Kosaku Yosida
Functional Analysis
Sixth Edition
Springer-Verlag
Berlin Heidelberg New York 1980
KSsaku Yosida
1157-28 Kiijiwara, Kamakura, 247/Japan
AMS Subject Classification (1970): 46-XX
ISBN 3-540-10210-8 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-10210-8 Springer-Verlag New York Heidelberg Berlin
ISBN 3-540-08627-7 5. Auflage Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-08627-7 5th edition Springer-Verlag New York Heidelberg Berlin
Library of Congress Cataloging in Publication Data, Yosida, Kbsaku, 1909 -
Junctional analysis (Grundlehrcn der math e mat is ch en Wisscnschaften; 123).
Bibliography: p. Includes index 1. Functional analysis, 1. Title. 11. Scries.
UA32OJ6 1980. 515.7. 80-18567.
ISBN (j-387-10210-8 (U.S.)
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" ' by SpriHgei Verlay. Berlin Heidelberg 1965, 1971, 1974, 1978, 1980
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Preface to the First Edition
The present book is based on lectures given by the author at the
University of Tokyo during the past ten years. It is intended as a
textbook to be studied by students on their own or to be used in a course
on Functional Analysis, i.e., the general theory of linear operators in
function spaces together with salient features of its application to
diverse fields of modern and classical analysis.
Necessary prerequisites for the reading of this book are summarized,
with or without proof, in Chapter 0 under titles: Set Theory, Topo-
logical Spaces, Measure Spaces and Linear Spaces. Then, starting with
the chapter on Semi-norms, a general theory of Banach and Hilbert
spaces is presented in connection with the theory of generalized functions
of S. L. Sobolev and L. Schwartz. While the book is primarily addressed
to graduate students, it is hoped it might prove useful to research mathe-
maticians, both pure and applied. The reader may pass, e. g., from
Chapter IX (Analytical Theory of Semi-groups) directly to Chapter XIII
(Ergodic Theory and Diffusion Theory) and to Chapter XIV (Integration
of the Equation of Evolution). Such materials as "Weak Topologies
and Duality in Locally Convex Spaces” and "Nuclear Spaces” are
presented in the form of the appendices to Chapter V and Chapter X,
respectively. These might be skipped for the first reading by those who
are interested rather in the application of linear operators.
In the preparation of the present book, the author has received
valuable advice and criticism from many friends. Especially, Mrs.
K. Hille has kindly read through the manuscript as well as the galley
and page proofs. Without her painstaking help, this book could not
have been printed in the present style in the language which was
not spoken to the author in the cradle. The author owes very much
to his old friends, Professor E. Hille and Professor S. Kakutani of
Yale University and Professor R. S. Phillips of Stanford University for
the chance to stay in their universities in 1962, which enabled him to
polish the greater part of the manuscript of this book, availing himself
of their valuable advice. Professor S. Ito and Dr. H. Komatsu of the
University of Tokyo kindly assisted the author in reading various parts
VI
1 Tciace
ol the g.dky piool, iчи ।e< 11пр errors and improving the presentation.
In .ill i*l them, the aulhoj expresses his warmest gratitude.
I li.inlv. .nr .il .i* due io Professor F. K. Schmidt of Heidelberg Uni-
irr.ily .nid b* Piolessor T. Kato of the University of California at
Brikelry win* constantly encouraged the author to write up the present
book, Finally, the author wishes to express his appreciation to Springer-
Vrilag for their most efficient handling of the publication of this book.
Tokyo, September 1964
Kosaku Yosida
Preface to the Second Edition
In the preparation of this edition, the author is indebted to
Mr. Floret of Heidelberg who kindly did the task of enlarging the Index
to make the book more useful. The errors in the second printing are cor-
rected thanks to the remarks of many friends. In order to make the book
more up-to-date, Section 4 of Chapter XIV has been rewritten entirely
for this new edition.
Tokyo, September 1967
Kosaku Yosida
Preface to the Third Edition
A new Section (9. Abstract Potential Operators and Semi-groups)
pertaining to G. Hunt’s theory of potentials is inserted in Chapter XIII
of this edition. The errors in the second edition are corrected thanks to
kind remarks of many friends, especially of Mr. Klaus-Dieter Bier-
stedt.
Kyoto, April 1971
K6saku Yosida
Preface to the Fourth Edition
Two new Sections "6. Non-linear Evolution Equations 1 (The
Komura-Kato Approach)" and "7. Non-linear Evolution Equations 2
(The Approach Through The Crandall-Liggett Convergence Theorem)”
are added to the last Chapter XIV of this edition. The author is grateful
to Professor Y. Komura for his careful reading of the manuscript.
Tokyo, April 1974
KdsAKu Yosida
Preface to the Fifth Edition
Taking advantage of this opportunity, supplementary notes are added
at the end of this new edition and additional references to books have
been entered in the bibliography. The notes are divided into two cate-
gories. The first category comprises two topics: the one is concerned with
the time reversibility of Markov processes with invariant measures, and
the other is concerned with the uniqueness of the solution of time depen-
dent linear evolution equations. The second category comprises those
lists of recently published books dealing respectively with Sobolev Spaces,
Trace Operators or Generalized Boundary Values, Distributions and
Hyperfunctions, Contraction Operators in Hilbert Spaces, Choquet’s
Refinement of the Krein-Milman Theorem and Linear as well as Non-
linear Evolution Equations.
A number of minor errors and a serious one on page 459 in the fourth
edition have been corrected. The author wishes to thank many friends
who kindly brought these errors to his attention.
Kamakura, August 1977	Kosaku Yosida
Preface to the Sixth Edition
Two major changes are made to this edition. The first is the re-
writing of the Chapter VI,6 to give a simplified presentation of Miku-
sinski’s Operational Calculus in such a way that this presentation does
not appeal to Titchmarsh's theorem. The second is the rewriting of the
Lemma together with its Proof in the Chapter XII, 5 concerning the
Representation of Vector Lattices. This rewriting is motivated by a
letter of Professor E. Coimbra of Universidad Nova de Lisboa kindly
suggesting the author’s careless phrasing in the above Lemma of the
preceding edition.
A number of misprints in the fifth edition have been corrected thanks
to kind remarks of many friends.
Kamakura, June 1980	KOsaku Yosida
Contents
0. Preliminaries................................................1
1.	Set Theory .............................................. 1
2.	Topological Spaces....................................... 3
3.	Measure Spaces...........................................15
4.	Linear Spaces............................................20
I.	Semi-norms..................................................23
1.	Semi-norm^ and Locally Convex Linear Topological Spaces . 23
2.	Norms and Quasi-norms...................................30.
3.	Examples of Normed Linear Spaces.........................32
4.	Examples of Quasi-normed Linear Spaces...................38
5.	Pre-Hilbert Spaces.......................................39
6.	Continuity of Linear Operators...........................42
7.	Bounded Sets and Bornologic Spaces.......................44
8.	Generalized Functions and Generalized Derivatives .... 46
9.	В-spaces and F-spaces ...................................52
10.	The Completion ..........................................56
11.	Factor Spaces of a B-space ..............................59
12.	The Partition of Unity...............................  .	60
13.	Generalized Functions with Compact Support...............62
14.	The Direct Product of Generalized Functions..............65
II.	Applications of the Baire-Hausdorff Theorem.................68
1.	The Uniform Boundedness Theorem and the Resonance
Theorem ...................................................68
2.	The Vitali-Hahn-Saks Theorem.............................70
3.	The Termwise Differentiability of a Sequence of Generalized
Functions...................................................72
4.	The Principle of the Condensation of Singularities ....	72
5.	The Open Mapping Theorem................................ 75
6.	The Closed Graph Theorem.................................77
7.	An Application of the Closed Graph Theorem (Hormander’s
Theorem)..............................................    ,	80
III.	The Orthogonal Projection and F. Riesz’ Representation Theo-
rem ..............................................................81
t. Tlie Orthogonal Projection..............................81
2.	"Nearly Orthogonal" Elements............................84
Contents	IX
3.	The Ascoli-ArzeH Theorem...............................86
4,	The Orthogonal Base, Bessel’s Inequality and Parseval’s
Relation.......................................................86
5.	E. Schmidt’s Orthogonalization..............................88
6.	F. Riesz’ Representation Theorem............................90
7.	The Lax-Milgram Theorem.....................................92
8.	A Proof of the Lebesgue-Nikodym Theorem.....................93
9.	The Aronszajn-Bergman Reproducing Kernel....................95
10.	The Negative Norm of P. Lax.................................98
11.	Local Structures of Generalized Functions..................100
IV.	The Hahn-Banach Theorems.......................................102
1.	The Hahn-Banach Extension Theorem in Real Linear Spaces 102
2.	The Generalized Limit .....................................103
3.	Locally Convex, Complete Linear Topological Spaces . . . 104
4.	The Hahn-Banach Extension Theorem in Complex Linear
Spaces .......................................................105
5.	The Hahn-Banach Extension Theorem in Normed Linear
Spaces........................................................106
6.	The Existence of Non-trivial Continuous	Linear Functionals 107
7.	Topologies of Linear Maps . . . ...........................110
8,	The Embedding of X in its Bidual	Space	X".......112
9.	Examples of Dual Spaces....................................114
V.	Strong Convergence and Weak Convergence........................119
1.	The Weak Convergence and The Weak* Convergence . . . 120
2.	The Local Sequential Weak Compactness of Reflexive B-
spaces. The Uniform Convexity.................................126
3.	Dunford's Theorem and The Gelfand-Mazur	Theorem	.	.	.	128
4.	The Weak and Strong Measurability.	Pettis’	Theorem	.	.	.	130
5.	Bochner’s Integral.........................................132
Appendix to Chapter V. Weak Topologies and Duality in Locally
Convex Linear Topological Spaces...............................136
1.	Polar Sets.............................................136
2.	Barrel Spaces .............................................138
3.	Semi-reflexivity and Reflexivity.......................139
4.	The Eberlein-Shmulyan Theorem..........................141
VI.	Fourier Transform and Differential Equations...................145
1.	The Fourier Transform of Rapidly	Decreasing Functions . 146
2.	The Fourier Transform of Tempered	Distributions.149
3.	Convolutions...............................................156
4.	The Paley-Wiener Theorems. The One-sided Laplace Trans-
form ........................................................ 161
5.	Titchmarsh’s Theorem ......................................166
6.	Mikusiiiski’s Operational Calculus.........................169
7.	Sobolev’s Lemma............................................173
Contents
8.	Garding’s Inequality.....................................175
9.	Friedrichs’Theorem ......................................177
10.	The Malgrange-Ehrenpreis Theorem.........................182
11.	Differential Operators with Uniform Strength.............188
12.	The Hypoellipticity (Hormander’s Theorem)................189
VII.	Dual Operators ...............................................193
1.	Dual Operators...........................................193
2.	Adjoint Operators .......................................195
3.	Symmetric Operators and Self-adjoint Operators...........197
4.	Unitary Operators. The Cayley Transform..................202
5.	The Closed Range Theorem.................................205
VIII.	Resolvent and Spectrum.......................................209
1.	The Resolvent and Spectrum...............................209
2.	The Resolvent Equation and Spectral Radius...............211
3.	The Mean Ergodic Theorem.................................213
4.	Ergodic Theorems of the Hille Type Concerning Pseudo-
resolvents .................................................215
5.	The Mean Value of an Almost Periodic Function............218
6.	The Resolvent of a Dual	Operator.........................224
7.	Dunford's Integral.......................................225
8.	The Isolated Singularities of a Resolvent ...............228
IX.	Analytical Theory of Semi-groups.............................231
1.	The Semi-group of Class	(Co).............................232
2.	The Equi-continuous Semi-group of Class (Co) in Locally
Convex Spaces. Examples of Semi-groups......................234
3.	The Infinitesimal Generator of an Equi-continuous Semi-
group of Class (Co).........................................237
4.	The Resolvent of the Infinitesimal Generator A...........240
5.	Examples of Infinitesimal Generators.....................242
6.	The Exponential of a Continuous Linear Operator whose
Powers are Equi-continuous..................................244
7.	The Representation and the Characterization of Equi-con-
tinuous Semi-groups of Class (Co) in Terms of the Corre-
sponding Infinitesimal Generators...........................246
8.	Contraction Semi-groups and Dissipative Operators .... 250
9.	Equi-continuous Groups of Class (Co). Stone’s Theorem . . 251
10.	Holomorphic Semi-groups..................................254
11.	Fractional Powers of Closed Operators . .................259
12.	The Convergence of Semi-groups. The Trotter-Kato Theorem 269
13.	Dual Semi-groups. Phillips’	Theorem......................272
X.	Compact Operators...........................................274
1.	Compact Sets in £f-spaces................................274
2.	Compact Operators and Nuclear Operators..................277
Contents	XI
3.	The Rellich-Garding	Theorem...............................281
4.	Schauder’s Theorem	...................................282
5.	The Riesz-Schauder	Theory................................283
6.	Dirichlet’s Problem.......................................286
Appendix to Chapter X. The Nuclear Space of A. Grothendieck 289
XI,	Normed Rings and Spectral Representation......................294
1,	Maximal Ideals of a Normed Ring...........................295
2,	The Radical. The Semi-simplicity..........................298
3.	The Spectral Resolution of Bounded Normal Operators . . 302
4.	The Spectral Resolution of a Unitary Operator.............306
5.	The Resolution of the Identity............................309
6.	The Spectral Resolution of a Self-adjoint Operator .... 313
7.	Real Operators and Semi-bounded Operators. Friedrichs’
Theorem ...................................................316
8.	The Spectrum of a Self-adjoint Operator. Rayleigh's Prin-
ciple and the Krylov-Weinstein Theorem. The Multiplicity
of the Spectrum............................................319
9.	The General Expansion Theorem. A Condition for the
Absence of the Continuous Spectrum.........................323
10.	The Peter-Weyl-Neumann Theorem............................326
11.	Tannaka's Duality Theorem for Non-commutative Compact
Groups........................................................332
12.	Functions of a Self-adjoint Operator......................338
13.	Stone’s Theorem and Bochner’s Theorem.....................345
14.	A Canonical Form of a Self-adjoint Operator with Simple
Spectrum......................................................347
15.	The Defect Indices of a Symmetric Operator. The Generalized
Resolution of the	Identity...............................349
16.	The Group-ring L’ and Wiener’s Tauberian Theorem , . . 354
XII.	Other Representation	Theorems in	Linear Spaces................362
1.	Extremal Points. The Krein-Milman Theorem...............362
2.	Vector Lattices.........................................364
3.	В-lattices and F-lattices...............................369
4.	A Convergence Theorem of Banach.........................370
5.	The Representation of a Vector Lattice	as	Point	Functions	372
6.	The Representation of a Vector Lattice	as	Set Functions	.	375
XIII.	Ergodic Theory and	Diffusion	Theory.........................379
1.	The Markov Process with an Invariant Measure .............379
2.	An Individual Ergodic Theorem and Its Applications . . . 383
3.	The Ergodic Hypothesis and the H-theorem..................389
4.	The Ergodic Decomposition of a Markov Process with a
Locally Compact Phase Space................................393
5.	The Brownian Motion on a Homogeneous Riemannian Space 398
6.	The Generalized Laplacian of W. Feller ...................403
7.	An Extension of the Diffusion Operator....................408
8.	Markov Processes and Potentials.........................  410
9.	Abstract Potential Operators and Semi-groups..............411
XII
Contents
XIV.	The Integration of the Equation of Evolution................418
1	Integration of Diffusion Equations in L2(Rm}...........419
2.	Integration of Diffusion Equations in a Compact Rie-
mannian Space.............................................425
3.	Integration of Wave Equations in a Euclidean Space Rm 427
4.	Integration of Temporally Inhomogeneous Equations of
Evolution in a	B-space..................................430
5.	The Method of	Tanabe and	Soboi.evski..................438
6.	Non-linear Evolution Equations 1 (The Komura-Kato
Approach).................................................445
7.	Non-linear Evolution Equations 2 (The Approach through
the Crandall-Liggett Convergence Theorem).................454
Supplementary Notes...............................................466
Bibliography......................................................469
Index.............................................................487
Notation of Spaces................................................501
0. Preliminaries
It is the purpose of this chapter to explain certain notions and theo-
rems used throughout the present book. These are related to Set Theory,
Topological Spaces, Measure Spaces and Linear Spaces.
1. Set Theory
Sets, x G X means that x is a member or element of the set X; x £ X
means that x is not a member of the set X. We denote the set con-
sisting of all x possessing the property P by {%; P). Thus {у; у ~ is
the set {%} consisting of a single element x. The void set is the set with
no members, and will be denoted by 0. If every element of a set X is also
an element of a set Y, then X is said to be a subset of У and this fact
will be denoted by У £ У, or У □ X. If ЗЕ is a set whose elements are
sets X, then the set of all x such that x G X for some X G 3E is called the
union of sets X in 3E; this union will be denoted by U X. The inter-
section of the sets X in T is the set of all x which are elements of every
X G X; this intersection will be denoted by П X. Two sets are dis-
joint if their intersection is void. A family of sets is disjoint if every
pair of distinct sets in the family is disjoint. If a sequence
oo
of sets is a disjoint family, then the union U Xn may be written in
П-1
OO
the form of a sum £ Xn.
Mappings. The term mapping, function and transformation will be
used synonymously. The symbol /: X -> У will mean that f is a single-
valued function whose domain is X and whose range is contained in У;
for every x 6 X, the function f assigns a uniquely determined element
f (*) == у g У. For two mappings f : X -> У and g : У -> Z, we can
define their composite mapping gf: X -> Z by (g/) (%) = g(f(x)). The
symbol f (M) denotes the set {/ (%); x G M} and f (M) is called the image
of M under the mapping /. The symbol /-х (2V) denotes the set {%; f(x)£N}
and /—1(2V) is called the inverse image of N under the mapping f. It is
clear that
Уг = /(/-1 (VJ) for all Ух С /(X), and £ /^(/(X^) br all Xt С X.
1. Yosida, Functional Analysis
2
0. Preliminaries
If / : X -> Yt and for each у C / (A-) there is only one x £ X with / (%) = y,
then / is said to have an inverse (mapping) or to be one-to-one. The inverse
mapping then has the domain /(X) and range X; it is defined by the
equation x = /-1 (у) = /-1 ({y}).
The domain and the range of a mapping / will be denoted by D (/) and
R(f), respectively. Thus, if / has an inverse then
/-1 (/(*)) = x for D(J), and /(/“Чу)) = У f°r all У С B(/).
The function / is said to map X onto Y if /(X) = Y and into Y if /(X) С У.
The function / is said to be an extension of the function g and g a restriction
of / if D (/) contains D (g), and / (x) = g (x) for all x in D (g).
Zora’s Lemma
Definition. Let P be a set of elements a, b, .. . Suppose there is a
binary relation defined between certain pairs (a, b} of elements of P,
expressed by a -< b, with the properties:
a a,
if a -< b and b	a, then a = b,
if a b and b	c, then a -< c (transitivity).
Then P is said to be partially ordered (or semi-ordered} by the relation
Examples. If P is the set of all subsets of a given set X, then the set
inclusion relation (Л £ B) gives a partial ordering of P. The set of all
complex numbers z ~ x + iy, w = и + iv, . . . is partially ordered by
defining z w to mean x и and у < v.
Definition. Let P be a partially ordered set with elements a, b, . . .
If a c and b <( c, we call c an upper bound for a and b. If furthermore
c -< d whenever d is an upper bound for a and b, we call c the least upper
bound or the supremum of a and b, and write c = sup(«, b} or a V b.
This element of P is unique if it exists. In a similar way we define the
greatest lower bound or the infimum of a and b, and denote it by inf {a, b}
or а Л b. If a V Ъ and а Д b exist for every pair (a, b} in a partially
ordered set P, P is called a lattice.
Example. The totality of subsets M of a fixed set В is a lattice by
the partial ordering -< M2 defined by the set inclusion relation
M. Q M2.
Definition. A partially ordered set P is said to be linearly ordered (or
totally ordered} if for every pair (a, b) in P, either a b or b -<( a holds.
A subset of a partially ordered set is itself partially ordered by the rela-
tion which partially orders P; the subset might turn out to be linearly
ordered by this relation. If P is partially ordered and S is a subset of P,
ап m С P is called an upper bound of .S' if s m for every s g S. An
mt P is said to be maximal if 6 В and m p together imply m p.
2, Topological Spaces
3
Zorn’s Lemma. Let P be a non-empty partially ordered set with the
property that every linearly ordered subset of P has an upper bound
in P. Then P contains at least one maximal element.
It is known that Zorn’s lemma is equivalent to Zermelo’s axiom of
choice in set theory.
2. Topological Spaces
Open Sets and Closed Sets
Definition. A system т of subsets of a set X defines a topology in X
if r contains the void set, the set X itself, the union of every one of its
subsystems, and the intersection of every one of its finite subsystems.
The sets in т are called the open sets of the topological space (X, r) ; we
shall often omit r and refer to X as a topological space. Unless otherwise
stated, we shall assume that a topological space X satisfies Bausdorffs
axiom of separation:
For every pair (%Х1 %2) of distinct points xlr x2 of X, there exist disjoint
open sets Glf G2 such that aq E Cq, x2 E G2.
A neighbourhood of the point x of X is a set containing an open set which
contains x. A neighbourhood of the subset M of X is a set which is a
neighbourhood of every point of M. A point x of X is an accumulation
point or limit point of a subset M of X if every neighbourhood of x con-
tains at least one point m E M different from x.
Definition. Any subset M of a topological space X becomes a topolo-
gical space by calling “open” the subsets of M which are of the form
M Л G where G’s are open sets of X. The induced topology of M is called
the relative topology of M as a subset of the topological space X.
Definition. A set M of a topological space X is closed if it contains
all its accumulation points. It is easy to see that M is closed iff1 its
complement Mc ~ X — M is open. Here A — В denotes the totality of
points x E A not contained in B. If M G X, the intersection of all closed
subsets of X which contain M is called the closure of M and will be denoted
by Ma (the superscript “я” stands for the first letter of the German:
abgeschlossene Hiille).
Clearly Ma is closed and M Q Ma; it is easy to see that M = Ma iff
M is closed.
Metric Spaces
Definition. If А', У are sets, we denote by X x Y the set of all ordered
pairs {x, y) where x £ X and yE У; Xx У will be called the Cartesian
product of X and У. X is called a metric space if there is defined a func-
1 iff is the abbreviation for "if and only if”.
1*
4
0. Preliminaries
tion d with domain XxX and range in the real number field J?1 such
that
%2)	0 and d(xlr x2) = 0 iff хг = x2l
 d(xlf x2) = d(x2,x1),
,d(xlt %3) £ d(xlf x2) + d(#2, %3) (the triangle inequality).
d is called the metric or the distance function of X. With each point x0
in a metric space X and each positive number r, we associate the set
S(x0; r) = [x G A"; d(x, %0) < r) and call it the open sphere with centre %0
and radius r. Let us call “open” the set M of a metric space X iff, for
every point xQ G M, M contains a sphere with centre xQ. Then the totality
of such “open” sets satisfies the axiom of open sets in the definition of the
topological space.
Hence a metric space Xis a topological space. It is easy to see that a
point x0 of X is an accumulation point of M iff, to every e > 0, there exists
at least one point m =/= x0 of M such that d(m, %0) < e. The и-dimensional
Euclidean space Rn is a metric space by
/ »	\l/2
& — Vi)a . where x - (лр . . .,xn) and у = (yp . . yn).
v=l	/
Continuous Mappings
Definition. Let f: X —> У be a mapping defined on a topological
space X into a topological space Y. f is called continuous at a point xQ £ X
if to every neighbourhood U of f (%0) there corresponds a neighbourhood
V of xQ such that f{V} C U. The mapping / is said to be continuous if it is
continuous at every point of its domain D(f) = X.
Theorem. Let X, У be topological spaces and f a mapping defined
on X into У. Then f is continuous iff the inverse image under / of every
open set of У is an open set of X.
Proof. If / is continuous and U an open set of У, then V — ^(U)
is a neighbourhood of every point %0C X such that /(x0)G U, that is,
У is a neighbourhood of every point x0 of V. Thus V is an open set of X.
Let, conversely, for every open set U Э/(ж0) of У, the set V — f-^U)
be an open set of X. Then, by the definition, / is continuous at x0 E X.
Compactness
Definition. A system of sets Ga, a E A, is called a covering of the set
X if X is contained as a subset of the union иа€Л Ga- A subset M of a
topological space X is called compact if every system of open sets of X
which covers M contains a finite subsystem also covering M,
In view of the preceding theorem, a continuous image of a compact set
is also compact.
Proposition 1. Compact subsets of a topological space are necessarily
closed.
2. Topological Spaces
5
Proof. Let there be an accumulation point x0 of a compact set M of a
topological space X such that x0 £ M. By Hausdorff’s axiom of separa-
tion, there exist, for any point mfM, disjoint open sets GmXo and GXoWl
of X such that mE GmXo, л0{- GXoW. The system {GmXi>; M} surely
covers M. By the compactness of M, there exists a finite subsystem
{GmiiX<>; i = 1, 2, . .., и} which covers M. Then Q Gx^mi does not
1=1
intersect M. But, since x0 is an accumulation point of M, the open set
n G^>w< 3 must contain a point m C M distinct from x0. This is a
1=1
contradiction, and M must be closed.
Proposition 2. A closed subset Mr of a compact set M of a topological
space X is compact.
Proof. Let {Ga} be any system of open sets of X which covers Mv
Мг being closed, = X — M1 is an open set of X. Since С M,
the system of open sets {G«} plus Л/f covers M, and since M is compact, a
properly chosen finite subsystem {G^; i = 1, 2, . . ., n} plus Mf surely
covers M. Thus {Gei; i = 1, 2, . . ., я} covers Mv
Definition. A subset of a topological space is called relatively compact
if its closure is compact. A topological space is said to be locally compact if
each point of the space has a compact neighbourhood.
Theorem. Any locally compact space X can be embedded in another
compact space У, having just one more point than X, in such a way that
the relative topology of X as a subset of У is just the original topology
of X. This У is called a one point compactification of X.
Proof. Let у be any element distinct from the points of X. Let {U} be
the class of all open sets in X such that Uc = X — U is compact. We
remark that X itself E {G}. Let У be the set consisting of the points of X
and the point y. A set in У will be called open if either (i) it does not
contain у and is open as a subset of X, or (ii) it does contain у and its
intersection with X is a member of {[/}. It is easy to see that У thus
obtained is a topological space, and that the relative topology of X
coincides with its original topology.
Suppose {У}be a family of open sets which covers У. Then there must
be some member of {V} of the form UQ V {y}, where C'o- {f/}. By the
definition of {C7}> Uq is compact as a subset of X. It is covered by the
system of sets УЛХ with V E {У}. Thus some finite subsystem:
Lj A X, V2 А X, . . ., Vn Г\ X covers Uff. Consequently, Vv V2, . . ., Vn
and Uo 'A {y} cover У, proving that У is compact.
Tychonov’s Theorem
Definition. Corresponding to each a of an index set A, let there be given
a topological space Xa. The Cartesian product Ц Xa is, by defini-
A
6
0. Preliminaries
tion, the set of all functions / with domain A such that /('x)E Xa for
every xfA. We shall write / = fl f(a) and call /(a) the rx-th coordi-
n
nate of /. When A consists of integers (1, 2, . . ., n), Ц Xk is usually
fe = l
denoted by X X2 X • • -X Xn. We introduce a (weak) topology in the
product space TI Xa by calling “open” the sets of the form П Ga,
a£A	<x€A
where the open set Ga of Xa coincides with Xa for all but a finite set of oc.
Tychonov’s Theorem. The Cartesian product X = П Xa of a
system of compact topological spaces Хя is also compact.
Remark. As is well known, a closed bounded set on the real line Rl is
compact with respect to the topology defined by the distance d(x, y) =
x — у | (the Bolzano-Weierstrass theorem). By the way, a subset M
of a metric space is said to be bounded, if M is contained in some sphere
S(xQ, r) of the space. Tychonov’s theorem implies, in particular, that a
Parallelopiped:
— oo <	< х^ <	< oo (i = 1, 2, . . ., n)
of the n-dimensional Euclidean space Rn is compact. From this we see
that Rn is locally compact.
Proof of Tychonov’s Theorem. A system of sets has the finite inter-
section property if its every finite subsystem has a non-void intersection.
It is easy to see, by taking the complement of the open sets of a covering,
that a topological space X is compact iff, for every system {Мя; a E A}
of its closed subsets with finite intersection property, the intersection
П is non-void.
Let now a system {S} of subsets S of X = JJ Xa have the finite
intersection property. Let {TV} be a system of subsets N of X with the
following properties:
(i)	{S} is a subsystem of {TV},
(ii)	{TV} has the finite intersection property,
(iii)	{TV} is maximal in the sense that it is not a proper subsystem of
other systems having the finite intersection property and containing
{S} as its subsystem.
The existence of such a maximal system {.V} can be proved by Zorn’s
lemma or transfinite induction.
For any set N of {TV} we define the set Na = {/(oc); ft TV} С Хл.
We denote then by {Nx} the system {TVa; N E {TV}}- Like {TV}, {TVa}
enjoys the finite intersection property. Thus, by the compactness of Xa,
there exists at least one point p„ E X~ such that E П TV“. We have
to show that the point p = Ц ря belongs to the set П TVa.
2. Topological Spaces
7
But since />Ло belongs to the intersection n^7V“e, any open set Gx<i
of Xai which contains p„t intersects every 6 {AgJ. Therefore the open
set
= {x; x = П xa with y,o £ GJ
of X must intersect every N of {jV}. By the maximality condition (iii)
of {TV}, G[a"J must belong to {TV}. Thus the intersection of a finite number
of sets of the form G(ao) with a01 A must also belong to {TV} and so such a
set intersect every set N £ {TV}. Any open set of X containing p being
defined as a set containing such an intersection, we see that p = П pa
a£A
must belong to the intersection П Na.
N£{N}
Urysohn’s Theorem
Proposition. A compact space X is normal in the sense that, for any
disjoint closed sets Fx and F2 of X, there exist disjoint open sets Gx and
G2 such that Fx g Gv F2 g G2.
Proof. For any pair (x, y) of points such that x g F1, у £ F2, there
exist disjoint open sets G(x, y) and G(y, x) such that x£ G(x, y),
у £ G (у, x). F2 being compact as a closed subset of the compact space X,
we can, for fixed x, cover F2 by a finite number of open sets С(уг, x),
G(y2, x), . . G{y„(x}, x). Set
n(x)
Gx = U G(yy, x) and G(x) = П G(x, yy).
7=1	J=1
Then the disjoint open sets Gx and G(x) are such that F2 g Gx, xE
Fr being compact as a closed subset of the compact space X, we can cover
Fj by a finite number of open sets G (xx), G (x2), . . ., G (xft). Then
Gj = U G{xd and G2 = Г) Gx.
7=1	;=1	1
satisfy the condition of the proposition.
Corollary. A compact space X is regular in the sense that, for any
non-void open set G{ of X, there exists a non-void open set G'9 such that
(G'f g Gx.
Proof. Take = (G})c and F2 = {x} where x 6 G{. We can then take
for G'2 the open set G2 obtained in the preceding proposition.
Urysohn’s Theorem. Let A, В be disjoint closed sets in a normal space
X. Then there exists a real-valued continuous function /(/) on X such
that
0 <	 1 on X, and = 0 on A, f(t) - 1 on B.
Proof. We assign to each rational number r = k/2n (k = 0, 1, . . ., 2”),
an open set G (r) such that (i) A g G (0), В = G (l)c, and (ii) G (r)a g G (r')
whenever r < r’. The proof is obtained by induction with respect to n.
8
0. Preliminaries
For n — 0, there exist, by the normality of the space X, disjoint open sets
Go and G} with A g Go, В g Gr We have only to set Go ~ - G (0). Suppose
that G(r)*s have been constructed for r of the form A/2”-1 in such a
way that condition (ii) is satisfied. Next let k be an odd integer > 0.
Then, since (k— l)/2” and (k + 1)/2W are of the form й'/Й”-1 with
0 k’ 2”-1, we have G ((k - l)/2“)e g G((k + l)/2”). Hence, by the
normality of the space X, there exists an open set G which satisfies
G((k — l)/2n)“ g G, Ga Q G((k 4- l)/2”). If we set G(£/2”) = G, the induc-
tion is completed.
Define /(/) by
/(/) = 0 on G (0), and f(t) = sup r whenever t € G(0)c.
Then, by (i), f(t) = 0 on A and = 1 on B. We have to prove the
continuity of /. For any t0£ X and positive integer n, we take r with
Ж) < r < /Oct) + 2-w-1. Set G = G(r) A G(r — %~nyc (we set, for
convention, G(s) = 0 if s < 0 and G(s) = X if s > 1). The open set G
contains i0. For, /(£0) < r implies tgEG(r), and (r — 2-“-1) </(Zo)
implies t0 6 G(r — 2~”-1)c g G (r — 2~”)“c. Now t£G implies t£G(r)
and so /(£) r; similarly t£G implies t G G(r — 2~”)aC QG(r — 2-”)c so
that r—2“w Therefore we have proved that |/(0—/(A>) I = 1/2*
whenever t G G.
The Stone-Weierstrass Theorem
Weierstrass’ Polynomial Approximation Theorem. Let /(%) be a real-
valued (or complex-valued) continuous function on the closed interval
[0,1]. Then there exists a sequence of polynomials Pn (%) which converges,
as n->oo, to /(%) uniformly on [0, 1]. According to S. Bernstein, we
may take
fl
P.W = X „cp fip/n) ^(1 - %)”-/.	(1)
Proof. Differentiating (% 4-	nCp xp yn~p with respect to
£=o
n
x and multiplying by x, we obtain nx(x + y)”^1 =	/> nCp xp yn~p.
£=0
Similarly, by differentiating the first expression twice with respect to % and
multiplying by %2, we obtain n (« — 1) %2 (x y)w-2 = p {p — 1) nCp xp
Thus, if we set
=„C?	(2)
we have
rt	n
rp(x) I , 2? рГр (x) = nx, £ p(p — 1) rp(x) = n(n — 1) X2. (3)
p0	/i-0	/> 0
2. Topological Spaces
9
Hence
n	n	n	n
£ (p—nx)2 rp(x) = n2x2 J? rp(x) — 2nx r prp(x) + Ё p2rp(x)
p=0	p = 0	p=0
= n2x2 — 2nx  nx 4- [nx + n[n — 1) x2)
— nx(l — x).
(4)
We may assume that |/(x) | M < oo on [0, 1]. By the uniform
continuity of f(x), there exists, for any e > 0, a <5 > 0 such that
/(%)—| < e whenever |x — x' [<<5.	(5)
We have, by (3),
n
p=0
/(*) — £ f[ptIn) Tp(x)
P=Q
For the first term on the right, we have, by rp[x) 0 and (3),
n
£	8 Гр[х) = g .
£ = 0
For the second term on the right, we have, by (4) and |/(%) | M,
£.\<2M Z rp(x)<^-2 £ (p~nx)2rp[x)
— ip-nX\>Sn M ’ — n2S2 p=0^	’ M 1
2Mx(l~x) M
=-----v0 (as»^oo).
The Stone-Weierstrass Theorem. Let X be a compact space and С (X)
the totality of real-valued continuous functions on X. Let a subset В of
С (X) satisfy the three conditions: (i) if /, g С В, then the function pro-
duct f • g and linear combinations л/ + /?g, with real coefficients a,/?,
belong to В, (ii) the constant function 1 belongs to В, and (iii) the uniform
limit /oo of any sequence {/H} of functions £ В also belongs to B. Then
В = С [X) iff В separates the points of X, i.e. iff, for every pair (sx, s2) of
distinct points of X, there exists a function x in В which satisfies
x(si) %(s2).
Proof. The necessity is clear, since a compact space is normal and so,
by Urysohn’s theorem, there exists a real-valued continuous function x
such that x(sx) # x(s2).
To prove the sufficiency, we introduce the lattice notations:
(/ V g) ($) =max(/(s),g(s)), (/ Л g) (s) = min(/(s), g(s)), |/| (s) = |/(s)|.
By the preceding theorem, there is a sequence {Pw} of polynomials such that
< i/«
for — n < t n.
10
(). Preliminaries
Непсе ||/(л)|— /-*„(/(s)) < l/я if —n < /(s) * . n. This proves, by (iii),
that |/| С В if B, because any function /(s) В C G(A') is bounded on
the compact space X. Thus, by
/ v s = f--p + ILtM and /Ag=/ + ₽_lZ^f,
we see that В is closed under the lattice operations V and A .
Let h € С (X) and sp s2 6 X be arbitrarily given such that лх /- s2.
Then we can find an fS1S1£B with /SiSi(sx) = Л(зх) and/JiSi(s2) = h(s2).
To see this, let g 6 В be such that g (sx) g (s2), and take real numbers a
and /9 so that /SiSj = <%g + satisfies the conditions: /S1Sj(sx) = A(sx)
and /slS1 (s2) = h {s2).
Given e > 0 and a point t £ X. Then, for each s £ X, there is a neigh-
bourhood U (s) of s such that /s/(«) > h(u) — e whenever и £ U (s). Let
U ($х), U (s2), ,U (s„) cover the compact space X and define
= V V fSni-
Then fi^B and /{(w) > h(u) — e for all и £ X. We have, by = &(£),
Hence there is a neighbourhood V (/) of t such that /((w) <
h(u) -f- e whenever u £ V(t). Let V	V (t2), . . ., V (tk) cover the com-
pact space X, and define
Mtk-
Then fQ_B and f(u] > h(u}— e for all X, because /(Дм) > h{u)~ e
for и С X. Moreover, we have, for an arbitrary point и £ X, say и t V(Zj,
/(«) X /(ДМ) < А(и) 4- e.
Therefore we have proved that | / (u) — h (u) | < e on X.
We have incidentally proved the following two corollaries.
Corollary 1 (Kakutani-Krein). Let X be a compact space and С(X)
the totality of real-valued continuous functions on X. Let a subset В
of С (X) satisfy the conditions: (i) if /, g £ B, then / V g, / A g and the
linear combinations af + fig, with real coefficients oc, belong to Bt
(ii) the constant function 1 belongs to B, and (iii) the uniform limit
of any sequence {/„} of functions Q В also belongs to B. Then В = C(X)
iff В separates the points of X.
Corollary 2. Let X be a compact space and С (X) be the totality of
complex-valued continuous functions on X. Let a subset В of С (X)
satisfy the conditions: (i) if f, gC B, then the function product /  g and
tin1 linear combinations ocf + ^g, with complex coefficients a, /9, belong
to Bi, (ii) the constant function 1 belongs to B, and (iii) the uniform
limit of any sequence {/„} of functions С В also belongs to B. Then
В (' (X) iff В satisfies flic conditions: (iv) В separates points of X
and (v) if /(s)( B, then its complex conjugate function /{.sj also belongs
to B.
2, Topological Spaces
11
Weierstrass’ Trigonometric Approximation Theorem. Let X be the
circumference of the unit circle of R2. It is a compact space by the usual
topology, and a complex-valued continuous function on X is represented
by a continuous function /(x), —oo <' x < oo, of period 2 л. If we take,
in the above Corollary 2, for В the set of all functions representable by
linear combinations, with complex coefficients of the trigonometric
functions
einx (W = 0,±l,±2,...)
and by those functions obtainable as the uniform limit of such linear
combinations, we obtain Weierstrass’ trigonometric approximation theo-
rem-. Any complex-valued continuous function /(%) with period 2 л can
be approximated uniformly by a sequence of trigonometric polynomials
of the form cn einx.
Completeness
A sequence {x„} of elements in a metric space X converges to a limit
point x £ X iff lim d[xn, x) = 0. By the triangle inequality d(xn, xm)
x) + d(x, xm), we see that a convergent sequence in X satisfies
Cauchy’s convergence condition
lim d(xn, xm) = 0.	(1)
>oo
Definition. Any sequence in a metric space X satisfying the above
condition (1) is called a Cauchy sequence. A metric space X is said to be
complete if every Cauchy sequence in it converges to a limit point 6 X.
It is easy to see, by the triangle inequality, that the limit point of
{%„}, if it exists, is uniquely determined.
Definition. A subset M of a topological space X is said to be non-
dense in X if the closure Ma does not contain a non-void open set of X.
M is called dense in X if Ma = АГ. M is said to be of the first category if M
is expressible as the union of a countable number of sets each of which is
non-dense in X; otherwise M is said to be of the second category.
Baire’s Category Argument
The Baire-Hausdorff Theorem. A non-void complete metric space is of
the second category.
Proof. Let {Af„} be a sequence of closed sets whose union is a complete
metric space X. Assuming that no Mn contains a non-void open set, we
shall derive a contradiction. Thus My is open and = X, hence My
contains a closed sphere Sy = {x; d (хг, x) < whose centre xx may be
taken arbitrarily near to any point of X. We may assume that 0 < гг<. 1/2.
By the same argument, the open set contains a closed sphere
0. Preliminaries
12
S2 = {x;d{x2,x)^ r2} contained in 5\ and such that 0 < r2 < 1/22.
By repeating the same argument, we obtain a sequence {S„} of closed
spheres S„ = {x; d(xn, x) A rn} with the properties:
0 <	< 1/2”, S)l+1CS„, SwAM„ = 0 (n = l,2, ...).
The sequence {x„} of the centres forms a Cauchy sequence, since, for any
n < m, xm£Sn so that d(xnt xm) A rM < 1/2”. Let X be the limit
point of this sequence {xn}. The completeness of X guarantees the exist-
ence of such a limit point x^. By d(xnt A d(x„, xm) 4- d(xm, xK} A
r„ T d{xmt Xqo) -> rn (as m —> oo), we see that £ Sn for every n: Hence
OO
Xqq is in none of the sets Mn, and hence хж is not in the union U Mn X,
contrary to С X.
Baire’s Theorem 1. Let M be a set of the first category in a compact
topological space X. Then the complement Mc - X — M is dense in X.
