} of a Banach space E is
w-uniform if and only if {xn} is shrinking. Indeed, by Ch. I, § 13, theorem
13.1, \xn} is a norm-basis of ?. Let feE* and e>0 be arbitrary and
consider the w-neighbourhood V= Vf.,@) of 0 in ?. Then, if {х„}
is w-uniform, there exists a positive integer N — N(V) such that
<8{xeSE,n>N), whence
/-I/W,-
<?
{n>N) and thus [/„] = ?*¦ Conversely, if {xn} is shrinking, then, by
the above remark, {xn} is a a(E,E*) = a(E, [/„])-uniform basis of ?,
which completes the proof.

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234. — Weak* bases in conjugate Banach spaces. Studia Math. 21, 75—81 A961).
235. — On Banach spaces reflexive with respect to a linear subspace of their
conjugate space, II. Math. Annalen 145, 64—76 A962).
236. — Ona theorem of I. M. Gelfand. UspehiMatem. Nauk 17,1 A03), 169—176
A962) [Russian].
237. — Basic sequences and reflexivity of Banach spaces. Studia Math. 21,
351—369A962).
238. — Some characterizations of symmetric bases in Banach spaces. Bull. Acad.
Polon. Sci. 10, 185—192 A962).
239. — On Banach spaces reflexive with respect to a linear subspace of their
conjugate space, III. Rev. math, pures et appl. 8, 139—150 A963).
240. — Weak* bases in conjugate Banach spaces, II. Rev. math, pures et appl. 8,
575—584 A963).
241. — Bases in Banach spaces, I. Studii si cercet. mat. 14, 539—585 A963)
[Romanian].
242. — Bases in Banach spaces, II. Studii si cercet. mat. 15, 157—208 A964)
[Romanian].
243. — Bases in Banach spaces, III. Studii si cercet. mat. 15, 675—725 A964)
[Romanian].
244. — Bases and quasi-reflexivity of Banach spaces. Math Annalen 153,199—209
A964).
245. — On the basis problem in topological linear spaces. Rev. math, pures et
appl. 10, 453—457 A965).
246. — Best approximation in normed linear spaces by elements of linear sub-
spaces. Edit. Acad. R. S. Romania, Bucuresti 1967 [Romanian]. English
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247. — On metric projections onto linear subspaces of normed linear spaces.
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248. — On a theorem of N. K. Bari and I. M. Gelfand. Arch. Math. 19, 508—510
A968).
249. — On the constants of basic sequences in Banach spaces. Studia Math. 31,
125—134A968).
250. — Some remarks and problems on bases in Banach spaces. Proc. Conf. on
"Abstract Spaces and Approximation" held in Oberwolfach, July 1968,
pp. 130—139. Stuttgart: Birkhauser Verlag 1969.
251. — Some remarks on domination of sequences. Math. Annalenl84, 113—132
A970).
252. Sobczyk, A.: Projections in Minkowski and Banach spaces. Duke Math.
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253. Solomyak, M. Z.: On the orthogonal basis in Banach space. Vestnik Lenin-
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254. Szlenk, W.: The non-existence of a separable reflexive Banach space universal
for all separable reflexive Banach spaces. Studia Math. 30, 53—61 A968).
255. Taylor, A. E.: A geometric theorem and its applications to biorthogonal
systems. Bull. Amer. Math. Soc. 53, 614—616 A947).
256. Timan, A. F.: Theory of approximation of functions of a real variable.
Moscow 1960 [Russian]. English translation, New York 1963.
257. Titchmarsh, E. С: The theory of functions. Oxford: Clarendon Press 1932.
258. Veic, В. Е.: On some characteristic properties of unconditional bases.
Doklady Akad. Nauk SSSR 155, 509—512 A964) [Russian].
259. — On some stability properties of bases. Doklady Akad. Nauk SSSR 158,
13—16A964) [Russian].
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stability. Izv. Vyss. Uc. Zaved. Matematika 2 D5), 7—23 A965) [Russian].
260a. — On some characteristic properties of unconditional bases and stability
theorems. Izv. Vyss. Ucebn. Zaved. Matematika 4 D7), 24—36 A965)
[Russian].
261. Vinokurov, V. G.: On biorthogonal systems passing through given subspaces.
Doklady Akad. Nauk SSSR 85, 685—687 A952) [Russian].