Proof. We have to show that, for any non-void open set G, Mc inter-
oo
sects G. Let M = U Mn where each Mn is a non-dense closed set. Since
«=1
Мг = is non-dense, the open set intersects G. Since X is regular
as a compact space, there exists a non-void open set G1 such that
G® C G A . Similarly, we can choose a non-void open set G2 such that
G2 4 G( A M2. Repeating the process, we obtain a sequence of non-void
open sets {Gn} such that
c?+i £ G„ Л M°+1	(» = 1,2,...).
The sequence of closed sets {G„} enjoys, by the monotony in n, the finite
intersection property. Since X is compact, there is an X such that
00
x £ Г1 G“. x£ G“ implies x E G, and from x g G„+1 QGn^ M%+1
n=l
oo
[n = 0, 1, 2, . . .; Go = G), we obtain x t П = Mc. Therefore we
n = l
have proved that G A Mc is non-void.
Baire’s Theorem 2. Let {%„(/)} be a sequence of real-valued continuous
functions defined on a topological space X. Suppose that a finite limit:
lim xn (7) = x (t)
n—ЮО
exists at every point t of X. Then the set of points at which the function
x is discontinuous constitutes a set of the first category.
Proof. We denote, for any set M of X, by Мг the union of all the
open sets contained in M; ЛГ will be called the interior of M.
OO
Put P„(E) = {teX; И)-*„(/) I	0}. G(s) = U P'„{e).
tn = l
co
Then we can prove that G = П G(.1/w) coincides with the set of all
H I
points at wind) x(L) is continuous. Suppose x(i) is continuous at t = /0.
2. Topological Spaces
13
oo
We shall show that Zo£ (T	Since lim xn (Z)=%(Z), there
№1	п—ЮО
exists an m such that J x (Zo) — xm (Zo) |	e/3. By the continuity of x {t) and
xm (t) at t = Zo, there exists an open set Ute 3 tG such that | x (Z) — x (t0) | < e/3
| xm (t) — xm (Zo) |	e/3 whenever t G Thus t £ Ut<j implies
*(*) — xm(Z) |	[%(£) — x(t0) | + |x(Z0) — *„(Z0) | + ]%m(Z0) — xm(t) | < e,
which proves that t0£ Ргт(е) and so Zo£ G(e). Since e > 0 was arbitrary,
ОС
we must have t0£ П G(l/w).
oo
Let, conversely, t0£ П	Then, for any e > 0, t0£ G(e/3) and
W = 1
so there exists an m such that Zo £ Р„(е/3). Thus there is an open set
Ut<j3t0 such that 16 Ut* implies |%{Z)—Hence, by the
continuity of xmP) and the arbitrariness of e> 0,x(i) must be continuous
at I = t0.
After these preparations, we put
Fm(s) =	|%TO(Z) — %M+A(Z) | e {k = 1, 2, . . .)}.
OO
This is a closed set by the continuity of the xn (t) 's. We have X = U Fm (e)
by lim xn (Z) = x (Z). Again by lim xn (Z) = x (t), we have Fm (e) g Pm (e) .
n—КЮ
OO
Thus Fw(e) С Р\п(е) and so U ГЦе) £G(«). On the other hand, for
m = l
OO
any closed set F, (F — F^ is a non-dense closed set. ThusX~ U Flm(E)
m = l
oo
= U (Fm (e) — F*m (e)) is a set of the first category. Thus its subset
m=l
G(e)c = X — G(e) is also a set of the first category. Therefore the set
of all the points of discontinuity of the function x (Z), which is expressible
OO	00
as X — Q G(l/w) = U G(1/h)c, is a set of the first category.
Theorem. A subset M of a complete metric space X is relatively com-
pact iff it is totally bounded in the sense that, for every e > 0,there exists
a finite system of points . • ., mn of M such that every point m of
M has a distance < e from at least one of mt, m2,. . ., mn. In other words,
M is totally bounded if, for every e > 0, M can be covered by a finite
system of spheres of radii < e and centres G M.
Proof. Suppose M is not totally bounded. Then there exist a positive
number e and an infinite sequence {mw} of points £ Af such that d [mit mj}
e for i /. Then, if we cover the compact set Ma by a system of open
spheres of radii < e, no finite subsystem of this system can cover Ma.
Гог, this subsystem cannot cover the infinite subset {mJ QMQ Ma.
Thus a relatively compact subset of X must be totally bounded.
14
0. Preliminaries
Suppose, conversely, that M is totally bounded as a subset of a com-
plete metric space X. Then the closure Ma is complete and is totally
bounded with M. We have to show that Ma is compact. To this purpose,,
we shall first show that any infinite sequence {Д;} of Ma contains a sub-
sequence {p[ny} which converges to a point of Мл. Because of the total
boundedness of M, there exist, for any e > 0, a point qE E Ma and a sub-
sequence {pn} of {/>„} such that d{pn>, qe) < s/2 for n = 1, 2, . . .; conse-
quently, d(pn>, pm>) d(pn>, qe) 4- d (qE, pm>) < E for n, m = 1, 2, . . . We
set e = 1 and obtain the sequence {Д'}, and then apply the same rea-
soning as above with e = 2-1 to this sequence {Д-}. We thus obtain a
subsequence {pn"} of {pn'} such that
[Pn’> P^ < 1, d (pn-, pm„) < 1/2	(я, w = 1, 2, . . .).
By repeating the process, we obtain a subsequence {^„(fc-n)} of the sequence
such that
d(pn^), pm№y) < 1/2A	(n, m = 1, 2, . . .).
Then the subsequence {/>(«)-} of the original sequence {pn}, defined by the
diagonal method:
P[n)'-- Pn™ >
surely satisfies lim d (ptny> P(my) = 0. Hence, by the completeness of
>oo
Ma, there must exist a point p E Ma such that lim d{p[ny, p) = 0.
We next show that the set Ma is compact. We remark that there
exists a countable family {F} of open sets F of X such that, if U is any
open set of X and x E U Г\ Ma, there is a set F E {F} for which x E F C U.
This may be proved as follows. Ma being totally bounded, Ma can be
covered, for any e > 0, by a finite system of open spheres of radii e
and centres E Ma. Letting e = 1, 1/2, 1/3, . . . and collecting the coun-
table family of the corresponding finite systems of open spheres, we
obtain the desired family {F} of open sets.
Let now {U} be any open covering of Ma. Let {F*} be the subfamily
of the family {F} defined as follows: F £ {F*} iff F C {F} and there is
some U E {LT} with F £ U. By the property of {F} and the fact that
{C7} covers Ма, we see that this countable family {F*} of open sets covers
Ma. Now let {F*} be a subfamily of {17} obtained by selecting just one
U E {F} such that F £ U, for each F E {F*}. Then {F*} is a countable
family of open sets which covers Ma. We have to show that some finite
subfamily of {F*} covers Ma. Let the sets in {U*} be indexed as Ult
n
U2, . . . Suppose that, for each n, the finite union U Uj fails to cover
j=i
/	n \
Ma. Then there is some point xn E \M — U Uk\. By what was proved
3. Measure Spaces
15
above, the sequence {%„} contains a subsequence which converges
to a point, say xMl in Ma. Then xx G UN for some index N, and so
xn € UN for infinitely many values of n, in particular for an n > N. This
(n
M — и U it
fc = l
Hence we have proved that Ma is compact.
3. Measure Spaces
Measures
Definition. Let S be a set. A pair (S, S3) is called a а-ring or a a-
additive family of sets Q S if $8 is a family of subsets of S’ such that
SC S3,	(1)
implies Bc = (S — B) G 93,	(2)
Bj G 93 (7 = 1,2,.. .) implies that U Bj G 93 [a-additivity).	(3)
7=1
Let (S, 33) be a о-ring of sets c S. Then a triple (S, 33, m) is called a
measure space if m is a non-negative, o-additive measure defined on 93:
m(B) iS 0 for every В G 93,	(4)
/ 00	\	00
m Bj j = X m(Bj) for any disjoint sequence {By} of sets G 93
v=i /	;=i
[countable- or a-additivity of m),	(5)
S is expressible as a countable union of sets By G 93 such that m [Bp
 ; 00 (7 = 1, 2, . . .) (в-finiteness of the measure space (S, 93, m)).	(6)
This value m[B] is called the m-measure of the set B.
Measurable Functions
Definition. A real- (or complex-) valued function x(s) defined on Sis
said to be Ъ-measurable or, in short, measurable if the following condition
is satisfied:
For any open set G of the real line Л1 (or complex (7)
plane C1), the set {s; x(s) G 6} belongs to 93.
II is permitted that x (5) takes the value 00.
Definition. A property P pertaining to points 5 of S is said to hold m-
tilmost everywhere or, in short m-a. e., if it holds except for those s which
hu m a set G 93 of ж-measure zero.
A real- (or complex-) valued function x(s) defined m-a.e. on S and
>a( is lying condition (7) shall be called a Щ-measurable function defined
m a.c. on S or, in short, a ^-measurable function.
0. Preliminaries
Egorov’s Theorem. If В is a 93-measurable set with m[B} < oo and if
{/„(s)} is a sequence of SB-measurable functions, finite ш-а. e. on B, that
converges w-a. e. on В to a finite SB-measurable function / (s), then
there exists, for each e > 0, a subset E of В such that m(B - - E) Й e
and on E the convergence of /„(s) to /(s) is uniform.
Proof. By removing from B, if necessary, a set of m-measurc zero,
we may suppose that on B, the functions /„(s) are everywhere finite, and
converge to /($) on B.
OO
The set Bn = П {s£B; |/(s) — /д(з)
e} is SB-measurable and
OQ
Bn C Bk if n < k. Since lim = /(s) on B, we have В ~ U Bn.
ti>-oo	л ™ I
Thus, by the o-additivity of the measure m, we have
w(B) = m{B1 + (B2 — BJ + (B3 — B2) d--------------}
= m (Bj) T ж (B2 — Bj) + m (B3 - B2) + •  
= ж(Вг) + (w(B2) — т(Вг)) + (w(B3) — m(B2)) + • • 
- lim tn (Bn),
rt—>oo
Hence lim tn (B - - Bn) - 0, and therefore, from a sufficiently large E
n—их
on, m(B — Bk) < г/ where r/ is any given positive number.
Thus there exist, for any positive integer k, a set Ck £ В such that
w{Cft) e/2fe and an index Nk such that
/ (s) — /„ (s) < l/2ft for n > Nk and for s € В — Ck.
Let us set E — В — U Ck. Then we find
k=i
m(B-E)^ £т(Сь) <: Д fi/2* =
and the sequence fn (s) converges uniformly on E.
Integrals
Definition. A real- (or complex-) valued function %($) defined on S
is said to be finitely-valued if it is a finite non-zero constant on each of
a finite number, say и, of disjoint !B-measurable sets Bj and equal to zero
onS— U Bj. Let the value of %(s) on By be denoted by Xj.
j=i
Then x (s) is m-integrable or, in short, integrable over S if Л1 Xj|ot(Bj) <oo,
and the value £ Xj m(Bj) is defined as the integral of %{$) over S with
respect to the measure m; the integral will be denoted by f x(s) m(ds)
s
3. Measure Spaces
17
or, in short, by f x(s) or simply by f x($) when no confusion can be
s
expected. A real- (or complex-) valued function x (s) defined m-a. e. on S is
said to be m-integrable or, in short, integrable over S if there exists a
sequence {x„(s)} of finitely-valued integrable functions converging to
x(s) w-a. e. and such that
lim / I xn (s) — xk (s) ] m (ds) = 0.
It is then proved that a finite lim f xn (s) m (ds) exists and the value
ЮО §
of this limit is independent of the choice of the approximating sequence
{xn (s)}. The value of the integral f x(s) m(ds) over 5 with respect to the
s
measure m is, by definition, given by lim J x„(s) m(ds). We shall
n—5
sometimes abbreviate the notation f x(s) m (ds) to f x (s) m (ds) or to
p(s).	5
Properties of the Integral
i)	If x($) and y(s) are integrable, then ax(s) T fiy(s) is integrable
and f (<xx(s) 4- /3y(s)) m(ds) = oc f x(s) m(ds) 4- /2 / y(s)m(ds).
s	s	s
ii)	x(s) is integrable iff |x(s)| is integrable.
iii)	If x(s) is integrable and x(s) 0 a. e., then f x(s) m(ds) 0,
s
and the equality sign holds iff x (s) — 0 a. e.
iv)	If x(s) is integrable, then the function X(B) = [ x(s) m(ds) is
в
/ CO \	00
o-additive, that is, X £ Bjj = £ X (Bf) for any disjoint
sequence {B;-} of sets C S3- Here fx(s)m(ds)= f C B(s) x(s) m(ds),
в	s
where CB(s) is the defining function of the set B, that is,
CB(s) = 1 for and CB (s) = 0 for s G S — B.
v)	X (B) in iv) is absolutely continuous with respect to m in the sense
that т (В) = 0 implies X (B) = 0. This condition is equivalent to
the condition that lim X(B) = 0 uniformly in В 93.
The Lebesgue-Fatou Lemma. Let {xn ($)} be a sequence of real-valued
inlcgrable functions. If there exists a real-valued integrable function
* (s) such that %(s)	x„(s) a. e. for n = 1, 2, . . . (or %(s) < xn(s) a. e.
lor n — 1, 2, . . then
f I lim xn (s)1) m (ds) lim f xn (s) m (ds)
S *-OO	/	П—иОО 5
Vimhla, h’unctlunul Лгт1ун1н
18
0. Preliminaries
under the convention that if lim x„(s) for lim x„(s)\ is not integrable,
__	X n->oc J
we understand that lim f xn(s) tn(ds) ~—оо/or lim f xn(s) m(ds) = ooj.
S	\ h—кю S	J
Definition. Let (S, 23, tn) and (S', 23', tn') be two measure spaces. We
denote by 23x23' the smallest o-ring of subsets of SxS' which contains
all the sets of the form BxB', where В £ 23, B' £ SB'. It is proved that
there exists a uniquely determined cr-finite, u-additive and non-negative
measure tn x tn' defined on S3 X SB' such that
(wX w') (Bx5') - - m(B) m (B').
mXtn' is called the product measure of tn and mf. We may define the
S3 X SB'-measurable functions x (s, s') defined on S X S', and the tn x m'-
integrable functions x(s, s'). The value of the integral over SxS' of an
tn x tn'-integrable function x(s, s') will be denoted by
J"/ x(s, s') (mxtn') (ds ds') or ff x(s, s') tn (ds) m' (ds),
SxS'	SxS'
The Fubini-Tonelli Theorem. A 23 X 23'-measurable function x(s, s') is
mXm'-integrable over SxS' iff at least one of the iterated integrals
f f/ |%(s, s')| m(ds)\m' (ds') and fff |^(s, s')| tn (rfs')l tn(ds)
s' Is	J	s Is'	J
is finite; and in such a case we have
f f x (s, s') tn (ds) tn' (ds') = J J f x (s, s') tn(ds)\ m' (ds')
SxS'	S' (s	f
x(s, s') tn' (ifs')l m(ds).
Topological Measures
Definition. Let S be a locally compact space, e.g., an к-dimensional
Euclidean space Rn or a closed subset of Rn. The Baire subsets of S are
the members of the smallest cr-ring of subsets of S which contains every
compact G^-set, i.e., every compact set of S which is the intersection of
a countable number of open sets of S. The Borel subsets of S are the
members of the smallest <r-ring of subsets of S which contains every
compact set of S.
If S is a closed subset of a Euclidean space Rn, the Baire and the Borel
subsets of S coincide, because in every compact (closed bounded)
set is a G5-set. If, in particular, S is a real line 7?1 or a closed interval on
R1, the Baire (= Borel) subsets of S may also be defined as the members
of the smallset cr-ring of subsets of S which contains half open intervals
(a, b].
Definition. Let .S' be a locally compact space. Then a non-negative Baire
(Borel) measure on S is a o-additive measure defined for every Baire
3. Measure Spaces
19
(Borel) subset of S such that the measure of every compact set is finite.
The Borel measure m is called regular if for each Borel set В we have
m(B\ = inf m(U)
where the infimum is taken over all open sets U containing B. We may
also define the regularity for Baire measures in a similar way, but it
(urns out that a Baire measure is always regular. It is also proved that
each Baire measure has a uniquely determined extension to a regular
Borel measure. Thus we shall discuss only Baire measures.
Definition. A complex-valued function /(s) defined on a locally
compact space S is a Baire function on S if /“*(£) is a Baire set of S for
every Baire set В in the complex plane C1. Every continuous function
is a Baire function if S is a countable union of compact sets. A Baire
I unction is measurable with respect to the cr-ring of all Baire sets of S.
The Lebesgue Measure
Definition. Suppose S is the real line A1 or a closed interval of 7?1.
I cl F (%) be a monotone non-decreasing function on S which is continuous
110111 the right: F(x) = inf F(y). Define a function m on half closed
mlci vals {a, by m((a, 6]) = F (6) — F (a). This m has a uniquely deter-
mined extension to a non-negative Baire measure on S. The extended
mciv.iire m is finite, i.e.,	< oo iff F is bounded. If m is the Baire
inc;r,iire induced by the function F(s) = s, then m is called the Lebesgue
fm-asurc. The Lebesgue measure in Rn is obtained from the и-tuple of the
<nir dimensional Lebesgue measures through the process of forming the
piodnct measure.
Concerning the Lebesgue measure and the corresponding Lebesgue
mh-gral, we have the following two important theorems:
Theorem 1. Let M be a Baire set in Rn whose Lebesgue measure [ЛГ |
r I inite. Then, if we denote by В © C the symmetric difference of the
.rl />' and C:BqC = BUC — В C\ C, we have
Iim | (M -|- h) © M | = 0, where M h = {x£R”"t x = m -J- h,m£ M}.
Иск m + h = (mr +	+ h„) for № =	h =
,	/ »	\l/2
(Лhn) and | h | = ( J? hA .
Theorem 2. Let G be an open set of Rn. For any Lebesgue integrable
1111ict ion /(%) in G and e > 0, there exists a continuous function Ce(%) in
t, 'inch that {m G; CK(x) 0}“ is a compact subset of G and
/ |/(%) “ C«W| dx < «•
20
0. Preliminaries
Remark. Let m be a Baire measure on a locally compact space S.
A subset Z of S is called a set of m-measure zero if, for each e > 0, there
is a Baire set В containing Z with m (f?) < e. One can extend m to the
class of m-measurable sets, such a set being one which differs from a
Baire set by a set of m-measure zero. Any property pertaining to a
set of m-measure zero is said to hold m-ahnost everywhere (m-a. e.).
One can also extend integrability to a function which coincides m-a. e.
with a Baire function.
4. Linear Spaces
Linear Spaces
Definition. A set X is called a linear space over a field К if the
following conditions are satisfied:
X is an abelian group (written additively),	(1)
A scalar multiplication is defined: to every element'
x € X and each ос £ К there is associated an element of
X, denoted by ocx, such that we have
я (% -I- y) = ocx 4- ixy (a 6 К; x, у 6 X),
(a -|- /?) x = ocx + fix (л, p g K ', x g X),
(pep) x = oc(px) (ос, P £ К; x£ X),
1 • x = x (1 is the unit element of the field X).
In the sequel we consider linear spaces only over the real number
field A1 or the complex number field C1. A linear space will be said to be
real or complex according as the field К of coefficients is the real number
field A1 or the complex number field C1. Thus, in what follows, we mean
by a linear space a real or complex linear space. We shall denote by
Greek letters the elements of the field of coefficients and by Roman
letters the elements of X. The zero of X (= the unit element of the
additively written abelian group X) and the number zero will be denoted
by the same letter 0, since it does not cause inconvenience as 0 • x =
(ex-— oc) x ~ ocx — oc x — 0. The inverse element of the additively written
abelian group X will be denoted by —x; it is easy to see that — x — (— l)x.
Definition. The elements of a linear space X are called vectors (of X).
The vectors xr, x2,   -, xn of X are said to be linearly independent if the
n
equation £ ocj Xj = 0 implies = «j = • • • = 0. They are linearly
7 = 1
dependent if such an equation holds where at least one coefficient is
different from 0. If X contains n linearly independent vectors, but every
system of (w | 1) vectors is linearly dependent, then X is said to be of
4. Linear Spaces
21
п-dimension. If the number of linearly independent vectors is not finite,
then X is said to be of infinite dimension. Any set of n linearly indepen-
dent vectors in an и-dimensional linear space constitutes a basis for X
n
and each vector % of X has a unique representation of the form x = £	у,-
j=i
in terms of the basis ylf y2, . . ., yn. A subset M of a linear space X is
called a linear subspace or, in short, a subspace, if whenever x, y^M,
the linear combinations ocx -f- fly also belong to M. M is thus a linear
space over the same coefficient field as X.
Linear Operators and Linear Functionals
Definition. Let X, Y be linear spaces over the same coefficient field
K. A mapping T: x > y - 'T (x') 'I'x defined on a linear subspace D
of X and taking values in У is said to be linear, if
T{pcxx 4- px2) =oc(Tхг} + P(Tx2).
The definition implies, in particular,
7-0 = 0, T(—x) = - (Tx).
We denote
I)=D(T),{y£Y;y = Tx,x£D(T)} = R(T),{x£D(T);Tx = ty = N(T)
and call them the domain, the range and the null space of T, respectively.
T is called a linear operator or linear transformation on D (7) QX into
У, or somewhat vaguely, a linear operator from X into У. If the range
R (T) is contained in the scalar field К, then 7 is called a linear functional
on D(T). If a linear operator 7 gives a one-to-one map of D(T) onto
A’(T), then the inverse map T^1 gives a linear operator on 7? (7) onto
/>(7):
7-1 Tx = x for x £ D (T) and 7 7~Ty = у for у e R (T).
Г 1 is the inverse operator or, in short, the inverse of T. By virtue of
Г(хг — x2) = Txr — 7x2, we have the following.
Proposition. A linear operator 7 admits the inverse Т~г iff Tx = 0
implies x = 0.
Definition. Let Tx and T2 be linear operators with domains D (7J
and D(T2) both contained in a linear space X, and ranges 2?(7t) and
A’(72) both contained in a linear space Y. Then Tr = 72 iff D(T^ =
D(72) and Trx = T2x for all x^ d\t^ = D(T2). If D(T\) £D(T2} and.
Ttx = T2x for all x £ D (7*), then T2 is called an extension of Tv and 7г
-i restriction of T2; we shall then write TY C T2.
Convention. The value T(x) of a linear functional 7 at a point
v t D(T) will sometimes be denoted by (x, T>, i. e.
7(x) = (Xi T).
22
0. Preliminaries
Factor Spaces
Proposition. Let M be a linear subspace in a linear space X. We say
that two vectors xv x2£ X are equivalent modulo M if (Zj — x^ £ M and
write this fact symbolically by хг = x2 (mod M). Then we have:
(i)	x = x (mod M),
(ii)	if xt - x2 (mod M), then x2 = хг (mod M),
(iii)	if xr = x2 (mod M) and x2 = x3 (mod M), then	— x3 (mod M).
Proof, (i) is clear since x — x — 0 6 M. (ii) If (хг — x2) £ M, then
(x2 — %) = — (zx — x2) £ M. (iii) If (xt ~x2) EM and (x2 — x3) 6 M,
then (%j	x3) = (%i	z2) Ч- (%2	z3) €
We shall denote the set of all vectors £ X equivalent modulo M to a
fixed vector x by l-x. Then, in virtue of properties (ii) and (iii), all vectors in
£x are mutually equivalent modulo M. is called a class of equivalent
(modulo M) vectors, and each vector in gx is called a representative of the
class £x. Thus a class is completely determined by any one of its repre-
sentatives, that is, у £ implies that %y = gx. Hence, two classes gx,
are either disjoint (when у Q ^x) or coincide (when у G $x). Thus the entire
space X decomposes into classes $x of mutually equivalent (modulo M)
vectors.
Theorem. We can consider the above introduced classes (modulo M)
as vectors in a new linear space where the operation of addition of classes
and the multiplication of a class by a scalar will be defined through
Proof. The above definitions do not depend upon the choice of repre-
sentatives x, у of the classes £x, gy respectively. In fact, if (xr — x)£M,
(.У1 — y)£M, then
(*i + У1) ~ (* + y) = (*i - x) + (yi — у) e m,
(axY — ax') = a(x1 — x) £ M.
We have thus proved £Ж1+л = $x+y and = $aX) and the above defini-
tions of the class addition and the scalar multiplication of the classes are
justified.
Definition. The linear space obtained in this way is called the factor
space of X modulo M and is denoted by X[M.
References
Topological Space'. P. Alexandroff-H. Hopf [1], N. Bourbaki [1],
J. L. Kelley [1].
Measure Space: P. K. Halmos [1], S. Saks [1].
1. Semi-norms and Locally Convex Linear Topological Spaces 23
I. Semi-norms
The semi-norm of a vector in a linear space gives a kind of length for
the vector. To introduce a topology in a linear space of infinite dimension
suitable for applications to classical and modem analysis, it is sometimes
necessary to make use of a system of an infinite number of semi-norms.
11 is one of the merits of the Bourbaki group that they stressed the
importance, in functional analysis, of locally convex spaces which are
defined through a system of semi-norms satisfying the axiom of separa-
tion. If the system reduces to a single semi-norm, the correspond-
ing linear space is called a normed linear space. If, furthermore, the
space is complete with respect to the topology defined by this semi-
norm, it is called a Banach space. The notion of complete normed linear
spaces was introduced around 1922 by S. Banach and N. Wiener inde-
pendently of each other. A modification of the norm, the quasi-norm in
the present book, was introduced by M. Frechet. A particular kind of
Innit, the inductive limit, of locally convex spaces is suitable for discussing
t lie generalized functions or the distributions introduced by L. Schwartz,
as a systematic development of S. L. Sobolev’s generalization of the
notion of functions.
1. Semi-norms and Locally Convex Linear Topological Spaces
As was stated in the above introduction, the notion of semi-norm is of
Inndamental importance in discussing linear topological spaces. We shall
begin with the definition of the semi-norm.
Definition 1. A real-valued function p (%) defined on a linear space X
r. called a semi-norm on X, if the following conditions are satisfied:
P(x + У) = P(x) + P(y) (subadditivity),	(1)
p (pcx) — ]л j p (x).	(2)
Example 1. The «-dimensional Euclidean space Rn of points x =
(i!, . . ., xn) with coordinates xlt x2, . . ., xn is an «-dimensional linear
 pace by the operations:
x + у = (%! 4- ylf x2 + y2, . . ., xn + y„),
ocx = (azj, ocx2,   ., cxxn).
In this case p(x) = max |xy| is a semi-norm. As will be proved later,
1	< n
/ « \l/g
/>(.»:) - I £ I with q > 1 is also a semi-norm on Rn.
Proposition 1. A semi-norm p(x) satisfies
/>(0)=0,	(3)
p(x\ x2) > | p (xj) — p(x2) |, in particular, p(x) 0.	(4)
24
I. Semi-norms
Proof, p (0) = /> (0 • x) = 0 • p (x) = 0. We have, by the sub additivity,
p (xj — x2) + P (%z) = P (%i) and hence p (xx — x2) p (xj — p (x3). Thus
p{xr — x2) = | —11 • p (x2 — x-j) p (x2) — P(x-l) and so we obtain (4).
Proposition 2. Let p (x) be a semi-norm on X, and c any positive
number. Then the set M = {xEX;fi(x} c} enjoys the properties:
МЭ0,	(5)
M is convex: x, у £ M and 0 < <x < 1 implies
ocx + (1 — а) у G M,	(6)
M is balanced (6quilibr6 in Bourbaki’s terminology):
x M and |tx |	1 imply xx G M,	(7)
M is absorbing: for any x € X, there exists a > 0
such that a-1xG M,	(8)
p(x) = inf occ (inf = infimum = the greatest lower
bound).	(9)
Proof. (5) is clear from (3), (7) and (8) are proved by (2). (6) is proved
by the subadditivity (1) and (2). (9) is proved by observing the equi-
valence of the three propositions below:
[dT1 x G M] \j> (л-1 x) <: с] [p (x)	«с].
Definition 2. The functional
M*) = inf x	(9')
is called the Minkowski functional of the convex, balanced and absorbing
set M of X.
Proposition 3. Let a family {py (x); у G Г} of semi-norms of a linear
space X satisfy the axiom of separation:
For any x0 0, there exists pyi>(x) in the family such
that^d(x0)^ 0.	°	(10)
Take any finite system of semi-norms of the family, say pVi (x), p7t (x),...,
. . ., pyn (x) and any system of n positive numbers £x, e2, . . ., e„, and set
U = {x G X; py. (x) Ej {j = 1, 2, . . ., «)}.	(11)
U is a convex, balanced and absorbing set. Consider such a set U as a
neighbourhood of the vector 0 of X, and define a neighbourhood of any
vector x0 by the set of the form
xQ 4- U = {y^X;y = x0 + u,u£U}.	(12)
1. Semi-norms and Locally Convex Linear Topological Spaces 25
Consider a subset G of X which contains a neighbourhood of each of its
point. Then the totality {G} of such subsets G satisfies the axiom of open
sets, given in Chapter 0, Preliminaries, 2.
Proof. We first show that the set Go of the form Go = {x E X; pY (x) < c}
is open. For, let %oEGo and pY(x0) = Д < c. Then the neighbourhood
of x0, x0 + U where U = {x £ X; PY (x) 2-1 (c — /3)}, is contained in Go,
because и E U implies pY(x0 + u) pvG'o) + pY(u) < /J + (c — /?) = c.
Hence, for any point %()E X, there is an open set я0 + Go which con-
tains x0. It is clear, by the above definition of open sets, that the union
of open sets and the intersection of a finite number of open sets are also
open.
Therefore we have only to prove Hausdorff’s axiom of separation:
If xx =/= x2, then there exist disjoint open sets Gx and G2 such that
«jEGp x2^G2.	(13)
In view of definition (12) of the neighbourhood of a general point %0,
it will be sufficient to prove (13) for the case xr = 0, %2	0. We choose,
by (10), pYs(x) such that pYi(x2) = oc > 0. Then Gx = {%E X;PYl(x) < a/2}
is open, as proved above. Surely Gx Э 0 = xv We have to show that Gx
and G2 = %2 4“ have no point in common. Assume the contrary and
1 < l t there exist a Gx Л G2. у E G2 implies у = x2 -|- g = x2 — (—g) with
some gE Gj and so, by (4), ^(y) рУг(х2) — p (— g) ос — 2"1 oc = a/2,
because —g belongs to Gx with g. This contradicts the inequality
/уДу) < a/2 implied by у E GP
Proposition 4. By the above definition of open sets, X is a linear
topological space, that is, X is a linear space and at the same time a
t opological space such that the two mappings	X \ (x,y)-> x + у
and KxX —> X : (oc, x) —> ocx are both continuous. Moreover, each semi-
norm PY(x) is a continuous function on X.
Proof. For any neighbourhood U of 0, there exists a neighbourhood
I ‘ of 0 such that
V -|- V = {w E X; w = wx vz where zq, E V} C U,
.nice the semi-norm is subadditive. Hence, by writing
(% + y) —	(%0 +	y0) = (x - %0) + (y — y0),
wr sec that the mapping	(x, у)	-> x + у is continuous at x —	x0,	у = y0.
I'or any neighbourhood	U of	0 and any scalar oc 7^ 0, the	set	ocU =
I \ ( A; x ~ оси, и E U} is also	a neighbourhood of 0. Thus,	by	writing
ocx — ocQxo — oc(x — %0) + (oc — oc0) xq,
we see by (2) that (oc, x) -> ocx is continuous at oc = a0, x = x0.
I'he continuity of the semi-norm PY(x) at the point x = is proved
hV \/G(x) I <	- xo)-
26
I. Semi-norms
Definition 3. A linear topological space X is called a locally convex,
linear topological space, or, in short, a locally convex space, if any of its
open sets Э 0 contains a convex, balanced and absorbing open set.
Propositions. The Minkowski functional Pm{%} of the convex, ba-
lanced and absorbing subset M of a linear space X is a semi-norm on X.
Proof. By the convexity of M, the inclusions
х1(рм(х) + e)eM, у1(рм(у) + e) G M for any e > 0
imply
Pm (*) + «_____ .	____x__	I	(У) + £___________У	r M
Рм(х} + Рм(у) +	’	рм(х) + e Pm W + Puty) +	’ Рм{у} +
and so Рм{х + У) Рм (x) + Рм (у) + 2e. Since £ > 0 was arbitrary,
we obtain the subadditivity of pM(x). Similarly we obtain pM(ocx) =
Iя | Pm (x) since M is balanced.
We have thus proved
Theorem. A linear space X, topologized as above by a family of semi-
norms py{x) satisfying the axiom of separation (10), is a locally convex
space in which each semi-norm py(x) is continuous. Conversely, any
locally convex space is nothing but the linear topological space, topolo-
gized as above through the family of semi-norms obtained as the Min-
kowski functionals of convex balanced and absorbing open sets of X.
Definition 4. Let /(%) be a complex-valued function defined in an open
set Q of Rn. By the support (or carrier) of /, denoted by supp(/), we mean
the smallest closed set (of the topological space Q) containing the set
{x^Q; /(%) =/= 0}. It may equivalently be defined as the smallest closed
set of Q outside which f vanishes identically.
Definition 5. By Ск(Хр, 0 k oo, we denote the set of all complex-
valued functions defined in Q which have continuous partial derivatives
of order up to and including k (of order < oo if k = oo). By Cq(X2), we
denote the set of all functions g Ck{Q) with compact support, i.e., those
functions С Ck {Q) whose supports are compact subsets of Q. A classical
example of a function C (7?*) is given by
/ n \l/2
/(%) = exp (([x |2 -— l)-1) for |x| = [ (%b . . ., x«)| = \^x?j < 1,
(14)
= 0 for
%[ > 1.
The Space &*(&)
Ck is a linear space by
(/1 + /2) W = Ш	M (*) = ocf(x).
For any compact subset К of Q and any non-negative integer m < Л
(m < co when k == oo), we define the semi-norm
sup |D'/W|,
1. Semi-norms and Locally Convex Linear Topological Spaces 27
where sup — supremum = the least upper bound and
3*i + '• +    +
De fix) = , , . ,---=—— /(%,,%,, . . ., %_) ,
1 ' '	dx** dx‘‘   • дх‘я 1 ' 1 1	’ n/ >
1 *
я
|s | — | (^1» ^2’ ’  • > $n) I == J™. Sy •
J = I
Then Ck(Q) is a locally convex space by the family of these semi-norms.
We denote this locally convex space by (£*(.0). The convergence
lim Д = f in this space (£*(£?) is exactly the uniform convergence
А—изо
.lim Dsfh{x) = Dsf(x) on every compact subset К of Q, for each s
with |s| k (|s[ < oo if k = oo). We often write (£(£?) for ®°° (£).
Proposition 6. @*(Q) is a metric space.
Proof. Let Ky Q K2 Q    Q Kn Q   ‘ be a monotone increasing
00
sequence of compact subsets of Q such that Q = U Kn. Define, for
each positive integer h} the distance
k
d» (f, g) = X.2- 1>кл„ (/-?)•(!+ Рк^ (/ ~ g))-1 •
w = 0
Then the convergence lim /s = f in (42) is defined by the distance
5—изо
00
d(J, g) = X 2-» 4(/, g) • (1 + dh(j, g))-*.
Л —1
We have to show that dh(f, g) and d(f, g) satisfy the triangle inequality.
The triangle inequality for dh (/, g) is proved as follows: by the sub-
additivity of the semi-norm	we easily see that dh(f,g) =
satisfies the triangle inequality dh(f, g) tfA(/, k) + dh(k, g), if we can
prove the inequality
|a — ft | • (1 + |a — /3 |)-1 [a — у | (1 + |a — у j)-1
+ lx -£1(1 + lx -£|)-‘
for complex numbers л,/J andy; the last inequality is clear from the in-
equality valid for any system of non-negative numbers a, and y:
(« + £)(!+«+ АИ «(1 +	+ /3(1 + Ft1-
The triangle inequality for d (J, g) may be proved similarly.
Definition 6. Let X be a linear space. Let a family {XJ of linear
subspaces Хл of X be such that X is the union of Xa’s. Suppose that each
Хя is a locally convex linear topological space such that, if Xai £ ХЯ1,
then the topology of ХЛ1 is identical with the relative topology of ХЯ1
as a subset of X„t. We shall call "open,J every convex balanced and
absorbing set U of X iff the intersection U Хл is an open set of Xa
containing the zero vector 0 of Xa, for all Xa. If X is a locally convex
28
1. Semi-norms
linear topological space whose topology is defined in the stated way,
then X is called the (strict) inductive limit of X/s.
Remark. Take, from each XA, a convex balanced neighbourhood
of 0 of Хл. Then the convex closure U of the union V = U U*, i.e.,
a
U = w g X; и
w
= S fa Vj, Vj e V, 0 (; = 1, 2,.
/=1

with arbitrary finite n >
surely satisfies the condition that it is convex balanced and absorbing in
such a way that U A Xa is a convex balanced neighbourhood of 0 of Xx,
for all Xa. The set of all such U's corresponding to an arbitrary choice of
Ua’s is a fundamental system of neighbourhoods of 0 of the (strict) inductive
limit X of X^s, i.e., every neighbourhood of 0 of the (strict) inductive
limit X of X'a s contains one of the U's obtained above. This fact justifies
the above definition of the (strict) inductive limit.
The Space
С£°(£?) is a linear space by
(/i + /2) (*) = /i(*) + /jW. («/) (*) =
For any compact subset К of Q, let (£) be the set of all functions
/ G Cq° (£?) such that supp (/) £ K. Define a family of semi-norms on
&) by
Pk,M = sup \Dsf{x) where m < 00.
%K (Q) is a locally convex linear topological space, and, if Кг C X2,
then it follows that the topology of ФХ1 (Q) is identical with the relative
topology of ®д-ДЙ) as a subset of ®Kj(P). Then the (strict) inductive
limit of ®K(f2)'s, where К ranges over all compact subsets of Q, is a
locally convex, linear topological space. Topologized in this way, C^(_Q)
will be denoted by ®(D). It is to be remarked that,
/>(/) =sup |/{x)|
xCQ
is one of the semi-norms which defines the topology of © (£). For, if we
set U = {/ £ C£° (£?) ;/>(/) ^ 1}, then the intersection U A ©^(/2) is given
by UK = {f e 1 Pk (f) = sup \f(x) J 1}.
x£K
Proposition 7. The convergence lim fh = 0 in ® (£?) means that the
h—юо
following two conditions are satisfied: (i) there exists a compact subset
К of A such that supp(/^) С К (A = 1, 2, . . and (ii) for any differential
operator D*, the sequence {/)S/A(x)} converges to 0 uniformly on K. .
1. Semi-norms and Locally Convex Linear Topological Spaces 29
Proof. We have only to prove (i). Assume the contrary, and let there
exist a sequence of points C-f-1 having no accumulation points in
12 and a subsequence {А,(#)}	{/л (x)} such	0- Then the
semi-norm
co
p(f] =	2 sup	|/(#)//** (яw) where the mono-
tone increasing sequence of compact subsets Kj of
Q satisfies U K, = Q and € Kk — Kk_r
j=i
(й=1,2, ...),7fo=0
defines a neighbourhood U = {/ £ C£° (£2); p (f)	1} of 0 of Ф(Р).
However, none of the /Ajt’s is contained in U.
Corollary. The convergence lim fh = / in ф (42) means that the
following two conditions are satisfied: (i) there exists a compact subset
К of£? such that supp(/A) QK{h = l,2, and (ii) for any differential
operator Ds, the sequence DJ/A(x) converges to Dsj(x) uniformly on K.
Proposition 8 (A theorem of approximation). Any continuous function
/ ( C’q (Д”) can be approximated by functions of C£° (Д”) uniformly on Rn.
Proof. Let 6} (.x) be the function introduced in (14) and put
0a{x) — k~l e^{xja), where a. > 0 and ha > 0 are such that
ра(х)^=1.	(15)
VVr I hen define the regularization /a of /:
A,(x) = f f(x — y) 6a(y) dy = f f(y) 0a(x — y) dy, where
Rn	Rn	(16)
' У = (*1~ У1. ж2 — У2, • • x„~ y„)-
I hr integral is convergent since / and 0a have compact support. Moreover,
\hh г
fa{x) = J fly) 9a(x — y) dy,
supp(/)
I hr support of ja may be taken to be contained in any neighbourhood of
I he sup]>(/) if we take a > 0 sufficiently small. Next, by differentiating
Hiider I he iniegra] sign, we have
/>* Л(«) I Ax} - f f(y) Овх в Ax -y)dy ,	(17)
зи
I. Semi-norms
and so /л is in Cg°(/?*). Finally we have by f 8a{x — y) dy = 1,
Rn
|/«(*) — /(*)|ё / |/(y) — f{x)\ 0a(x — y) dy
R”
f |/(y) ~f(x)\ea(x — y) dy
+	/	|/(y) — f{x)\6a{x~ y) dy.
The first term on the right is e; and the second term on the right
equals 0 for sufficiently small a > 0, because, by the uniform continuity
of the function / with compact support, there exists an a > 0 such that
\f(y) —Hx)\ > e implies |y— x\ > a. We have thus proved our Pro-
position.
2. Norms and Quasi-norms
Definition 1. A locally convex space is called a normed linear space,
if its topology is defined by just one semi-norm.
Thus a linear space X is called a normed linear space, if for every
x С X, there is associated a real number ||x ||, the norm of the vector x,
such that
||x||	0 and ||x|| = 0 iff x — 0,	(1)
||x + У || is ||*|| + ||y|| {triangle inequality),	(2)
|a*|| = |л | • ||x ||.	(3)
The topology of a normed linear space X is thus defined by the distance
^(*. У) = II* — У||-	(4)
In fact, d(x, y) satisfies the axiom of distance'.
d (x, y)	0 and d (x, y) = 0 iff x = у,
d {x, y) <	d {x, z) + d {z, y) (triangle inequality),
d (*, y) = d (y, x).
For, d {x, y) = [ | x — у ] | = { у — x j | = d (y, %) and d{x,y)	—	11 x — у 11	=
II* — z + z — у ||	|jx — z I + ||z — у I) = d{x, z) + d(z,	y)	by	(1),	(2),
(3) and (4).
The convergence lim d (xn, x) = 0 in a normed linear space X will be
denoted by s-lim xn = x or simply by xn > x, and we say that the se-
quence {%„} converges strongly to x. The adjective “strong” is introduced
to distinguish it from the "weak” convergence to be introduced later.
2. Norms and Quasi-norms
31
Proposition 1. In a normed linear space X, we have
(5)
s-lim <xn xn = ocx if lim ocn = oc and s-lim x„ = x, (6)
fk-НЗО	n—KJQ	fl—>OQ
s-lim (%№ + yn) — x + у if s-lim xti = x and s-lim y„ = у. (7)
fl—ЮО	>O0	fl—ИЗО
Proof. (5), (6) and (7) are already proved, since X is a locally convex
pace topologized by just one semi-norm p(x) = ||% ||. However, we shall
give a direct proof as follows. As a semi-norm, we have
(8)
and
||x — у || ^ j ||% || — || у
hence (5) is clear, (7) is proved by
the boundedness of the sequence {%„} we obtain (6).
(x + y) — (xn + yj I x=
. From j|a% —	|
+ |art|  ||% — xn [| and
Definition 2. A linear space X is called a quasi-normed linear space,
11, lor every x G X, there is associated a real number
ol the vector x, which satisfies (1), (2) and
11—x |] -  ||x||, lim ||a„% || = 0 and
‘ ’	an—>0
lim
| x [ I, the quasi-norm
(3')
Proposition 2. In a quasi-normed linear space X, we have (5), (6)
..nd (7).
Proof. We need only prove (6). The proof in the preceding Propo-
•.11 ion shows that we have to prove
lim |k„|| = 0 implies that lim ||axn|| = 0 uniformly
п-кю	n-изо
in oc on any bounded set of a.
(9)
I he following proof of (9) is due to S. Kakutani (unpublished). Consider
the functional pn[pc) — !|a%M|| defined on the linear space Z?1 of real
nmnbcrs normed by the absolute value. By the triangle inequality of
/>„(>) and (3'), p„(oc) is continuous on Rl. Hence, from lim p„(pc) = 0
fl—ИЗО
implied by (3') and Egorov's theorem (Chapter 0, Preliminaries, 3. Mea-
•inr Spaces), we see that there exists a Baire measurable set A on the
real line Z?1 with the property:
the Lebesgue measure |Л | of A is > 0 and lim />„(«) = 0
h-к»	(IQ)
uniformly on A .
Sim e Ihe Lebesgue measure on the real line is continuous with respect to
li.inslafions, we have, denoting by В 0 C the symmetric difference
/n/с /me,
(/1 | o) о A | о
as
tx%„|| = 0.
cr —> 0.
32
1. Semi-norms
Thus there exists a positive number cr0 such that
cr [ iC (jQ implies | (A ст) 0 A [ < |A|/2, in particular, \(A + a) A A\> 0.
Hence, for any real number cr with |ct| aQ, there is a representation
о - = tx — <x with л £ A, <x E A ,
Therefore, by pn(a) = pn[x — a') АЛл) + /’«(«О, we see that
lim pn (ст) - 0 uniformly in cr when I ст I A ct0.
П—Н50
Let M be any positive number. Then, taking a positive integer
and remembering pn{ka) kpn(G), we see that (9) is true for [aJ M.
Remark. The above proof may naturally be modified so as to apply
to complex quasi-normed linear spaces X as well.
As in the case of normed linear spaces, the convergence lim I x—xn\ =0
Л—ЮО
in a quasi-normed linear space will be denoted by s-lim x„ = x, or
ft—Э-OG
simply by xn-> x; we shall then say that the sequence {x„} converges
strongly to x.
Example. Let the topology of a locally convex space X be defined by a
countable number of semi-norms pn(x} (n = 1, 2, . . .). Then X is a
quasi-normed linear space by the quasi-norm
OO
||x||=	/>я(х) (1 + pn(x))~\
For, the convergence lim />„(xA) = 0 (-и = 1, 2, . . .) is equivalent to
Л—xo
s-lim xh = 0 with respect to the quasi-norm || x || above.
Л—>00
3. Examples of Normed Linear Spaces
Example 1.	C (S). Let S be a topological space. Consider the set C (S)
of all real-valued (or complex-valued), bounded continuous functions
x(s) defined on S. C (S) is a normed linear space by
(x + y) (s) = x(s) 4- y(s), (txx) (5) = ax(s), (|x || = sup |x(s)| .
In C (S), s-lim xn — x means the uniform convergence of the functions xn (s)
to x(s).
Example 2.	L^S, S3, m}, or, in short, 7/(S) (1	< o©). Let L^(S)
be the set of all real-valued (or complex-valued) 41 -measurable functions
x(s) defined w-a. e. on S such that ]x(s) is m-integrable over S. If (S) is
a linear space by
(x 4- y) (s) — x(s) + y(s), (ar) (s) = ax(s).
3. Examples of Normed Linear Spaces
33
For, (x(s) + у (5)) belongs to 24 (S) if % (5) and у (s) both belong to Lp (S),
as may be seen from the inequality j x (s) + y/s) 2^ (| x (s) + {У ($) l^) •
We define the norm in LP(S) by
The subadditivity
I % (s) j* m {ds)^p.
+ (/ I у К w (dstyVP >
(1)
(2)
called Minkowski’s inequality, is clear for the case p = 1. To prove the
general case 1 < p < сю, we need
Lemma 1. Let 1 < / < 00 and let the conjugate exponent p' of p be
defined by	1	1
? + F = r	(3>
Then, for any pair of non-negative numbers a and b, we have
ap . bp'
where the equality is satisfied iff a =	11 .
cp 1
Proof. The minimum of the function /(c) = — +	~ c for c > 0
' ' ' p p	—
is attained only at c = 1, and the minimum value is 0. By taking
c	we see that the Lemma is true.
The proof of (2). We first prove Holder s inequality
/ |x(s) y(s)| <; (/	(Jly(s)H1^
(5)
/lor convenience, we write f z(s) for f z($) m(t/s)V
\	s	/
Го this end, we assume that A = (f j# (s) and В = (f |y
.u<! both 7^ 0, since otherwise x(s) у (s) — 0 a.e. and so (5) would be true.
Now, by taking a = |a;(s)[/^ and b = |y(s)|/Bin (4) and integrating, we
obtain
fk(s)y(5)I^ 1 Ap ,1 Bp' ,	....	..	...
< -- -л + 35 —7 == 1 which implies (5).
A В — P Ap P Bp	r
Next, by (5), we have
/j%(s) 4- У(s)I* f Jx(s) 4- y(s)[?_1 • {%(s)|
+ / \x(s) + У W-1' Ms)
(f |x(s) + У (s) (f ^(s)!^
4 (/ (Из) +
which proves (2) by p'(p — .1) = p.
3 Yonldft, Functional Analysis
34
I. Semi-norms
Remark 1. The equality sign in (2) holds iff there exists a non-negative
constant c such that x(s) = cy(s) m-a.e. (or y(s) = cx(s) m-a.e.). This is
implied from the fact that, by Lemma 1, the equality sign in Holder's
inequality (5) holds iff |x(s)| = c- |y(s)l1/^-1) (or |y(s)| = c- |x(s)J1/**—*)
are satisfied m-a.e.
Remark 2. The condition ||x|j = (J	= 0 is equivalent to the
condition that x(s) = 0 m-a.e. We shall thus consider two functions of
L^(S) as equivalent if they are equal m-a.e. By this convention, ZZ(S)
becomes a normed linear space. The limit relation s-lim xn = x in IP (S)
rt—ЮО
is sometimes called the mean convergence of p-th order of the sequence of
functions x„(s) to the function x(s).
Example 3.	L°°(S). A ^-measurable function x{s) defined on 5 is
said to be essentially bounded if there exists a constant <x such that
x (s) | < a m-a.e. The infimum of such constants л is denoted by
vrai max I x (s) I or essential sup I x (s) I.
sES 1	1	s€5	1
L°°(S, 23, m) or, in short, Z.°°(S) is the set of all®-measurable, essentially
bounded functions defined m-a.e. on 5. It is a normed linear space by
(% + y) (s) = x(s) + У ($), («%) (s) = ax (s),
s€S
under the convention that we consider two functions of L°°(S) as equi-
valent if they are equal m-a.e.
Theorem 1. Let the total measure m(S) of S be finite. Then we have
lim f |x(s)[^ m(^$)y^ = vrai max |x(s)| for x(s)eL°°(S). (6)
Proof. It is clear that / f ]%(s)|^ m(ds)\lli> m (S)2^ vrai max |%(s)l
\s	/
so that lim J | # (s) vrai max |x(s
By the definition of the
vrai max, there exists, for any s > 0, a set В of m-measure > 0 at each
point of which |x(s)[ > vrai max | x (s) | — e. Hence
sCS
f | x (s) m(</s)y^
Si m(B)1^ (vraimax [x(s)| — e). Therefore lim (f	vraimax
p—>oo	s€S
|x (s) | — e, and so (6) is true.
Example 4. Let, in particular, S be a discrete topological space con-
sisting of countable points denoted by 1, 2,...; the term discrete
means that each point of S = {1, 2,. ..} is itself open in S. Then as linear
subspaces of C ({1, 2, . . .}), we define (c0), (c) and {lp), 1 p < сю.
(c0): Consider a bounded sequence of real or complex numbers {£„}.
Such a sequence {£„} defines a function x(n) — defined and continuous
on the discrete space S -- {1,2,...}; we shall call x -- {£„} a vector
3. Examples of Normed Linear Spaces
35
with components £n. The set of all vectors x = {£„} such that
lim £„ = 0 constitutes a normed linear space (c0) by the norm
||x|| — sup[x(n) | = sup \$„).
П	n
(c): The set of all vectors % = {£„} such that finite lim £„ exist,
constitutes a normed linear space (c) by the norm ||x|j = sup
-sup|fH|.
(Z^), 1 p < oo: The set of all vectors x = {f„} such that
OQ
--	< oo constitutes a normed linear space (1Р) by the norm
»=j
/ CO	\lip
lx II = ч . As an abstract linear space, it is a linear subspace
v*=i	/
of С ({1, 2, . . .}). It is also a special case of LP(S, 58, m) in which
«({!}) = »({2}) = •   = 1.
(Z°°) = (w): As in the case of L°°(S), we shall denote by (Z°°) the
linear space C({1, 2, . . .}), normed by ||x|| ~ sup [x(w)j = SUP |fn|-
n	n
(Z,XJ) is also denoted by (m).
The Space of Measures. Let 83 be a <r ring of subsets of S. Consider
(lit: set A {S, 58) of all real- (or complex-) valued functions <p(B) defined
tnt 23 such that
9? (B) l^oofor every В E 58,	(7)
JS BA = £(p (Bj) for any disjoint sequence {Bj} of sets E 58.	(8)
. I (A, 88) will be called the space of signed (or complex) measures defined
on (S, 58).
Lemma 2. Let <p E A (S, 58) be real-valued. Then the total variation of
71 on .S' defined by
r(?>; 5) = V(<p; S) +	S)|	(9)
r. tiiiite; here the positive variation and the negative variation of q> over
H ( s8 are given respectively by
V (<p; B) = sup <p (Вг) and V(tp; B) = inf 9>(Вг).	(10)
В, £8	Bi£B
Proof. Since <p(0) = 0, we have K(tp; B) iS 0 L(<Pi Suppose
f h;il I7 {9?; S) — 00. Then there exists a decreasing sequence {Bn} of sets
1 >8 such that
У (ff> '> %„) = сю, I?? (Bn) j n — 1.
I he proof is obtained l>y induction. Let us choose BY~S and assume
(led (lie sets B2, B!t, . . ., Bk have been defined so as to satisfy the above
i ondilions. By the first condition with n -= k, there exists a set ВE 58
36
I. Semi-norms
such that В £ Bk, [97 (B) [ > [99 (BA)) + k. We have only to set Bk+1 — В
in the case V (99; B) = oo and Bk+1 = Bk — В in the case V (q>; B) < oo.
For, in the latter case, we must have V{gp',Bk — B) = oo and
I99(Bk — B)|	[99 (В)I — \<p(Bft) ] k which completes the induction.
By the decreasing property of the sequence {B„}, we have
00	00
s- n B„= J?(S-Bn)
»=1	n=l
— (S—Bt) + (Bj—B2) + (B2 — B3) 4	+(ВЯ—B»+i) + • • •
so that, by the countable additivity of 99,
(OO \
S- П	+ 9>(B2-Bs) + ••
» = 1	/
= [?(S) -pfB,)] + ^(B,) -<p(B2)]
+ [<P (B2) — y(Bs)] +   
= ® (S) — lim 99 (BH) = 00 or — oo,
n—>00
which is a contradiction of (7).
Theorem 2 (Jordan’s decomposition). Let 99 £ Л (S, ЯЗ) be real-valued.
Then the positive variation V (95; B), the negative variation V (97; B)
and the total variation У (99; В) are countably additive on B. Moreover,
we have the Jordan decomposition
<p{B) = V(<p; B) + V(<p; B) for any В 6 58.	(11)
Proof. Let {BM} be a sequence of disjoint sets 6 58. For any set BE SB
such that В C Bn, we have <p (В) = £ <p(B Г\ Bn) У £ V (g ; B„)
П=1	Я=1
and hence У (99; £ B„) V (97; Bn). On the other hand, if C„G 58
(00	\	/ 00	\
9?; 2? Bj > 99 2? Cn j
H = 1 /	= I /
00	__ /	00	\ co ____
= 2 <p(Cn) and so V(93; £ Вм V(99; ВД Hence we have pro-
ved the countable additivity of V (99; B) and those of V(tp; B) and of
V (9?; B) may be proved similarly.
To establish (11), we observe that, for every C G 58 with С С B, we
have 99(C) = 99(B)—<p (B — C) 99 (B)—^(99; B) and so V{gp‘,B)^
<p (В) — У (<jp; B). Similarly we obtain V (99; B) 97 (В) — V(<p; B). These
inequalities together give (11).
Theorem 3 (Hahn's decomposition). Let 99 C A (S, 58) be a signed
measure. Then there exists a set P C 58 such that
97 (B)	0 for every В € 58 with В С P,
9₽(B) iC 0 for every В € 58 with В C Pc ~ S - P.
3. Examples of Normed Linear Spaces
37
The decomposition S = P + (S — P) is called the Hahn decomposition
of S pertaining to <p.
Proof. For each positive integer n we choose a set Bn £ 93 such that
(p(Bn) V(<p; S) — 2~n. Hence by (11), we have
У(^Ви)^-2~” and	(12)
The latter inequality is obtained from V (pp; S—Bn) = V (tp; S) — V (<p; Bn)
and V ((p; Bn) (p(Bn). We then put
CO co
P = lim Bn = и n Bn.
*oa	n—k
Then S - P = Йт (S - B„) = П U (S - Bn) Q U (S - B„) for
«—>oo	A = 1 n=k _____	ft—к
every k, and therefore, by the cr-additivity of V (<p; B),
_	oo _
V(p; S-P) £ s	< 2-<м>,
which gives V(<p; S  - P) = 0. On the other hand, the negative variation
И (<p; B) is a non-positive measure and so, by (12) and similarly as above,
|K(»>;P)|i Um |K(^;B.)| = 0,
«—*<30
which gives 7(<p; P) = 0. The proof is thus completed.
Corollary. The total variation V (<p; S) of a signed measure (p is defined
’У
V {<p; S) = sup f % (s) (ds)
supj^(s)] <1 5
(13)
where x(s) ranges through 95-measurable functions defined on 5 such that
sup | JV(s) I 1.
Proof. If we take x (s) = 1 or = — 1 according as s £ P or s G S — P,
then the right hand side of (13) gives Vfy; S). On the other hand, it is
easy to see that
I f x(s)<p(ds) g sup |x(s)J • f V(tp; ds) — sup |x(s) |  V(tp; S)
'5	s	S	5
and hence (13) is proved.
Example 5.	A (S, ®). The space A (S, S3) of signed measures <p on S3
is a real linear space by
(oqg?! + oc2<pz} (B) = aj^jfB) + <x2(pz(B), BG S3.
is a normed linear space by the norm
S) = SUp
SUp|r(5)f£l
f x(s) $(ds)
s
(14)
38
I. Semi-norms
Example 6.	The space A (S, 23) of complex measures g? is a complex
linear space by
(Л1 <Pi + я2 Фг) ft) = Л1 Pi (s) 4- a2 go.a(B), В E 58 with complex	л2.
It is a normed linear space by the norm
||g? j| = sup
sup|*(s)|gl
f x(s)cp(ds) ,
5
(10)
where complex-valued ©-measurable functions %(s) defined on S are
taken into account. We shall call the right hand value of (15) the total
variation of <p on S and denote it by V (cp; S).
4.	Examples of Quasi-normed Linear Spaces
Example 1.	Qg* (X2). The linear space ©*(/?), introduced in Chapter 1,1,
is a quasi-normed linear space by the quasi-norm ||%|| = d(xt 0), where
the distance d(x, y) is as defined there.
Example 2.	M(S, 23, m). Let m(S) < oo and let M(S, 23, m) be the
set of all complex-valued 23-measurable functions x (s) defined on S and
such that |x(s)| < oo ?к-а.е. Then M(S, 23, m) is a quasi-normed linear
space by the algebraic operations
(% + y) (s) = %(s) + y(s), (a%) (s) = ax(s)
and (under the convention that x = у iff %(s) = y(s) w-a.e.)
KU = / |*(s)[ (1 + l^(s)l)-1 m(ds).
s
(1)
The triangle inequality for the quasi-norm | | x 11 is clear from
_±l±A1_ < N + И < H
1 + [a + Д| "1 + |a| +	— 1 + |a|
\P\
1 + И'
The mapping {a, x) -> ocx is continuous by the following
Proposition. The convergence s-lim xn = x in M(S, 23, m) is equi-
n—ЮО
valent to the asymptotic convergence (or the convergence in measure) in S
of the sequence of functions {xn (s)} to x (s):
For any e > 0, lim m {s £ S; | x (s) — %M($) | > e} = 0.	(2)
п—КЮ
Proof. Clear from the inequality
Y^w(Ba) < ||x|| < w(Bd) +	m(S— Вл), B6~{s£S', |x(s)|^(3}.
Remark. It is easy to see that the topology of M (S, 23, m) may also
be defined by the quasi-norm
||% || = inf tan”1 [e + m{s E S: |x(s)|	f}].	(Г)
₽>0
5. Pre-Hilbert Spaces
39
Example 3.	Фк (.£?). The linear space introduced in Chapter 1,1,
is a quasi-normed linear space by the quasi-norm |[%|| = d{x, 0), where
the distance d(x, y) is defined in Chapter I, 1.
5.	Pre-Hilbert Spaces
Definition 1. A real or complex normed linear space X is called a
f>re-Hilbert space if its norm satisfies the condition
|[% + у j|2 + ||% — у ||2 = 2(||% |j2 -J- 11у ||2).
(1)
Theorem 1 (M. Frechet-J. von Neumann-P. Jordan). We define, in
a real pre-Hilbert space X,
(*,y) = 4	+ у
Then we have the properties:
]%-y||2).
(%%,y) = a (%,у) (%eя1),
(% + у, 2) = (%, z) + (у, z),
у) = (у. ,
(x, %) = ||%112.
(2)
(3)
(4)
(5)
(6)
Proof. (5) and (6) are clear. We have, from (1) and (2),
(%, z) + (y, z) = 4_ 1 (11 % 4- z||2 — ||% — z |[2 -j- ||y + z ||2 — ||y — z ||2)
I f we take у ~ 0, we obtain (%, z) = 2 (, z), because (0, z) = 0 by (2).
Hence, by (7), we obtain (4). Thus we see that (3) holds for rational
numbers a of the form a = m/2”. In a normed linear space, ||«% + y||
and ||%% — у [| are continuous in a. Hence, by (2), (a%, y) is continuous in
л. Therefore (3) is proved for every real number a.
Corollary {J. von Neumann-P. Jordan). We define, in a complex
normed linear space X satisfying (1),
(%, y) = (%, y)j + t(%, i'y\,
where i = ]/—• !, (%, y)T = 4-1(||% + у ||2 — ||% — y||2).
<8)
I hen, we have (4), (6) and
(<%%, y) = a(%, y) (txCC1),	(S')
— —
(%, y) = (y, %) (complex-conjugate number).	(o')
40
I. Semi-norms
Proof. X is also a real pre-Hilbert space and so (4) and (3') with real a
hold good. We have, by (8), (у, х)г = (x, y)lf (ix, iy}1 = (x, y)x and hence
(y,	= (— iiy, гх)г — — (iy, x)x = — (%, ty)r Therefore
(y, xj = (y, x)r + i(y, ix)t = (x, y)x — i(x, iy\ = (%, y).
Similarly, we have
(ix, y) = (ix, y)x + i (ix, iy)t = — (x, ty)x + i (x, y)x = i (x, y),
and therefore we have proved (3'). Finally we have (6), because
(x, x)x = |[x||2 an(l (x'	= 4~1 (| 1 + г|2 — 11 — i |2) ||x ||2 = 0.
Theorem 2. A (real or) complex linear space X is a (real or) complex
pre-Hilbert space, if to every pair of elements x, у С X there is associated
a (real or) complex number (x, y) satisfying (3'), (4), (5') and
(x, x) > 0, and (x, x) — 0 iff x = 0.	(9)
Proof. For any real number <x, we have, by (3'), (4) and (5')
(x + a(x, y) y, x + a(x, y) y) = ||x||2 + 2« | (x, y)p
+ a2 |(x, y)|2 j|y [|2	0, where ||x|| = (x, x)1/2
so that we have [ (x, у)]4 — ||x||2 | (x, у) ||y ||2	0. Hence we obtain
Schwarz’ inequality
|(x, y)|	J|x|| • [|y||,	(10)
where the equality is satisfied iff x and у are linearly dependent.
The latter part of (10) is clear from the latter part of (9).
We have, by (10), the triangle inequality for |[x f]:
||x + У||2 = (x + y, x + y) = 11%||2 + (x, y) + (y, x) + l|y ||2
(IM! + 1MI)2-
Finally, the equality (1) is verified easily.
Definition 2. The number (x, y) introduced above is called the scalar
product (or inner product} of two vectors x and у of the pre-Hilbert space
X.
Example 1.	L2(S, 53, m) is a pre-Hilbert space in which the scalar
product is given by (x, y) = f x(s) у (s) m(ds).
s
Example 2,	The normed linear space (Z2) is a pre-Hilbert space in
00	_
which the scalar product is given by ({£„}, {ty,}) =	£„ ijn.
n=l
Example 3.	Let £ be an open domain of Rn and 0	k < oo. Then the
totality of functions f EC* (£?) for which
ypi	у
oo, where dx = dxTdx2 • •  dxn
(И)
is the Lebesgue measure in Rn,
5. Pre-Hilbert Spaces
41
constitutes a pre-Hilbert space Hk (£) by the scalar product
(/,?)* = £ tD’f(x}-D‘g(X}dx.
JJ |
(12)
Example 4.	Let Q be an open domain of Rn and 0 < k < oo. Then
C'o (Q) is a pre-Hilbert space by the scalar product (12) and the norm (11).
We shall denote this pre-Hilbert space by
Example 5.	Let G be a bounded open domain of the complex 2-plane.
Let Л2(С) be the set of all holomorphic functions f(z) defined in G and
such that
|/|| = (ff dxdy^l2< oo, (2 = x + ty). (13)
Then A2(G) is a pre-Hilbert space by the norm (13), the scalar product
(Л = ff f(z)g(z)dxdy	(14)
G
and the algebraic operations
(/ + g) (*) = /(*) + g(^)> («/) W = ^/(z).
Example 6.	Hardy-Lebesgue class H-L2. Let H-L2 be the set of
all functions /(2) which are holomorphic in the unit disk {2; [2] < 1}
of the complex 2-plane and such that
sup (f |/(re10)|2 d&) <00.	(15)
0<r<l \o	/
OO
Then, if f(z) = £ c„zn is the Taylor expansion of /,
n~0
1	1	00 2л
F(r)=^-f |/(rC«)|2 d9=± Xfr.n c„ r”+” ««-—)• rffl
is monotone increasing in r, 0 < r < 1, and bounded from above. Thus
it is easy to see that
[ 1 / гЛ	W2	/ 00	\V2
I/ = SUP ЛЛ/(^е)|М = Лл«12	<16>
0<r<l L \o	/ J	\«=0	/
is a norm which satisfies condition (1), since (Z2) is a pre-Hilbert space.
Remark. Let a sequence {c„} 6 (I2) be given, and consider
/(г) = f(rei0) = 1 cnzn = Scnr” e™e, z <1.
n=0	w=0
By Schwarz' inequality, we have
42
I. Semi-norms
e <1.
1 such that
00
and so cnzn is uniformly convergent in any disk | z |	q with 0
Thus /(z) is a holomorphic function in the unit disk |z
(15) holds good, that is, /(z) belongs to the class H-LX
Therefore we have proved
Theorem 3. The Hardy-Lebesgue class /7-7.2 is in one-to-one corre-
spondence with the pre-Hilbert space (Z2) as follows:
00
n-0
in such a way that
oo	oo
= £ cnzn ++ {c„}, g(z) = JE dnzn {dn} imply
fl=O	n=0
(oo	\l/2
2 №)
n=0	/
Hence, as a pre-Hilbert space, H-IX is isomorphic with (Z2).
6.	Continuity of Linear Operators
Proposition 1. Let X and Y be linear topological spaces over the
same scalar field K. Then a linear operator T on D (T) £ X into У is
continuous everywhere on D(T") iff it is continuous at the zero vector
x = 0.
Proof. Clear from the linearity of the operator T and T • 0 = 0.
Theorem 1. Let X, Y be locally convex spaces, and {/>}, {y} be the
systems of semi-norms respectively defining the topologies of X and Y.
Then a linear operator T on D (T) QX into У is continuous iff, for every
semi-norm ?€{?}, there exist a semi-norm p E {p} and a positive number
/? such that
q(Tx) fip(x) for ah x£D(T).	(1)
Proof. The condition is sufficient. For, by T • 0 = 0, the condition
implies that T is continuous at the point x = 0 E D (T) and so T is con-
tinuous everywhere on D(T).
The condition is necessary. The continuity of T at x = 0 implies that,
for every semi-norm q E {?} and every positive number «, there exist a
semi-norm p E {p} and a positive number d such that
x € D (T) and p(x) d imply q(Tx) e.
Let x be an arbitrary point of D (T), and let us take a positive number Л
such that /.р(х)Хд- Then we have p(Ax) <5, AzCD(T) and so
q(T[Ax')) < e. Thus q(Tx) < е/Я. Hence, if p(x) 0, we can take л
arbitrarily large and so q(Tx) = 0; and if p (x) 0, we can takel = (%)
and so, in any case, we have q{Tx) /!/>(%) with /3 = e/d.
6. Continuity of Linear Operators
43
Corollary 1. Let X be a locally convex space, and / a linear functional
on D (f) g X. Then f is continuous iff there exist a semi-norm p from the
system {p} of semi-norms defining the topology of X and a positive
number fl such that
|/(z)| g /?£>(%) for all D(f).	(2)
Proof. For, the absolute value | л ] itself constitutes a system of semi-
norms defining the topology of the real or complex number field.
Corollary 2. Let X, Y be normed linear spaces. Then a linear operator
T on D (T) g X into У is continuous iff there exists a positive constant
fl such that
\\Tx\\	/5||%|| for all %C D(T).	(3)
Corollary 3. Let X, Y be normed linear spaces. Then a linear operator
T on D (T) g X into Y admits a continuous inverse T-1 iff there exists a
positive constant у such that
|| Tx || g* у
x || for every x £ D (T).
(4)
Proof. By (4), Tx = 0 implies x — 0 and so the inverse Т~л exists.
The continuity of T”1 is proved by (4) and the preceding Corollary 2.
Definition 1. Let T be a continuous linear operator on a normed linear
space X into a normed linear space У. We define
||T || = inf/?, where В = {/?; || Tx ||	/? ||x|| for all x £ X}.	(5)
By virtue of the preceding Corollary 2 and the linearity of T, it is easy
to see that
= sup
| T x 11 = sup 11T x 11.
IWI=1
(6)
|| T |[ is called the norm of T. A continuous linear operator on a normed
linear space X into У is called a bounded linear operator on X into У,
since, for such an operator T, the norm
T x 11 is bounded when x ranges
over the unit disk or the unit sphere {% С X;
x|[ X 1} of X.
Definition 2. Let T and S be linear operators such that
D (T) and D (S) g X, and R (T) and R (S) g У.
Then the sum T + S and the scalar multiple ос T are defined respectively
by
(T + -S') (x) = Tx 4- Sx for xQD (T) T\ D(S), (ос T) (x) = oc(Tx).
Let T be a linear operator on D (T) g X into У, and S a linear operator
on D (S) g У into Z. Then the product S T is defined by
(ST)x = S(Tx) for x£{x;x£D(T) and TxCD(S)}.
T + S,ocT and ST are linear operators.
44
I. Semi-norms
Remark. ST and TS do not necessarily coincide even if X = У = Z.
An example is given by Tx = tx(t)„ Sx{t) = ]/(—l)-^x (t) considered as
linear operators from Z2^1) into Z2^1). In this example, we have the
commutation relation (ST — TS) x(t) = j/— 1 x(t).
Proposition 2. If T and S are bounded linear operators on a normed
linear space X into a normed linear space У, then
|z + s|| ||r|| + ||s||, ||«z|| = н ||t|
(7)
If T is a bounded linear operator on a normed linear space X into a
normed linear space У, and S a bounded linear operator on У into a
normed linear space Z, then
|ST[|	||S|| • Ill'll.
(8)
Proof. We prove the last inequality; (7) may be proved similarly.
ЦВТ*I] ||S|| ||Tx]|	||S|[ ||T|| ||x|| and so ||SZ|| g ||S|| ||Т|
Corollary. If T is a bounded linear operator on a normed linear space
X into X, then
(9)
where T” is defined inductively by Гй = TT”-1 (n = 1, 2, . . .; T° = I
which maps every x onto x itself, i.e,, lx = x, and I is called the identity
operator).
7.	Bounded Sets and Bornologic Spaces
Definition 1. A subset В in a linear topological space X is said to be
bounded if it is absorbed by any neighbourhood U of 0, i.e., if there exists
a positive constant such that or1 В C U. Here аг1 В = [% 6X; x —x yb,
b£B}.
Proposition. Let X, Y be linear topological spaces. Then a continuous
linear operator on X into У maps every bounded set of X onto a bounded
set of У.
Proof. Let В be a bounded set of X, and V a neighbourhood of 0 of У.
By the continuity of T, there exists a neighbourhood U of 0 of X such
that T • U — {Tu; и £ U} Q V. Let tx > 0 be such that В QxU. Then
T • В QT(ocU) = oc(T • U) £ a V. This proves that T • В is a bounded
set of У.
Definition 2. A locally convex space X is called bornologic if it satisfies
the condition:
If a balanced convex set M of X absorbs every bounded
set of X, then M is a neighbourhood of 0 of X.	(1)
Theorem 1.	A locally convex space X is bornologic iff every semi-
norm on X, which is bounded on every bounded set, is continuous.
7. Bounded Sets and Bornologic Spaces
45
Proof. We first remark that a semi-norm p (x) on X is continuous iff
it is continuous at x — 0. This we see from the subadditivity of the semi-
norm: p(x — y) \p{x) — p(y)| (Chapter I, 1, (4)).
Necessity. Let a semi-norm p(x) on X be bounded on every bounded
set of X. The set M = {%£ X; p(x) 1} is convex and balanced. If В
is a bounded set of X, then sup/>(£) = a < <x> and therefore В C »M.
b€B
Since, by the assumption, X is bornologic, M must be a neighbourhood
of 0. Thus we see that p is continuous at x ~ 0.
Sufficiency. Let M be a convex, balanced set of X which absorbs
every bounded set of X. Let p be the Minkowski functional of M. Then p
is bounded on every bounded set, since M absorbs, by the assumption,
every bounded set. Hence, by the hypothesis, p(x} is continuous. Thus
Мг = {x£ X; p(x) < 1/2} is an open set Э 0 contained in M. This proves
that M is a neighbourhood of 0.
Example 1. Normed linear spaces are bornologic.
Proof. Let X be a normed linear space. Then the unit disk S =
{x E X; [[ x 11	1} of X is a bounded set of X. Let a semi-norm p {x} on X
be bounded on S, i.e., sup p (x) = <x < oo, Then, for any у 0,
P (У) = P

Thus p is continuous at у = 0 and so continuous at every point of X.
Remark. As will be seen later, the quasi-normed linear space M (S, S3)
is not locally convex. Thus a quasi-normed linear space is not necessarily
bornologic. However we can prove
Theorem 2.	A linear operator T on one quasi-normed linear space into
another such space is continuous iff T maps bounded sets into bounded
sets.
Proof. As was proved in Chapter I, 2, Proposition 2, a quasi-normed
linear space is a linear topological space. Hence the “only if" part is al-
ready proved above in the Proposition. We shall prove the "if” part.
Let T map bounded sets into bounded sets. Suppose that s-lim xk = 0.
k—>oo
Then lim ||xA|| = 0 and so there exists a sequence of integers {nk}
А—юо
such that
lim nk - oo
while
lim nk [|xA
= 0.
We may take, for instance, nk as follows:
nk the largest integer |[хл|[ 1/2 if хл # 0,
k if xk — 0.
46
I. Semi-norms
Now we have 11 nk xk | ] = | ] xk + xk + •   + xk 11 nk 11 xk 11 so that
s-lim nkxk = 0. But, in a quasi-normed linear space, the sequence
{nk хД, which converges to 0, is bounded. Thus, by the hypothesis,
{T (nA %ft)} = {nk TxA} is a bounded sequence. Therefore
s-lim Txk = s-lim «L1 (T (nk xA)) = 0,
Л—k—изо
and so T is continuous at x = 0 and hence is continuous everywhere.
Theorem 3.	Let X be bornologic. If a linear operator T on X into a
locally convex linear topological space У maps every bounded set into a
bounded set, then T is continuous.
Proof. Let У be a convex balanced neighbourhood of 0 of У. Let p be
the Minkowski functional of V. Consider q(x) = p(Tx). q is a semi-norm
on X which is bounded on every bounded set of X, because every bounded
set of У is absorbed by the neighbourhood V of 0. Since X is bornologic,
q is continuous. Thus the set {% G X; Tx G Va} = {xf X; q(x) 1} is a
neighbourhood of 0 of X. This proves that T is continuous.
8.	Generalized Functions and Generalized Derivatives
A continuous linear functional defined on the locally convex linear
topological space ® introduced in Chapter I, 1, is the "distribution”
or the "generalized function” of L. Schwartz. To discuss the generalized
functions, we shall begin with the proof of
Theorem I.	Let В be a bounded set of ф(/2). Then there exists a
compact subset К of Q such that
supp (у) £ К for every <p G В,	(1)
sup | D]cp (%) | < oo for every differential operator D*.	(2)
Proof. Suppose that there exist a sequence of functions {tpi} С В and
a sequence of points such that (i):	has no accumulation point
in.12, and (ii):	(/>»)	0 (i = 1, 2, . . .). Then
OO
p((p) = Z г |<рШ|/|<?\(А)|
t — 1
is a continuous semi-norm on every Ф^(£2), defined in Chapter I, 1.
Hence, for any e > 0, the set {<p E (f2)	~ £} is a neighbourhood
of 0 of (Q). Since Ф(12) is the inductive limit of ^к(£2)'5, we see that
{<p G Ф (12)p((p) e} is also a neighbourhood of 0 of ® (Q). Thus p is
continuous at 0 of ®(£) and so is continuous on © (12). Hence p must be
bounded on the bounded set В of ® (12). However, p (<pf) i {i 1, 2, . ..).
This proves that we must have (1).
8. Generalized Functions and Generalized Derivatives
47
We next assume that (1) is satisfied, and suppose (2) is not satisfied.
Then there exist a differential operator ZF’ and a sequence of functions
{99J S В such that sup | ZF° (x) | > i (i = 1, 2, . . Thus, if we set
6K
p (93) = sup I (f) (x)
xQK
for 9? e ©а (£),
/>(93) is a continuous semi-norm on an(^ /’(<?’») > 1 (r = 1, 2, . ..).
Hence {93J С В cannot be bounded in ©K(I2), and a fortiori in ©(Z2).
This contradiction proves that (2) must be true.
Theorem 2.	The space ©(D) is bornologic.
Proof. Let q(<p) be a semi-norm on ©(D) which is bounded on every
bounded set of © (Q). In view of Theorem 1 in Chapter I, 7, we have only
to show that q is continuous on © (D). To this purpose, we show that q is
continuous on the space ©^(D) where К is any compact subset of Q.
Since © (Q) is the inductive limit of ©a(D)'s, we then see that q is con-
tinuous on © (12).
But q is continuous on every ©а (D). For, by hypothesis, q is bounded
on every bounded set of the quasi-normed linear space ®K(D), and so, by
Theorem 2 of the preceding section, q is continuous on ©a(D). Hence q
must be continuous on ©(/2).
We are now ready to define the generalized functions.
Definition 1. A linear functional T defined and continuous on ©(D)
is called a generalized function, or an ideal function or a distribution inD;
and the value T (<p) is called the value of the generalized function T at
the testing function q> £ © (D).
By virtue of Theorem 1 in Chapter I, 7 and the preceding Theorem 2,
we have
Proposition 1. A linear functional T defined on © (X2) is a generalized
function in Q iff it is bounded on every bounded set of ©(D), that is,
iff T is bounded on every set В C ©(D) satisfying the two conditions (1)
and (2).
Proof. Clear from the fact that T (90) is continuous iff the semi-norm
I 7’(<р) I is continuous.
Corollary. A linear functional T defined on C^°(D) is a generalized
function in D iff it satisfies the condition:
To every compact subset К of Q, there correspond a
positive constant C and a positive integer k such that
IГ (9?) I C sup J [x) I whenever 9? C©a(<?)-	(3)
Proof. By the continuity of T on the inductive limit ©(D) of the
©к (D)’s, we see that 7’ must be continuous on every ©a(D). Hence
1 het necessity of condition (3) is clear. The sufficiency of the condition
48
I. Semi-norms
(3) is also clear, since it implies that T is bounded on every bounded set
of Ф(12).
Remark. The above Corollary is very convenient for all applications,
since it serves as a useful definition of the generalized functions.
Example 1.	Let a complex-valued function /(%) defined a.e. in Q
be locally integrable in Q with respect to the Lebesgue measure
dx = dxx dx2 • • • dxn in Rn, in the sense that, for any compact subset К
of £2, J |/(%)j dx < oo. Then
к
ТДу} = $ f{x}<p{x} dx, <p£^{Q}.	(4)
Q
defines a generalized function '/} in Q.
Example 2.	Let m (B) be a cr-finite, ст-additive and complex-valued
measure defined on Baire subsets В of an open set Q of Rn. Then
TmW = f W m(dx), <p e ® {&},	(5)
Q
defines a generalized function Tm in Q,
Example 3.	As a special case of Example 2,
T{<p} = <p{p}t where /> is a fixed point of Q, <p t ф (£?),	(6)
defines a generalized function Tip inLC It is called the Dirac distribution
concentrated at the point p^Q. In the particular case p — 0, the origin
of Rn, we shall write T9 or Й for
Definition 2. The set of all generalized functions in 12 will be
denoted by Ф(42)'. It is a linear space by
(T + S) (?) = T(<p) + S(<p), («Т) (?) =«TM,	(7)
and we call Ф (12)' the space of the generalized functions in Q or the dual
space of Ф(Х2).
Remark. Two distributions and Tf are equal as functionals
(Tffy} ~ Tft(<p) for every <p £ ®(-^)) iff AW = fz(x) a-e- this fact is
proved, then the set of all locally integrable functions in 12 is, by f Tf,
in a one-one correspondence with a subset of ф (/2)' in such a way that
(A and A being considered equivalent iff A (x) = A W a-e-)
Л. + 7}, =	«1, = T*.	(7-)
In this sense, the notion of the generalized function is, in fact, a genera-
lization of the notion of the locally integrable function. To prove the
above assertion, we have only to prove that a locally integrable
function / is = 0 a.e. in an open set Q of Rn if f f(x) tp(x) dx = 0 for
в
all <p£ C“(X2). By introducing the Baire measure m(B) — f f(x) dx,
в
the latter condition implies that f <p (x) pi {dx} = 0 for all g>GC^°(^2),
n
8. Generalized Functions and Generalized Derivatives
49
which further implies that f <p(x) p (dx) = 0 for all у 6	(£2), by virtue
я
of Proposition 8 in Chapter I, 1. Let В be a compact Gd-set in f2:
OO
В = П Gn, whereG„ is an open relatively compact setin £2. By applying
M = 1
Urysohn’s theorem in Chapter 0, 2, there exists a continuous function
fn(x) such that
0 <; fn(x) 1 for x^Q, fn(x) = 1 for x C G“+2 and fn(x) = 0
for x € G“ — G„+1 (n = 1, 2, . . .),
assuming that {GK} is a monotone decreasing sequence of open relatively
compact sets of £2 such that G“+2 C G.„+1, Setting tp = fn and letting
n —> oo, we see that p(B) = 0 for all compact G^-sets В oiQ. The Baire
sets of Q are the members of the smallest ст-ring containing compact Gd-
sets of £2, we see, by the a-additivity of the Baire measure p, that p
vanishes for every Baire set of £2. Hence the density / of this measure p
must vanish a.e. in
We can define the notion of differentiation of generalized functions
through
Proposition 2. If T is a generalized function in £2, then
S(?>) = - T (jg),	(8)
defines another generalized function S in £2,
Proof. S is a linear functional on Ф (Q) which is bounded on every
bounded set of ®(й).