262. Wilansky, A.: The basis in Banach space. Duke Math. J. 18, 785—791 A951).
263. — Functional analysis. New York-Toronto-London: Blaisdell Publ. Co. 1964.
264. Williams, J. P.: A "metric" characterization of reflexivity. Proc. Amer. Math.
Soc. 18, 163—165A967).
265. Yamazaki, S.: On bases in Banach spaces. Sci. Papers Coll. Gen. Ed. Univ.
Tokyo 10, 163—169A960).
266. — Normed ring and unconditional bases in Banach space. Sci. Papers Coll.
Gen. Ed. Univ. Tokyo 14, 1—10 A964).
267. — Normed rings and bases in Banach spaces. Sci. Papers Coll. Gen. Ed.
Univ. Tokyo 15, 1 — 13 A965).
268. — Remark to "Normed rings and bases in Banach spaces". Sci. Papers
Coll. Gen. Ed. Univ. Tokyo 16, 25—26 A966).
269. Yano, S.: On a lemma of Marcinkiewicz and its applications to Fourier
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270. Yosida, K.: Functional analysis. Berlin-Heidelberg-New York: Springer
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272. Zippin, M.: On perfectly homogeneous bases in Banach spaces. Israel J.
Math. 4, 265—272 A966).
273. — A remark on bases and reflexivity in Banach spaces. Israel J. Math. 6,
74—79 A968).
274. Zygmund, A.: Trigonometric series. Vol. I. Cambridge: Cambridge Univ.
Press 1959.
275. Ciesielski, Z.: A construction of basis in C*1^/2). Studia Math. 33, 243—247
A969).
276. Marti, J. T.: Introduction to the theory of bases. Berlin-Heidelberg-New York:
Springer 1969.
afj 522
al+ (type) 315
(al+)*(type) 315
aP (type) 308
aP* (type) 308
а Я 626
A 9
(A) 217
A1 18
Л2 39
Л3 128
Л4 128
/1<M) 147
Л<"> 547
Л,-,,.,,,, 569
Л„,„ 575
А + В 377
b 99
bv 40,42
bv0 627
В 374
В„ 374
Л 373, 626
3SS 582
.#„ 547
М* 626
со{х„} 316
ГОЛ 22
codim? G 94
с 5 627
58
58
58
59
С5 59
С2к 24
d 361
dn(p,q) 375
dn(a) 509
Д, 509
® 458
Notation Index
233
230
Еь
Е„
Е"
?'
Es
Еи
Ег
Е2
Е,
№.
№i
№i
№i
№i
60
*1
$2
$Ъ
ll/l
/It
№;
Fr
G
Я1
/
r
Im
42, 74,
387
220
220
" 22, 58
•) 2
386
550
36
36
х-хЕ„
, x-x?,
x E2 x ¦¦
x E2 x ¦ ¦
x E2 x ¦¦
x E2 x ¦ ¦
31
31
32
32
L 269
el 27
G) 63
233
583
17
438
374
374
3
11'
1
•пДР
¦h
¦h
•v
• x
',374
539
539
123
80
539
Е„х-
303
J 273
376
К 1
486

660
(K2) 486
Кегм 216
Ж[х , 320
¦*"и".л> 472
II 220
/+ (type) 315
(h)*(type) 315
lim ? x,. 458
v- 'Тб
L(?,?) 43
L(?,E) 54
m4 500
M, 241
Mx(s) 241
Ma 508
Mf 563
Mj 499
M2 500
M3 500
M4 500
M5 500
M6 500
M7 500
M8 500
М(Е,(х„,/„)) 40,43
М(х,(х„,/„)) 40
Ж 79,458
Лр 571
ordp 185
Й? 458
571
(Р) 435
И 58
Р (type) 308
Р 53
По 53
Р|п) 53
Р,„» 499
р<<0 499
Р„т(а) 508
Р* (type) 308
р(±»ч 233
(.0?) 147
C) 147
) ПО
r = r(F) 115
г*@ 396
R 374
Notation Index
Rk
Rn
Rx
Re
53
374
374
3
5 643
sd —
sn
sl/i
¦V, i ) 461
25'
, 461
' 461
swc0 (type) 292
-o)* {type) 292
629
sign a 117
sd=
sk
sB
S"
Sy
S"
Sp
tn(a)
trb
т„
(«)
um
"{¦4 )
17(и)
sii, i i 499
53 '
51
526
131
596
:) 224
522
509
522
509
233
442
507
588
547
239
[/(?)=[/„_!(?) 239
%a
*!