Definition 3. The generalized functional S defined by (8) is called the
generalized derivative or the distributional derivative of T (with respect to
tj), and we write
О)
so that we have
(10)
Remark. The above notion is an extension of the usual notion of the
derivative. For, if the function / is continuously differentiable with
respect to xlr then we have
£TfM = -T,($)=-f... ffwdXn
Я
= f   ' f • <p(%) dx\   • dxn =
Я
as may be seen by partial integration observing that <p(x) vanishes iden-
tically outside some compact subset of Q.
4 YoHhiii, Fiiiictloiml Лпн1ун1н
50
I. Semi-norms
Corollary. A generalized function T in Q is infinitely differentiable
in the sense of distributions defined above and
(О’Г)(9.) = (-1)^Т(£»?>), where ; =	=	(11)
д^...д^пп
Example 1.	The Heaviside function H (x) is defined by
H{x) — 1 or = 0 according as x 0 or x < 0.	(12)
Then we have
zzT» = '1‘-	(12')
where is the Dirac distribution concentrated at the, origin 0 of R1.
In fact, we have, for any (p £ ф (7?1),
J \	oo	oo
Тн) M = ~ f H {X)<P W dx= — $ <p (x) dx=~ = <p(0) •
4	7	—oo	0
Example2.	Let /(x) have a bounded and continuous derivative in the
k
open set J?1 — U Xj of R1. Let s,- = f(xj + 0) — f(xj ~ 0) be the saltus
J-1
or the fump of f(x) at x ~ Xj. Since
f Кх)?'(х) dx = ^(P^ si + J f{x)(p(x)dx,
'	'	— oo	J	—oo
we have
^Tf^Tf. + ^s,d4,	(12")
where 6Xj is defined by (6).
Example 3.	Let /(x) = f(xlt x2, . . xn) be a continuously differen-
tiable function on a closed bounded domain Q C Rn having a smooth
boundary S. Define f to be 0 outside Q. By partial integration, we have
d Tr) W = - ff W s; f «dx
n
= J f(x)<p W cos (*, Xj) dS + J 0^-<p(x) dx,
where v is the inner normal to S, (v, xf) = (xh v) is the angle between v
and the positive Xy-axis and dS is the surface element. We have thus
— Tf^ TdfjSxi + Ts, where Ts(tp) = //(x) cos (v,Xj)<p(x) dS. (12'")
s
Corollary. If /(x) = /(xlf x2, . . ., xn) is C2 on £? and is 0 outside,
then, from (12'") and^- =	— cos v) we obtain Green s
'	'	} dv j Sxj x f
8. Generalized Functions and Generalized Derivatives
51
integral theorem
(ATl)(<f) = T^<r) +	(12"')
s	s
n
where /1 is the Laplacian JL" cP/dxf.
Proposition 3. If Г is a generalized function in Q and / £ C00 (Q), then
S(<p) = T(/«p),	(13)
defines another generalized function S in Q.
Proof. S is a linear functional on $>(<□) which is bounded on every
bounded set of ф\Q'\. This we see by applying Leibniz’ formula to ftp.
Definition 4. The generalized function S defined by (13) is called
the product of the function / and the generalized function T.
Leibniz* Formula. We have, denoting S in (13) by fT,
8xf	8XjT + dxj	(14)
because we have
-т(/®=г(Д’,)-г(^М
by Leibniz’ formula for	This formula is generalized as follows.
Let P(£) be a polynomial in fj, f2> . . .,%n, and consider a linear
partial differential operator P(D) with constant coefficients, obtained
by replacing by г-1 0/5*,-. The introduction of the imaginary coefficient
г-1 is suitable for the symbolism in the Fourier transform theory in
Chapter VI.
Theorem 3 (Generalized Leibniz’ Formula of L. Hormander). We
have
P(D')tfT)	P<UD) T.	(15)
s
where, for s = (sp s2, . . s„),
T*(a)(£)	p№)
/1 Э Vi /1	/1 3
s \г dx1) \t	\i &xn)
(16)
and
c I — c, I I cl
о * dp dg. . . . dft. .
(17)
Proof. Repeated application of (14) gives an identity of the form
PCW =S(P.f)Q.(D) T,	(18)
52
I. Semi-norxns
where Qs{£)’s are polynomials. Since (18) is an identity, we may substitute
/(x) = and T = where (%, $) = £ хз$з
j=i
in (18). Thus, by the symbolism
P(D) е^> = P(£) &*<*>*>,	(19)
we obtain
P(t + V) = 27 where £* =	•••£*“.
$
On the other hand, we have, by Taylor's formula,
P(f+ ч) =2/4fpwW 
Thus we obtain
e.w=^h).
9.	В-spaces and F-spaces
In a quasi-normed linear space X, lim |lxw — x || = 0 implies, by the
n—НЭО
triangle inequality
IK —xll + IIх —Ml*
that {xn} is a

Cauchy sequence, i.e., {xn} satisfies Cauchy’s convergence condition
lim 11 xn — xm 11 = 0.
>oo
(1)
Definition 1. A quasi-normed (or normed) linear space X is called an
F-sfiace (or a В-space] if it is complete, i.e., if every Cauchy sequence {x„}
of X converges strongly to a point x^ of X:
lim ||x« — xTO|| = 0.	(2)
n—>oo
Such a limit .i№, if it exists, is uniquely determined because of the triangle
inequality ||x —x']|	||x— x„j| + ||x„ —x'(|. A complete pre-Hilbert
space is called a Hilbert space.
Remark. The names /’ -space and В-space are abbreviations of Fre-
chet space and Banach space, respectively. It is to be noted that Bour-
baki uses the term Fr^chet spaces for locally convex spaces which are
quasi-normed and complete.
Proposition 1. Let Q be an open set of Rn, and denote by @(P) =
C°°(P) the locally convex space, quasi-normed as in Proposition 6 in
Chapter I, 1. This @(£?) is an F-space.
Proof. The condition lim — /OT|| = 0 in @(jQ) means that, for
any compact subset К of Q and for any differential operator Da, the
sequence {£)“/„ (x)} of functions converges, as n^oo, uniformly bn
K. Hence there exists a function /(x) C C°°(12) such that lim Dafu(x) =
n—НЮ
9. В-spaces and Л-spaces
53
D°A(x) uniformly on K. Da and К being arbitrary, this means that
lim ||/л — /II = 0 in @(D).
♦*—►00
Proposition 2. LP{S) = LP{S, Я9, m) is a В-space. In particular, L2(S)
and (Z2) are Hilbert spaces.
Proof. Let lim }|xrt — xml| = 0 in LP(S). Then we can, choose a
ntm—K?c 1
subsequence {xWfc} such that ] |	- xnie 11 < oo. Applying the tri-
angle inequality and the Lebesgue-Fatou Lemma to the sequence of
functions
yt ($) = | ($) | 4- 2: | x„t+l (s) —	(«) | € Lp (S),
Л = 1
we see that
/	00	\ ?
/ (limy^sH w(^s) lim	||*J| + -S 11xnk^ — x |[
q	!	i_)<x	\	Л=1	/
Thus a finite lim yt (s) exists a.e. Hence a finite lim xn (s) = хж (s) exists
t—изо	f-^o f+1
a.e. and x^ (s) £ Lp (S'), since | xn As) I lim yt (s) G Lp (5). Applying
+	l-KJO
again the Lebesgue-Fatou Lemma, we obtain
11 *co 11* = / (Нт
Therefore lim llx^.— x„.
nt (s) — (s) H w{ds) 2; II X»(+I - *JI •
'	M=A	/
| = 0, and hence, by the triangle inequality and
Cauchy's convergence condition lim i|x„ — xm
п,т—изо
| - 0, we obtain
lim IIxqo — xn|l <: lim I Xoq — xwJ| + lim |lxrt—%JI = 0.
n-+oo	A—>oo	A, ft—изо
Incidentally we have proved the following important
Corollary. A sequence {x„} G lA (S') which satisfies Cauchy's conver-
gence condition (1) contains a subsequence {%„*} such that
a finite lim = %„($) exists a.e.,	(s) £LP(S) and
A —► cc
s-lim xn = Xqo .	(3)
n—ЮО
Remark. In the above Proposition and the Corollary, we have assumed
in the proof that 1 p < ex?. However the results are also valid for the
case p = oo, and the proof is somewhat simpler than for the case
1 p < oo. The reader should carry out the proof.
Proposition 3. The space A2{G) is a Hilbert space.
Proof. Let {/„(г)} be a Cauchy sequence of A2(G). Since Л2(б) is a
linear subspace of the Hilbert space L2{G), there exists a subsequence
{/«л (z)} suc}1 that
a finite lim fn}Az) == ^{z) exists a.e., /оо G	and
А—ню *
lim f 1/^(2) —/n(z) |2 dx dy = 0.
n—ooG
54
I. Semi-norms
We have to show that (z) is holomorphic in G. To do this, let the sphere
| z — z01 q be contained in G. The Taylor expansion /„ (z) — fm (z) =
00
£ Cj(z — ZqY implies
f \fM—fm(z)\zdxdy
к-г0|йе
= / ( f 2:	£ с^е-гкв do} r dr = £ 2?r / |c, [2 r2j+1 dr
0 \o t	k	]	j 0
00
= 2тг ^р2;+2 |cj2 (2/+ 2)"1 л | c012 p2	(4)
= e21in Ы — fm Ы i2 •
Thus the sequence {/n(z)} itself converges uniformly on any closed sphere
contained in G. fn (z)’s being holomorphic in G, we see that Go (z) = lim fn (z)
must be holomorphic in G.
Proposition 4. M(S, S3, w) with m(S') < oc is an F-space.
Proof. Let {%„} be a Cauchy sequence in M(S, LI, ж). Since the con-
vergence in M(S, SB, rri) is the asymptotic convergence, we can choose a
sub-sequence {xHfc(s)} of {%rt(s)} such that
w (Bk) <^2~k for Bk = {s € S; 2~k J хПк¥1 (s) — xn* (s) [}.
The sequence x„k (s) = хП1 (s') +
is s-convergent to a function
we have Д |x„ +i(s) — xn.(s)
)=t
A-l
(*»3+1(s) ~хп№) (k = 1, 2,. . .)
__ 00
GM (S, SB, m), because, if 5 6 U Bj,
2-J <i 2l~f and m \ U В J
J=i	U = t 7
consequently we see, by letting
converges ж-a.e. to a func-
lim — Xqo |[ = 0 and so, by
ft'—^00
lim |x„ — ХооЦ = 0.
00	00
t 00, that the sequence (%
tion xw (s') G	m). Hence
lim 1|ян —	= 0, we obtain
«,tn-+co
The Space (s). The set (s) of all sequences {£„} of numbers quasi-nor-
med bv
00
Ш11 = x + If,-I)
J=1
constitutes an F-space by {<£„} + {??„} = {£„ 4- ??„}, a{£n} ={«£„}. The
proof of the completeness of (s) may be obtained as in the case of
M(S, Ю, rn). The quasi-norm
||{£я}|[ — inf tan-1 {e 4- the number of £n’s which satisfy |f„| > g}
€>0
also gives an equivalent topology of (s).
9. В-spaces and F-spaces
55
Remark. It is clear that C (S), (c0) and (c) are В-spaces. The complete-
ness of the space (lp) is a consequence of that of.Z^(S). Hence, by Theo-
rem 3 in Chapter I, 5, the space H-L2 is a Hilbert space with (/2).
Sobolev Spaces Wk,p (£2). Let _Q be an open set of Rn, and k a positive
integer. For l^/>< oo, we denote by	the set of all complex-
valued functions /(x) = /(%i, x2, . . ., xn) defined in Q such that / and
its distributional derivatives Dsf of order |$| = £ Ы k all belong
to Lp (12). Wk'p (12) is a normed linear space by
(A + A) (x) = /1(%) + 4W, M) (x) = a/(x) and
I/ k,P = I \ DSf(x) P dxYIP> dx = dxi dx2. . . d%n

under the convention that we consider two functions /х and /2 as the same
vector of Wk,p (£)) if (x) = /2 (x) a.e. in Q. It is easy to see that Ж*'2 (£?)
is a pre-Hilbert space by the scalar product
Proposition 5. The space Wk,p (Q) is a В-space. In particular, Wk (12) =
lTft’2(f2) is a Hilbert space by the norm ||/|[A — /1|Aj2 and the scalar
product (/, g)k = (/, g)ki2.
Proof. Let {/л} be a Cauchy sequence in Wk,p(£?). Then, for any diffe-
rential operator Ds with | s | Li k, the sequence {Ds /A} is a Cauchy sequence
in Lp (Q) and so, by the completeness of LP(Q}, there exist functions
/{s}€Lp(Q) (|s| k) such that lim f Ds-fh(x)~f^(x)pdx^G. By
x. AA *
virtue of Holder’s inequality in Chapter I, 3, applied to compact sets
of Q, we easily see that fh is locally integrable in Q. Hence, for any func-
tion q> G C£° (Q),
= f Dsfh(x)  <p(x) dx = (—l)|s| f fh(x) Ds<p(x) dx,
and so, again applying Holder's inequality, we obtain, by
lim f | /A (x) — /{0> (x) \p dx = 0,
Л *OQ
lim TD.fM = (—1)1'1 Tfm(D'<p) = ITT,^ .
Л—ЮО
Similarly we have, by lim f | Dsfh (x) — /{s) (x) dx = 0,
Дт TDSfh(q)} = 7>(<р).
Hence we must have	that is, the distributional deri-
vative Ds/(0) equals /(s). This proves that lim I fh —/t0) IL* = 0 and
Id I. M-n 1	I r J Л
Л—кю
is complete.
об
1. Semi-norms
10.	The Completion
The completeness of an F-space (and a В-space) will play an important
role in functional analysis in the sense that we can apply to such spaces
Baire's category arguments given in Chapter 0, Preliminaries. The follow-
ing theorem of completion will be of frequent use in this book.
Theorem (of completion). Let X be a quasi-normed linear space which
is not complete. Then X is isomorphic and isometric to a dense linear
subspace of an F-space X, i.e., there exists a one-to-one correspondence
x ++ x of X onto a dense linear subspace of X such that
(x-b y) = x + y, (ax) = a x, ||x[| = ||x]|.	(1)
The space X is uniquely determined up to isometric isomorphism. If X
is itself a normed linear space, then X is a B-space.
Proof. The proof proceeds as in Cantor’s construction of real numbers
from rational numbers.
The set of all Cauchy sequences {xn} of X can be classified according
to the equivalence {x,.} ~ {y„} which means that lim l|xn — y„|[ = 0.
«—►CO
We denote by {x„}' the class containing {xw}. Then the set X of all such
classes x — {x„}' is a linear space by
W' + {yn}' = {*„ + У.}', Oi{xn}' = {axM}'.
We have l|ixnil — l|xwl II l|xn —xw|| and hence lim ||хл I exists.
«—►CO
We put
IIW'II = hm II*»I 
w—кзо
It is easy to see that these definitions of the vector sum {x„}' + {yn}',
the scalar multiplication a{xn}' and the norm ||{xw}' || do not depend
on the particular representations for the classes {yw}', respectively.
For example, if {xft} {4}, then
lim llxjl < lim llxjl + lim 11xn —"X11 < lim |lxj
»—>CO	«—>00	«—>oo	«—>co
and similarly lim |lxjl lim llxjl, so that we have |[{xn}'|| =
ft—ЮО	tt—>OO
IKO'll-
To prove that 11 {xn}' 11 is a quasi-norm, we have to show that
lim ||a{x„}'I =0
1
and
lim
[|{*n}'||->0
«{xj’ || = °-
The former is equivalent to lim lim 11ахи II = 0 and the latter is
<x->0 «—K3O
equivalent to lim llaxJI = 0. And these are true because |laxl|iscon-
tinuous in both variables a and x.
10. The Completion
57
To prove the completeness of X, let = {{*2}} be a Cauchy
sequence of X. For each k, we can choose nk such that
I _ *№ 11 - ь-i
I лт лПк 11
if m > nk.
(2)
Then we can show that the sequence {xA} converges to the class containing
the Cauchy sequence of X:
.....(3)
To this purpose, we denote by x® the class containing
(4)
Then, by (2),
*2 || = lim 11*2 ~*2ll	(5)
m-xo
and hence
*2-4211^ ||*2	+ Il**-*J| + ||Я»-*2!1
I xk — xm 11 + k 1 + m 1.
Thus (3) is a Cauchy sequence of X. Let x be the class containing (3).
Then, by (5),
||% —||x —*211 + 11*2 — M g II* —*2 II + k 1
Since, as shown above,
* — *2 I
we prove that lim
Л-кю
lim ||x^ — ^2 || lim (|^ — *fc|| + k 1
* — *2 I — O’ and so bm | x — xk | = 0.
й 1	А—изо
The above proof shows that the correspondence
X Э x x = {*, x, . . ., x, . .	= x
is surely isomorphic and isometric, and the image of X in X by this
correspondence is dense in X. The last part of the Theorem is clear.
Example of completion. Let Q be an open set of Rn and k < oo. The
completion of the space Cq (£>) normed by
fl-O'/HIM1'2
\|sTSfeo	/
will be denoted by #§(42); thus Hq(Q) is the completion of the pre-
Hilbert space (Д) defined in Chapter 1,5, Example 4. Therefore (Q) is a
Hilbert space. The completion of the pre-Hilbert space Hk(Q) in Chap-
ter I, 5, Example 3 will similarly be denoted by Hk{Q).
The elements of Hq (X?) are obtained concretely as follows: Let {/A} be a
Cauchy sequence of Ct*(X?) with regard to the norm ||/||A. Then, by the
58
1. Semi-norms
completeness of the space L2(I2), we set; that (here exist functions
/w (i) E Z2 (fi) with [si — JL'sy S /< such that
lim f |/'s'(a:) — Ds !h(x) 2 dx =- 0 {dx d.xl dxv . . , dx„).
Л—К» д
Since the scalar product is continuous in the norm of /?(<&), we see, for
any test function 99 (%) £ C£° (P), that
T/(.) (9?) = lim (Dsfh, <p> = lim (~1)и Tf {Ds(f>)
h-+CQ	h^CQ
= (—l)’»l lim </»,	= (-l)w «’?’> -= («’7» (p).
Л—Н»
Therefore we see that £ L2{Q) is, when considered as a generalized
function, the distributional derivative of /(0): /(,) = Ds /(0).
We have thus proved that the Hilbert space is a linear subspace
of the Hilbert space PK*(X2), the Sobolev space. In general Hq{<2) is a
proper subspace of TLA(/2). However, we can prove
Proposition. = PT* (7?*).
Proof. We know that the space Wk {Rn) is the space of all functions
/(%)€ L2{Rn) such that the distributional derivatives -Ds/(%) with |s| =
£ Sj k all belong to L2 {Rn), and the norm in Wk (7?") is given by
Let / £ Wk (Rn) and define fN by
= «w («)/(%),
where the function ocN{x) £ C^(Rn) (N = 1, 2, . . .) is such that
<^(7 = 1 for|x|^2Vand sup	[ < oo.
Then by Leibniz’ formula, we have
Zy/(x)-D7w(x) = 0 for |x|^lV,
= a linear combination of terms
Dl aN (%) - Duf(x) with |м I 4- |Z [ k for [ x | > IV.
Hence, by Ds f g L2 (R*] for s 1 k, we see tKit lim
1V—►OO
IP7n- l>7|[o = 0
and so lim
jV—>OO
н/х-а=о.
Therefore, it will be sufficient to show that, for any /С	with
compact support, there exists a sequence {fa{x}} QC%>{Rn] such that
lim 11/л — /||* = 0. To this purpose, consider the regularization of /
4—HOO
(see (16) in Chapter I, 1):
/o(*)= f НУ) — У) dy, a > 0.
К*
11. Factor Spaces of a В-space
59
By differentiation, we have
D‘fM= f f(y)D^(x~y)dy = (~l^ f I^D^.(x~y)dy
A”*	Rn
= C°s(where 6ax(y) = 0a(x — y\)
fi"
Hence, by Schwarz’ inequality,
Rn
dx
Uen
We know that the inner integral on the extreme right tends to 0 as e -> 0
(see Theorem 1 in Chapter 0, 3),and hence lim f | Dsfa (%) — Dsf(x)\2dx
л—*0 д fl
= 0. Thus lim fa — / j k = 0- Therefore, the completion Hq (Rn\ of Cq (7?")
л—>0
with regard to the norm || l|A is identical with the space Wk(Rn).
Corollary. Hk (Rn) = Hk (Rn) = Wk (Rn).
11.	Factor Spaces of a B-space
Suppose that X is a normed linear space and that M is a closed linear
subspace in X. We consider the factor space XjM, i. e., the space whose
elements are classes modulo M. In virtue of the fact that M is closed,
all these classes g are closed in X.
Proposition. If we define

= inf x
xtf 1
(1)
then all the axioms concerning the norm are satisfied by l|£!
Proof. If £ = 0, then £ coincides with M and contains the zero vector
of X; consequently, it follows from (1) that
£ || = 0. Suppose, conversely,
that 11 £ 11 = 0. It follows then from jl) that the class contains a sequence
{x„} for which we have lim |x„j =0, and hence the zero vector of X
n—Ю0
belongs to the closed set £ of X. This proves that £ = M and hence is
the zero vector in XjM.
Next suppose £, ?? £ X/M. By definition (1), there exists for any e > 0,
vectors x £ £, у £ т} such that
во
L. Semi-norms
Hence ||x 4- y||	||x|| 4- ||у|| ^5 Ilf II ' II *7II + Chi other hand,
(x 4- y) g (f 4- y), and therefore 4- r; l| ^ ||% 4 у |1 by (1). Con-
sequently, we have |f4-^||^s||f||4-||r/ 4- 2e and so we obtain the
triangle inequality |£4-^||ls||f||4“|j?7 .
Finally it is clear that the axiom oc£
= |a | ||£ [J holds good.
Definition. The space X/M, normed by (1), is called a normed factor
space.
Theorem. If X is a В-space and M a closed linear subspace in X, then
the normed factor space XfM is also a B-space.
Proof. Suppose {£„} is a Cauchy sequence in XfM. Then {£„} contains
a subsequence {£„fc} such that ||£ni:+l — $nk || < 2~k~2. Further, by definition
(1) of the norm in XfM, one can choose in every class (f„ — £Hjle)
a vector yk such that
1^,-M + 2-‘’2 <
Let vrti4 The series xrti 4- Уг 4" У2 + ' ' converges in norm and conse-
quently, in virtue of the completeness of X, it converges to an element x
of X. Let $ be the class containing x. We shall prove that f — s-lim
M—»-oo
Denote by sk the partial sum 4- уг 4- y2 4- *   4- Ул of the above
series. Then lim )|x —
= 0. On the other hand, it follows from the
relations xKi £ yp G (^+t — fWp) that sft £ f„fc+v and so, by (1),
||f—ik—as
k 00.
Therefore, from the inequality 11£ — || ^ 11£ —	11 4- 1111 and
the fact that {£„} is a Cauchy sequence, we conclude that lim |£ — £J|
ft—ЮО
= 0.
12.	The Partition of Unity
4
To discuss the support of a generalized function, in the next section,
we shall prepare the notion and existence of the partition of unity.
Proposition. Let G be an open set of Rn. Let a family {U} of open
subsets U of G constitute an open base of G: any open subset of G is
representable as the union of open sets belonging to the family {U}.
Then there exists a countable system of o^en sets of the family { U'} with
the properties:
the union of open sets of this system equals G,	(1)
any compact subset of G meets (has a non-void inter-
section with) only a finite number of open sets of this
system.	(2)
Definition 1. The above system of open sets is said to constitute a
scattered open covering of G subordinate to {U}.
12. The Partition of Unity	61
Proof of the Proposition. G is representable as the union of a countable
number of compact subsets. For example, we may take the system of all
closed spheres contained in G such that the centres are of rational coor-
dinates and the radii of rational numbers.
Hence we see that there exists a sequence of compact subsets K,
such that (i) Kr £ KT+4 (r = 1, 2, . . .), (ii) G is the union of K/s and
(iii) each Kr is contained in the interior of Kr+1. Set
Uf = (the interior of К,+г) — Kr_2 and V, = Kr— (the interior of ,
where for convention we set KQ = y = the void set. Then Ur is open and
OO
Vf is compact such that G = U V,. For any point x E Vr, take an open
T = 1
set f7(x; r) € {17} such that x£ U(x; r) £ Ur. Since Fris compact, there
лг
exists a finite system of points x(1), x(2),..., x(W such that Vr C U U (x(1J; r).
Then, since any compact set of G meets only a finite number of Uf's, it is
easy to see that the system of open sets U (x{l); r) (r = 1, 2,...; 1 < i LJ hr)
is a scattered open covering of G subordinate to {U).
Theorem (the partition of unity). Let G be an open set of Rn, and let
a family of open sets {Gt; 1} cover G, i.e., G = U G». Then there
exists a system of functions {ay (x); / E J} of C£° (&“) such that
for each j £ J, supp (a,) is contained in some Gt,	(3)
for every j	g J,	0 «у (%)	1,	(4)
£ Xj (x)	— 1	for x G G.	(5)
Proof.	Let	x'°> £ G and take	a Gi	which contains x(0>.	Let the	closed
sphere	S	(x(0);	r)	of	centre x(0)	and	radius r be	contained	in	We
construct, as in (14), Chapter I, 1, a function /%>(*) € Co°(.R*) such that
for I x — x™ I < r, 0%, (x) = 0 for |x —xl0> |Jjr.
We put 1% = {x - ^>, (x) ф 0}. Then U%, £ Gt and U<'>, = 6,
and, moreover, supp (/?<'>,) is compact.
There exists, by the Proposition, a scattered open covering
subordinate to the open base ; x101 £ G, r > 0} of G. Let &(x) be
JV  *
any function of the family	which is associated with Ц-.
Then, since {Uf, j £ J} is a scattered open covering, only a finite number
of /3y(x)’s do not vanish at a fixed point x of G. Thus the sum s(x) =
£ fy(x) is convergent and is > 0 at every point x of G. Hence the func-
tions

satisfy the condition of our theorem.
62
I. Semi-norms
Definition 2. The system {(Xj (x); j € J} is called a partition of unity
subordinate to the covering {G,; i £ I}.
13.	Generalized Functions with Compact Support
Definition 1. We say that a distribution T E ф (.Q)' vanishes in an
open set U of Q if T (99) = 0 for every 99 £ ф {p} with support contained
in U. The support of Tf denoted by supp (7”), is defined as the smallest
closed set F of Q such that T vanishes in Q — F.
To justify the above definition, we have to prove the existence of the
largest open set of Q in which T vanishes. This is done by the following
Theorem 1.	If a distribution T E ф(Х2)' vanishes in each U± of a family
{Ui ', i E 1} of open sets of Q, then T vanishes in U = U UP
Proof. Let <p E Ф (£?) be a function with supp (99) C U. We construct
a partition of unity {ay (x); / € J} subordinate to the covering of Q
consisting of {Ut', i^I} and P —supp (99). Then9?= £ <Xj<p is a finite
sum and so T{<p) =	If the supp (ay) is contained in some Uit
T (ptjfp) ~ ‘0 by the hypothesis; if the supp (ay) is contained inf? — supp (93),
then ay 97 = 0 and so T (0,99) — 0. Therefore we have T(<p) = 0.
Proposition 1. A subset В of the space @(f?) is bounded iff, for any
differential operator IF and for any compact subset К of 22, the set of
functions {7T/(x); / 6 B) is uniformly bounded on K.
Proof. Clear from the definition of the semi-norms defining the topo-
logy of @(£).
Proposition 2. A linear functional T on ® (p) is continuous iff T is
bounded on every bounded set of ®(f2).
Proof. Since ®(J2) is a quasi-normed linear space, the Proposition is
a consequence of Theorem 2 of Chapter I, 7.
Proposition 3. A distribution T E Ф (22)' with compact support can
be extended in one and only one way to a continuous linear functional
on (£ (f?) such that TQ (/) = 0 if / E ® (p) vanishes in a neighbourhood
of supp (T).
Proof. Let us put supp (T) = К where К is a compact subset of Q.
For any point x° 6 К and s > 0, we take a sphere S (x°, e) of centre
x° and radius e. For any s > 0 sufficiently small, the compact set К is
covered by a finite number of spheres S (x°, e) with E K. Let {ay (x) E J}
be the partition of the unity subordinate to this finite system of spheres.
Then the function y(x) =	£	where K' is a compact
supp(ay) П 4= void
neighbourhood of К contained in the interior of the finite system of
spheres above, satisfies:
y>(x) € Co°(£?) and y(x) = 1 in a neighbourhood of K.
13. Generalized Functions with Compact Support	63
We define To{/) for /G C°°(D) by T0(f) = T(ipf). This definition is in-
dependent of the choice of v. For, if £ Cq0 (D) equals 1 in a neighbourhood
of K, then, for any /G C°°{D), the function (y - Vh) ©(D) vanishes
in a neighbourhood of К so that T (ipf) — T (Vi/) = T ((ip —	/) — 0.
It is easy to see, by applying Leibniz’ formula of differentiation to
ipft that {ipf} ranges over a bounded set of ©(D) when {/} ranges over
a bounded set of (£(£?). Thus, since a distribution Tg ф (Q)' is bounded
on bounded sets of 2) (D), the functional TQ is bounded on bounded sets
of ®(D). Hence, by the Theorem 2 of Chapter I, 7 mentioned above, To is
a continuous linear functional on C (D). Let / G (g (Q) vanish in a neighbour-
hood U(K) of K. Then, by choosing ayGC^fD) that vanishes in
(Q — U(K)), we see that To(/) = T(ipf) = 0.
Proposition 4. Let K' be the support of ip in the above definition of To.
Then for some constants C and k
\T^}\^C sup \Dif(x)\ forall /€C°°(D).
Proof. Since T is a continuous linear functional on 2) (D), there exist ,
for any compact set K' of Q, constants C and k' such that
[ T (q>) |	C sup	forall
(the Corollary of Proposition 1 in Chapter I, 8). But, for any g £ C°°(D),
we have tp = ipgE	Consequently, we see, by Leibniz’ formula of
differentiation, that
sup \D>(ipg) (%) | C" sup |D}g(%) |
with a constant C" which is independent of g. Setting g = / and k = kf,
we obtain the Proposition.
Proposition 5. Let S() be a linear functional on C°°(D) such that, for
some constant C and a positive integer k and compact subset К oiQ,
So (/) | ^ C sup | Dj f (%) | for all / C C°° (Q).
Then the restriction of So to C50 (D) is a distribution T with support con-
tained in K.
Proof. We observe that So (/) = 0 if / vanishes identically in a neigh-
bourhood of K. Thus, if ip G Cg° (Q) „equals 1 in a neighbourhood of K, then
50(/)=50(у./) forall /GC°°{D).
It is easy to see that if {/} ranges over a bounded set of ©(D), then, in
virtue of Leibniz’ formula, {ipf} ranges over a set which is contained in
a set of the form
{g G C°°(D); sup	I = Ck < oo}.
64
I. Semi-norms
Thus 50(y7) = ?’(/) is bounded on bounded sets of so that T is
a continuous linear functional on ${£?).
We have thus proved the following
Theorem 2.	The set of all distributions in Q with compact support is
identical with the space ®(P)' of all continuous linear functionals on
<£(£?), the dual space of (£(£?). A linear functional T on	belongs to
iff, for some constants C and k and a compact subset К of Q,
|7-(/)|gC sup \D’f(x}
for all
We next prove a theorem which gives the general expression of distri-
butions whose supports reduce to a single point.
Theorem 3.	Let an open set Q of Rn contain the origin 0. Then the
only distributions T € $(£)' with supports reduced to the origin 0 are
those which are expressible as finite linear combinations of Dirac's
distribution and its derivatives at 0.
Proof. For such a distribution T, there exist, by the preceding Theo-
rem 2, some constants C and k and a compact subset К of Q which con-
tains the origin 0 in such a way that
sup \ОЩх)\ for all
bls vex
We shall prove that the condition
7У/(0) — 0 for all / with |/j k
implies T (J) = 0. To this purpose, we take a function у C which is
equal to 1 in a neighbourhood of 0 and put
Л W = t (x) у» (x/e) .
We have T(f) = T (/e) since / = /e in a neighbourhood of the origin 0.
By Leibniz' formula, the derivative of /e of order k is a linear com-
bination of terms of the form |e I^D’y,  7У/ with \i\ 4- |? | ^ k. Since,
by the assumption, 07(0) = 0 for |z[ k, we see, by Taylor’s formula,
that a derivative of order |s| of Д is 0(е*+1~^) in the support of
yf(x/e). Thus, when e J, 0, the derivatives of Д of order k converge to 0
uniformly in a neighbourhood of 0. Hence T (f) = lim T (/e) = 0.
Now, for a general /, we denote by fk the Taylor's expansion of / up
to the order k at the origin. Then,фу what we have proved above,
Л/) = T(/J + T(/-/A) = T(/s) + о = T(/A).
This shows that T is a linear combination of linear functionals in
the derivatives of / at the origin of order Si k.
14. The Direct Product of Generalized Functions
65
14.	The Direct Product of Generalized Functions
We first prove a theorem of approximation.
Theorem 1. Let x = (zn x2, . . xn) £Rn, у = (yx, y2, . . ., ym) € Rm
and z = хку = (x1, x2, . . xn, ylt y2, . . ., yj 6 Then, for any
function rp (z) = <p (x, у) C C£° (2?'rt+*”), we can choose functions (x) £ Cq° (!?”)
and functions (y) £	(R”’} such that the sequence of functions
<?i(2) = <Pi(x> У) = £	(x) Vij (у)	(1)
tends, as i —> oo, to tp (z) = <p (x, y) in the topology of Ф (Rn+™},
Proof. We shall prove Theorem 1 for the case n = ж = 1. Consider
Ф(х, у, t) = (2	f /??(£, ^) exp (—((x — £)2 + (y—?/)2)/42)^сбу,
£ > 0; Ф (x, y, 0) = <p (x, y).	(2)
We have, by the change of variables — (f — x)/2 |/£,	= {y — y)/2 j/if,
Ф (x, y, t) = (|A)~2 f f tp (x + 2^ j/t, у + 2 ]/t) d^dr^.
—oo — oo
OO 00
Hence, by f J e~i‘~T!aidg1dt}1 = л,
—oo —co
Ф{х,у,1)—(р{х>у) I (jAr)-2 J J j(p(x + 2g ]/t,y + 2 ц j/I)— <p{x, y)j
— OO —oo
X e-^did-q
s^-1/ ff + Я I-
Since the function <p is bounded and is integrable in R2, we see
that the first term on the right tends to zero as T f oo. The second term
on the right tends, for fixed T > 0, to zero as t j. 0. Hence we have
proved that НшФ(х, у, t} — <p(x, y) uniformly in (%, y).
Next, since suppfqs) is compact, we see, by partial integration,
Г+М(Ж y, i)	z>Oi
dxm8yk doJ  y	1
= , t = 0.
8xm 8yk	-
Thus we see as above that
+ У, /	аЖ + Ар ЛГ,у -г ! . / v
lim---- „ - г— - —р.—r ~ uniformly in {x, y).	(3)
UO 8xm8yk	8xm8yk	J
It is easy to see that Ф(х, у, i) for / > 0 given by (2) may be extended
as a holomorphic function of the complex variables x and у for |x| < oo,
5 Yuichi, KiiJictloiial Лпи1ун1н
1. Senn not nis
66
jy | < oo. Hence, for any given у > 0, the. function Ф(х, v, /•) for fixed
t > 0 may be expanded into Tavlor’s scries
oo m
Ф(*,у.0 = v Xc,(z)x>-S
m=0 s^O
which is absolutely and uniformly convergent for |x| < у, ]y[ у and
may be differentiated term by term:
аш+АФ(х, y,t} «> «}
8xm 8yV~ ~	& Cs ( )	‘
Let {ij be a sequence of positive numbers such that | 0. By the above
we can choose, for each tif a polynomial section Д(х, у) of the series
£ cs(t)^ym~s such that
ет=о s=o
lim Pi (x, y) = <p (x, y) in the topology of @ (R2),
that is, for any compact subset К of R2, lim DsPi(x, у) = Ds(p(x, у)
i—>oo
uniformly on К for every differential operator Ds. Let us takep (%) 6	(R1)
and c(y) G C^f-R1) such that p(x) a(y) = 1 on the supp (95 (x, y)). Then
we easily see that <^(х, y) = £>(%) n(y) Pi{x, y) satisfies the condition of
Theorem 1.
Remark. We shall denote by © (Rw) x © (Rm) the totality of functions
C © (Rw+m) which are expressible as
k
2Мх)щ(у) with ®(RW), %(y)€ ©(Rw).
The above Theorem 1 says that © (Rw) x © (Rm) is dense in ©(R"+w)
in the topology of ©(Rn+w). The linear subspace © (RH) X © (K*1) of
© (Rw+*”) equipped with the relative topology is called the direct product
ot^R*) and ©(Rw).
We are now able to define the direct product of distributions. To indi-
cate explicitly the independent variables x = (xlt %2, . . ., xj of the
function (p(x) E © (Rn), we shall write (©ж) for ©(R*). We also write
(©у) for ©(Rm) consisting of the functions y(y), у = (yx, y2, . . ., ym).
Likewise we shall write (ФжХу) for © (R”+w) consisting of the functions
% (x> y) • We shall accordingly write Tw for the distribution T G © (Rft)' =
(©*)' in order to show that T is to be applied to functions (x) of x.
Theorem 2. Let 7\) G 6\У) € (®>У- Then we can define in one
and only one way a distribution W = W^xy^ {%XXyY such that
W(u(x) y(y)) = TM(u{x))Sly}(y(y}') for	(®r), '(4)
№(<р(х,уУ) ^ Sly}\T^((f) (x,y))) = T[x}(S{y)(<p{x>y})) for (p€(®xxy) (5)
(Fubini’s theorem).
14. The Direct Product of Generalized Functions
67
Remark. The distribution W is called the direct product or the tensor
product of and S(y), and we shall write
Ж = TM x SM = SM X Tw.	(6)
Proof of Theorem 2. Let SB = {<p{x, y)} be a bounded set of the space
(Фжху). For fixed y(0), the set {9? (%, ym); <p £ SB} is a bounded set of
(фх). We shall show' that
{у (Л; у	= Tw (*> У(0))), <:P € SB}	(7)
is a bounded set of (®y(o)). The proof is given as follows.
Since SB is a bounded set of (Фх x y), there exist a compact set Kx C Rn
and a compact set Kv C Rm such that
supp (95) g {(x, y) 6 Rn+ftt; x 6 Kx, у € Ky} whenever 9? 6 SB .
Hence y!oJ 6 Ky implies <p(x, y(0)) = 0 and ^(yf0}) =	(<p(x, y(0))) = 0.
Thus
supp (у) C Ky whenever 9? E 9B.	(8)
We have to show that, for any differential operator Dy in Rm,
sup |Dyy>(y) I <00 where y)(y) =	<?€ 93.	(9)
To prove this, we take, e.g., Dyi = d/dyr Then, by the linearity of
У (У1 + ^> Уй> • - • , Ут) ~ У (У1> Уй>  •  , Ут)
h
(у(*. У1 + h, уг, . . ., ут) — <р{х, Ур у8, . . ут)'
	л	" г
When <р ranges over SB, the functions £ { } of x, with parameters у E Rm
and h such that [Л[ rg 1, constitute a bounded set of This we see
from the fact that 93 is a bounded set of (®xxy)- Hence we see, by letting
h—> 0 and remembering Proposition 1 in Chapter I, 8, that (9) is true.
Therefore, by the same Proposition 1, we see that the set of values
{Sw(ZwM*.y))); .?eS}	(10)
is bounded. Consequently, the same Proposition 1 shows that we have
defined a distribution PP(1)E (®жху)' through
Ж01 (?) = 5M (TM (y (x, y))].	(11)
Similarly we define a distribution W(2) £ (ф^ху)' through
W'w(9>) = rw(sM('?(^>'»)-	(13>
Clearly we have, for и £ (®J and v E (®y),
W(1)(u(x) v(y)) = Wi2)(u(x) v(y}) = Tw(«(x))  S{y}(y(y)).	(13)
Therefore, by the preceding Theorem 1 and the continuity of the
distributions and P02\ we obtain PF(1) = W(Zi. This proves our
Theorem 2 by setting W =	= P02'.
68
II. Applications of the Baire-Hausdorff Theorem
References for Chapter I
For locally convex linear topological spaces and Banach spaces, see
N. Bourbaki [2], A. Grothendieck [1], G. Kothe [1], S. Banach [1],
N. Dunford-J. Schwartz [1] and E. Hille-R. S. Phillips [1]. For
generalized functions, see L. Schwartz [1], I. M. Gelfand-G. E. Silov
[1], L. Hormander [6] and A. Friedman [1].*
IL Applications of the Baire-Hausdorff Theorem
The completeness of a В-space (or an F-space) enables us to apply
the Baire-Hausdorff theorem in Chapter 0, and we obtain such basic
principles in functional analysis as the uniform boundedness theorem, the
resonance theorem, the open mapping theorem and the closed graph theorem.
These theorems are essentially due to S. Banach [1]. The termwise
differentiability of generalized functions is a consequence of the uniform
boundedness theorem.
1. The Uniform Boundedness Theorem and the Resonance
Theorem
Theorem 1 (the uniform boundedness theorem). Let X be a linear
topological space which is not expressible as a countable union of closed
non-dense subsets. Let a family {Ta; a E A) of continuous mappings be
defined on X into a quasi-normed linear space У. We assume that, for
any a E A and x, у E X,
||Га(х + у)||	||Гах|| + ||Tay|| and ||Ta(ax)|| = ||«Гах||
for oc > 0.
If the set {Tax’} a^A] is bounded at each x£ X, then s-lim Tax = 0
x—>0
uniformly in a E A.