#2
*3
F'
V"
V'"
yW
F1
K(C,
wc0
(wco.
wk(t)
W
461
461
461
461
270
270
270
270
114
,,e) 224
(type) 292
)* (type) 292
398
365
Notation Index
661
25
13
328
3i
40
27
П5
82
42
42
63
у»,
504
41
}
Г(п) 238
Г(?) 63
^« Ю
дГ 407
4М 407
0(G1,G2) 223
e(G!,G2) 224
191
436
63
582
328
> 68
>^ 68
~ 69
к 69
S 69
~ 79
-< 110
= 112,539
s 387, 539
I 215
II 215
[]x 269
(((( 175
llxll. 574
9Г* 82
82
29
29
504
I 3
л :
to
Я
Щ
a
Jd
a(n
,/"
°B
Ф-
E^E**
:?->G
: RX-*R
458
>; + i)
. 499
i 499
) 53
! 53
51
115
?->F*
ф„(Е) 110
113
175
m 374
579
114
199
))))
@)
((((„
))))„
(O)u
—>
(M)
176
176
550
551
551
292
298
147
(M)-lim 147
(M)-
147

Author Index
Akutowicz, E. 201
Alaoglu, L. 625
Albinus, G. 633
Altman, M. §. 206, 629, 630, 633
Arsove, M. G. 202, 204, 206, 209, 210
Arzela, С 248, 376
Ascoli, G. 623
Auerbach, H. 623
Babenko, K. I. 205, 629, 633, 634
Baire, R. 33, 34, 438, 439
Banach, S. 2, 19, 20, 24, 29, 32, 116,
200, 201, 202, 203, 204, 207, 208,
209, 349, 469, 623
Bari, N. K. 208, 629, 630, 633, 640
Baric, L. W. 207, 209
Berkson, E. 208
Bessaga, С 202, 203, 204, 205, 206,
207, 209, 210, 623, 625, 637, 638,
640,641,644
Boas, R. P. 205
Bohnenblust, H. F. 208, 220, 623,644
Bourbaki, N. 200
Byckov, В. М. 637
Canturija, Z. A. 629, 630
Ceitlin, Ya. M. 638
Ciesielski, Z. 201
Darboux, F. 245
Davies, R. O. 212
Davis, P. 205
Davis, W. J. 206, 624, 642, 644
Day, M. M. 200, 209, 622, 623 624,
638, 642
Dean, D. W. 644
Dieudonne, J. 200
Dixmier, J. 118, 149
Duffin, R. J. 206
Dunford, N. 625
Dvoretzky, A. 303, 306
Eachus, J. J. 206
Edelstein, I. 642, 643
Edwards, R. E. 204, 209, 210
Ellis, H. W. 201
Ezrohi, I. A. 211
Fan, Ky 205
Fichtengolz, G. 245
Foguel, S. R. 202
Foias, С 629
Fubini, G. 426
Fullerton, R. E. 628, 629, 632, 638
Gaposhkin, V. F. 208, 624, 629, 630,
633, 641
Garling, D. J. H. 643
Gelbaum, B. R. 202, 209, 210, 211,
624, 629, 632, 633, 640
Gelfand, I. M. 208, 211, 637, 638,640
Gil de Lamadrid, J. 211,640
Goffman, С 210
Gohberg, I. C. 623, 629, 630
Gram, J. P. 212,428,431
Grinblium, M. M. 202, 203, 205, 207,
208, 623, 624, 638, 640
Grober, V. M. 637
Grothendieck, A. 151, 170, 171, 641
Guraril, V. I. 203, 622, 623, 630, 643
Gurevich, L. A. 204, 205, 208, 211,
624, 638, 640
Halperin, I. 201
Hardy, G. H. 414
Harsiladze, F. I. 194,195,212
Helly, E. 127, 309
Helson, H. 634
Hildebrandt, Т. Н. 638
Hilding, S. H. 206, 208
Holub, J. R. 204,625,626,627,631,
636, 638, 640
Ishii, J. 201
Istratescu, V.