Proof. For a given e > 0 and for each positive integer n, consider
Xn = pc X' sup{||«-17'aa:|I + Ци-1 Ta(— x)||} g A Each set Xn is
I atA	J
closed by the continuity of Ta. By the assumption of the boundedness of
oo
{l| Tax ||; a £ A}, we have X = U Xn. Hence, by the hypothesis on
n=l
X, some ХПо must contain a neighbourhood U = x0 + V of some point
%0 E X, where V is a neighbourhood of 0 of X such that V — — V.
Thus x E V implies sup	4- x) || e. Therefore we have
я€Л
II Та(По1X) || = НГДио^Хо + X — *ой||	+ *) ||
I
4- 11По 1ТД(—х0) || ig 2e for xQV, a£A.
Thus the Theorem is proved, because the scalar multiplication ocx in a
linear topological space is continuous in both variables oc and %.
* See also Supplementary Notes, p. 466.
1. The Uniform Boundedness Theorem and the Resonance Theorem 69
Corollary 1 'the resonance -theorem). Let {Ta; a£ Л} be a family of
bounded linear operators defined on a В-space X into a normed linear
space Y. Then the boundedness of {|| Tax ||; a 6 A} at each x £ X implies
the boundedness of {||Ta||;	Л}.
Proof. By the uniform boundedness theorem, there exists, for any
8 > 0, a <5 > 0 such that ||x|| gj <5 implies sup || Tax | e. Thus
sup ||Te || e/д.	a£A
a(A
Corollary 2. Let {TM} be a sequence of bounded linear operators defined
on aB-space X into a normed linear space У. Suppose that s-lim Tnx — Tx
exists for each x С X. Then T is also a bounded linear operator on X into
У and we have
11 T11 lim 11Tn 11.
(1)
Proof. The boundedness of the sequence {|| Tnx ||} for each x£ X is
implied by the continuity of the norm. Hence, by the preceding Corol-
lary, sup ||TW|| < oo, and so ||7,„%|| sup ||ТЯ|| • || %|| (n = 1, 2, . ..).
Therefore, again by the continuity of the norm, we obtain
1*11»
which is precisely the inequality (1). Finally it is clear that T is linear.
Definition. The operator T obtained above is called the strong limit
of the sequence {ВЗ and we shall write T = s-lim Tn.
We next prove an existence theorem for the bounded inverse of a
bounded linear operator.
Theorem 2 (C. Neumann). Let T be a bounded linear operator on a
В-space X into X. Suppose that ||Z — T|| < 1, where I is the identity
operator: I  x = x. Then T has a unique bounded linear inverse T~x
which is given by C. Neumann s series
T-1x = s-lim (/ + (Z-T) + (Z-T)2 + • •• 4- (Z-T)”)%, x£X. (2)
n—>oo
Proof. For any x 6 X, we have
oo
The right hand side is convergent by |Z — T|| < 1. Hence, by the com-
k
pleteness of X, the bounded linear operator s-lim (Z — T)n is defined.
k—xo w=0
70
II. Applications of the Bairc-Hausdorff Theorem
It is the inverse of T as may be seen from
k	/ k
T • s-lim 2; (I - T)n x = s-lim (Z— (Z —?'))( 2? (Z - T)n x
Л—юо я==0	А-кю	u \«-0
~ x — s-lim (Z —	41 x _ x ,
A—>oo
/ k	\
and the similar equation s-lim ( 2? (Z — T)n } Tx = x.
k-xx \n=0	/
2.	The Vitali-Hahn-Saks Theorem
This theorem is concerned with a convergent sequence of measures,
and makes use of the following
Proposition. Let (S, 59, ж) be a measure space. Let 53O be the set
of all В C S3 such that m(B) < oo. Then by the distance
d(Bp B2) — m(B1 G B2), where BY G B2 = Bx \J B2 — Br C\ B2,	(1)
59O gives rise to a metric space (59O) by identifying two sets Br and B2
of 53O when т(Вг G B2) == 0. Thus a point В of (59O) is the class of sets
BXE 53O such that m(B G Bx) = 0. Under the above metric (1), (53O)
is a complete metric space.
Proof. If we denote by CB(s) the defining function of the set B:
Cg (s) = 1 or 0 according as s 6 В or s 6 В,
we have
d[Bx,B2) = f |CBi(s) — CBt{s) | m(ds).	(2)
s
Thus the metric space (53O) may be identified with a subset of the B-space
Ll(S, 59, ж). Let a sequence {CBn(s)} with BnE 53O satisfy the condition
lim d(B„,Bk)~ lim f |CBn(s) — CB (s) | m{ds) = 0.
n,k—>oo	л,А—>oo 5
Then, as in the proof of the completeness of the space L1 (S, 53, m), we can
choose a subsequence {€#„,($)} such that lim CBn,(s) = C(s) exists
ж-а.е. and lim f ]C(s) —CBn, (s) I m(ds) = 0. Clearly C(s) is the
n'—*oo
defining function of a set B^ G 59O, and hence lim ^(B^, Bn) = 0.
n—>oo
This proves that (59O) is a complete metric space.
Theorem (Vitali-Hahn-Saks). Let (S, 03, m) be a measure space with
m(S) < oo, and {2ЭТ(В)} a sequence of complex measures such that the
total variations |2n| (S) are finite for n — 1, 2, . . . Suppose that each
2n(B) is ж-absolutely continuous and that a finite lim 2„(B) — 2(B)
n—>cc
exists for every B£ 03. Then the ж-absolute continuity of 2n(B) is uniform
in n, that is, lim ж(В) = 0 implies lim ЛП(В) = 0 uniformly in n. And
2(B) is ог-additive on 03.
I
2. The Vitali-Hahn-Saks Theorem
71
Proof. Each лп defines a one-valued function An(B) on (S30) by
An(B) = Лп(В), since the value ЛДВ) is, by the ш-absolute continuity
of Я„(В), independent of the choice of the set В from the class B. The
continuity of Ян (В) is implied by and implies the ш-absolute continuity of
4(5).
Hence, for any e > 0, the set
Fk (e) = [B; sup | Лк (В) - 4+„ (В) | Л
I
is closed in (S30) and, by the hypothesis lim ЯМ(В) — Я(В), we have
n юс
oo
(S30) = U Fk(£)- The complete metric space (S30) being of the second
category, at least one FAo(s) must contain a non-void open set of (S30).
This means that there exist a BQ E (S30) and an щ > 0 such that
d(B, Bo) < rj implies
SUP |4.(B) — 4+„(B)| e.
Я^1
On the other hand, any В E 33O with m(B) can be represented as
В = Bx — B2 with d(Blf Во) ?y, d(B2, BQ) 5- т]. We may, for example,
take	Bo, B2 — BG — В A Bo. Thus, if ш(В) S ?? and k > kQ,
we have
|4(B)|^|4(B)[+ |4o(B)-4(B)
14(^)1 + 14(B,)-4(BJ
+ I4(B2)-4(B2)|
14(B)
Therefore, by the ш-absolute continuity of Xko and the arbitrariness
of e > 0, we see that m(B) -> 0 implies ЯМ(В) —> 0 uniformly in n. Hence,
in particular, m (B)	0 implies Я (В)	0. On the other hand, it is clear
/' n	\	n
that Я is finitely additive, i.e., Я( £ B} I =	Я(В;). Thus, by
\;=i	/	;=i
lim Я(В) = 0 proved above, we easily see that Я is ог-additive since
m(S) < oo.
Corollary 1. Let {Яя(В)} be a sequence of complex measures on S such
that the total variation |Я„ | (S) is finite for every n. If a finite lim Яя(В)
n—ЮО
exists for every В £ S3, then the a-additivity of {Яя(В)} is uniform in nt
in the sense that, lim Яя(Вл) = 0 jmiformly in n for any decreasing
fc-юо
oo
sequence {Bft} of sets € S3 such that П Bk = 0.
Proof. Let us consider
00
ж(В)= ^2-4,(B)
where Л.(В) = |Я,.| (B)/|A,.| (S) .
12
II Applications ol the Вдп< I! iisdortt theorem
I'he (j-additivity of m is a consequence ol that of the 2/s, and we have
0 5^ m(B) 5^ 1. Each p.} and hence each Я; is ш-absolutcly continuous.
Thus, by the above Theorem, we have lim Лм(/*л)	0 uniformly in n,
k f-CXJ
because lim m(B^ = 0.
Corollary 2. Я(В) in the Theorem is a-additive and m-absolutely
continuous even if m(S) = oo.
3.	The Termwise Differentiability of a Sequence
of Generalized Functions
The discussion of the convergence of a sequence of generalized func-
tions is very simple. We can in fact prove
Theorem. Let {7n} be a sequence of generalized functions E
Let a finite lim Tn (99) — T (99) exist for every 99 E ® (£?). Then T is also a
n—>OO
generalized function 6 ® (£?)'. We say then that T is the limit in ®(£?)'
of the sequence {Tn} and write T = lim T„(® (£)').
n—>00
Proof. The linearity of the functional T is clear. Let К be any compact
subset of Q. Then each Tn defines a linear functional on the F-space
®к-(£?). Moreover, these functionals are continuous. For, they are bounded
on every bounded set of %K(Q) as may be proved by Proposition 1 in
Chapter I, 8. Thus, by the uniform boundedness theorem, T must be a
linear functional on %K(Q) which is bounded on every bounded set of
Hence T is a continuous linear functional on every	Since
®(£?) is the inductive limit of $x(^)'s, T must be a continuous linear
functional on ® (£?).
Corollary (termwise differentiability theorem). Let T — lim Fw(®) (£?)').
Then, for any differential operator Z);, we have DjT = lim	(5) (£?)').
Proof, lim Tn = T	implies that lim Tn ((—1)1J,Z);9?) —
T((—1)IJ1/)J9?) for every 99 E ® (/2). Thus we have
(DjT) (99) = lim (DjTn) (99) for every 99 E ®(/2).
n-^00
4.	The Principle of the Condensation of Singularities
The Baire-Hausdorff theorem may be applied to prove the existence of
a function with various singularities. For instance, we shall prove the
existence of a continuous function without a finite derivative.
Weierstrass’ Theorem. There exists a real-valued continuous function
x(/) defined on the interval [0, 1] such that it admits a finite differential
quotient x' (/0) at no point /0 of the interval [0, 1/2].
4. The Principle of the Condensation of Singularities
73
Proof. A function admits finite upper- and lower-derivatives on
the right at t = t0 iff there exists a positive integer n such that
sup A-1
2“1>A>0
*(*o + k')~ %(Zo)|
Let us denote by Mn the set of all functions x(t) £ С [0, 1] such that
x(t) satisfies the above inequality at a certain point t0 of [0, 1/2]; here
the point t0 may vary with the function #(£). We have only to show that
oo
each Mn is a non-dense subset of С [0,1]. For, then, the set C [0,1] — 4J Mn
»=i
is non-void since; by the Baire-Hausdorff theorem, the complete metric
space C [0, 1] is not of the first category.
Now it is easy to see, by the compactness of the closed interval [0, 1/2]
and the norm of the space C [0, 1], that Mn is closed in C [0, 1]. Next,
for any polynomial z (/) and e > 0, we can find a function у (Z) C C[0,1] —Mn
such that sup |z(£)— у (t) | = | [ z — у 11 < e. We may take, for ex-
ample, a continuous function у (Z) represented graphically by a zig-zag
line so that the above conditions are satisfied. Hence, by Weierstrass’ poly-
nomial approximation theorem, Mn is non-dense in C [0, 1].
S.	Banach [1] and H. Steinhaus have proved a principle of conden-
sation of singularities given below which is based upon the following
Theorem (S. Banach). Given a sequence of bounded linear operators
{7\J defined on a В-space X into a normed linear space У„. Then the set
x£X\ lim ||Tnx
К XX
oo
either coincides with X or is a set of the first category of X.
Proof. We shall show that В — X under the hypothesis that В is of
the second category. By the definition of B, we have lim sup [|A-1Tn% | (
k—>OO
= 0 whenever x E В. Thus, for any e > 0,
oo
В Q U Bk where
вк =	X; sup 11 k~1 Tnx I
l
Each Bk is a closed set by the continuity of the Tn’s. Hence, if В is of the
second category, then a certain Bko contains a sphere of positive radius.
That is, there exist an %0£X and a d > 0 such that ||x — x0|	<5
implies sup 11 k^1 Tnx || e. Hence, *by putting x — x0 = y, we obtain,
for ||y|| <L д, ||йо1Т„у||	||^o17'„x|| + ||^о1Т„хо|| = 2£- We have
thus
sup || 7\tz||	2e,
and so В = X
74
II. Applications of the Baire-Hausdorff Theorem
Corollary (the principle of the condensation of singularities). Let
{Tp,q} (# = 1, 2, . . .) be a sequence of bounded linear operators defined
on a В-space X into a normed linear space Yp (p = 1, 2, . . .). Let, for
each p, there exist an xp G X such that lim [I TPqxp I = oo. Then the set
q—ню	v
В =	E X; lim 11TP qx I1 -- oo for all /> = 1,2,...}
I 7-ню	J
is of the second category.
Proof. For each p, the set Bp = {x£X\ lim	< oo} is,
by the preceding Theorem and the hypothesis, of the first category,
oo
Thus В = X — Bp must be of the second category.
The above Corollary gives a general method of finding functions with
many singularities.
Example. There exists a real-valued continuous function x (0 of
period 2 л such that the partial sum of its Fourier expansion'.
q	17
fq(%; t) = £ (ak cos kt + bk sin kt) = — / x(s) К (s, t) ds,
k=0	„	*
(1)
where Kg (s, t) ~ sin((<? -h 2-1)(s — Z))/2 sin 2-1 (s — t),
satisfies the condition that
lim [ — 00 on a set P C [0, 2тг] which is of the
7->oo	(2)
power of the continuum.
Moreover, the set P may be taken so as to contain any countable sequence
ft} C [0, 2 л].
Proof. The totality of real-valued continuous functions x (t) of period
2 л constitutes a real В-space C2n by the norm ||x|| = sup | %(/)].
As may be seen from (1), fq(x; t) is, for a given /0C [0, 2 л], a bounded
linear functional on C2n. Moreover, the norm of this functional £0)
is given by
1 c
— J Kq(s, t0) | ds — the total variation of the function
x	' (3)
— f Kg (s, i0) ds.
—n
It is also easy to see that, for fixed /0, (3) tends, as q oo, to oo.
Therefore, if we take a countable dense sequence {^} C [0, 2 л], then,
by the preceding Corollary, the set
В = {%еС2я; lim \fq(x;t)
q->oo
= oo for t = ip i2, .. .
5. The Open Mapping Theorem
75
is of the second category. Hence, by the completeness of the space C^,
the set В is non-void. We shall show that, for any x £ B, the set
P = u € [0, 2jt] ; lim \fq(x', t) I = oo
is of the power of the continuum. To this purpose, set
Fm,q = {*€ [0, 2л;]; \fq{x-, /)| <Z m},
By the continuity of x(t) and the trigonometric functions, we see
that the set Fm q and hence the set Fm are closed sets of [0, 2 л]. If we
oo
can show that U Fm is a set of the first category of [0, 2 л], then the
(oo \
[0,2л]— U Fm|3{73 would be of the second category.
tn=l / J
Being a set of the second category of 0, 2л], P cannot be countable and
so, by the continuum hypothesis, P must have the power of the conti-
nuum.
Finally we will prove that each Fw is a set of the first category of
[0, 2 л]. Suppose some FWo be of the second category. Then the closed
set Fmc of [0, 2 л] must contain a closed interval [a, /5] of [0, 2 л]. This
implies that sup |/?(%; t) | mQ for all t E [a, /?], contradicting the fact
that the set P contains a dense subset {/•} of [0, 2 л].
Remark. We can prove that the set P has the power of the con-
tinuum without appealing to the continuum hypothesis. See, for
example, F. Hausdorff [1].
5. The Open Mapping Theorem
Theorem (the open mapping theorem of S. Banach). Let T be a con-
tinuous linear operator on an F-space X onto an F-space Y. Then T maps
every open set of X onto an open set of Y.
For the proof, we prepare
Proposition. Let X, Y be linear topological spaces. Let T be a con-
tinuous linear operator on X into Y, and suppose that the range R(T)
of T is a set of the second category in Y. Then, to each neighbourhood
U of 0 of X, there corresponds some neighbourhood V of 0 of Y such that
V C (TU)a.
Proof. Let W be a neighbourhood of 0 of X such that W — — W,
И7 4- W C U. For every x 6 X, xjn > 0 as n-> oo, and so x £ nW for
oo	oo
large n. Hence X = U (nW), and therefore R(T) = U T(nW). Since
n=l	H=1
R (T) is of the second category in Y, there is some positive integer n0 such
that (T (w0IF))a contains a non-void open set. Since (T (n PT))a = n(T (РИ))"
76
11. Applications of the 13aire-Hausdorff Theorem
and since н(7 (И))" and (7'(W))e are homeomorphic1 to each other,
(/'(IF))“ also contains a non-void open set. Let y0 = TxQj %0E W, be
a point of this open set. Then, since the mapping x-> — xQ 4- x is a
homeomorphic mapping, we see that there exists a neighbourhood V
of 0 of У such that V C — y0 4- (T (Ж))л. The elements of —y0 4- T (PL)
are expressible as — y0 4- Tw = T(w ~ xQ) with w £ W. But w — x0 E
IV 4- W Q U, since W is a neighbourhood of 0 of X such that
W = — W and W 4- W QU.
Therefore, —y0 4- T(PL) QT(U) and hence, by taking the closure
-y0 4- (T(PF))a С (T (U))“ and so V Q ~yQ 4h (T(W))a Q (T(U))a = (T U)<\
Proof of the Theorem. Since У is a complete metric space, it is of the
second category. Thus, by the preceding Proposition, the closure of the
image by T of a neighbourhood of 0 of X contains a neighbourhood of 0
of У.
Let X£t Ye denote the spheres in X, Y respectively, with centres
at the origins and radii 8 > 0. Let us set — s/2* (i = 0, 1, 2, . .
Then by what we have stated above, there exists a sequence of positive
numbers such that
lim ^ = 0 and Ym Q(TXei)a (i - 0, 1, 2, . . .).	(1)
Let у E У% be any point. We shall show that there is an x E X2e<j such
that Tx = y. From (1) with i — 0, we see that there is an x0 E XEo such
that ||y~ r%oli < Since (у — Tx0) E YVi, we see again from (1)
with i = 1 that there is an xt E with 11 у — T x0 — T хг 11 < ?;2.
Repeating the process, we find a sequence {%J with E XCi such that
We have
(n = 0, 1, 2,...).
n	n
11 x^ I I
k — tn + 1	Л = т+1
so the sequence
e0>
and
is a Cauchy sequence. Hence, by the complete-

«
n
ness of X, we see that s-lim Jt" xk = x E X exists. Moreover, we have
n—>oo A=0
||% || = lim
n—КЮ
n	/ oo	\
lim 11 xk 11	2~k) s0 = 2 s,
n—*oo A=0	\ft=0	/
We must have у ~ Tx since T is continuous. Thus we have proved that
any sphere X2Co is mapped by T onto a set which contains a sphere У^.
After these preliminaries, let G. be a non-void open set in X and
x E G. Let U be a neighbourhood of 0 of X such that x 4- U C G. Let V
1 A one-to-one mapping M of a topological space onto a topological
space S2 is called a homeomorphic mapping if M and M-1 both map open
sets onto open sets.
6. The Closed Graph Theorem
77
be a neighbourhood of 0 of Y such that TU^V. Then T G 2 T (x + U) =
Tx + TU 2Tx + V. Hence T G contains a neighbourhood of each one
of its point. This proves that T maps open set of X onto open sets of У.
Corollary of the Theorem. If a continuous linear operator T on an
F-space onto an F-space gives a one-to-one map, then the inverse operator
is also a linear continuous operator.
6. The Closed Graph Theorem
Definition 1. Let X and Y be linear topological spaces on the same
scalar field. Then the product space X x Y is a linear space by
{^i> У1}	{#2, У2} ~ {-^i "b ^2» У1	у} — {л%, лу}.
It is also a linear topological space if we call open the sets of the form
Gi X G2 = {{x- y} • x£Gi, y£G2},
where G2 are open sets of X, Y respectively. If, in particular, X and
Y are quasi-normed linear spaces, then XxY is also a quasi-normed
linear space by the quasi-norm
||{*>y}|| = (INI2 + lbll2)1/2-	U)
Proposition 1. If X and Y are В-spaces (F-spaces), then Xx Y is also
a В-space (F-space).
Proof. Clear since s-lim {xn, ул} = {x, y} is equivalent to s-lim xn = x>
n—>oo
s-lim yn = у.
n—ЮО
Definition 2. The graph G(T) of a linear operator T on D (T) С X into
Y is the set {{%, T x}; x £ D (F)} in the product space XxY. Let X, Y be
linear topological spaces. Then T is called a closed linear operator when
its graph G(T) constitutes a closed linear subspace of Xx Y. If X and Y
are quasi-normed linear spaces, then a linear operator T on D (T) Q X
into Y is closed iff the following condition is satisfied:
{%„} QD(T), s-lim xn = x and s-lim T xn — у imply that
H—>00	n—>OO
%EO(T) and Tx = y.
Thus the notion of a closed linear operator is an extension of the notion
of a bounded linear operator. A linear operator T on D (F) £ X into Y is
said to be closable or pre-closed if the closure in X x Y of the graph G (T)
is the graph of a linear operator, say S, on D (S) Q X into Y.
Proposition 2. If X, Y are quasi-normed linear spaces, then T is
closable iff the following condition is satisfied:
C D(F), s-lim xn = 0 and s-lim Txn = у imply that у = 0. (3)
«-►oo	n—►oo
78
II. Applications of the Baire-Hausdorff Theorem
Proof. The ‘"only if” part is clear since the closure in Xx Y of G(T)
is a graph G (S) of a linear operator S and so у = S • 0 = 0. The “if” part
is proved as follows. Let us define a linear operator S by the following
condition and call S the smallest closed extension of T:
x£ D(S) iff there exists a sequence {xn} Q D{T) such that
s-lim xn ~ x and s-lim T xn = у exist; and we define	(4)
>oo	n—>oc
У-
That the value у is defined uniquely by x follows from condition (3).
We have only to prove that S is closed. Let wn £ D (S), s-lim wn = w and
n—ЮО
s-lim S wn = u. Then there exists a sequence {%n} C D (T) such that
n—>oo
||^л —	||Swrt—Тхп\\^п~г (n — 1, 2, . . .). Therefore
s-lim xn = s-lim wn — w, s-lim Txn = s-lim Swn — u, and so w G D (S),
Л—>oo	n—>oo	n—>00	n—W3O
S • w = u.
An example of a discontinuous but closed operator. Let X = У —
С [0, 1]. Let D be the set of all x (t) € X such that the derivative x' (t) £ X;
and let T be the linear operator on D (T) == D defined by Tx = x'. This
T is not continuous, since, for xn (t) = f1,
11 %П 11   1 > I | xn 11  
sup
14 (01 = sup
11 = n

But T is closed. In fact, let {%„} C D (T), s-lim xn = x and s-lim Txn = y.
n—>oo	n—>oo
Then x'n(t) converges uniformly to y(/), and xn(t) converges uniformly
to x(t). Hence x(t) must be differentiable with continuous derivative
у (t). This proves that x € D (T) and Tx = y.
Examples of closable operators. Let Dx be a linear differential operator
At = , 2 cj (x) Dj	(5)
I7|
with coefficients Cj{x) 6 Ck(Q)y where Q is an open domain of Rn. Consider
the totality D of functions / (x) E L2 (JQ) А Ck (Q) such that Dxf (%) E L2 (jQ).
We define a linear operator T on D (T) = D Q L2 (£?) into L2 (L?) by
(Tf) (x) = Dxf(x). Then T is closable. For, let {fh} Q D be such that
s-lim Д = 0, s-lim Dx fh = g. Then, for any (%) £ Cq(£?), we have, bv
Л—юо	h—кю
partial integration,
dx = f f (x) • D'x<p (x) dx,	(6)
where D'x is the differential operator formally adjoint to Dx\
D'x<p (x) =	(-l)W D’(Cj (x) v (x)).
(7)
6. The Closed Graph Theorem
79
The formula (6) is obtained, since the integrated term in the partial
integration vanishes by <?(%)€ Cq(£?). Hence, by the continuity of the
scalar product in L2(Q), we obtain, by taking f — fh and letting h —> oo
in (6),
J ъ (x) * 9? (%) dx = f 0 • D'x<p(x) dx = 0.	(8)
Q	Я
Since (p(x) 6 Cq(Q) was arbitrary, we must have g(x) = 0 a.e., that is,
g — 0 in L2(Q).
Proposition 3. The inverse of a closed linear operator on D (T)
С X into Y, if it exists, is a closed linear operator.
Proof. The graph of T~T is the set {{Fx, x}; x £ D(F)} in the product
space Y x X. Thus the Proposition is proved remembering the fact that
the mapping {%, y} -> {y, x) of X X У onto Y xX is a homeomorphic one.
We next prove Banach's closed graph theorem'.
Theorem 1. A closed linear operator defined on an F-space X into an
F-space У is continuous.
Proof. The graph G (T) of T is a closed linear subspace of the F-space
ХхУ. Hence, by the completeness of XxY, G(T) is an F-space. The
linear mapping U defined by U{x, Tx} = x is a one-to-one, continuous
linear transformation on the F-space G(T) onto the F-space X. Hence,
by virtue of the open mapping theorem, the inverse U^1 of Uis continuous.
The linear operator У defined by V{%, Tx} = Tx is clearly continuous
on G (F) onto R (F) С У. Therefore, T =V C7-1 is continuous on X into У.
The following theorem of comparison of two linear operators is due to
L. Hormander:
Theorem 2. Let X^i = 0, 1, 2; Xo = X) be В-spaces and 7\ (i = 1, 2)
be linear operators defined on В(Т^) Q X into X.. Then, if 7\ is closed
and T2 closable in such a way that C D(T2), there exists a con-
stant C such that
|Т2%П = ^(li^i%l|2 "b ll%l|2)1/2 f°r aU B^FJ. (9)
Proof. The graph G(F1) of 7\ is a closed subspace of XxXv Hence
the mapping
С(Л) Xх’ Т1Х}~>	(1°)
is a linear operator on the В-space G^Tj) into the В-space X2. We shall
prove that this mapping is closed. Suppose that {xn, Тгхп} s-converges
in G(Tr) and that T2xn s-converges in X2. Since T\ is closed, there is an
element D(FX) such that x = s-lim xn and 7\x = s-lim TYxn. By the
n—>OO	W—КЮ
hypothesis, we have %ЕР(Г2), and, since T2 is closable, the existing
s-lim T2xn can only be T2x. Hence the mapping (10) is closed, and so,
я—ню
by the closed graph theorem, it must be continuous. This proves (9).
80
II. Applications of the Baire-Hausdorff Theorem
7- An Application of the Closed Graph Theorem
(Hormander’s Theorem)
Any distribution solution w 6 £2 of the Laplace equation
Zh/ — f L2
is defined by a function which is equal to a C°° function after correction
on a set of measure zero in the domain where / is C°°. This result is known
as Weyl’s Lemma and plays a fundamental role in the modern treatment
of the theory of potentials. See H. Weyl [1]. There is an abundant
literature on the extensions of Weyl’s Lemma. Of these, the research
due to L. Hormander [1] seems to be the most far reaching. We shall
begin with his definition of hypoellipticity.
Definition!. Let Q be an open domain of A”. A function и (%),
is said to belong to Zfoc (£?) if J | и (%) |2 dx < oo for any open subdomain
Q'
Q' with compact closure in Q. A linear partial differential operator P (D)
with constant coefficients:
P (Z)) = p (A А Д. A	-1 A), where
' 1	\г г дх2	г дхп/	Ц)
P(f) =	f2, . . ., fn) is a polynomial in . .,
is said to be hypo elliptic, if every distribution solution	°f
P (D) и — f is C°° after correction on a set of measure zero in the sub-
domain where / is C°°.
Theorem (Hormander). If P(D) is hypoelliptic, then there exists, for
any large positive constant Clf a positive constant C2 such that, for any
solution £ = £ + it] of the algebraic equation P(C) = 0, we have
Proof. Let U be the totality of distribution solutions и 6 L2 (&') of
jP(D) и — 0, that is, the totality of и £ L2(£?') such that
f и • P'(D) <p dx = 0 for all <p £	,	(3)
я'
where the adjoint differential operator P' (D) of P(D) is defined by
P'(f) = P(-^,-^2................................(4)
We can prove that U is a clo&d linear subspace of L2 (£?'). The linearity
of U is obvious from the linearity of the differential operator P (D). Let
a sequence of U be such that s-lim uh = и in L2(Qr). Then, by the
h—юо
continuity of the scalar product in Ь2(П'), we have
0 = lim f uh • Pf [D) cp dx — f и • P' (D) qdx — 0, i.e., и £ (7.
Л-КЮ
1. The Orthogonal Projection
81
Thus U is a closed linear subspace of L2 (£?'), and as such a B-space.
Since P(D) is hypoelliptic, we may suppose that every и G U is C°°
in£?'. Letbe any open subdomain with compact closure ini?'. Then,
for any и E U, the function du/dxk is C°° in £2' (k = 1, 2, . . ., n). By the
argument used in the preceding section, the mapping

(k = 1, 2, . . n)
is a closed linear operator. Hence, by the closed graph theorem, there
exists a positive constant C such that
и 2 dx,
for all и £ U.
If we apply this inequality to the function u(x) = ег(^\ where £ =
f 4- irj = (%i + irjlt f2 + irj2, -	+ iVn) is a solution of P(£) = 0
n
and <x,	(x3, C7>, we obtain
7 = 1
|C*|2- f e~2<*,1?} dx C J e~2<x,r}>dx.
k=1 n;	n'
Therefore, when q is bounded, it follows that |£| must be bounded.
Remark. Later on, we shall prove that condition (2) implies the
hypoellipticity of P(D). This result is also due to Hormander. Thus, in
particular, we see that Weyl’s Lemma is a trivial consequence of Hor-
n
mander’s result. In fact, the root of the algebraic equation — G = 0
7 = 1
satisfies (2).
References for Chapter II
S. Banach [1|, N. Bourbaki 2], N. Dunford-J. Schwartz [1],
E. Hille-R. S. Phillips [1] and L. Hormander [6].
III. The Orthogonal Projection and F. Riesz* Representation
Theorem
1. The Orthogonal Projection
In a pre-Hilbert space, we can introduce the notion of orthogonality
of two vectors. Thanks to this fact, a Hilbert space may be identified
with its dual space, i.e., the space of bounded linear functionals. This
result is the representation theorem of F. Riesz [1], and the whole theory
of Hilbert spaces is founded on this theorem.
6 Yoaida, Functional Лпа1уя1я
82 111. I he Orthogonal Projection and b'. Ries/.’ I<<• presentation Theorem
Definition 1. Let %, у be vectors of a pre-I lilbei 1 space X. We say that
x is orthogonal to у and write x J_ у if ( v, y) 0; if x | у then у | x,
and x | x iff x = 0. Let M be a subset of a pre-Hilbert space X, We
denote by M-1- the totality of vectors С X orthogonal to every vector m
of M,
Theorem 1.	Let M be a closed linear subspace of a Hilbert space X.
Then M1- is also a closed linear subspace of X, and M-1- is called the
orthogonal complement of M. Any vector x £ X can be decomposed
uniquely in the form
x = m + n, where m E M and n E M1-.	(1)
The element m in (1) is called the orthogonal projection of x upon M and
will be denoted by P^x‘>Pm is called the projection operator or the
projector upon M. We have thus, remembering that M С (M1-)1-,
x =	+ PM±x> that is, I = PM +	.	(1')
Proof. The linearity of M1 is a consequence of the linearity in x of
the scalar product (x, y). M1 is closed by virtue of the continuity of the
scalar product. The uniqueness of the decomposition (1) is clear since a
vector orthogonal to itself is the zero vector.
To prove the possibility of the decomposition (1), we may assume
that M X and x Ё M, for, if x E M, we have the trivial decomposition
with m = x and n = 0. Thus, since M is closed and x Ё M, we have
d = inf 11 x — m 11 > 0.
Let {mnj С M be a minimizing sequence, i.e., lim llx — mn II = d. Then
{wn} is a Cauchy sequence. For, by 11 a + b 112 4- 11 a — b 112 = 2 (|[ a 1|2 +|1 b 112)
valid in any pre-Hilbert space (see (1) in Chapter I, 5), we obtain
2
n
n
n
mA||2— 4<Z2)
(since (mn + mk)/2 E M)
—> 2 (d2 + d2) — 4t/2 = 0 as k, n —> oo.
By the completeness of the Hilbert space X, there exists an element m £ X
such that s-lim mn = m. We have MsinceM is closed. Also, by the
continuity of the norm, we have — m|| = d.
Write x = m + (x — m). Putting n = x — m, we have to show that
n £ M1-. For any m' E M and any real number a, we have (m + am') E M
1. The Orthogonal Projection
83
and so
d2	11 я — m — a m' 1|2 = (n ~~ oc m', n — a m')
= 11 n 1|2 — a (n, m') — oc (m’, n) + a2 | | ni 112.
This gives, since ||w|| = d, 0	— 2ocRe(n, m') + a2 ||m' ||2 for every
real a. Hence Re(n, mf) = 0 for every m' £ M. Replacing m' by im ,
we obtain Im(n, m') — 0 and so (n, m') = 0 for every M.
Corollary. For a closed linear subspace M of a Hilbert space X, we
have M =	= (M-1-)1.
Theorem 2.	The projector P = PM is a bounded linear operator such
that
P — P2 {idempotent property of P),	(2)
(Px, y) == (x, Py) (symmetric property of P).	(3)
Conversely, a bounded linear operator P on a Hilbert space X into X
satisfying (2) and (3) is a projector upon M = R(P)-
Proof. (2) is clear from the definition of the orthogonal projection.
We have, by (Г) and Рмх | PM±y,
{PMX, y) = {PMX, PMi.y + PMy) = (PMx, PMy)
= (PmX + Рм-i-X, PMy) = (x, PMy).
Next let у = x 4- z, x 6 M, z С and w = и + v, uQ M, v£M±t then
у 4- w == (x + u) + (z + v) with (x + w) 6 M, (z + v) 6 M1- and so,
by the uniqueness of the decomposition (1), PM (y 4- w) = PMy 4- PM w\
similarly we obtain Pm fay) = ocPMy. The boundedness of the operator
PM is proved by
INI2 =	+ Pm± XII2 = (PMx + PMxx, PMx + PMxx)
= ||P^!I2+ !1Ли±*||2^ ЦЗД!2-
Thus, in particular, we have
(4)
The converse part of the Theorem is proved as follows. The set
M = R (P) is a linear subspace, since P is a linear operator. The condition.
% E Mis equivalent to the existence of a certain у £ X such that x = P • y,
and this in turn is equivalent, by (2), to x — Py = P2y ~ Px. Therefore
x£ M is equivalent to x = Px. M is a closed subspace; for, xn^Mf
s-iim xn = у imply, by the continuity of P and xn — Pxn, s-lim xn =
я—K>0	n—ЮС
s-lim Pxn = Py so that у — Py.
n—¥<X>
We have to show that P = PM. If x С M, we have Px = x = PM • x;
and if v E M-1-, we have РмУ = Moreover, in the latter case, (Py, Py) =
6*
84	111. 'I he Orthogonal Projection and b\ Kirsz* K< presentation Theorem
(y, P2y) = (y,Py) = 0 and so Py — О. i'heivforr we obtain, for any
у ex,
Py = P(PMy + PMJty) = РРмУ I /’/wiv
= РмУ + 0, i.e., Py = PMy.
Another characterization of the projection operator is given by
Theorem 3.	A bounded linear operator P on a Hilbert space X into X
is a projector iff P satisfies P = P2 and j | P11	1.
Proof. We have only to prove the “if” part. Set M = R{P) and
N (P) — {y; Py = 0}. As in the proof of the preceding Theorem 2,
M is a closed linear subspace and x 6 M is equivalent to % = Px. N is
also a closed linear subspace in virtue of the continuity of P. In the
decomposition x — Px + (I — P)x, we have Px£M and (I — P) x 6 X.
The latter assertion is clear from P (I — P) = P — P2 = 0.
We thus have to prove that X = Mr. For every x^X, у = Px — x^X
by P2 = P. Hence, if, in particular, x 6 N1, then Px = x 4- у with
(x, y) = 0. It follows then that	||Px 112 = 11x (|2 4- ||y||2 so that
у = 0. Thus we have proved that	implies x = Px, that is,
TV1- QM — R(P). Let, conversely, zQM = R(P), so that z = Pz. Then
we have the orthogonal decomposition z = у x, yeX, N-1-, and
so z = Pz = Py 4- Px = P% = x, the last equality being already
proved. This shows that M = R(P) СЛ71. We have thus obtained
M = X\ and so, by N = (AN-)-1, X = M1.
2.	“Nearly Orthogonal” Elements
In general, we cannot define the notion of orthogonality in a normed
linear space. However, we can prove the
Theorem (F. Riesz [2]). Let X be a normed linear space, and M be
a closed linear subspace. Suppose M X. Then there-exists, for any
€ > 0 with 0 < £ < 1, an xe С X such that
(1)
"	M
The element xE is thus “nearly orthogonal” to M.
Proof. Let у £ X — M. Since Mis closed, dis (у, M) = inf lly — ш|| =
т С M
0. Thus there exists an w£CM>such that ||y —me|| a ^14-
The vector xe = (y — me)j||у — satisfies ||%e || = 1 and
£
1 — £
a
= 1 — £ .
1 — £.
3. The Ascoli-Arzela Theorem
85
Corollary 1. Let there exist a sequence of closed linear subspaces Mn
of a normed linear space X such that Mn Q Mn+l and Mn Mn+1
(n ~ 1, 2, . . .). Then there exists a sequence {yn} such that
Упб Mn, ||y„|| = 1 and dis(yn+1,Mn) 1/2 (n= 1, 2, ...). (2)
Corollary 2. The unit sphere S — {%£ X\ ||x||	1} of a В-space X
is compact iff X is of finite dimension.
Proof. If x2, . . ., xn be a base for X, then the map (alf a2, . . an) ->
n
-> аэхз °f Rn onto X is surely continuous and so it is open by the
7 = 1
open mapping theorem. This proves the “if” part. The “only if” part
is proved as follows. Suppose X is not of finite dimension. Then there
exists, by the preceding Corollary 1, a sequence {yw} satisfying the condi-
tions: ||yM || = 1 and 11	— yn ||	1/2 for m > n. This is a contradiction
to the hypothesis that the unit sphere of X is compact.
3.	The Ascoli-Arzela Theorem
To give an example of a relatively compact infinite subset of a
В-space of infinite dimension, we shall prove the
Theorem (Ascoli-Arzela). Let S be a compact metric space, and
C (S) the В-space of (real- or) complex-valued continuous functions x (s)
normed by |1 x || = sup | x (s) Then a sequence {#rt (s)} С C (S) is relatively
s(5
compact in C (S) if the following two conditions are satisfied:
xn(s) is equi-bounded (in n), i.e., sup sup |*n(s) | < oo, (1)
niSl s € s
xn(s) is equi-continuous (in n), i.e.,	(2)
lim sup | xn (s') — xn (s")
<5 J-0
Proof. A bounded sequence of complex numbers contains a convergent
subsequence (the Bolzano-Weierstrass theorem). Hence, for fixed s, the
sequence {%M(s)} contains a convergent subsequence. On the other hand,
since the metric space S is compact, there exists a countable dense subset
{srt} Q S such that, for every s > 0, there exists a finite subset
{sn; 1 < n k(e)} of {$„} satisfying the condition
sup inf dis(s,	(3)
s£S
The proof of this fact is obtained as follows. Since S is compact, it is
totally bounded (see Chapter 0, 2). Thus there exists, for any <5 > 0, a
finite system of points € 5 such that any point of S has a distance <5
from some point of the system. Letting <5 = 1, 2-1, 3’1, . . . and collecting
the corresponding finite systems of points, we obtain a sequence {sw}
with the stated properties.
86 III. The Orthogonal Projection and F. Riesz’ Representation Theorem
We then apply the diagonal process of choice to the sequence {xM(s)},
so that we obtain a subsequence {%n,(s)} of {%M(s)} which converges for
s = slf s2, . . ., Sb . . . simultaneously. By the equi-continuity of {%w(s)},
there exists, for every e > 0, a d = <5(s) > 0 such that dis (s', s') d
implies | %„($') — xn(s") | rg e for n = 1, 2, . . . Hence, for every
there exists a j with / k (e) such that
|	($) — Xm> (S) | ^ | Xn> (s) — Xn. (Sj) |	|	&) — Xm> (Sj) |
I' ($y)	(s) I 2e -J- ]Хл' (Sy) Xm' (Sy) | .
Thus lim max|xw'(s)—(s) I 2s, and so lim ||%„—|| = 0.
n,m—кю	s	n,m—xxj
4.	The Orthogonal Base. Bessel’s Inequality and
Parseval’s Relation
Definition 1. A set S of vectors in a pre-Hilbert space X is called
an orthogonal set, if x 1 у for each pair of distinct vectors x, у of S. If,
in addition, ||%|| = 1 for each x£ S, then the set S is called an ortho-
normal set. An orthonormal set S of a Hilbert space X is called a complete
orthonormal system or an orthogonal base of X, if no orthonormal set of X
contains S as a proper subset.
Theorem 1. A Hilbert space X (having a non-zero vector) has at
least one complete orthonormal system. Moreover, if S is any ortho-
normal set in X, there is a complete orthonormal system containing S as
a subset.
Proof (by Zorn's Lemma). Let S be an orthonormal set inX. Such a set
surely exists; for instance, if x 7^ 0, the set consisting only of %/||x|| is
orthonormal. We consider the totality {S} of orthonormal sets which
contain S as a subset. {S} is partially ordered by writing Sx S2 for
the inclusion relation Sj £ S2. Let {S'} be a linearly ordered subsystem
of {S}, then 1J S' is an orthonormal set and an upper bound of
{S'}. Thus, by Zorn's Lemma, there exists a maximal element So of {S}.