212
Author Index
663
James, R. С 203,211,622,624,625, Orlicz, W. 201,203,207,318,633,
628, 640, 641 637, 638
Joiner, H. 627 Oskolkov, K. I. 643
Kaczmarz, S. 203
Kadec, M. I. 203, 208, 212, 623, 637,
639, 640, 642, 643
Kaplansky, I. 200
Karlin, S. 208, 209, 623, 624, 625,
629, 632, 633, 637, 642, 644
Klee, V. 639
Kothe, G. 200
Kottman, С 624
Kozlov, V. Ya. 207, 622
Krasnoselskii, M. A. 201,479,623
Krein, M. G. 206, 208, 623, 639
Kwapien, S. 640
Landau, E. 426
Lebesgue, H. 295, 301, 425, 440
Levin, A. Yu. 623
Lichnerowicz, A. 200
Lindenstrauss, J. 625, 631, 634, 638,
641,643,644
Littlewood, J. 414
Liusternik, L. A. 206, 208, 211
Lorentz, G. G. 630
Lovaglia, A. R. 625
Lozinski, S. M. 194, 195, 212
Luzin, N. 436
Marcinkiewicz, J. 633
Markus, A. S. 623, 629, 630
Markushevich, A. I. 57, 202, 203, 205
Mazur, S. 2,207,209,281,318,447
McArthur, С W. 206, 210, 638, 644,
645
McCarthy, С. А. 634
McGivney, R. J. 203, 209, 639
McWilliams, R. D. 209
Milman, D. P. 207, 210, 623, 628
Milman, V. D. 205, 208, 628, 630
Mityagin, B. S. 202, 639
Morse, A. P. 625
Miintz, С 50
Nagy, B. Sz. 206, 208, 639
Naimark, M. A. 201
Neubauer, G. 208
Newns, F. 202
Nguen, Van Khue 638
Nikolskil, S. M. 624
Nikolskil, V. N. 202, 203, 211, 212
Paley, R. E. A. C. 205, 208, 633
Pelczynski, A. 202, 204, 205, 206, 207
209, 210, 624, 625, 626, 629, 630,
632, 634, 637, 638, 639, 640, 641,
642, 643, 644
Petunin, Yu. I. 623
Pollard, H. 206, 208
Pontryagin, L. S. 623
Ptak, V. 626
Retherford, J. R. 206, 208, 209, 210,
211, 212, 624, 629, 630, 631, 636,
638,640,641,642,644,645
Riesz, F. 622
Riesz, M. 342,344,409,413
Rolle, M. 246
Ruckle, W. 203, 208, 209, 211, 625,
626, 627, 630, 639, 640, 643
Ruston, A. F. 623
Rutickii, Ya. B. 201
Rutman, M. A. 207, 210,
639
Schaefer, H. H. 628, 638
Schafke, F. W. 205
Schatten, R. 172
Schauder, J. 200, 201
Schmidt, E. 24,212,428,
431
Schonefeld, S. 201
Schwartz, J. 634
Semadeni, Z. 211
Shimogaki, T. 201
Sobczyk, A. 642
Sobolev, V. I. 211
Solomiak, M. Z. 203, 623
Steckin, S. B. 212
Steinhaus, H. 203
Sternbach, L. 639
Szego,G. 634
Szlenk, W. 625, 630
Taylor, A. E. 623
Timan, A. F. 623
Tychonov, A. 452
Ulam, S. 207

664
Author Index
Vaher, F. S. 201
Veic, В. Е. 206, 629, 630, 633, 638,
639
Vinokurov, V. G. 202, 636
Waterman, D. 210
Weierstrass, K. 24, 249,
345
Wiener, N. 205, 208
Wilansky, A. 208, 209, 624
Williams, J. P. 625
Yamazaki, S. 203, 639, 640
Yano, S. 633
Yudin, A. 639
Zippin, M. 203, 204, 641, 643, 644
Zygmund, A. 414
Subject Index
Antilinear 9
Approximation property 171
A. s. с f. = associated sequence of coeffi-
coefficient functionals 17,151
Automorphism 4
Averaging projection 589
Banach-Mazur theorem 2, 499
Banach-Steinhaus theorem 348
Base of a cone 320
Basic sequence 27
—, block 66
— of type/+ 369
—, unconditional 505
Basis 1, 144
—, absolute 625
—, absolutely convergent 621
—, (b)-Schauder 158
—, Besselian 337
—, block-universal 388
—, bounded 21
—, bounded weak ( = bw-) 145
—, bounded weak* ( = bw*-) 145
—, boundedly complete 284
—, bw-Schauder 153
—, bw*-Schauder 153
—, complementably universal 374
—, conditional 396
—, (e)-Schauder 158
—, (F, {en})-Besselian 354
—, (F,{en})-Hilbertian 354
—, Haar, ofLp([0,l]) 16
—, Hilbertian 338
—, hyperorthogonal 558
—, fc-boundedly complete 284
—, fe-shrinking 268
—, monotone 214
—, natural, of co,/p 11
—, non-absolute 632
—, non-monotone 213
—, normal 252
—, normalized 21