This orthonormal set So contains S and, by the maximality, it must be
a complete orthonormal system.
Theorem 2. Let S = {xa \ a € A} be a complete orthonormal system
of a Hilbert space X. For any f £ X, we define its Fourier coefficients
(with respect to S)
f»={f,Xa).	(1)
Then we have Parseval’s relation
Proof. We shall first prove Bessel’s inequality
(2)
(2')
4. The Orthogonal Base. Bessel's Inequality and Parseval’s Relation 87
Let <xlt a2> • • • > an be any finite system of <x’s. For any finite system
of complex numbers	, can, we have, by the orthonormality of
2
n
Hence the minimum of / — ca. xt
attained when c
n
ai
2
, for fixed a2, . . ., ocn, is
at = f«, (/ = 1, 2, . . ., n). We have thus
7 = 1
n
2, and hence
7=1
2^II/F
(4)
«7
By the arbitrariness of alt oc2> . . ocn, we see that Bessel's inequality (2')
is true, and fx 0 for at most a countable number of a's, say
n
л2, . . ., ocn> . . . We then prove that / = s-Em £ f x . First, the
1	n—>ocj=l ’
is a Cauchy sequence, since, by the orthonormality
2
n
which tends, by (4) proved above, to Oas Л->оо. We set s-lim	= f,
n—>ooj=l ’
and shall prove that (/ — /') is orthogonal to every vector of S. By the
continuity of the scalar product, we have
П—ЮС
«ft X<*k> X<*j

== 0 — 0 = 0.
(f t > % a) — lim (f fajc хЛк, xa
Thus, by the completeness of the orthonormal system S = {#a}, we
must have (/ — /') = 0. Hence, by (4) and the continuity of the norm,
we have
n
2
n
n—>oo
я—>oo j=l
a£A
Corollary 1. We have the Fourier expansion
oo
n
(5)
88 HI. The Orthogonal Projection and E. Kiesz’ Representation Theorem
Corollary 2. Let I2 (A) be the space L2(A, m) where m ({<%}) = 1 for
every point oc of A. Then the Hilbert space X is isometrically isomorphic
to the Hilbert space /2(Л) by the correspondence
X Э/<>{/«}€ И)
(6)
in the sense that
pint.
v •
Example.
system in the Hilbert space L2(0, 2л).
Proof. We have only to prove the completeness of this system. We
have, by (3),
n 1
|2л
is a complete orthonormal
/ = -n
j = —n
j — — n
2
н
2
n
If /£L2(0, 2л) is continuous and with period 2л, then the left hand
side of the inequality above may be taken arbitrarily small by virtue of
Weierstrass’ trigonometric approximation theorem (see Chapter 0, 2).
Thus the set of all the linear combinations G is dense, in the sense
of the norm, in the subspace of L2(0, 2л) consisting of all continuous
functions with period 2 л. Such a subspace is also dense, in the sense of
the norm, in the space Z2(0, 2л). Therefore, any function / С L2(0, 2л),
orthogonal to all the functions of
- pint
)/2л
£2(0, 2 л). This proves that our system of functions
orthonormal system of L2(0, 2 л).
must be a zero vector of
eint is a complete
5. E. Schmidt’s Orthogonalization
Theorem (E. Schmidt’s orthogonalization). Given a finite or countably
infinite sequence {у,} of linearly independent vectors of a pre-Hilbert
space X. Then we can construct an orthonormal set having the same
cardinal number as the set {y} and spanning the same linear subspace
as {*,}.
Proof. Certainly 0. We define ур y2, . . . and ulf u2, . . . recurrent-
ly as follows:
У1 = xi>
У2 ~ X2 (X2> Wl) Wl>
ui = У1/1Ы1-
иг = Уя! | У z 11 >
n
Уп+I = xn+l — J? (xn+1,Uj) u},
7 = 1
W»4-l = y» + i/||y» + i ||>
5. E. Schmidt's Orthogonalization
89
This process terminates if {x,} is a finite set. Otherwise it continues
indefinitely. Observe that yn 0, because x2, . . ., xn are linearly
independent. Thus un is well defined. It is clear, by induction, that each
un is a linear combination of x2, . . ., xn and that each xn is a linear
combination of ulf u2, . . ., un. Thus the closed linear subspace spanned
by the u’s is the same as that spanned by the x’s.
We see, by Ц^Ц = 1, that y2 [ and hence u2 [ uv Thus, by
|uj || = 1, y3 | uA and hence u3 [ u±. Repeating the argument, we see
that is orthogonal to u2, u3, ..., un, . . . Thus, by ||u21| = 1, we have
Уз I u2 and so I u2- Repeating the argument, we finally see that
uk 1 whenever k
m. Therefore constitutes an orthonormal set.
Corollary. Let a Hilbert space X be separable, i.e., let X have a dense
subset which is at most countable. Then X has a complete orthonormal
system consisting of an at most countable number of elements.
Proof. Suppose that an at most countable sequence {u,} of vectors
С X be dense in X. Let xT be the first non-zero element in the sequence
the first which is not in the closed subspace spanned by xv
and xn the first which is not in the closed subspace spanned by
xv x2, . . ., xn_T. It is clear that the as and the xs span the same closed
linear subspace. In fact, this closed linear subspace is the whole space X,
because the set {л;} is dense in X, Applying Schmidt’s orthogonalization
to {xj, we obtain an orthonormal system {u;} which is countable and spans
the whole space X.
This system {щ} is complete, since otherwise there would exist a
non-zero vector orthogonal to every u3 and hence orthogonal to the
space X spanned by w.'s.
Example of Orthogonalization. Let S be the interval (a, 6), and con-
sider the real Hilbert space L2(S, ®, m), where ЯЗ is the set of all Baire
subsets in (a, b). If we orthogonalize the set of monomials
1, s, s2, s3, . . ., s\ . . .,
we get the so-called Tchebyschev system of polynomials
PQ(s) = constant, Pi(s), P2(s)> рз(*)> • • •>	• • •
which satisfies
ь
f Pi (s) Pj (s) m (ds) = dij
a
(— 0 or 1 according as i f or i = /).
In the particular case when a = — 1, b — 1 and m(ds) = ds, we obtain
the Legendre polynomials', when a = —oo, b = oo and m(ds) = e~s* ds,
we obtain the Hermite polynomials and finally when a = 0, b = oo and
m(ds) = e~s ds, we obtain the Laguerre polynomials.
90 III. The Orthogonal Projection and F. Riesz’ Representation Theorem
It is easy to see that, when —oo < a < Ъ < oo, the orthonormal
system {Pj (s)} is complete. For, we may follow the proof of the complete-
ness of the trigonometric system (see the Example in the preceding
section 4); we shall appeal to Weierstrass" polynomial approximation
theorem, in place of Weierstrass’ trigonometric approximation theorem.
As to the completeness proof of the Hermite or Laguerre polynomials,
we refer the reader to G. Szego [1] or to K. Yosida [1].
6.	F. Riesz’ Representation Theorem
Theorem (F. Riesz’ representation theorem). Let X be a Hilbert space
and / a bounded linear functional on X. Then there exists a uniquely
determined vector of X such that
/(*) = (x,yf) for all x^X, and Ц/|| = ||У/[|-	(1)
Conversely, any vector у E X defines a bounded linear functional fy on
X by
fy (*) = (*, У) for all %E X, and ||/y|| = ||y]|.	(2)
Proof. The uniqueness of yy is clear, since (x, z) = 0 for all x £ X
implies z — 0. To prove its existence, consider the null space N = N(f) =
{x £ X; f(x) = 0} of /. Since / is continuous and linear, N is a closed
linear subspace. The Theorem is trivial in the case when N = X; we
take in this case, yy = 0. Suppose N X. Then there exists a y0 0
which belongs to A'1 (see Theorem 1 in Chapter III, 1). Define
37= (/(Уо)/1|Уо1|2)Уо-	(3)
We will show that this yy meets the condition of the Theorem. First,
if x^N, then f(x) = (x,yf) since both sides vanish. Next, if x is of
the form x = ay0, then we have
(*> 37) = («Уо> 3>) =(a3'o.-j^2 Уо)= «/(Уо) = /(аУо) = f(x)-
Since f(x) and (%, yf) are both linear in x, the equality f(x) = (x, yy),
x E X, is proved if we have proved that X is spanned by N and y0. To
show the last assertion, we write, remembering that /(yy)	0,
..V,
V f(57)V^+/(3'/)yj"
The first term on the right is an element of 2V, since
We have thus proved the representation f(x} = (%, yy).
6. F. Riesz’ Representation Theorem
91
Therefore, we have
||/||= sup |/(*)| = sup |(x, yz)|^ sup ||x||  l|»ll = 1МГ
1И151	Ikllfii	IMIS1
and also ||/|| = sup |/(x)|	|/(У//||У/||)| = Кст . Vf) = ||У/||-
Hence we have proved the equality ||/|| = ||yy||.
Finally, the converse part of the Theorem is clear from [ fy (x) | =
|(*,y)|11*11 • IMI-
Corollary 1. Let X be a Hilbert space. Then the totality X' of bounded
linear functionals on X constitutes also a Hilbert space, and there is a
norm-preserving, one-to-one correspondence f +> уf between X' and X.
By this correspondence, X' may be identified with X as an abstract set;
but it is not ahowed to identify, by this correspondence, X' with X as
linear spaces, since the correspondence f <-> is conjugate linear:
(«i/i + a2/a)	(«1УЛ + «2ул),	(4)
where oclt oc2 are complex numbers.
Proof. It is easily verified that X' is made into a Hilbert space by
defining its scalar product through (Д, /2) = (y^, уд), so that the state-
ment of Corollary 1 is clear.
Corollary 2. Any continuous linear functional T on the Hilbert space
X' is thus identified with a uniquely determined element t of X as
follows:
H/) = /(0 for all fEXf.
Proof. Clear from the fact that the product of two conjugate linear
transformations is a linear transformation.
Definition. The space Xf is called the dual space of X. We can thus
identify a Hilbert space X with its second dual X" = (X')' in the above
sense. This fact will be referred to as the reflexivity of Hilbert spaces.
Corollary 3. Let X be a Hilbert space, and Xf its dual Hilbert space.
Then, for any subset F of X' which is dense in the Hilbert space X', we
have
1*011= sup
|/(*o) I’ X0^ X .
(6)
Proof. We may assume that %0 7^ 0, otherwise the formula (6) is tri-
vial. We have (%0, *0/||xoll) = 11*0 ll> anc^ so ^btere exists a bounded linear
functional /0 on X such that 11/011 — 1, /0 (%G) = | %0 L Since f (%0) = (%0, yy)
is continuous in yf, and since the correspondence f yy is norm-preserving
we see that (6) is true by virtue of the denseness of F in X'.
Remark. Hilbert’s original definition of the “Hilbert space” is the
space (Z2). See his paper [1]. It was J. von Neumann [1] who gave an
92 Ill. I he Orthogonal Project ion and I*. Riesz ‘ Representation Theorem
axiomatic definition (see Chapter 1,9) <>t the Ihlbert space assuming
that the space is separable. F. Riesz | 1 ] proved the above representation
theorem without assuming the separability of the. Hilbert space. In this
paper, he stressed that the whole theory of the Hilbert space may be
founded upon this representation theorem.
7.	The Lax-Milgram Theorem
Of recent years, it has been proved that a variant of F. Riesz’ repre-
sentation theorem, formulated by P. Lax and Л. N. Milgram [1], is a
useful tool for the discussion of the existence of solutions of linear partial
differential equations of elliptic type.
Theorem (Lax-Milgram). Let X be a Hilbert space. Let B(x, y) be a
complex-valued functional defined on the product Hilbert space X X X
which satisfies the conditions:
Sesqui-linearity, i.e.,
В + a2 x2, у) = <хгВ (xv у) + oc2B (x2, у) and (1)
в (x. Р1У1 +	У2)	(x> У1) + @2B (x- У2) >
Boundedness, i.e., there exists a positive constant у such that
B(x, y)| <Ly ||x|| • ||y||,	(2)
Positivity, i.e., there exists a positive constant <5 such that
B(xt x) <5 11 x ||2.
(3)
Then there exists a uniquely determined bounded linear operator S with
a bounded linear inverse S-1 such that
(%, у) = В (x, Sy) whenever x and у £ X, and 11S 11	d-1, 11S-1 | у. (4)
Proof. Let D be the totality of elements у 6 X for which there exists
an element y* such that (%, y) = B(x, y*) for all x 6 X. D is not void,
because 0 C D with 0* — 0 • y* is uniquely determined by y. For, if w
be such that В (x, w) = 0 for all x, then w = 0 by 0 = В (w, w) <5 11 w 112.
By the sesqui-linearity of (x, y) and B(x, y), we obtain a linear operator
S with domain D(S) = D: Sy = y*. S is continuous and ||Sy||^
d-1 11 у 11, у £ D (S), because
<5 ||Sy ||2 B(Sy, Sy) = (Sy, y) j|Sy11 • ||y||.
Moreover, D — D (S) is a closed linear subspace of X. Proof: if yn 6 D (S)
and s-lim yn = y^, then, by the continuity of S proved above, {Syn}
n—>OG
is a Cauchy sequence and so has a limit z = s-lim Syn. By the continuity
of the scalar product, we have lim (%, y„) = (x, y^). We have also, by
n ЮО
8. A Proof of the Lebesgue-Nikodym Theorem
93
(2), lim B(x, S y„) = B(x, z). Thus, by (x, y„) = B(x, S yn), we must
n—к»
have (x,	= В (x, z) which proves that y^ E D and Sy^ = z.
Therefore the first part of the Theorem, that is, the existence of
the operator S is proved if we can show that D(S) = X. Suppose
D (S) X, Then there exists a E X such that 0 and w0 € D (S) -1-.
Consider the linear functional F (z) = В (z, w0) defined on X. F (z) is
continuous, since | F (z) | = | В (z, wQ) | rg у | [ z 11 • 11 w011. Thus, by F. Riesz’
representation theorem, there exists a w'Q E X such that B(z, w0) =
F (z) = (2, w'o) for all z £ X. This proves that w'o E D (S) and Swq = w0.
But, by <3 || Ko 112 < в (Wq, ~ (wo> wo) = 0, we obtain = 0 which
is a contradiction.
The inverse S-1 exists. For, Sy = 0 implies (x, у) = В (x, Sy) = 0
for all x E X and so у = 0. As above, we show that, for every у E X there
exists a y' such that (z, у') = В (z, y) for all z 6 X. Hence у = Syr and so
в y)
S-1 is an everywhere defined operator, and, by (z, S~ry) | =
у 11z 11 ’ 11У 11 > we see that 11S-111 rg у.
Concrete applications of the Lax-Milgram theorem will be given in
later chapters. In the following four sections, we shall give some examples
of the direct application of F. Riesz’ representation theorem.
8.	A Proof of the Lebesgue-Nikodym Theorem
This theorem reads as follows.
Theorem (Lebesgue-Nikodym). Let (S, S3, m) be a measure space,
and?(B) be a а-finite, cr-additive and non-negative measure defined on 23.
If v is m-absolutely continuous, then there exists a non-negative, m-
measurable function p (s) such that
v (B) = f p(s) m (ds) for all
в
BE 23 with v(B) <00.	(1)
Moreover, the “density” p(s) of r(B) (with respect to m(B)) is uniquely
determined in the sense that any two of them are equal m-a.e.
Proof (due to J. von Neumann [2]). It is easy to see that q(B) —
m(B) + v(B) is a o-finite, cr-additive and non-negative measure defined
00
on 53. Let {B„} be a sequence of sets E 23 such that S = U Bn, Bn C Bn+1
Л = 1
and q (Bn) < 00 for n = 1, 2, ... If we can prove the theorem for every
В Q Bn (for fixed n) and obtain the density pn (s), then the Theorem is
true. For, we have only to take p(s) as follows:
p(s) = Pi(s) for s€ Bv and />(s) =/>„+i(s)
for s € Bn+i — Bn
(n=l,2, ...).
94 III. The Orthogonal Projection and F. Riesz' Representation Theorem
Therefore we may assume that q (S) < oo without losing the gene-
rality. Consider the Hilbert space L2(5, 93, g). Then
/W = /%(s) v(ds), xe L2(S, 93, q),
s
gives a bounded linear functional on L2(S, 93, g), because
|/W | f |*($) | v(ds) |x(s) |2 v(ds)^112 (J 1 • v(ds)^1/2
ll< • *(S)1/2>
where ||x ||e = ( J |x(s) |2 @(ds)^2 . Thus, by F. Riesz’ representation
theorem, there exists a uniquely determined у £ L2(S, 93, q) such that
J x (s) v (ds) = f x(s) у (s) q (ds) = f x(s) у (s) m (ds) + f x(s) у (s) v (ds)
S	S	S	s
holds for all x £ L2 (S, 93, g). Taking x as non-negative functions and con-
sidering the real part of both sides, we may assume that y(s) is a
real-valued function. Hence
Jx(s)(l — y(s))v(ds) = f x(s) у (s) m(ds) if x(s) € L2(S, 93, g) (2)
5	s
is non-negative.
We can prove 0 у (s) < 1 g-a.e. To this purpose, set E± = {s; у (s) < 0}
and E2 = {s; y(s) 1}. If we take the defining function C£i(s) of Ex
for x (s) in (2), then the left hand side is 0 and hence f у (s) m (ds) 0.
Thus we must have m(E^ = 0, and so, by the ти-absolute continuity
of v, v(ET) = 0, g(£\) = 0. We may also prove q(E2) = 0, by taking
the defining function C£i(s) for x(s) in (2). Therefore 0 y(s) < 1
g-a.e. on S.
Let x(s) be 93-measurable and 0 g-a.e. Then, by g(S) < oo, the
“truncated functions” xn(s) — min(x(s), n) belong to L2(S, 93, g)
(n = 1, 2, . . and hence
f x„(s) (1 — y(s)) v(ds) = f xn(s) y(s) m(ds) (n=l,2,...).	(3)
s	s
Since the integrals increase monotonely as n increases, we have
lim f xn(s) (1 — у (s)) v(ds) = lim f xn(s) у (s) m(ds) = L ^'oo. (4)
П-+ОО s	$
Since the integrands are 0 g-a.e., we have, by the Lebesgue-Fatou
Lemma,
Цт	(1 — у(s)) v{ds) = I x(s) (1 — y(s)) v(ds),
»->oo	У
n-*oo
(5>
9. The Aronszajn-Bergman Reproducing Kernel
95
under the convention that if %(s) (1— у (s)) is not v-integrable, the right
hand side is equal to oo; and the same convention for % (s) у ($). If x (s) у (s)
is ж-integrable, then, by the Lebesgue-Fatou Lemma,
L f lim (xn (s) у ($)) m (ds} = f x(s) у (s) m (ds).	(6)
This formula is true even if x (s) у (s) is not ж-integrable, under the con-
vention that then L = oo. Under a similar convention, we have
f x(s)(l-y(s))v(ds).	%(7)
s
Therefore, we have
J x(s) (1 — y(s)) v(ds) = f x(s) y(s) m(ds) for every x(s) which
5	5	(8)
is SB-measurable and J> 0 @-a.e.,
under the convention that, if either side of the equality is ==z oo then the
other side is also = oo.
Now we put
x(s) (1 —y(s)) =s(s), y(s) (1 —y(s))-1 =
Then, under the same convention as in (8), we have
f z(s) v(ds) = f z(s) p(s) m(ds) if я($) is SB-measurable
s	5	(9)
and 0 p-a.e.
If we take the defining function CB(s) of В (E S8 for z(s), we obtain
v (B) = f fi(s) m (ds) for all В E S3.
в
The last part of the Theorem is clear in view of definition (1).
Reference. For a straightforward proof of the Lebesgue-Nikodym theo-
rem based upon Hahn’s decomposition (Theorem 3 in Chapter I, 3) see
K. Yosida [2]. This proof is reproduced in Halmos [1], p. 128. See also
Saks [1] and Dunford-Schwartz [1].
9.	The Aronszajn-Bergman Reproducing Kernel
Let A be an abstract set, and let a system X of complex-valued func-
tions defined on A constitute a Hilbert space by the scalar product
(/.?) = (/(«).? («)).• (1)
A complex-valued function К (a, b) defined on A X A is called a reproduc-
ing kernel of X if it satisfies the condition:
For any fixed b, K(at b) E X as a function of at	(2)
/(6) = (/(л)- K (a- b))« and henCe /(&) = (K	b)> f(aY)a-	(3)
96	111 I'hc ()rt hogonal Projection and F. Kiesz’ Representation Theorem
As for the existence of reproducing kernels, we have
Theorem 1 (N. Aronszajn [1], S. Bergman 11 j). X has a reproducing
kernel К iff there exists, for any y0 E Л, a positive constant Cn, depending
upon y0, such that
|/(y0)|^CrJ|/|| for all fex.	(4)
Proof. The "only if” part is proved by applying Schwarz’ inequality
to /(y0) = (/(*). K(x> Уо)Х:
I/(Уо) I Н/П • (K{x, y0), K(x, у0))У2 =||/|| K(y0, y0)™.	(5)
The "if” part is proved by applying F. Riesz’ representation theorem to
the linear functional Fyo(f) ~ f(yQ) of f £ X. Thus, there exists a uniquely
determined vector gVo(v) of X such that, for every / E X,
fM = FyAf) = (f(x),gyAx)\.
and so gye(%) = K(x, yQ) is a reproducing kernel of X. The proof shows
that the reproducing kernel is uniquely determined.
Corollary. We have
sup |/(y0) I = К (y0, y0)1/2,	(6)
Ml sa
the supremum being attained by
f0(x)=6K(x,y0)IK{y0,y0)112, lel-1.	(7)
Proof. The equality in the Schwarz’ inequality (5) holds iff /(%) and
К(x, yQ) are linearly dependent. From the two conditions f (x) = tx К (x, y0)
and Il/П = 1, we obtain
1 = |a | (K (%,y0),K (x, y0))i/2= |a |K (y0, y0)1/2, that is, |a | = К (y0, y0)~1/2.
Hence the equality sign in (5) is attained by /0(%).
Example. Consider the Hilbert space ?12(G). For any /Е A2(G) and
яЕ G, we have (see (4) in Chapter I, 9)
|/(z0)|25S (яг2)-1 f |/(z) \2dxdy (z = x 4- iy).
|z—£0|^r
Thus A2 (G) has the reproducing kernel which will be denoted by KG (z, z').
This KG (z, z') is called Berg-man s kernel of the domain G of the complex
plane. The following theorem of Bergman illustrates the meaning of
KG(z, z) in the theory of conformal mapping.
Theorem 2. Let G be a simply connected bounded open domain of the
complex plane, and z0 be any point of G. By Riemann’s theorem, there
exists a uniquely determined regular function w = f(z’t zQ) of z which
gives a one-to-one conformal map of the domain G onto the sphere
j w | < qg of the complex w-plane in such a way that
9. The Aronszajn-Bergman Reproducing Kernel
97
Bergman's kernel KG(z\ г0) is connected with f0(z; zQ) by
/о (2 > Zo) = KG (z0 > 2o)-1 / KG ; 2o) di’	(8)
zo
where the integral is taken along any rectifiable curve lying in G and
connecting zQ with z.
Proof. We set
A | (G) = {/ (я); f (z) is holomorphic in G, f (z) C A 2 (G), / (z0) = 0 and /' (z0)=1},
and consider, for any f g Л2 (G), the number
||/'||2 = f |/' (z) |2 dx dy, z = x + iy.	(9)
G
If we denote by z — (p (w) the inverse function of w = fo(z> 2o)> then, for
any /Е A2 (G),
|[/'||2 = fj |/'2 |y>'(w) |2 du dv, w = и + iv.
[w|<eo
For, by the Cauchy-Riemann partial differential equations
-- У Vi --- Уи>
we have
dx dy = du dv = {xu y„ — yu xv) du dv = (4 + у2) du dv
J C\U, V)
— | <p (w) |2 du dv.
Let /С X^(G), aRd ^(w) = /(^(w)) be expanded into power series:
00
F(w) = f((p(w)) = w 4-	for |w| < qg.
rt=2
oo
Then F' (w) = /' (<p (w)) <p' (w) ~ 1 + £ n cn and so
я = 2
oo
n=2
Therefore minimum II/' I =V^Qg, and this minimum is attained iff
fiAKG) "	r
F(») = К<Р(™У) = w. that is, iff f(z) = /0(г; z0).
We set, for any /€ Л?(б), g(z) = f(z)/ \ f ||. Then ||g' || = 1. If we put
A2(G) = {g(z) ; g(z) is holomorphic in G, g(z0) = 0, g' (г0) > 0
and ||g'|| = 1},
7 Yonldft, Functional Analy«iB
98 HI I hr Ort hogonal Projection and I*'. Rirsz’ Representation Theorem
then the above remark shows that
maximum g' (z0) = l/\\/'o || = (j/л t»<.) 1.
and this maximum is attained iff g(z) is equal to
£o (z) — /0 (z, Zq)I 11 /о 11 = /0 (z , Л()) I j/л ()(i .
Hence, by (7), we obtain
Ш	= *KG(z; z0)IKG^; z0)"2, |A | = 1.
Hence, by putting z = z0, we have
(Л|/ярс)-1 = KG{z0\ z0)/KG(z0; z0)112 = KG(z0; z0)112,
and so we have proved the formula
^^^KG(z;z0)IKG(z0-,z0).
10. The Negative Norm of P. Lax
Let Hq (£?) be the completion of the pre-Hilbert space Cg° (L?) endowed
with the scalar product (99, ip)s and the norm
(9’>V’)s= JE f D}<p{x)D’y(x)dx, ||<p||s = (99, <p)s1/2.	(1)
Any element b E Hq (Q) = L2 (Й) defines a continuous linear functional
A on (£?) by
fb(w) = (w,b)0, wtH‘0(Q).	(2)
For, by Schwarz’ inequality, we have
|(w,г>)0|^ !M|0- IWk• ll6llo-
Therefore, if we define the negative norm of b£ #o(£?) — L2(Q) by
IHII-S =	sup |/6(w)|= sup |(w, &)0|,	(3)
then we have
IINI-s^ l!Nlo-	(4)
and so, by ||b||_, 2г |(w/||w||s, d)0|,
|(w, t)0| IMkx IHl-s-	(5)
Hence we may write
||6|l_s = ЦДЩ = sup I (w, b)01 for any b G Hg(£}).	(3')
We shall prove
Theorem 1 (P. Lax [2]). The dual space of the space Н^(£2)
may be identified with the completion of the space (/?) == L2 (£2) with
respect to the negative norm.
10. The Negative Norm of P. Lax
99
For the proof we prepare
Proposition. The totality F of the continuous linear functionals on
Hq (/2) of the form fb is dense in the Hilbert space HSQ PY, which is the dual
space of Hq (/2).
Proof. F is total on Hq(12) in the sense that, for a fixed
the simultaneous vanishing /&(w) = 0, b^H^p), takes place only if
w = 0. This is clear since any element w £ Hq (Z2) is also an element of
Now if F is not dense in the .Hilbert space HqPY- then there exists
an element TV 0 of the second dual space ZZq(/2)" = (HqPYY such
that T (fb) = 0 for all fb C F. By the reflexivity of the Hilbert space
HsopY there exists an element t^H^p) such that T(f) — f(t) for all
/Е HSQPY- Thus T(fb) = fb(t) = 0 for all b E #{}(Х2). This implies, by the
totality of F proved above, that t = 0, contrary to 7	0.
Corollary. We have, dually to (3'),
||wl|s = sup |(w, b)Q | for any w € Hq (X2).	(6)
Proof. Clear from Corollary 3 in Chapter III, 6, because F —
{fb;be HqP)} is dense in H^pY-
Proof of Theorem 1. Clear from the facts that i) F is dense in the
dual space Hq (£?)' and ii) F is in one-to-one correspondence to the set
Hq(X2) = £2(£?) preserving the negative norm, i.e.,
F3fb<+ b£H*P) and ||A|l-s= UNI—
We shall denote by Hqs (12) the completion of Hq (12) with respect to
the negative norm ||d||_s. Thus
hsqPY = HqsP)-
(7)
For any continuous linear functional / on #o(12), we shall denote by
/) the value of / at w E Hq pY Thus, for any b E Hq (12), we may write
fb W = (^, *)o = fb> = O> by, w E Hq p),	(8)
and have the generalized Schwarz’ inequality
|O? fc>| gj || & ||„s,
(9)
which is precisely (o).
Now we can prove
Theorem2 (P. Lax [2]). Any continuous linear functional g(6) on
Hq s(Z2) can be represented, by a fixed element w E HqP), as
g(b)=gw(b)^=<^b),	(10)
We have, in particular,
Hq PY = Hq5 p) , Hq S PY = HSqP)-	(П)
100 ILL. Ihe ()i t hogoii.il I'roj<( (ion and I*. Kirs/ Kepi escalation Theorem
Proof. If &£//{}(£?), then ч«”, /Л //,(«’’)  (/e, 6)0. Since F =
{fb; b C Hq(£2)} is dense in the Hilbert space	we know, by (9),
that (w, by ~ (b, w)q with a fixed 70 G	‘fines a linear functional gw
continuous on a dense subset /*' of 11^(12)'. Ihc norm of this functional
gw on F will be denoted by ||gw||s. Then, by (6),
||gw||s= sup |(6, w)o| = sup |(rr, 6)0| = ||w||s.	(12)
IMI-.SI	Ml .-si
We may thus extend, by continuity, the functional gw on F to a con-
tinuous linear functional on the completion of F (with respect to the
negative norm), that is, gw can be extended to a continuous linear func-
tional on Hq(£2)' = Hqs(£2) ; we denote this extension by the same letter
gw. We have thus
li&,|| = sup |gw(6)l = ||w||s.	(13)
ll»ll-.gl
Hence, in view of the completeness of the space the totality G of
the continuous linear functionals gw on Hqs (£2) may be considered as
a closed linear subspace of Hqs (£?)' by the correspondence gw <-> w. If this
closed subspace G were not dense in Hqs (£))', then there exists a continuous
linear functional /7^ 0 on Hqs (£2)' such that f(gw) ~ 0 for all gw£ G. But ,
since the Hilbert space Hqs(£2) is reflexive, such a functional / is given
by /(gJ=g„(/0),/0€Hos(12),and so by (13) f0 must be equal to 0, con-
trary to the fact / 7^ 0. Therefore we have proved Hqs (£2)' = HSQ(£2).
Remark. The notion of the negative norm was introduced by P. Lax
with the view of applying it to the genuine differentiability of distribution
solutions of linear partial differential equations. We shall discuss such
differentiability problems in later chapters. It is to be noted that the
notion of the negative norm is also introduced naturally through the
Fourier transform. This was done by J. Leray [1] earlier than Lax.
We shall explain the point in a later chapter on the Fourier transform.
11. Local Structures of Generalized Functions
A generalized function is locally the distributional derivative of a
function. More precisely, we can prove
Theorem (L. Schwartz [1]). Let be a generalized function in
£2 Q B?. Then, for any compact subset К of £?, there exist a positive
integer mQ = m0 (T, K) and a function / (%) — f (%; T, K, mQ) £ L2 (K)
such that
-——-----------dx whenever о? £	(P).	(1)
’ • fa?
11. Local Structures of Generalized Functions
101
Proof. By the Corollary in Chapter 1,8, there exist a positive con-
stant C and a positive integer m such that
|Z(9?)|^C sup	whenever (p C •	(2)
Thus there exists a positive number d such that
/’»(<?’)= sup	6, 9 6®я(й), implies |T>)| 5S 1. (3)
We introduce the notation
ая __ a8n
дх8 дх8 дх'2 - • • дх* '
1 л	71
and prove that there exists a positive constant e such that, for
m + 1,
f \дто<р(х)/8хто\2 dx <p€ $#(&), implies /^(92) d.
This is proved by repeated application of the following inequality
(4)
m0 =
(5)
/	|dy>(x1(. .Х{_!, у, xi+1, . .xn)/8y\dy
Kr\(-oo,z<)
= lfl/2f f	y,xi+ll..-,х„)18у |2 dy\llz
where t is the diameter of K, i.e., the maximum distance between two
points of the compact set K.
Consider the mapping 99 (%) -> y(x) — d™* (p (x) jdxm* defined on
into	As may be seen by integration, ip(%) — 0 implies 99 (x) = 0.
Hence the above mapping is one-to-one. Thus T(<p), <p£ <£)K(Q), defines
a linear functional S(^), ip(x) = 8ma(p[x)/dxm°, by S(^) = 7Д99). By (3)
and (5), S is a continuous linear functional on the pre-Hilbert space X
consisting of such ips and topologized by the norm
Thus there exists, by F. Riesz’ representation theorem, a uniquely deter-
mined function f(x) from the completion of X such that
T(<p) = S(ip) = f (^<р(х)/8хм°) • f(x) dx for all <p£
к
Actually, the completion of X is contained in L2(K) as a closed linear
subspace, and so the Theorem is proved.
References for Chapter III
For general account of Hilbert spaces, see N. I. Achieser-
I. M. Glasman [1], N. Dunford-J. Schwartz [2], B. Sz. Nagy [1],
F. Riesz-B. Sz. Nagy [3] and M. H. Stone [1].
102
IV. 1 he Hahn-Banach Theorems
IV. The Hahn-Banach Theorems
In a Hilbert space, we can introduce the notion of orthogonal coor-
dinates through an orthogonal base, and these coordinates are the values
of bounded linear functionals defined by the vectors of the base. This
suggests that we consider continuous linear functionals, in a linear topolo-
gical space, as generalized coordinates of the space. To ensure the exist-
ence of non-trivial continuous linear functionals in a general locally
convex linear topological space, we must rely upon the Hahn-Banach
extension theorems.
1.	The Hahn-Banach Extension Theorem in Real Linear Spaces
Theorem (Hahn [2], Banach [1]). Let X be a real linear space and
let p (x) be a real-valued function defined on X satisfying the conditions:
p (x 4- y) rg p (x) + p(y) (subadditivity),	(1)
P(pcx) = ocp{x) for oc 0.	(2)
Let M be a real linear subspace of X and /0 a real-valued linear functional
defined on M:
jQ(pcx 4- fiy) = ocfQ(x) 4- fif0(y) for x, y^M and real oc, ft. (3)
Let /0 satisfy /0 (x) 5g p (x) on M. Then there exists a real-valued linear
functional F defined on X such that i) F is an extension of /0, i.e.,
F (x) = /0 (%) for all x E M, and ii) F (x) 5g p (x) on X.
Proof. Suppose first that X is spanned by M and an element %0 £ M,
that is, suppose that
X = {% = m 4- m E M, oc real}.
Since x0 E M, the above representation of x E X in the form x — m 4- oc xit
is unique. It follows that, if, for any real number c, we set
F(x) = F(m 4- ocx0) = f0(m) 4- occ,
then F is a real linear functional on X which is an extension of /0. We
have to choose c such that F (x) ^p (x), that is, /0 (m) 4- осc p (m 4- oc xQ).
This condition is equivalent to the following two conditions:
fbimjoc) 4- c />(x0 + m/oc) for
/oW(—a)) ~ c	P(—x0 +Уп1(—а))
ос > 0,
for ос < 0.
To satisfy these conditions, we shall choose c such that
— %o) = ° 5g p{m" 4- #o)	for all m', m" M.
Such a choice of c is possible since
/oW + /o(w") =fAm‘ + w")	+ w”) =/>(w'~ x0 + m” + ,r0)
j>(m' — x0) 4- p(m" 4 %0);
2. The Generalized Limit
103
we have only to choose c between the two numbers
sup t/о (w') — (m' — *o) ] and inf [/> (m" + x0) — /0 (m") ].
Consider now the family of all real linear extensions g of /0 for which
the inequality g (x) 5^ p (%) holds for all x in the domain of g. We make
this family into a partially ordered family by defining h g to mean that h
is an extension of g. Then Zorn’s Lemma ensures the existence of a
maximal linear extension g of /0 for which the inequality g (x) p (x)
holds for all x in the domain of g. We have to prove that the domain
D(g)of g coincides with X itself. If it does not, we obtain, taking D(g) as
M and g as fQ, a proper extension F of g which satisfies F (x) p (%) for
all x in the domain of F, contrary to the maximality of the linear exten-
sion g.
Corollary. Given a functional p(x) defined on a real linear space X
such that (1) and (2) are satisfied. Then there exists a linear functional f
defined on X such that
(4)
Proof. Take any point X and define M — {%; x = ocxo, oc real}.
Set /0(a%0) = ocp(x^. Then /0 is a real linear functional defined on M.
We have fQ(x) 5^ p(x) on M. In fact, ocp(x^ = p(ocx^ if oc > 0, and
if oc < 0, we have ocp (x0) < — exp ( -%0) = p (pcx^ by 0 = p (0) < p (x0) -f-
p{— %o)- Thus there exists a linear functional / defined on X such that
/(%) = /0(%) on M and /(%) p (%) on X. Since —/(%) = /(—%) 5^ p (—%),
we obtain —p(—x] f(x) p{x)-
2.	The Generalized Limit
The notion of a sequence {%n} of a countable number of elements xn
is generalized to the notion of a directed set of elements depending on a
parameter which runs through an uncountable set. The notion of the
limit of a sequence of elements may be extended to the notion of the
generalized limit of a directed set of elements.
Definition. A partially ordered set A of elements oc, fl, . . . is called a
directed set if it satisfies the condition:
For any pair oc, fl of elements of A, there exists а у C A
such that oc y, P <7-	(1)
Let, to each point a of a directed set A, there be associated a certain set
of real numbers f(oc). Thus f(oc) is a, not necessarily one-valued, real
function defined on the directed set A. We write
lim /(a) = a (a is a real number)
(xQA
104
IV. 1 he Hahn-Banach I hroirms
if, for any £>0, there exists an a0G/1 such th.it a0 < oc implies
|/(a) — a | ^ £ for all possible values of / al a. Wc say, in such a case, that
the value a is the generalized limit or Moore-Smith limit of /(a) through
the directed set A.
Example. Consider a partition A of the real interval [0, 1]:
0 = t0 <Z t^ < • • • < tn = 1 •
The totality P of the partition of [0, 1] is a directed set A by defining
the partial order A -<( A' as follows: If the partition Af is given by
0 = tg <Z < • • • < t'm = 1, then A A' means that n 5^ m and that
every ti is equal to some t'j. Let x (Z) be a real-valued continuous function
defined on [0, 1], and let f(A) be the totality of real numbers of the form
H—1
(tj^x — ty x(tf-), where is any point of
Pp ^'4-1] •
Thus f(A) is the totality of the Riemann sum of the function %(£) per-
i
taining to the partition A, The value of the Riemann integral f x(t) dt
о
is nothing but the generalized limit of /(d) through P.
As to the existence of a generalized limit, we have the
Theorem (S. Banach). Let x(oc) be areal-valued bounded function
defined on a directed set A, The totality of such functions constitutes a
real linear space X by
(x + y) (a) = x(a) + y(a), (£%) (a) = fix(a).
We can then define a linear functional, defined on X and which we shall
denote by LIM % (a), satisfying the condition
а£А
lim x (oc)	LIM x (a) 5^ lim x (oc),
a£A	tx£A	a{A
where
lim x (a) = sup inf x (/?), lim x (fi) = inf sup x (fi).
«ЕЛ	a.	*£A	oc ap$
Therefore LIM x (oc) = lim x (oc) if the latter generalized limit exists.
а^Л	af:A
Proof. We put p(x) = lim x(oc). It is easy to see that this p(x)
octzA
satisfies the condition of the Hahn-Banach extension theorem. Hence,
there exists a linear functional / defined on X such that —p(—x)
f(x) < p(x) on X. Wc can easily prove that lim x(oc) = — p(—x) so
a€A
that we obtain the Theorem, by putting LIM x(a) = f(x).
a£A
3.	Locally Convex, Complete Linear Topological Spaces
Definition. As in numerical case, we may define a directed set {яа} in
a linear topological space X. is said to converge to an element x of
4. The Hahn-Banach Extension Theorem in Complex Linear Spaces 105
X, if, for every neighbourhood U (%) of x, there exists an index a0 such
that xa 6 U (%) for all indices oc >- a0. A directed set {%л} of X is said to
be fundamental, if every neighbourhood U (0) of the zero vector 0 of X
is assigned an index л0 such that (xa—x$} 6 U (0) for all indices ос, (Fp^oc^.
A linear topological space X is said to be complete if every directed
fundamental set of X converges to some element x С X in the sense
above.
Remark. We can weaken the condition of the completeness, and
require only that every sequence of X which is fundamental as a directed
set converges to an element x С X; a space X satisfying this condition is
said to be sequentially complete. For normed linear spaces, the two defini-
tions of completeness are equivalent. However, in the general case, not
every sequentially complete space is complete.
Example of a locally convex, sequentially complete linear topological
space. Let a sequence {/*(%)} of	satisfy the condition
lim (fh — fk) = 0 in <£)(£?). That is, by the Corollary of Proposition 7
h,k—HX)
in Chapter I, 1, we assume that there exists a compact subset К of Q for
which supp(/^) С К (h = 1, 2, . . .) and lim (D4 sfh(x) —	= 0
h,k-+oo
uniformly on К for any differential operator Ds. Then it is easy to see,
by applying the Ascoli-Arzela theorem, that there exists a function /Е Ф (-G)
for which lim Dsfk(x) = Dsf(x) uniformly on К for any differential
A—
operator Ds. Hence lim Д = f in Ф (£>) and so ® (£?) is sequentially
Л—>oo
complete. Similarly, we can prove that ® (£?) is also sequentially complete.
As in the case of a normed linear space, we can prove the
Theorem. Every locally convex linear topological space X can be
embedded in a locally convex, complete linear topological space, in
which X forms a dense subset.