Basis of type al+ 315
Basis of type (a/+)* 315
— of type a P 308
— of type a P* 308
— of type а Я 626
— of type Я* 626
— of type /+ 315
— of type (l+)* 315
— of type P 308
— of type P* 308
— of types wc0 292
— of type (swc0)* 292
— of type s3S 629
— of type wc0 292
— of type (wc0)* 292
—, orthogonal 555
—, p-Besselian 354
—, p-Hilbertian 354
—, perfectly homogeneous 609
—, permutatively homogeneous 641
—, polynomial 184
—, positive 261
—, retro-basis 279
—, Schauder, ofC([0,l]) 13
—, Schauder 152
—, semi-shrinking 625
—, shrinking 268
—, strict polynomial 184
—, strictly co-orthogonal 641
—, strictly monotone 214
—, strictly hyperorthogonal 558
—, strictly orthogonal 555
—, subsymmetric 563
—, symmetric 574
—, T-uniform 644
—, unconditional 396
—, uniform 622
—, universal 373
—, unit vector, of c0, V 11
—, w-Schauder 153
—, w*-Schauder 153
—, w-uniform 645
—, w*-uniform 645

666
Subject Index
Basis, weak (w-) 145
—, weak* (w*-) 145
—, weakly closed 300
— (weakly closed)* 301
— with a discontinuity of the uncondi-
unconditionally 636
— with a simple discontinuity of the
unconditionality 636
— with respect toMc?* 147
— a(E, [/„])-uniform 645
— oo-Besselian 354
— oo-Hilbertian 354
Basis problem 2
Biorthogonal system 23
, ?-complete 24
, irregular 25
, regular 25
, total 472
Block basic sequence 66
Block perturbation 30
Block subspace 506
Bounded weak topology
( = bw-topology) 145
Bounded weak* topology
(= b w*-topology) 145
Canonical mapping of E into E** 113
of ? into V*, where V cz E*
114
Cartesian product of two bases 29
of an infinity of bases 82
of two equivalence classes 82
Characteristic (of a subspace of a con-
conjugate space) 115
Circled 22
Coefficient functionals 17, 151
, associated sequence of 17, 151
Compact, conditionally 297
—, countably 297
—, sequentially 297
Complemented subsequence 373
Complete (sequence) 24
, {an}-complete 78
of order p 78
Complex structure 4
Complexification 5
Condition (A) 217
Cone 320
— associated to a basis 320
— associated to a biorthogonal
system 472
—, generating 327
Cone, minihedral 473
—, normal 328
—, regular 479
—, solid 321
Conjugate function 344
Coordinate space 131
Disjoint support condition 264
Distance between two basic sequences
207
— between two classes of related bases
207
Domination 68
—, affine 79
—, permutative 79
—, strict 68
Dual (of a sequence space), oc-dual 526
—, ?-dual 135
—, y-dual 131
—, (j-dual 596
Dual norm 280
Duality properties 112
, strong 112
, weak 112,151
Eberlein-Smulian theorem 297
Element of best approximation 175
Endomorphism 54
Equivalent bases 68
, affinely 79
, permutatively 79
Equivalent sequences 68
, fully 69
, strictly 68
Equivalent spaces 2
Existence problem 213
, restricted 213
Extremal point 321
Extremal subset 321
Field of scalars 1
, extension of 5
, restriction of 2
Formal expansion 25
Fundamental parallelotope of Hilbert
475
Gram-Schmidt orthogonalization
procedure 24,212,428,431
Haar functions 13
Haar system with respect to a basis
598
Subject Index
667
Hahn-Banach theorem 55, 229, 272,
280, 293, 297, 310, 330, 440, 445, 448
Hyperbase of a cone 329
Hyperoctahedron 255
Hyperparallelepiped 255
Inclination 63
Index of a sequence 63
— of a space 63
—, unconditional, of a sequence 504
Infinite power (of an equivalence class)
82
Involution 9
Isomorphic 2
Isomorphically universal space 389
KB-lineal 559
Khinchin inequality 425, 644
KMR property 157,210
for bases 157
for Schauder bases 157
Krein-Milman-Rutman theorem 98
157, 212, 249
Length function 16
Levelling length function 16
Limit point 297
Linear manifold 329
Linearly independent 50
,c0- 356
,(F,{e.}y 356
, finitely 50
, I2- 347
, V- 356
, strongly 57
, unconditionally со- 470
, со- 50
Locally isomorphic 323
(M)-convergent 147
Matrix, orthogonal 402
—, symmetric 402
— which preserves bases 135
Matrix norm 507
Maximal theorem 414
Minimal (sequence) 50
Minkowski functional 22, 377
Minkowski-Weyl theorem 482
Monotonize 250
—, strictly 250
Multiplier 40
— of an element 40
— of a space 40
Multiplier algebra for a basis 142
«-independent (system) 221
n-subsystem 221
n-th main triangle projection 509
Near 84, 106
-,(F,{e.}> 355
—, KLr 106
—, N- 106
—, p- 355
— PH- 106
—, PW- 106
—, quadratically 346
—, strictly 106
—, strongly KL- 107
—, weakly 106
—, weakly (F,{en}y 355
—, weakly p- 355
—, weakly quadratically 346
—, weakly oo- 355
—, oo- 355
Norm 375
—, semi-monotone 328
Norm with respect to a basis, K-
175
— with respect to a basis, NK- 551
— with respect to a basis, NT- 550
— with respect to a basis, NTK- 551
— with respect to a basis, T- 175
— with respect to a basis, TK- 176
— with respect to a basis, weak K-
183
— with respect to a basis, weak N T-
= weak NK- = weak NTK- 554
— with respect to a basis, weak T-
182
— with respect to a basis, weak TK-
183
Norm of a sequence 63
, unconditional 504
, symmetric (of a basis) 582
Norm of a space with a basis, symmetric
(with respect to the basis) 582
Opening 224
Operator of finite rank 170
Order of a polynomial 185
Orthogonal (elements, subspaces) 215
—.strictly 215
Paley-Wiener theorem 84
Part of a permutation 637
Partial order relation induced by a
cone 328

668
Subject Index
Partial sum operators 25, 158
, associated sequence of 25,
158
Permutation 361
Pliicker-Grassman coordinates 225
Polynomial 175
Polynomial complement 175
Property (p) 435
Property (u) 442
Rademacher functions 396
Rademacher sytem with respect to a
basis 598
Ray 321
—, extremal 321
Real Banach space associated to a
complex Banach space 3
Real-linear 4
Related bases 207
Riesz convexity theorem 409
for vector-valued functions
413
Semi-monotone (norm) 328
Separable 1, 144
—, sequentially w*- 210
Sequence space 131
associated to a basis 131
, normal 527
, symmetric 596
, oc-perfect 526
, y-perfect 131
, (j-perfect 596
Sequentially w*-separable 210
Stability theorems 84
Stable property 84, 107
, KLr 107
, N- 107
,PH- 107
,PW- 107
, strictly 107
, strictly KLr 107
Stable property, strongly 107
, weakly 107
Stable sequence 108
,(F, {<>„})- 356
, p- 356
, quadratically 349
, weakly 108
, weakly p- 356
, weakly quadratically 349
, weakly oo- 356
, oo- 356
Stationary point 244
Strictly positive functional 329
Subspace 2
Support 264
Symmetric constant of a basis 582
Symmetric gauge function 630
Symmetric space 582
System, Lozinsky-Harsiladze 194
— sub-Г 193
—, sub-Л 193
— Г 194
— Л 194
Tensor product of two bases 173
Total, sequence of functionals 32
— subspace of E* 114
— system of linear manifolds in E"
223
(/-stable E-complete biorthogonal
system 470
Unconditionally Cauchy series,
weakly = a(E,E*)- 432
, a(E*,E)- 432
Unconditionally convergent series 458
Walsh functions 398
Weak basis theorem 209
{yj-neighbourhood 106
Л sequence 198
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