We omit the proof. The reader is referred to the literature listed in
J. A. Dieudonne [1]. Cf. also G. Kothe [1].
4. The Hahn-Banach Extension Theorem in Complex Linear Spaces
Theorem (Bohnenblust-Sobczyk). Let X be a complex linear space
and p a semi-norm defined on X. Let M be a complex linear subspace of
X and / a complex linear functional defined on M such that \f(x
on M. Then there exists a complex linear functional F defined on X
such that i) F is an extension of /, and ii) | F (x) | p (x) on X.
Proof. We observe that a complex linear space is also a real linear
space if the scalar multiplication is restricted to real numbers. If /(%) =
g(%) + ih(x), where g(%) and h[x) are the real and imaginary parts of
/(%) respectively, then g and h are real linear functionals defined on M.
106
IV. 1 he I l.ihn-B.in.ich I h<*<и this
Thus
|g(*)|	\f(x)\ g p(x)
Since, for each x £ M,
and |/z (x) |	|/(\) | * />(\) on M.
g(ix) + ih(ix) = f{ix) = if(x) = »(g(x) I ih(x)) h(x) + ig(x),
we have
Л(х) = — g(ix)
for all x ( M.
By the Theorem in Chapter IV, 1, we can extend g to a real linear func-
tional G defined on X with the property that G (x) <=> /> (л) on X. Hence
—G(x) = G(~x) S p(—x) — />(%), and so |G(%) j <, p(x). We define
F (x) — G (x) — iG (ix).
Then, by F(ix) = G (tx) — iG (—x) — G (ix) 4- iG (x) = iF (x), we easily
see that F is a complex linear functional defined on X. F is an extension
of /. For, x E M implies that
F(%) = G(x) —iG(ix) = g(x)~ig(ix) = g(%) + ih(x) = f(x).
To prove |F(*) |	P(x)> we write F(x)=re~*° so that |F(%)
eldF(x) =^F(el6x) is real positive; consequently |F(^)| = |G(el0%)
р(егех) = |ete| p(x) — p(x).
5. The Hahn-Banach Extension Theorem in Normed Linear Spaces
Theorem 1. Let X be a normed linear space, M a linear subspace of X
and Д a continuous linear functional defined on M. Then there exists a
continuous linear functional f defined on X such that i) / is an extension
of /1, and ii) Ц^Ц = ||/||.
Proof. Set p(x) — ЦДЦ • [IхThen p is a continuous semi-norm
defined on X such that |Д (x) | p (x) on M. There exists, by the Theorem
in the preceding section 4, a linear extension / of Д defined on the whole
space X and such that |/(%)|	p(x). Thus ||/|| sup p(x) = ЦДЦ.
On the other hand,since f is an extension of Д, we must have ||/|| ^ ЦДЦ
and so we obtain ||/i|| = Ц/Il-
An Application to Moment Problems
Theorem 2. Let X be a normed linear space. Given a sequence of
elements {%„} С X, a sequence of complex numbers {aw} and a positive
number y. Then a necessary and sufficient condition for the existence of
a continuous linear functional / on X such that	(i = 1, 2, . . .)
and 11 /11 у is that the inequalities
6. The Existence of Non-trivial Continuous Linear Functionals 107
hold for any choice of positive numbers n and complex numbers
A, A>  • •, A-
Proof. The necessity of this condition is clear from the definition of
||/||. We shall prove the sufficiency. Consider the set
where n and /9 are arbitrary} .
n	m
For two representations z= £ Ргх± =	of the same element
4 = 1	4=1
г E we have, by the condition of the Theorem,
Thus a continuous linear functional /r is defined on by
n
£ We have only to extend, by the preceding Theorem 1, /x to a
i-l
continuous linear functional f on X with f
Remark. As will be shown in section 9 of this chapter, any continuous
linear functional /on C [0, 1] is representable as
i
/ (я) = J % (t) m (dt)
о
with a uniquely determined Baire measure m on the interval [0, 1]. Thus,
if we take	(/ = 1, 2, . . .), Theorem 2 gives the solvability
condition of the moment problem:
i
J t^m^dt) =af (? = 1,2,...).
0
6. The Existence of Non-trivial Continuous Linear Functionals
Theorem 1.	Let X be a real or complex linear topological space, x0 a
point of X and p (x) a continuous semi-norm on X. Then there exists a
continuous linear functional F on X such that F (%0) = p (%0) and
| F (x) | p (я) on X.
Proof. Let M be the set of all elements ocxQt and define / on M by
f(ocxo) = ocp (%0). Then / is linear on M and |/(ля0) | = \<xp (я0) | —- p (<%%0)
there. Thus there exists, by the Theorem in Chapter IV, 4, an extension
F of f such that F (%) | 5g p (%) on X. Hence F (x) is continuous at x = 0
with />(%), and so, by the linearity of F, F (%) is continuous at every
point of X.
Corollary 1. Let X be a locally convex space and x0 7^ 0 be an element
of X. Then there exists a continuous semi-norm p on X such that
108
IV. The Hahn-Banach Theorems
P(x0)	0- Thus, by Theorem 1,
tional /0 on X such that
there exists a continuous linear func-
/o (*o) — P C^o) and
/oW |	/’W on X.
Corollary 2, Let X be a normed linear space and 0 be any ele-
ment of X. Then there exists a continuous linear functional /0 on X such
that
/о (*o) = 11 xo 11 and ||/o|| = 1-
Proof. We take ||я|| for />(%) in Corollary 1. Thus ||/0||	1 from
/0(%) | g ||я||. But, by /0(%0) = ||x0[|, we must have the equality
l/oll = 1-
Remark. As above, we prove the following theorem by the Theorem
in Chapter IV, 1.
Theorem 1'. Let X be a real linear topological space, %0 a point of X
and p (x) a real continuous functional on X such that
p (x + У) = P (x) + p (y) and p(pcx) = ocp(x) for oc 0.
Then there exists a continuous real linear functional F on X such that
F(x0) = P (xo) and ~P {~~x) ^F(x) X? P (*) on X.
Theorem 2.	Let X be a locally convex linear topological space. Let M
be a linear subspace of X, and f a continuous linear functional on M.
Then there exists a continuous linear functional F on X which is an
extension of /.
Proof. Since / is continuous on M and X is locally convex, there exists
an open, convex, balanced neighbourhood of 0, say U, of X such that
M C\ U implies |/(%) |	1. Let p be the Minkowski functional of U.
Then p is a continuous semi-norm on X and U = {%; p(x) < 1). For
any x£M choose oc > 0 so that oc > p{x). Then p(xjoc) < 1 and so
f(xjoc) |^1, that is |/(x) | oc. We thus see, by letting oc | p(x), that
f(x) [ P (x) on M. Hence, by the Theorem in Chapter IV, 4, we obtain
a continuous linear functional F on X such that F is an extension of f an d
|F (x) [ gZ p(x) on X.
Theorem 3	(S. Mazur) . Let X be a real or complex, locally convex linear
topological space, and M a closed convex balanced subset of X. Then,
for any x0 С M, there exists a continuous linear functional f on X such
that /0 (%0) > 1 and | /0 (x) | rg 1 on M.
Proof. Since M is closed, there exists a convex, balanced neighbourhood
V of 0 such that M Г\ (x0 + V) = 0. Since V is balanced and convex,
/ V\	/	F\	/ V\
we have [M + у)	(x0 + у) = 0. The set (x0 + у 1 being a neighbour-
/ y\
hood of x0, the closure U of (M + -к ) does not contain x0. Since M Э C,
the closed convex balanced set U is a neighbourhood of 0, because U
6. The Existence of Non-trivial Continuous Linear Functionals 109
contains -g- as a subset. Let p be the Minkowski functional of U. Since U
is closed, we have, for any x0C U, > 1 and p(x) 1, if x£ U.
Thus there exists, by Corollary 1 of Theorem 1, a continuous linear
functional /0 on X such that /0 (x0) = p (x0) > 1 and | /0 (%) | X p (x) on X.
Hence, in particular, | /0 (%) |	1 on M.
Corollary. Let M be a closed linear subspace of a locally convex
linear topological space X. Then, for any x0 С X — M, there exists a
continuous linear functional/0 on X such that /0(%0) > 1 and /0(%) = 0
on M. Moreover, if X is a normed linear space and if dis (%0, M) > d,
then we may take | fG | j X ljd.
Proof. The first part is clear from the linearity of M. The second
part is proved by taking U = {x; dis (x, M) 5g tZ} in the proof of
Theorem 3.
Remark. As above, we prove the following theorem by virtue of
Theorem 1'.
Theorem 3' (S. Mazur). Let X be a locally convex real linear topolo-
gical space, and M a closed convex subset of X such that M Э 0. Then,
for any %0 Ё M, there exists a continuous real linear functional /0 on X
such that /0 (%0) > 1 and /0 (x)	1 on M.
Theorem 4	(S. Mazur). Let X be a locally convex linear topological
space, and M a convex balanced neighbourhood of 0 of X. Then, for
any x0 £ M, there exists a continuous linear functional /0 on X such that
/o(*o) SUP |/o(*) I-
x^M
Proof. Let p be the Minkowski functional of M. Then p (%0)	1 and
P(x) X 1 on M. p is continuous since M is a neighbourhood of 0 of X.
Thus there exists, by Corollary 1 of Theorem 1, a continuous linear func-
tional /0 on X such that
/о(*o) =	(%o)	1 and |/o(*)|	1 on M.
Theorem 5	(E. Helly). Let X be a В-space, and /ъ/2, . . .,/яЬе a
finite system of bounded linear functionals on X. Given n numbers
Л1, <x2, . . ., ocn. Then a necessary and sufficient condition that there exists,
for each e > 0, an element xe С X such that
fi(xE) = ос{ (i = 1, 2, ..., ri) and ||%e
is that the inequality
holds for any choice of n numbers ft, ft, . . ., fin.
Proof. The necessity is clear from the definition of the norm of a
continuous linear functional. We shall prove the sufficiency. We may
110
IV. l lie Hahn-Banach 1 heorems
assume, without losing the generality, that /’s are linearly independent;
otherwise, we can work with a linearly independent subsystem of {Д}
which spans the same subspace as the original system {/J.
We consider the mapping %->92 (я) = (/j (%), /2(я),. .	/„(%)) of X
onto the Hilbert space l2(n) consisting of all vectors x — (fb f2, • • •> Sn)
/ *	\l/2
normed by 11 x 11 — I | |2I . By the open mapping theorem in
Chapter II, 5, the image (p(Se) of the sphere Se = {%£ X; ||%|| у 4- s}
contains the vector 0 of l2[n) as an interior point for every e > 0. Let
us suppose that (ab a2, .  ., ocn) does not belong to 9? (Se). Then, by
Mazur's theorem given above, there exists a continuous linear functional
F on l2(n) such that
F((ab «2,	sup |F(g>(x))|.
Since l2(n) is a Hilbert space, the functional F is given by an element
n
(Pi> &>>’••> &) € F (n) in such a way that F ((аь a2, . .., an)) — £ ос,&.
7=1
Thus
n
n
for
But the supremum of the right hand side for 1| x 11 у + e is =
n
fjpj , and this contradicts the hypothesis of the Theorem.
7=1
7.	Topologies of Linear Maps
Let X, Y be locally convex linear topological spaces on the same scalar
field (real or complex number field). We denote by L(X, Y) the totality
of continuous linear operators on X into Y. L (X, Y) is a linear space by
(ocT 4- /3S) x = aTx 4- fiSx, where T, S£L(X, У) and x£X.
We shall introduce various topologies for this linear space L(X, У).
i)	Simple Convergence Topology. This is the topology of convergence
at each point of X and thus it is defined by the family of semi-norms
of the form
t> (T) = p(T; xlt x2, ••-,%„;?) = sup q (Гxj},
'	1^7<«
where xlf x2, . . ., xn are an arbitrary finite system of elements of X and q
an arbitrary continuous semi-norm on У. L (X, У) endowed, with this
topology will be denoted by LS(X, У). It is clearly a locally convex
linear topological space.
ii)	Bounded Convergence Topology. This is the topology of uniform
convergence on bounded sets of X. Thus it is defined by the family of
7. Topologies of Linear Maps
111
semi-norms of the form
=p(T;B;q) = sup^T*)
where В is an arbitrary bounded set of X and q an arbitrary continuous
semi-norm on Y. L (Xt У) endowed with this topology will be denoted by
Lh(X, Y), It is clearly a locally convex linear topological space.
Since any finite set of X is bounded, the simple convergence topology
is weaker than the bounded convergence topology, i.e., open sets of
LS(X, У) are also open sets of Lb(X, У), but not conversely.
Definition 1. If X, Y are normed linear spaces, then the topology of
Ls (X, У) is usually called the strong topology (of operators); the one of
Lb(X, Y) is called the uniform topology (of operators).
Dual Spaces. Weak and Weak* Topologies
Definition 1'. In the special case when У is the real or complex number
field topologized in the usual way, L (X, У) is called the dual space of X
and will be denoted by X'. Thus X' is the set of all continuous linear
functionals on X. The simple convergence topology is then called the
weak* topology of X'; provided with this topology, X' will sometimes be
denoted by X^,» and we call it the weak* dual of X. The bounded con-
vergence topology for X' is called the strong topology of X'; provided
with this topology, X' is sometimes denoted by X's and we call it the
strong dual of X.
Definition 2. For any x E X and x' £ X', we shall denote by <%, x'")
or x' (x) the value of the functional x' at the point x, Thus the weak*
topology of X', i.e., the topology of X^* is defined by the family of semi-
norms of the form
P (*') = P(*': *1, x2,..., x„) = sup | <Xy, x') I,
l^j<n
where xlt x2, . . ., xn are an arbitrary finite system of elements of X.
The strong topology of X', i.e., the topology of X's is defined by the family
of semi-norms of the form
P (x’) = p(x’;B) = sup I (x, x'y
x£B
where В is an arbitrary bounded set of X.
Theo	rem 1. If X is a normed linear space, then the strong dual space
X'$ is a В-space with the norm
||/|| = sup
IMIsi
Proof. Let В be any bounded set of X. Then sup ||&|| = fl < oo,
ьев
and hence ||/|j^a implies p (J; B) = sup |/(Z>) ] X sup |/(x) | a/h
kb	11*11 £0
1 12
IV. The Hahn-Banach Ihcoicins
On the other hand, the unit spheir 5 — (v; || t1| • . 1} of X is a bounded
set, and so ||/|| — p (/; S). This proves that the topology of X's is equi-
valent to the topology defined by the norm ||/||.
The completeness of X's is proved as follows. Let a sequence {/n}
of X's satisfy lim \f„ — fm II - 0. Then, for any x ( A', |/„(%) I g
n,m—>oo
||/л — /mil • |%|| -> 0 (as n, m —> oo), and hence a finite lim fn(x) — f(x)
n->oo
exists. The linearity of / is clear. The continuity of / is proved by ob-
serving that lim fn(x) = f[x) uniformly on the unit sphere S. Incident-
n->oo
ally we have proved that lim ||/n — /II = 0.
n—>oo
Similarly we can prove
The	orem 2. If X, Y are normed linear spaces, then the uniform topo-
logy (of operators) Lb (X, У) is defined by the operator norm
11T j | = sup 11T x 11.
Definition 3. We define the weak topology of a locally convex linear
topological space X by the family of semi-norms of the form
p(x) = p(x-, x'i’ 4>
••*») = Sup ।	। (
ISiS»
where x'lf x2i . . ., xn are an arbitrary finite system of elements of X'.
Endowed with this topology, X is sometimes denoted by Xw.
8.	The Embedding of X in its Bidual Space X"
We first prove
Theorem 1	(S. Banach). Let X be a locally convex linear topological
space, and X' its dual space. A linear functional /(%') on X' is of the form
/(*') = <x0> X'>
with a certain %0E X iff f(x') is continuous in the weak* topology of XL
Proof. The "only if’ part is clear since |<Lc0, %'> | is one of the semi-
norms defining the weak* topology of XL The "if” part will be proved
as follows. The continuity of f(x') in the weak* topology of X' implies
that there exists a finite system of points x1} x2, . . xn such that
|/(%') | < sup %'y |. Thus
fi(x') = <%£, %'> = 0 (i = 1, 2, . . ., n) implies /(%') = 0.
Consider the linear щар L: X' —> Z2 (w), defined by
£ (%i) = L (x2) implies L (%i — x^ = 0 so that Д (^q ~ *2) — 0 (г = 1, 2,..., n)
and hence / (%^ — %g) — 0- Hence we may define a continuous linear map F
8. The Embedding of X in its Bidual Space X"
113
defined on the linear subspace L (X') of Z2 (n) by
F{L{x'y) = F^x')........f„(X'))=/(X').
This map can be extended to a continuous linear functional defined on
the whole space I2 (n); the extension is possible since Z2 (w) is of finite
dimension (easier than using the Hahn-Banach extension theorem in
infinite dimensional linear spaces). We denote this extension by the same
letter F, Writing
n
(У1.У2. •  ->Уп) = ХУ1е,, where e,-= (0, 0, . .	0, 1, 0, 0, . .0)
yvith 1 at the /-th coordinate,
we easily see that
Р(У1,У2.
Therefore
n
• > Уп) ~ У] > (Xj — F (^).
J T
n	n	/ n
f	1j	% У ' — \	%
j=l	i=l	\у*=1
Corollary. Each x0 £ X defines a continuous linear functional /0 (%')
on X' by fQ(x') = O0, %'>. The mapping
%0
/0 — J
of X into (X's)'s satisfies the conditions
J (X1 4“ #2) — J %! + J %2> J (tx%) — <xj(x).
Theorem 2.	If X is a normed linear space, then the mapping J is
isometric, i.e., 111*11 = :l*l
Proof. We have | /0 (%') | = [ <%0, я'> |	11 xQ 11 • | x' so that 11 /01 j
||*0||. On the other hand, if xQ^ 0, then there exists, by Corollary 2 of
Theorem 1 in Chapter IV, 6, an element С X' such that <%0,	—
|%011 and 11Xq 11 = 1. Hence /0 (*q) — <x0,	= j [x011 so that 11/01 [	11 %011.
We have thus proved |1J x 11 = 11 x .
Remark. As the strong dual space of X', the space (X'), is a B-space.
Hence a normed linear space X may be considered as a linear subspace
of the В-space (X')' by the embedding x—>Jx. Therefore the strong
closure of /X in the В-space (X')' gives a concrete construction of the
completion of X.
Definition 4. A normed Unear space X is said to be reflexive if X may
be identified with its second dual or the bidual (X')' by the correspon-
dence x<> J x above. We know already (see Chapter III, 6) that a Hilbert
space is reflexive. As remarked above, (X')' is a В-space and so any re-
flexive normed linear space must be a B-space.
8 Yoeida, functional Апа1ун1я
114
IV. The Hahn-Banach Theorems
Theorem 3.	Let X be a В-space and Xq any bounded linear func-
tional on Xg Then, for any e > 0 and any finite system of elements
/1, /2, • • •, fn of there exists an %0E X such that
|*oII ||*o || + e and /,• (x0) = Xq (/,)(/= 1, 2,	.
Proof. We apply Helly’s theorem 5 in Chapter IV, 6. For any system
of numbers /3lf ., fin, we have
n
J=1
and hence, again by Helly’s theorem, we obtain an xQ E X with the
estimate ||я0|| 5g у + e = ||*o || + £ and %o(/y) =	(/ = 1, 2, . . ., w).
Corollary. The unit sphere S = {я E X;
5g 1} of a В-space X is
dense in the unit sphere of (X')' in the weak* topology of (X's)r.
9.	Examples of Dual Spaces
Example 1. (c0)' — (I1).
To any /Е (c0)', there corresponds a uniquely determined У/ =
{’?»} € P) such that, for all x = {£,} € (c0).
00
<X/> = ^n7}n and ||/l| = ]|yz||.	(1)
And conversely, any у = {??„}£ (Z1) defines an (c0)' such that, for
any x = {£„}€ (c0),
OO
<ж./у>	and Ц/yll = ||y||.	(1')
Proof. Let us define the unit vector ek by
A-l
6A=(0,0, ...,0,l, 0,0,...) (A = l, 2, ...).
k
For any x = {£„} E (c0) and / E (c0)\ we have, by s-lim Д' gnen = %.
k—KX) n=l
z' A	k
/У —	\	f / — lim JX £n Tjni ?]n = f (^w).
woo \n = 1	/	А—юо n=l
Let т]п = еп I Yin I for т]п 0, and еп = op for т]п = 0. Take %(“o) —
{fn} € (£o) in such a way that = eg1 for n nQt and fn = 0 for n > n0.
Then ||%(Bo)||	1 and so ||/|| = sup |<x,/>|	|<%(Ио), /> | = J? |??„|.
IM|gi	«-1
Thus, by letting w0->oo, we see that yf = {?]„} E (Z1) and ||yy|| =
i|^l^ll/||.
— JL
9. Examples of Dual Spaces
115
If, conversely, у — {т]п} € (Z1), then
oo
in Tin
n = l
for all
= {£„} C (c0), and so у defines an fy P (c0)' and |
Example 2. (c)' — (I1).
For any x — {£n} 6 (c), we have the representation
k
x = T s-lim (£n	4o) en, where г о	lim q (1, 1,1,...).
k—ЮО n = l	n—>oo
Thus, for any /€ (c)', we have
<(%, fy —-	/У +
oo
(£n
(2)
where y'o = <e0, fy and 7]n = <ел, /> (n — 1, 2, . . .). As above, we may
take x(”o) = {£„} 6 (c0) £ (c) which satisfies
«0
I %(Wo) 11 5g 1, fо = lirn = o and <x(w°\ /> = S | T}n | *
П-+ОО	n=l
Hence, by |/> |	| %(По)|| ||/||, we see that {^л}°°€ (Z1). We set
oo
% — Ji T]„ = T]o. Then, by (2), we have
n = l
oo
<x, /> = f %	q„, where x = {£,} € (c) and £0 = lim
«=1	n—КЮ
(2')
Let r/n = en t]„ | for t)„ # 0, and e„ = oo for r/n = 0 (w = 0, 1, 2,. ..).
Take x — {£„} € (c) such that
in = e»1 if м n0, and i„ = ей1 if я > и0-
Then I x|| sg 1,	= lim £„ = £q\ and <*,/>= l%l+— W +
1	я—>oo	’	n=l
oo	oo
«о1 r}n. Hence, we must have | ??0 | + J? | Yjn | Ц/1|.
n = H0 + l	n = l
oo
If, conversely, у — {^rt}o° is such that ||y|| — |??0| + J? \^n \ < oo,
w=l
then
oo
% • lim £„+ 2^ i„r]ni where x = {£„}“ € (c),
n—>oo
oo
defines an fy 6 (c)' such that ||/y 11	| ??01 +	| Tjn |.
#1=1
Therefore, we have proved that (c)' — (Z1) as explained above.
Example 3. LP(S, 93, m)' = Lq(S, Sb, m) (1	/> < oc and />-1+
q-1 = 1). To any /Е LP(S)', there corresponds a yf£ Lq(S) such that
<^>/> = J x(s)
m (ds) for all x£Lp (S) and 11 /11 = 11 yy 11, (3)
and conversely, any у £ Tff(S) defines an fy E Lp(S)f such that
^x,fyy = J #(s) y(s) m(ds) for all xf Lp (S) and
s
1411 = lb
(3')

s
8*
116
IV. The Hahn-Banach Theorems
oo	n
Proof. Let S = U В2 with 0 < т(ВЛ < oo and set = U B,.
3	3	j~i 3
For a fixed nt the defining function CB (s) of the set В Q B(w) is E Lp (S).
Thus the set function уз (В) = <CB, f> is a-additive and w-absolutely
continuous in В Q B^n\ By the differentiation theorem of Lebesgue-
Nikodym (see Chapter III, 8), there exists a yn(s) E L1 (B(w),	, m)
such that
уз (B) = f yn (s) m (ds) whenever В C B(w),
в
the family of sets being defined by £0w)= {В Л В(и); В E S}.
Therefore, by setting у (s) = yn (s) for s E B(n), we have
<CB,/> = f у (s) m(ds) for В e ®(n) (« = 1,2,...).
В
Hence, for any finitely-valued function x with support in some B<n>,
(xt fy = f x(s) y(s) m(ds),
s
Let % E Lp (S) and put
(4)
xn (s) == % (s) if
x(s)
и and s 6 B'n ,
= 0 otherwise.
We decompose the set {2; |z| n} of the complex plane into a finite
number of disjoint Baire sets Mnk t (t = 1, 2, . . ., dk n) of diameters
1/k. Set, for the xn(s) E L°°(S, ®, m),
xn,k (s) — a constant zt such that z E (the closure M“>ki) and | z | = inf | w |
w£Mntk,i
whenever xn
Then |%nA(s) I I %n(s) I and lim %n^(s) = %n(s) and so, by
sgue-Fatou Lemma, s-lim xn k = xn (n = 1, 2, . . Thus,
k—>00	’
the Lebesgue-Fatou Lemma,
f> = lim {Xnj,, f) = lim / xn_k (s) у (s) m (ds)
Л—НХЗ	Л—ЮО s
the Lebe-
again by
(5)
= f lim xnk{s) • y(s) m(ds) = f %rt(s) y(s) m(ds).
s ft^co ’	s
Since s-lim xn = x, we see that <%, /> = lim (xn, fy. We put, for any
П-+ОО	n—КЮ
complex number z, a(z) = e~te if z = rew and a(0) = 0. Then ||%||
II (W • a(y)) II and so
||/Ц ||*|l <W a(y)./> = f |*»(s)
s
у (s) I m (ds),
Thus, by the Lebesgue-Fatou Lemma, ||/|| • ||%|| J |*($)| y($)| m(ds)
s
and so the function %(s)y(s) belongs to EB(S). Therefore, letting
9. Examples of Dual Spaces
117
n —> oo in (5), we obtain
(x, fy = f %(s) У (s) m (^s) whenever x E Lp (S).
s
We shall show that у E Lq(S). To this purpose, set
y»(s)=y(s) if |y(s)| s£BM,
= 0 otherwise.
Then yn E Lq(S) and, as proved above,
||/|| • ||*||	<1*1  л(у),/> = f |*(s)
s
|y(s)| m(ds)
/ l*(s)il y«(s)l т№-
s
If we take x (s) = | yn (s) \q!p and apply Holder’s equality, we obtain
Hence ||/||=||Ул||—|yn(s) |* m(ds)^g} with the understanding
that, when p — 1 we have ||/|[	||yw|| = essential sup |yn (s) .
Therefore, by letting w—>oo and applying the Lebesgue-Fatou Lemma,
we see that y£.Lq (S) and | \f 11	11 у 11. On the other hand, any у E L"(S)
defines an /Е Lp (S)f by <x, fy = f x(s) y(s) m(ds) as may be seen by
s
Holder’s inequality, and this inequality shows that ||/||	| у |.
Remark. We have incidentally proved that
(lp)f = (lq) (1 p < oo and p-1 4- q-1 = 1).
Example 4.	Let the measure space (S, SB, ж) with m(S) <oo have the
property that, for any В E SB with 0 < m(B) = d<oo and positive inte-
ger n, there exists a subset Bn of В such that д (n + l)-1 m (Bn) t5n-1.
Then no other continuous linear functionals E Af (S, SB, m)' than the zero
functional can exist.
Proof. Any xEl1^ SB, ж) belongs to M(S> SB, ж) and the topology
of L1 (S, SB, ж) is stronger than that of M (S, SB, ж). Thus any f E M(S, SB,
m)', when restricted to the functions of L1 (S, SB, ж), defines a continuous
linear functional /0E SB, ж)'. Thus there exists а у E L°°^S, SB, ж)
such that
(x, fy = (xf foy = f x (s) у (s) ж (ds) whenever %E L1 (S, SB, ж). *
Since L1 (S, SB, ж) is dense in M(S, SB, ж) in the topology of M(St SB, ж),
t he condition f 0 implies that /0	0. Thus there exists an £ > 0
such that В — {s ; |y (s) |	£} has its measure ж (В) — <5 > 0. Let Bn Q В
118
IV. The Hahn-Banach Theorems
be as in the hypothesis, and let y(s)=re'° for s£B. Set %rt(s)=^“1G
for s £ Bn and xn(s) = 0 otherwise. Then 2n(s) = nxn(s) converges to 0
asymptotically, that is, s-lim zn = 0 in M(S, SB, m). But
n—Ю0
lim <zn,f> = lim <z„,/o> = Hm I zn(s) y(s) m(ds) de > 0,
n~>OC	n—+OO	n—K5O s
contrary to the continuity of the functional f.
Example 5.	L°°(S, SB, m)'.
Let an /6 L°°(S, SB, m)' be given and set, for any B£i&,f(CB)
= (B) where CB (s) is the defining function of the set B. We have then:
Bi A B2 = 0 implies ip(B1 4- B2) = ^C^i) +	(6)
that is ip is finitely additive,
the real part ipY{B) and the imaginary part y>2(B)
^p(B) are of bounded total variation, that is, sup |^(B)|	(7)
в
< oo (i = 1, 2),
ip is m-absolutely continuous, that is m(B) = 0 implies
y>(B) = 0.
(8)
The condition (6) is a consequence of the linearity of /, and (7) and (8)
are clear from ip (B) | < 11 /1 • 11CB .
For any x C B°° (S, SB, m), we consider a partition of the sphere
{2; |zI ||% ||} of the complex plane into a finite system of disjoint Baire
sets Alt A2> . . ., An of diameters < e. If we set B: = {s £ S; %(s) C Аг),
then, no matter what point we choose from At (i = 1, 2, . . ., n), we
have
and so
Thus, if we let n —> 00 in such a wav that £ | 0, we obtain
n
f(x) = lim J? a-iij) (Bi),	(9)
i = l
n
independently of the manner of partition {2; I2I 11% I } = £ A± and
choice of points a’s. The limit on the right of (9) is called Radon s integral
of %(s) with respect to the finitely additive measure ip. Thus
/(%) = f %(s) ip(ds) (Radon’s integral) whenever %E L°°(S, SB, m), (10)
s
and so
sup
ess.sup|r(s)| gl
J %(S) V(^S)
S
(11)
9. Examples of Dual Spaces
119
Conversely, it is easy to see that any ip satisfying (6), (7) and (8) defines
an /Е L°° (S, 33, m)' through (10) and that (11) is true.
Therefore, we have proved that jL°°(S, 53, m)f is the space of all set
functions ip satisfying (6), (7) and (8) and normed by the right hand side
of (11), the so-called total variation of ip.
Remark. We have so far proved that LP{S, S3, w) is reflexive when
1 < /> < oo. However, the space L1 (S, 53, m) is, in general, not reflexive.
Example 6.	C (SY.
Let S be a compact topological space. Then the dual space C (S)' of the
space C (S) of complex-valued continuous functions on S is given as
follows. To any /EC(S)\ there corresponds a uniquely determined
complex Baire measure p on S such that
f(x) = f x(s) pt (ds) whenever x С C (S),	(12)
s
and hence
sup
sup|x(s)| ^1
f x(s) p(ds)
s
= the total variation of pt on S. (13)
Conversely, any Baire measure p on S such that the right side of (13)
is finite, defines a continuous linear functional / 6C(S)' through (12)
and we have (13). Moreover, if we are concerned with a real functional /
on a real В-space C (S), then the corresponding measurep is real-valued;
if, moreover / is positive, in the sense that f(x) 0 for non-negative
functions x (s), then the corresponding measure p is positive,i.e., p (B)	0
for every В 6 S3-
Remark. The result stated above is known as the F. Riesz-A. Markov-
S. Kakutani theorem, and is one of the fundamental theorems in topo-
logical measures. For the proof, the reader is referred to standard text
books on measure theory, e.g., P. R. Halmos [1] and N. Dunford-
J. Schwartz [1].
References for Chapter IV
For the Hahn-Banach theorems and related topics, see Banach [1],
Bourbaki 2] and Kothe i 1 • It was Mazur [2] who noticed the impor-
tance of convex sets in normed linear spaces. The proof of Helly’s theorem
given in this book is due to Y. Mimura (unpublished).
V. Strong Convergence and Weak Convergence
In this chapter, we shall be concerned with certain basic facts per-
taining to strong-, weak- and weak* convergences, including the com-
parison of the strong notion with the weak notion, e.g., strong- and weak
measurability, and strong- and weak analyticity. We also discuss the
120
V. Strong Convergence and Weak Convergence
integration of B-space-valued functions, that is, the theory of Bochner’s
integrals. The general theory of weak topologies and duality in locally
convex linear topological spaces will be given in the Appendix.
1. The Weak Convergence and The Weak* Convergence
Weak Convergence
Definition 1. A sequence {xn} in a normed linear space X is said to
be weakly convergent if a finite lim / (xn) exists for each / E X's; {x„} is
n—>oo
said to converge weakly to an element x^ X if lim / (xw) = f (x^) for
n—>oo
all / E X's. In the latter case, x^ is uniquely determined, in virtue of
the Hahn-Banach theorem (Corollary 2 of Theorem 1 in Chapter IV, 6); we
shall write w-lirn xn = xx or, in short, xn -> x^ weakly. X is said to be
n—ЮО
sequentially weakly complete if every weakly convergent sequence of X
converges weakly to an element of X.
Example. Let {xw(s)} be a sequence of equi-bounded continuous func-
tions of C [0, 1] which is convergent to a discontinuous function z (s) on
[0, 1]. Then, since C [0, 1]' is the space of Baire measures on [0, 1] of
bounded total variation, we see easily that {xrt(s)} gives an example of
a weakly convergent sequence of C [0,1] which does not converge weakly
to an element of C [0, 1].
Theorem 1. i) s-lim xn = x^ implies w-lim xn — x^, but not conver-
«—>oo	п-юо
sely. ii) A weakly convergent sequence {xn} is strongly bounded, and, in
particular, if ze/-lim xn = x^, then {IIxn 11} is bounded and llx^ll^
1* и "’ II	n—>oo
lim ||xn||.
n—ЮО
Proof, i) The first part is clear from | / (xn) — f (x^) |	11 f 1| • |1 xn — хж 11.
The second part is proved by considering the sequence {xn} in the Hilbert
space (I2):
xn = {£^} where — dntn (= 1 or 0 according as n — m or not).
^nrjn with some {rin} E (I2)', consequently, z^-lim xn
П—>OO
For, the value of a continuous linear functional E (I2)' at x = {fM} is given
by J
ft = l
does not converge strongly to 0 because ||xw|| = 1 (n = 1, 2, . . .).
ii) Consider the sequence of continuous linear functionals Xn defined on
the jB-space X's by Xn(f) = <xw, />, and apply the resonance theorem in
Chapter II, 1.	' «Д
Theorem 2	(Mazur). Let w-lim xn — x^ in a normed linear space A\
n->oo	n
Then there exists, for any e > 0, a convex combination £ ocjXj
/	n
n
^oo
1. The Weak Convergence and The Weak* Convergence
121
n
Proof. Consider the totality of elements of the form £(XjXj
n
with (Xj 0, ocj = 1. We may assume that 0 E by replacing Xqq
3	г
and Xj by (xqq—x-J and (xj—Jtq), respectively. Suppose that 11	— и 11 > e
for every и E Mr The set M = {v £ X; ||r — и ||	e/2 for some и E Mr}
is a convex neighbourhood of 0 of X and — v 11 > e/2 for all v E M.
Let p (y) be the Minkowski functional of M. Since = /L1 u0 with
p(uQ) — 1 and 0 < /? < 1, we must have /’-(%oo) —	>	1. Consider a
real linear subspace Xi = {% E X; % = у w0, —oo < у < oo} and put
Д (%) == у for x — у uq£ Xr This real linear functional Д on X2 satisfies
l(x) ^p (x) on Xr Thus, by the Hahn-Banach extension theorem in Chap-
ter IV, 1, there exists a real linear extension / of defined on the real
linear space X and such that f (x) p (x) on X. M being a neighbourhood
of 0, the Minkowski functional p(x) is continuous in x. Hence f is a con-
tinuous real linear functional defined on the real linear normed space X,
We have, moreover,
supV(*) sup f(x) sup P(x) = 1<P 1 = f(P 1 w0) = /(хте).
arCAfj	x£M	x^M
Therefore, it is easy to see that x^ cannot be a weak accumulation point
of M1} contrary to xK — w-lim xn,
n—>oo
Theorem 3.	A sequence {xn} of a normed linear space X converges
weakly to an element x^ E X iff the following two conditions are satis-
fied : i) sup
ni
|xwl < oo, and ii) lim f(xn) = /(x^) for every / from any
П >OC
strongly dense subset D' of X'.
Proof. We have only to prove the sufficiency. For any g E X' and
e > 0, there exists an f£D' such that |g — f || < e. Thus
|g(*n) — g(*oo)|	/W| + I/(*«)—/(*«>) I + |/(*oo)—g(*oo)|
= ® 11 ХП 11 + I / (^») / (*oo) I 4" ® 11XOO 11 >
and hence lim | g (хя) — g (x^ |	2 e sup |1 xn 11. This proves that
oo^w^l
lim g(x„) = g(xoa).
n—>oo
Theorem 4.	A sequence {хя} in L1 (S, S3, m) converges weakly to an ele-
ment x^L1 (S, S3, ж) iff {||хя 11} is bounded and a finite lim f xn (s) m (ds)
n—>00 Jg
exists for every В E 33-
Proof. The "only if” part is clear since the defining function CB(s)
of В E S3 belongs to £°° (S, S3, m) = L1 (S, S3, m)'.
The proof of the "if” part. The set function ip (B) = lim f xn (s) m (d,s)
B£ S3, is u-additive and m-absolutely continuous by the Vitali-Hahn-Saks
122
V. Strong Convergence and Weak Convergence
theorem. Hence, by the differentiation theorem of Lebesgue-Nikodym,
there exists an x^ C L1 (S, S3, m) such that
lim f xn (s) ж (ds) = f (s) m (ds) for all В £ SB .
П—НЮ g	g
k
Thus, for any decomposition S = £ Bj with BjQ SB, we have
7=1
fi
%„(s) y(s) m(ds) = J xoo{s) y{s) m(ds), y{s} = £а,СБ {s).
Since such functions as y(s) constitute a strongly dense subset of the
space L°°(S, 93, m) — 1^(5, S3, m)r, we see that the “if” part is true by
Theorem 3.
Theorem 5.	Let {xn} converge weakly to x^ in ^(S, 93, ж). Then
{%„} converges strongly to iff {%„ (s)} converges to (s) in ж-measure
on every S3-measurable set В such that m(B) < oo.
Remark. {%n(s)} is said to converge to *oo(s) in ж-measure on B, if,
for any £ > 0, the ж-measure of the set {s£ В; |%„(s) —^(s)	£}
tends to zero as n -> oo (see the Proposition in Chapter I, 4). The space
(Z1) is an example of L1 (S, S3, ж) for which S — {1, 2, . . .} and ж ({n}) -= 1
for я = 1, 2, ... In this case, we have (Z1)' = (Z°°) so that the weak con-
vergence of {xn}, x„ =	$,я),...,	..), to = (Й001,	$,°°’,...)
implies that lim $.n) = (A = 1, 2,...), as may be seen by taking
/С (Z1)' in such a way that f(x) — {xt fy = %k for x = {fj 6 (Z1). Thus,
in the present case, {%rt} converges to in ж-measure on every 93-
measurable set В of finite ж-measure. In this way we obtain the
Corollary (I. Schur). In the space (Z1), if a sequence {%„} converges
weakly to an x^ C (Z1), then s-lim xn = x^.
n—>co
Proof of Theorem 5. Since the strong convergence in L1(S, S3, ж)
implies the convergence in ж-measure, the “only if” part is clear. We shall
prove the “if” part. The sequence {xn — converges weakly to 0, and so
lim f (xn (s) — x^ (s)) ж (ds) = 0 for every В E 8.	(1)
n—>oo g
Consider the sequence of non-negative measures
V„{B) = f Ix„(s) — «oo(s) m{ds), Be®.
В
Then we have
lim yn(Bk) — 0 uniformly in n, for any decreasing
A-+00
oo
sequence {Bk} of sets C 93 such that П Bk — 0.
If otherwise, there exists an £ > 0 such that for each k there exists some
1, The Weak Convergence and The Weak* Convergence 123
nk for which lim nk = oc and уПк (Bk) > 8. Consequently, we must have
Аню
f lRe(x„k(s) —xcc(s))^m(ds)> e/]/2 or f \Im (xnk(s) — ^(s)) | m(<Zs)
Bk _	fie
e[\/ 2, and so there must exist some B'k C Bk such that
f (xnk («) ~ Xoo («)) m (ds)
B'k
e/2/2 (k = 1,2,...),
contrary to the fact that, in virtue of (1), the ш-absolute continuity ol
the sequence of measures <yn(B} = [ (xn(s) —^(s)) w(^s) is uniform
в
in n (see the proof of the Vitali-Hahn-Saks Theorem in Chapter II, 2).
Next let Bo be any set of 53 such that ш(В0) < oc. We shall show that
lim y>„(B0) = 0.	(3)
п—ЮО
Suppose there exist an 8 > 0 and a subsequence {yn'} of such that
Vn'(Bo) > £ (и = 1, 2, . . .).
(4)
By the hypothesis that {(%M(s) — *co(s))} converges to 0 in ж-measure on
Bo, there exist a subsequence {(%„" (s) —	(s))} of {(хп, (s) — x^ (s))} and
some sets В” C Bo such that m (B”) gZ 2~wand | xn-> (s) — хК (s) | < e/ж (Bo)
oo
on (Bq — B'^). We put Bk — U B„. Then is a decreasing sequence
n — k
such that
(oo	\	oo	/ oo	\
nfijkz w(B")	2-*+1 (A = 1, 2, . . .) and so m П Bk = 0.
A = 1	/	n=k	\Л=1 /
Hence, by (1) and the Corollary of the Vitali-Hahn-Saks Theorem referred
to above, lim ipn(B^} = 0 uniformly in n. Therefore
k—КЮ
Vn" (Bo) yn- (B”) + em (Bo)-1 • m (Bo - - B") -> s) as n -> oo,
contrary to (4). This proves (3).
Now we take a sequence {B'k} of sets С 93 such that ж (B'k) < oc
oo
(k = 1, 2, . . .) and S = U B'k. Then
Л = 1
I
и Б'
k=i к
t
S- и В’
Л=1 к
By (3) the first term on the right tends to zero as n oc for fixed t, and
the second term on the right tends, by (2), to zero as £->oc uniformly
in n. Therefore we have proved that s-lim xn = x^ in L1(S, 53, ж).
л—КЮ
A similar situation in the case of the space $(£?)' is given by
Theorem 6. Let {Тл} be a sequence of generalized functions £ ® (£?)'.
If lim Tn = T in the weak* topology .of Ф (£?)', then lim Tn = T in
n—>OO	П-+ОО
the strong topology of
124
V. Strong Convergence and Weak Convergence
Proof. The strong topology of the space Ф(12)' is defined (see Defini-
tion 1 in Chapter IV, 7) through the family of semi-norms
/>в(В) = sup | T (99)], where S3 is any bounded set of ®(12).
The weak* topology of the space is defined through the family of
semi-norms
= sup I T(cp) where is any finite set of ® (£2).
Thus lim Tn = T in the weak* topology of Ф(12)' is precisely
lim Tn =	defined in Chapter II, 3.
Let 53 be any bounded set of © (12). Then there exists a compact
subset К in 12 such that supp (92) С К for any 99 E ® and sup (%) |
< 00 for any differential operator D-* (Theorem 1 in Chapter 1,8). Thus, by
the Ascoli-Arzela Theorem, 93 is relatively compact in §)я(12). We apply
the uniform boundedness theorem to the sequence {Tn — T} to the effect
that, for any s > 0, there exists a neighbourhood U of 0 of §)K (12) such that
sup ICG, — T) (<p) I < e.
n;tp£U
The compact subset 53 of Ф^(12) is covered by a finite system of sets of
the form 99, + U, where E S3 (i = 1, 2, ..., k). Hence
|(ГМ- Г) (^ + и) I |(ГП- T) fo) I + |(Trt- T) (и) I
I (TM — T) (90J I + £ for any и 6 U.
Since lim (Тл— T) (99J = 0 for i = 1, 2, . . ., k, we have
п-юс
lim (Tn — T) (99) = 0 uniformly in 99 C 53.
n—>00
This proves our Theorem.
Theorem 7. A reflexive В-space X is sequentially weakly complete.
Proof. Let a sequence {%„} of X be weakly convergent. Each xn defines
a continuous linear functional Xn on X’s by X„(V) =	x'}. Since
is a В-space (Theorem 1 in Chapter IV, 8), we may apply the resonance
theorem. Thus a continuous linear functional onX' is defined by a finite
lim Xn{x) which exists by hypothesis. Since X is reflexive, there exists
n—+oo
an %оо E V such that (x^, % У — lim Xn{x'} — lim	that is,
и-хзо	и—*oo
xco — ^-lim xn.
>00
Theorem 8. Let X be a Hilbert space. If a sequence {xn} of X converges
weakly to x^ E X, then s-lim xn — iff lim 1I xn 11 = 11 I
n—нэо	я—КЮ
1. The Weak Convergence and The Weak* Convergence 125
Proof. The “only if” part is clear from the continuity of the norm. The
"'if” part is clear from the equality
11 ^oo 1—
(%n	%n ^oo
In fact, the limit, as n -> oo, of the right hand side is ||||2 — | x^ 112 —
11 y 112 _L_ 11 у 112 — 0
I I -^oo || П 11 ЛОО I --- v*
Weak* Convergence
Definition 2. A sequence {/„} in the dual space X's of a normed
linear space X is said to be weakly* convergent if a finite lim fn (%) exists
П-КС
for every x 6 X; {fn} is said to converge weakly* to an element 6 X'
if lim fn (x) = f^ (x) for all x E X. In the latter case, we write w*-lim fn=f
n—>oo	л—кэо
or, in short, fn —>	weakly*.
Theorem 9.	i) s-lim fn — implies z#*-lim fn = but not con-
n—>oo	n—>oo
versely. ii) If X is a В-space, then a weakly* convergent sequence
{fn} 2 X' converges weakly* to an element	£ X' and 11 f^ 11	lim 11 fn |1.
n—>OO
Proof, (i) The first part is clear from | fn (x) — f^ (x) |	11 fn —	11 • 11 x 11.
The second part is proved by the counter example given in the proof of
Theorem 1. (ii) By the resonance theorem, we see that f (%) — lim fn (%)
n—>oo
is a continuous linear functional on X and 11 f^ 1| lim | fn .
Theorem 10.	If X is a В-space, then a sequence {fn} С X' converges
weakly* to an element X' iff (i) { fn I } is bounded, and ii) lim fn (x) —
n—>oo
/co W on a strongly dense subset of X.
Proof. The proof is similar to that of Theorem 3.
Strong and Weak Closure
Theorem 11.	Let X be a locally convex linear topological space, and M
a closed linear subspace of X. Then M is closed in the weak topology of X.
Proof. If otherwise, there exists a point x0E X — M such that x0 is
an accumulation point of the set M in the weak topology of X. Then,
by the Corollary of Theorem 3 in Chapter IV, 6, there exists a con-
tinuous linear functional /0 on X such that /0 (%0) = 1 and /0 (x) = 0 on M.
Hence Xq cannot be an accumulation point of the set M in the weak
topology of X.
126
V. Strong Convergence and Weak Convergence
2.	The Local Sequential Weak Compactness of Reflexive B-spaces.
The Uniform Convexity
Theorem 1.	Let X be a reflexive В-space, and let {xw} be any sequence
which is norm bounded. Then we can choose a subsequence {xrt,} which
converges weakly to an element of X,
We will prove this Theorem under the assumption that X is separable,,
since concrete function spaces appearing in applications of this Theorem
are mostly separable. The general case of a non-separable space will be
treated in the Appendix.
Lemma. If the strong dual X's of a normed linear space X is separable>
then so is X.
Proof. Let {x^} be a countable sequence which is strongly dense on the
surface of the unit sphere {x' E X's; 11 x' |1 = 1} of X's. Choose xn E X so
that || xn || = 1 and | <xn, хпУ |	1/2. Let M be the closed linear subspace
of X spanned by the sequence {xw}. Suppose M X and x0 E X — M.
By Corollary of Mazur's Theorem 3 in Chapter IV, 6, there exists an
Xq E X' such that || Xq 11 = 1, <x0, Xq> 0 and <x, Xq> = 0 whenever
xEAf. Thus <xn, Xo> = 0 (n = 1, 2, . . .), and so 1/2	| <x„, x^> |
|<Х» 4>~ <A»,*o>| + |<xw, Xq>| which implies that 1/2^ |(x„|l ||х^ —Xo|[
= ||4— 4||. This is a contradiction to the fact that {x^} is strongly
dense on the surface of the unit sphere of X'. Thus M — X, and so linear
combinations with rational coefficients of {xn} are dense in X. This proves
our Lemma.
Proof of Theorem 1. As we have remarked above, we assume that X
is separable and so (XX = X is separable also. By the preceding Lemma,
X's is separable. Let {x'n} be a countable sequence which is strongly dense
in X's. Since {xn} is norm bounded, the sequence {<xw, x^)} is bounded.
Thus there exists a subsequence {xwJ for which the sequence {<xWi, x^>}
is convergent. Since the sequence {<х^, x2y} is bounded, there exists a
subsequence {xMJ of {x^} such that {<хПа, x'2y} is convergent. Proceeding
in this way, we can choose a subsequence {xw.+i} of the sequence {xn.}
such that the sequence of numbers , x'->} converges for / =
l,2,...,i-f-L Hence the diagonal subsequence {xn?J of the original
sequence {xn} satisfies the condition that the sequence {<xWn, x'->} con-
verges for / = 1,2,... Thus, by Theorem 3 in the preceding section,
lim X , x'y exists and is finite for every x E X'. Hence, by Theo-
Я—ЮО
rem 7 of the preceding section, we see that te'-lim хПп exists,
n—>oo
Milman’s Theorem
We owe to D. P. Milman a theorem that a В-space is reflexive when
it is uniformly convex in the sense that, for any e > 0, there exists a
2. The Local Sequential Weak Compactness of Reflexive B-spaces 127
6 = <5 (e) > 0 such that ||x || < 1, ||y|| < 1 and ||% —y||	£ implies
||% + y||^2(l — 6). A pre-Hilbert space is uniformly convex as may be
seen from the formula
valid in such a space. It is known that, for 1 < /> < oo, the spaces Lp and
(Z^) are uniformly convex (see J. A. Clarkson [1]).
Theorem 2	(Milman [1]). A uniformly convex В-space X is reflexive.
Proof (due to S. Kakutani). Given an x'o' € (X')'s with |%q j| = 1.
Then there exists a sequence {/„} С X' with ||/w|| = 1> xo (/n) = 1 — n~T
[n = 1, 2, . . .). By Theorem 5 in Chapter IV, 6, there exists, for
every n, an xn E X such that
fi(x„) = x'o (Ji) (i = 1, 2, .. .,n) and |
xn || <;	+ n 1=l + w
Since
1 —W 1	(Л) = /«(*«)	||/n|| ||*«|1 = IWI 1 + П
we must have lim ||xw|| = 1-
n-^oo ‘
If the sequence {%„} does not converge strongly, there exists an s > 0
and пг <T mi <E < ж < • • • <	< • • • such that e <
11 Х*к ~ xmk
< W2 < W2 < * ’ * < nk < mk
I (Л = 1, 2, . . .). Thus, by lim ||%M|| — 1 and the uniform
n—>oo
convexity of X, we obtain lim ||xw +	|| < 2(1 — 6(e)) < 2. But,
ft—НЭО
since nh < mk, f„k(xn/) = f„k(x„k) = x'o (fnic) and so
2(1 — nk J) < 2xq (J„k) = fnk(xnk + x„k)	||/„fc11

Hence, by ||/„fc|| = 1, we obtain a contradiction lim ||X*fc Xfnk I! = 2.
Л >OO
We have thus proved the existence of s-lim xn = x0, and x0 satisfies
я- >oo
I koi
= 1, fi (x0) = 4 (/.) (г = 1, 2, . . .).
(1)
We show that the solution of the above equations in (1) is unique. Other-
wise, there exists an %0 7^ which satisfies the same equations. By the
uniform convexity, ||%0 + Xq || < 2. We also have fi(xQ + %0) = 2%q (Д)
(i = 1, 2, . . .). Thus
2(1 -i-1)	2x'o' (fi) = fi(x0 + x0) HA
and so Jx0 + x0|| lim 2(1 — i *) = 2 which is a contradiction.
Finally let /0 be any point of X’s. If we show that /0(я0) = xo (/o)>
then (X')' £ X and the reflexivity of X is proved. To prove that /0 (%0) =
4' (/o)> we take /0. /i> • •  > fn> • •  in place of Д, /2, . .., /w, ... above, and
hence we obtain i0 £ X such that
Ipoll = L Л(^о) = 4'(/J (^ = °, 1,	.);

128
V. Strong Convergence and Weak Convergence
we must have x0 = x0 by virtue of the uniqueness just proved above,
and so the proof of Theorem 2 is completed.
3.	Dunford’s Theorem and The Gelfand-Mazur Theorem
Definition 1. Let Z be an open domain of the complex plane. A map-
ping x (C) defined in Z with values in a В-space X is called weakly holo-
morphic in C in the domain Z if, for each f£X't the numerical function
of C is holomorphic in Z.
Theorem 1 (N. Dunford [2]). If #(C) is weakly holomorphic in Z,
then there exists a mapping x' (C) defined in Z with values in X such that,
for each Co € Z, we have
s-lim h-1 (x (Co + h) — x (£„)) = x’ (£0) .
л—И)
In other words, the weak holomorphic property implies the strong holo-
morphic property.
Proof. Let C be a rectifiable Jordan curve such that the closed bounded
domain C enclosed by C lies entirely in Z and Co € C — C. Let Zo be any
open complex domain Э Co such that its closure lies in the interior of C.
Then, by Cauchy’s integral representation, we have
2jii
о
Hence, if both Co + Л and Co T g belong to Zf
0’
h
d£.
c
By the assumption, the distance between Zo and C is positive. Hence, for
fixed f C X', the absolute value of the right hand side is uniformly boun-
ded when Co ,to + and Co + & range over Zo. Thus, by the resonance
theorem, we have
sup
h
Therefore, by the completeness of the space X, %(C) is strongly differenti-
able at every Co C Z.
Corollary 1 (Cauchy's integral theorem). The strong differentiability
of x (C) implies its strong continuity in £. Thus we can define the curvi-
linear integral f x (C) d£ with values in X. Actually we can prove that
c
f x (C) d£ = 0, the zero vector of X.
c
3. Dunford's Theorem and The Gelfand-Mazur Theorem
129
Proof. We have, by the continuity and the linearity of /6 X',
But the right hand side is zero, because of the ordinary Cauchy integral
theorem. Since /Е X' was arbitrary, we must have f x(£) d£ = 0 by
Corollary 2 of Theorem 1 in Chapter IV, 6.	c
From the above Corollary, we can derive other Corollaries, as in the
ordinary theory of functions of a complex variable.
Corollary 2 (Cauchy’s integral representation).
x (£0) = —7 I у—- d£ for any interior point £0 of C.
J Q — 4o
C
Corollary 3 (Taylor’s expansion). For any point £0 which is in the
interior of the closed domain C, the Taylor expansion of x (C) at C = Co
converges strongly in the interior of the circle with centre at £0, if this
circle does not extend outside of C:
oo
X(C) = V („!)-! (C-C0)”xw(C0), where
xw >> \ —f ________________j?-
(Co) ~ J (C - Co)”+I ?’
Corollary 4 (Liouville’s theorem). If x(C) is (strongly) holomorphic in
the whole finite plane: |£| < oo, and sup |%(C) | < oo, then %(£) must
reduce to a constant vector x(0).	1^1 <°°
Proof. If we take |£ | = r for the curve C, then, as r —> oo,
Hence the Taylor expansion of %(£) at £ — 0 reduces to the constant
term x(0) only.
We shall now apply Corollary 4 to the proof of the Gelfand-Mazur theo-
rem. We first give
Definition 2. A commutative field X over the field of complex numbers
is called a normed field, if it is also a В-space such that the following
conditions are satisfied:
j | e 11 = 1, where e is the unit of the multiplication in X,
11 xy 11	11 % 11 11 у 11, where xy is the multiplication in X.
Theorem 2 (Gelfand [2]-Mazur [1]). A normed field X is isometri-
cally isomorphic to the complex number field. In other words, every ele-
ment x of X is of the form x = <£ e where £ is a complex number.
f) Yonida, Functional Лпа1ук1н
130
V. Strong Convergence and Weak Convergence
Proof. Assume the contrary, and let there exist an x £ X such that
(x — $e) =£ 0 for any complex number £. Since A is a field, the non-zero
element (% — f e) has the inverse (x — f г)-1 £ X.
We shall prove that (% — Яг)-1 is (strongly) holomorphic in Я for
|Я I < oo. We have, in fact,
h~L((x — (A 4- h) г)-1 — (% — Я гр1)
= h"1 (% — (Я + A)гр1 {e — (% — (Я + Л)e) (x — Яг)-1}
= h~\x —(Я + h) г)-1 {г — e 4- h\x — Я гр1}
= (% — (Я 4 /p) 1 (% — Яг) 1.
__/ oo	\
On the other hand, for sufficiently small | h |, the series у-11 e + (Ay-1p L
where у = (x — Яг), converges by (1), and it represents the inverse
(y — Ле)-1 = у 1 (г — Лу1)-1, as may be seen by multiplying the series
by (y — he}. Hence, by the strong continuity in h of the series, we can
prove that (x — Яг)1 is (strongly) holomorphic in Я with the strong
derivative (% — Яг)-2.
Now, if (Я|	2 ||%11, then, as above, (x — Яг)-1 = — Я-1 (г — Я-1 x)-1
г 4-	(Я-1хр) and so
n=l	j
11 (ж — Ae) xl
r1,(l+ 2(W|->0 as |A|->oo.
\ Tt = l	/
Moreover, the function (% — Яг)-1, being continuous in Я, is bounded on
the compact domain of Я: |Я| P 2 |[%||. Hence, by Liouville's theorem,
(% — Яг)-1 must reduce to the constant vector %-1 = (% — 0г)-1. But,
since s-lim (% — Яг)-1 = 0 as proved above, we have arrived at a contra-
|A|—>oo
diction x-1 = 0, г = %-1 x ~ 0.
4. The Weak and Strong Measurability. Pettis’ Theorem
Definition 1. Let (S, 53, m) be a measure space, and %(s) a mapping
defined on S with values in a ZLspace X. %(s) is called weakly ^-mea-
surable if, for any / С X', the numerical function /(%(s)) = <x(s), /> of s
is 93-measurable. x (s) is said to be finitely-valued if it is constant 0 on
each of a finite number of disjoint S3-measurable sets B3 with m (Bj) < oo
and %(s) — 0 on S — U B3. x(s) is said to be strongly -measurable if
i
there exists a sequence of finitely-valued functions strongly convergent
to x(s) m-a.e. on S.
Definition 2. x (s) is said to be separably-valued if its range {% (s); s C S}
is separable. It is m-almost separably-valued if there exists a 93-measurable
set Bq of m-measure zero such that {x(s); x C -S' — /?0} is separable.
4. The Weak and Strong Measurability. Pettis' Theorem 131
Theorem (B. J. Pettis [1]). x(s) is strongly 53-measurable iff it is
weakly 53-measurable and ш-almost separably-valued.
Proof. The “only if” part is proved as follows. The strong 53-measu-
i ability implies the weak 53-measurability, because a finitely-valued func-
tion is weakly 53-measurable, and, by the strong 53-measurability of
x(s), there exists a sequence of finitely-valued functions xn(s) such that
s-lim xn{s) ~ x(s) except on a set BO£53 of ш-measure zero. Thus the
»—ЮО
union of the ranges of xn(s) (n = 1, 2, . . .) is a countable set, and the
closure of this set is separable and contains the range {%(s); s C S — Bo}.
The proof of the “if” part. Without losing the generality, we may
assume that the range {% ($);$£ 5} is itself separable. So we may assume
that the space X is itself separable; otherwise, we replace X by the
smallest closed linear subspace containing the range of %(s). We first
prove that 11x (s) || is itself 53-measurable. To this purpose, we shall make
use of a lemma, to be proved later, which states that the dual space X'
of a separable В-space satisfies the condition
there exists a sequence {/rt} С X' with ||/w||	1 such
that, for any /0С X' with ||/0||	1, we may choose a
subsequence {/nJ of {/„} for which we have lim fn> (x)
n—*<x>
= fQ (x) at every x £ X.
(1)
Now, for any real number л, put
/1 =	; ||^(s) || 5^ d] and Af = {s; |/(%($)) (	a}, where / С X'.
co
If we can show that A = П Aff, then, by the weak 53-measurability of
i (s), the function ||%(s) || is 53-measurable. It is clear that А С П Af.
But, by Corollary 2 of Theorem 1 in Chapter IV, 6, there exists, for
fixed s, an /0(Е X' with ||/0|| = 1 and fQ(x(s)') — ||x(s) ||. Hence the re-
verse inclusion A 2 (T Af is true and so we have A — Г1 Af. By
J	llfllsi 7
oo	oo
(he Lemma, we obtain П Af— П Af. and so A = П Af..
Il/ll^i 7 j=i h	J}
Since the range {%($); s G S} is separable, this range may, for any
positive integer n, be covered by a countable number of open spheres
Sjn (j =Д, 2, . . .) of radius 1/n. Let the centre of the sphere be
\} n. As proved above, 11 x (s)—xJ>n 11 is 53-measurable in s. Hence the
co
set B} n	Sjn} is 53-measurable and S = U Bjn. We set
»-i
^n ($) — ^i,n if s G B^n  B±n U Byn.
oo
we have — *n(s)|| < 1/n for every s £ S.
132
V. Strong Convergence and Weak Convergence
Since B'in is 93-measurable, it is easy to see that each xn(s) is strongly 93-
measurable. Therefore %(s), which is the strong limit of the sequence
{*»($)}, is also strongly S3-measurable.
Proof of the Lemma. Let a sequence {xn} be strongly dense in X.
Consider a mapping /^ ?>„(/)= {/(xj,/(x2), ...,/(%„)} of the \init
sphere S' = {/€ X’; ||/||	1} of X' into an n-dimensional Hilbert space
O	/ »	\l/2
lz(n) of vectors (f1( £2>..., normed by ||(f1(f2...f„) [| =(>£|£,|2j •
The space l2(n) being separable, there exists, for fixed n, a sequence
{fntk} (^ = 1» 2, . . .) of S' such that {(pn (fnk); k = 1, 2, . . .} is dense in the
image <pn(S') of S'.
We thus have proved that, for any /0 E S', we can choose a subsequence
= 1, 2,. . .) Such that |f„imn(xf) — /0(xf) | < 1/я(г= 1, 2,..
Hence lim fnm	= /0(%J (i = 1, 2,. ..), and so, by Theorem 10 in Chap-
n—>oo ’
ter V, 1, we obtain that lim fnmn(x) = fQ(x) for every x E X.
5. Bochner’s Integral
Let %(s) be a finitely-valued function defined on a measure space
(S, S3, m) with values in a В-space AT; let x(s) be equal to 0 on
BiE 93 (i = 1, 2, . . ., ri) where B/s are disjoint and m(Bt) < oo for
(n \
S — £ Bi I. Then we can
4 = 1	/
Г	П
define the m-integral J x(s) m(ds) of x(s) over S by £ xt m(B^. By
s
virtue of a limiting procedure, we can define the m-integral of more
general functions. More precisely, we have the
Definition. A function %(s) defined on a measure space (S, 93, m) with
values in a В-space X is said to be Bochner m-integrable, if there exists
a sequence of finitely-valued functions {%„($)} which s-converges to #(s)
ж-а.е. in such a way that
lim f 11 x (s) — xn (s) 11 m (ds) = 0.	(1)
n *oo g
For any set В £ 93, the Bochner m-integral of x(s) over В is defined by
f x(s) m(ds) = s-lim f CB(s) xn(s) m(ds), where CB is the
В	5	(2)
defining function of the set В.
To justify the above definition, we have to verify that the s-limit on
the right of (2) exists and that the value of this s-limit is independent of
the approximating sequence of functions {%„($)}.
Justification of the Definition. First, x(s) is strongly 93-measurable
and consequently the condition (1) has a sense, since |x(s) —xn (s) 11 is
5. Bochner’s Integral
133
53-measurable as shown in the proof of Petti’s Theorem. From the in-
equality
J xn (s) m (ds) — f xk (s) m (ds) = f (xn (s) — xk (s)) m (ds)
в	в	I в
< f 11 xn (s) — xk (s) 11 m (ds) < f 11 xn (s) — x (s) 11 m (^s)
в	s
+ J ||x(s) ~x*(s)|| m(ds)
s
and the completeness of the space X, we see that s-lim f xn(s) m(ds)
n—>OO jg
exists. It is clear also that this s-lim is independent of the approximating
sequence, since any two such sequences can be combined into a single
approximating sequence.
Theorem 1 (S. Bochner [1]). A strongly 53-measurable function %(s) is
Bochner w integrablc iff 11 x (s) 1| is m-integrable.
Proof. The “ only if" part. We have ||%(s)||^ ||xn(s)|| + 11 x (s) — xn(s) 1|.
By the m-integrability of ||%„(s) || and the condition (1), it is clear that
11 x (s) 11 is ж-integrable and
f 11 x (s) 11 m (ds) f 11 xn (s) 11 m (ds) + f 11 x (s) — xn (s) 11 m (ds).
в	в	в
Moreover, since
/ I ||x„ (s) || — ||хл (s) || I m(ds) f 11 xn (s) — xk(s) || m(ds),
В	в
we see from (1) that lim f ||xn(s) || m(ds) exists so that we have
Пг—ИЗО jg
f 11 x (s) | m (ds) lim f 11 xn (s) | m (ds).
В	П—+<Х) В
The "if" part. Let {xn(s)} be a sequence of finitely-valued functions
strongly convergent to %(s) m-a.e. Put
Уп («) = xn (s)
= 0 if
||xM(S)||^||x(S)||(l + 2-i),
ll^(s)||>||^(s)||(l + 2-i).
I hen the sequence of finitely-valued functions {y„(s)} satisfies ||y„(s) |
I t(s)| • (1 + 2-1) and lim lx(s)— yn (s) 11 = 0 ж-a.e. Thus, by the
n—юо
m~integrability of ||x(s) ||, we may apply the Lebesgue-Fatou Lemma to
(he functions x(s) — y„(s) [I | %($) | (2 + 2-1) and obtain
lim f 11 x (s) — yn (s) 11 m (ds) = 0,
я—>co
(hat is, %(s) is Bochner m-integrable.
Corollary 1. The above proof shows that
У 11 x (s) 11 m (ds) f x(s) m (ds) ,
в	в
134
V. Strong Convergence and Weak Convergence
and hence f x (s) ж (ds) is ж-absolutely continuous in the sense that
в
s-lim f x(s) wi(ds) = 0.
tn{B)—+0 ё
г	n г
The finite additivity f x(s) m (ds) = £ f x(s) m (ds) is clear and
n	>=1 Bj
В Bj
5 = 1
so, by virtue of the a-additivity of f ||a(s) | ж(б&), we see that
в
f x(s) m(ds) is cr-additive, i.e.,
в
oo
В = X Bj with m (Bj) < oo implies f x(s) m (ds)
*=1	oo
В Bj
5=1
n
= s-lim X J % (s) m (ds).
n^OOJ = l g.
Corollary 2. Let T be a bounded linear operator on a В-space X into a
В-space У. If %(s) is an X-valued Bochner ж-integrable function, then
Tx(s) is a У-valued Bochner ж-integrable function, and
f Tx(s) m(ds) = T f x(s) m(ds).
В	в
Proof. Let a sequence of finitely-valued functions {y„(s)} satisfy
lyn(s) ||	||*(s) || (1 + n r) and s-lim yn(s) = x(s) m-a.e.
n—>oo
Then, by the linearity and the continuity of T, we have f Tyn (s) ж (ds) =
в
T f yn(s) m(ds). We have, moreover, by the continuity of T,
в
|7'y„(s)||^ НГЦ. ||y„(s)||^ ||T||. ||x(S)||-(l + n-1) and
s-JimTyM(s) = Tx(s) ж-а.е.
n—>oo
Hence T x (s) is also Bochner ж-integrable and
f Tx(s) m (ds) = s-lim f T yn (s) ж (ds) = s-lim T f yn (s) ж (ds)
£	>00 p	П—>OO jj
= T f x(s) ж (ds).
в
Theorem 2 (S. Bochner [1]). Let S be an n-dimensional euclidean
space, 93 the family of Baire sets of S, and m(B) the Lebesgue measure
of B. If x(s) is Bochner ж-integrable, and it B(s0; oc) is the parallelepiped
with centre at s0C S and side length 2a, then we have the differentiation
theorem
s-lim (2a)’n f x (s) ж (ds) = x (s0) for ж-а.е. s0.
F(s0;«)
5. Bochner’s Integral
135
Proof. Put
(2a) n f x(s) m(ds) = D(x'> sQtoc).
P(s0;tx)
If {%„ ($)} is a sequence of finitely-valued functions such that 11 xn (s) 11
x(s) I • (1 -J- w-1) and s-lim xn(s) = x(s) then
П>(Х)
D (x; s0, a) — x(s0) = D (x — xk; s0, a) + D (x^; s0, a) — x(s0),
and so
lim 11D (x; s0, a) —x (s0) I
a|0 1
5g lim Z) (I % — xk s0, a)
«;o	1	1
+ lim	s0, л) — xft(s0)|
a 10
I M^o) — * (s0) 11 •
The first term on the right is, by Lebesgue's theorem of differentiation of
numerical functions, equal to | (s0) —xk(s0) 11 ж-а.е. The second term
on the right is — 0 ж-а.е., since xk(s) is finitely-valued. Hence
lim 11D (%; s0, oc) — x (s0) 11	2 11 xk (s0) — x (s0) 1| for ж-а.е. s0.
a->0
Therefore, by letting k oo, we obtain Theorem 2.
Remark. Contrary to the case of numerical functions, a B-space-
valued, o-additive, ш-absolutely continuous function need not necessarily
be represented as a Bochner ж-integral. This may be shown by a counter
example.
A Counter Example. Let S = [0, 11 and S3 the family of Baire sets
on 0, 1], and ж (В) the Lebesgue measure of В C ®. Consider the totality
ж [1/3, 2/3] of real-valued bounded functions f = £(0) defined on the
closed interval [1/3, 2/3] and normed by ||f || = sup |£(0) |. We define
в
an ж [1/3, 2/3]-valued function x(s) = f (0; s) defined on [0, 1] as follows:
the graph in s-y plane of the real-valued function у = ye (s),
which is the ^-coordinate f (0; s) of % (s), is the polygonal line
connecting the three points (0, 0), (0, 1) and (1, 0) in this order.
Then, if s s', we have Lipschitz condition'.
I (s —5 (* (5) — x (s0)
= sup
о
(s-s')-1(f(0;s)-^(0;s'))|^3.
thus, starting with the interval function (%(s) — x(s')) taking values in
m [1/3, 2^3], we can define a or-additive, ж-absolutely continuous set
function x(B) defined for Baire set В of [0, 1].
If this function x(B) is represented as a Bochner ж-integral, then,
by the preceding Theorem 2, the function %(s) must be strongly differen-
tiable with respect to s ж-а.е. Let the corresponding strong derivative
v'(s) be denoted by s) which takes values in m 1/3, 2/3]. Then, for
136
Appendix to Chapter V. Weak Topologies and Duality
every 0 6 [1/3, 2/3] and m-a.e. s,
0 = lim | j hr1 (x (s + Л) — % ($)) — x' (s) 11 Ji lim | hr1 (0; s + h) — f (0 ;$))
h—>0	h—>0
— ^(0; s) |.
This proves that f (0; s) must be differentiable in s m-a.e. for all 0 6
[1/3, 2/3]. This is contradictory to the construction of f (0; s).
References for Chapter V
S. Banach [1], N. Dunford-J. Schwartz [1] and E. Hille-R. S.
Phillips [1].
Appendix to Chapter V. Weak Topologies and Duality
in Locally Convex Linear Topological Spaces
The present book is so designed that the reader may skip this appen-
dix in the first reading and proceed directly to the following chapters.
1.	Polar Sets
Definition. Let X be a locally convex linear topological space. For
any set Af С X, we define its (right) polar set MQ by
M° — {xf 6 X'; sup | <%, %') |	1}.	(1)
x^M
Similarly, for any set Mf Q X', we define its (left) polar set QM' by
°M' = {x(EX;sup |<%,%'>|	1} = XA (M')Q,	(2)
x'QM'
where we consider X to be embedded in its bidual (X's)'.
A fundamental system of neighbourhoods of 0 in the weak topology
of X is given by the system of sets of the form °Af' where M' ranges over
arbitrary finite sets of X'. A fundamental system of neighbourhoods of
0 in the weak* topology of X' is given by the system of sets of the form
Af° where M ranges over arbitrary finite sets of X. A fundamental
system of neighbourhoods of 0 in the strong topology of X’ is given by
the system of sets of the form Af° where M ranges over arbitrary bounded
sets of X.
Proposition. MG is a convex, balanced set closed in the weak* topology
of V.	ч
Proof. For any fixed x£ X, the linear functional f(x) = <%, x} is
continuous in the weak* topology of X'. Thus MQ ~ П {m}° is closed
in the weak* topology of X'. The balanced convexity of MQ is clear.
1. Polar Sets
137
An Application of Tychonov’s Theorem
Theorem 1.	Let X be a locally convex linear topological space, and A
a convex, balanced neighbourhood of 0 of X. Then Л0 is compact in the
weak* topology of X'.
Proof. Let p (x) be the Minkowski functional of A. Consider, for each
x E X, a sphere Sx = {z E C; | z | p (%)} and the topological product
S = П Sx. S is compact by Tychonov’s theorem. Any element %' E X'
is determined by the set of values x (%) = <X x'y, x£X. Since %E
(/? (x) + e) A for any e > 0, we see that x' E X' implies <% x'y =
((P(x) + e)a, x'y with a certain a£A. Thus x E A° implies that
| x (%) | p (x) + e, that is, x' (x) E Sx. Hence we may consider A0 as a
subset of S. Moreover, it is easy to verify that the topology induced on
A0 by the weak* topology of X' is the same as the topology induced on
A0 in the Cartesian product topology of S = ]J Sx.
x€X
Hence it is sufficient to prove that X°is a closed subset of S. Suppose
у = У (x) is an element of the weak* closure of AQ in S. Consider any
E
xtX
0 and any я;1, x2 E X. The set of all и = Ц u(x) E S such that
x(~X
I M (*1) — У (xl) I < e> \u(x2) ~У(х2) I < e and IU (X1 + X2}—y(x1 + x2) | < £
is a neighbourhood of у in S. This neighbourhood contains some point
x E ^4° and, since x' is a continuous linear functional on X, we have
|y(*i + x2) — у (ж,) — у (я2) I |y + x2) — <%, + x2, я/>|
+ |<*i. х'У — у(xx) I + I<x2, x'> — у (x2) I < 3e.
This proves that у (%x + x2) = у (%г) + у (х2). Similarly, we prove that
= fiy(x), and so у defines a linear functional on X. By the fact
that у = JJ y(x) E S, we know that | у (x) | p (%). Since p (%) is con-
tinuous, y(x) is a continuous linear functional, i.e., у E X'. On the other
hand, since у is a weak* accumulation point of A0, there exists, for any
e > 0 and a E A, an x’ E A0 such that |y (a) — (a, x'y | e. Hence
| У (л) |	| <X x'y | + e 1 + e, and so | у (a) Sj 1, that is, у E Л0.
Corollary. The unit sphere S* — {x' E X'; |1 x' 11	1} of the dual space
X' of a normed linear space X is compact in the weak* topology of X'.
An Application of Mazur’s Theorem
Theorem 2.	Let M be a convex, balanced closed set of a locally convex
linear topological space X. Then M — °(M°).
Proof. It is clear that M C °(M°). If there exists an %0E °(Af°) — Mt
then, by Mazur’s theorem 3 in Chapter IV, 6, there exists an x0 E X'
such that <%0, x'Qy > 1 and | (x,	1 for all x E M. The last inequality
shows that x^ E M° and'so x0 cannot belong to °(M°).
138
Appendix to Chapter V. Weak Topologies and Duality
2.	Barrel Spaces
Definition. In a locally convex linear topological space X, any convex,
balanced and absorbing closed set is called a barrel (tonneau in Bourbaki’s
terminology). X is called a barrel space if each of its barrels is a neighbour-
hood of 0.
Theorem 1.	A locally convex linear topological space X is a barrel
space if X is not of the first category.
Proof. Let T be a barrel in X. Since T is absorbing, X is the union of
closed sets nT = {nt',tQT}, where n runs over positive integers. Since
X is not of the first category, at least one of the (n T)'s contains an interior
point. Hence T itself contains an interior point %0. If x0 = 0, T is a neigh-
bourhood of 0. If x0 0, then —x0E T by the fact that T is balanced.
Thus —xG is an interior point of T with x0. This proves that the convex
set T contains 0 — (xG — %0)/2 as an interior point.
Corollary 1. All locally convex F-spaces and, in particular, all B-
spaces and &(Rn) are barrel spaces.
Corollary 2. The metric linear space %)K (Rn) is a barrel space.
Proof. Let {(pk] be a Cauchy sequence with respect to the distance
, y>) = v'2	, where = sup |Dj(p(x) |.
For any differential operator D3, the sequence {D3<pk(x)} is equi-contin-
uous and equi-bounded, that is,
lim sup |D3q)k(x1) —	(%2) j = 0 and sup |D3cpk (%) | < oo.
|жм—ж3|фО ASl
This we see from the fact that, for any coordinate xs,

sup
oo. Hence, by the Ascoli-Arzela theorem, there
exists a subsequence of {D3 cpk- (%)} which converges uniformly on K.
By the diagonal method, we may choose a subsequence {9?^ (%)} of {(pk (%)}
such that, for any differential operator D3, the sequence {D3q)k"(x}}
converges uniformly on K. Thus
lim D'Jq:k" (x) = D3<p (x) where 9? (x) = lim <pk„ (%),
A"—>oo	Л"->оо
and these limit relations hold uniformly on K. Hence the metric space
(Fw) is complete and so it is not of the first category.
Remark, (i) The above proof shows that a bounded set of %(Rn) is
relatively compact in the topology of %(Rn). For, a bounded set В of
® (Rn) is contained in some (Rn) where A is a compact set of R”, and,
moreover, the boundedness condition of В implies the equi-boundedness
and equi-continuity of {D3(p; 9? 6 B} for every D3. (ii) Similarly, we see
that any bounded set of &(Rtl) is a relatively compact set of &(Rn).
3. Semi-reflexivity and Reflexivity
139
Corollary 3. ® (Rn) is a barrel space.
Proof. Ф(ЛЛ) being an inductive limit of	when К ranges
over compact subsets of Rn, the present corollary is a consequence of
the following
Proposition. Let a locally convex linear topological space X be an
inductive limit of its barrel subspaces X*, oc£ A. Then X itself is a barrel
space.
Proof. Let V be a barrel of X. By the continuity of the identical
mapping Ta : x —> x of Xa into X, the inverse image (У) = V А Хл
is closed with V. Thus V A Xa is a barrel of Xa. Xa being a barrel space,
V А Хл is a neighbourhood of 0 of Xa. X being an inductive limit of Xa’s,
V must be a neighbourhood of 0 of X.
Theorem 2.	Let X be a barrel space. Then the mapping x Jx of X
into (X')', defined in Chapter IV, 8, is a topological mapping of X onto
JX, where the topology of JX is provided with the relative topology
of JX as a subset of (X')'.
Proof. Let B' be a bounded set of X's. Then the polar set (B')° —
{x" E (X')z; sup (<%', x"y | < 1} of B’ is a neighbourhood of 0 of (X')',
x'ZB'
and it is a convex, balanced and absorbing set closed in*(X')'« Thus
(B')° A X = °(B') is a convex, balanced and absorbing set of X. As a
(left) polar set, °(B') is closed in the weak topology of X, and hence
Ю(В') is closed in the original topology of X. Thus °(B') = (B')° A X is
a barrel of X, and so it is a neighbourhood of 0 of X. Therefore, the
mapping x—> Jx of X into (X')' is continuous, because the topology of
(X')' is defined by a fundamental system of neighbourhoods of 0 of the
form (B')°, where Bf ranges over bounded sets of X'.
Let, conversely, U be a convex, balanced and closed neighbourhood of
0 of X. Then, by the preceding section, U = °(A°).Thus J U = JX A (A0)0.
On the other hand, 17° is a bounded set of X'. since, for any bounded set
В of X, there exists an oc > 0 such that ocB Q U and so (aB)° J A0.
Hence (A0)0 is a neighbourhood of 0 of (Xs')'. Thus the image J U of the
neighbourhood A of 0 of X is a neighbourhood of 0 of J X provided with
the relative topology of J X as a subset of (X')
3.	Semi-reflexivity and Reflexivity
Definition 1. A locally convex linear topological space X is called
semi-reflexive if every continuous linear functional on X' is given by
x'y, with a certain x £ X.	(1)
Thus X is semi-reflexive iff
w
(•2)
140
Appendix to Chapter V. Weak Topologies and Duality
Definition 2. A locally convex linear topological space X is called
reflexive if
X = (x'sys.	(3)
By Theorem 2 in the preceding section, we have
Proposition 1. A semi-reflexive space X is reflexive if AT is a barrel space.
It is also clear, from Definition 2, that we have
Proposition 2. The strong dual of a reflexive space is reflexive.
Theorem 1.	A locally convex linear topological space X is semi-
reflexive iff every closed, convex, balanced and bounded set of X is
compact in the weak topology of X.
Proof. Let X be semi-reflexive, and T a closed, convex, balanced and
bounded set of X. Then, by Theorem 2 in Section 1 of this Appendix,
T = °(T°). T being a bounded set of A, T° is a neighbourhood of 0 of X'.
Thus, by Theorem 1 in Section 1 of this Appendix, (T°)° is compact in the
weak* topology of (X')'. Hence, by the semi-reflexivity of X, T = °(T°)
is compact in the weak topology of X.
We next prove the sufficiency part of Theorem 1. Take any x" £ (X's)'.
The strong continuity of x" on X's implies that there exists a bounded
set В of X such that
|<x', x"y \	1 whenever x € B°, that is, x" G (B°)°.
We may assume that В is a convex, balanced and closed set of X. Thus,
by the hypothesis of Theorem 1, В is a compact set in the weak topology
of X. Hence В = Bwa where Bwa denotes the closure of В in the weak
topology of X. Since Xw is embedded in (X')'w* as a linear topological
subspace, we must have (B°)° 2 Bwa ~ B. Therefore we have to show
that x" is an accumulation point of В in (X$)^*. Consider the mapping
x-><p(x) = {<%,	(%, х'пУ} of X into I2 (n), where x'l r ...x'n£ X'.
The image (p(B) is convex, balanced and compact, since В is convex,
balanced and weakly compact. If {(x.p x"), ...» (x', %”)} does not belong
to 92(B), then, by Mazur's theorem, there would exists a point
{clf . . ., cn} С I2 (n) such that sup £ c^b, x-)	1 and /2? (x'i> x”)] > h
b£B i	\ i	/
proving that 2? cixi € B° and x" cannot belong to B00.
г
Theorem 2.	A locally convex linear topological space X is reflexive
iff it is a barrel space and every closed, convex, balanced and bounded set
of X is compact in the weak topology of X. In particular, §>(Rn) and
(£(Bn) are reflexive.
Proof. The sufficiency part is proved already. We shall prove that
the first condition of Theorem 2 is necessary.
4. The Eberlein-Shmulyan Theorem
141
Let T be a barrel of X. We shall prove that T absorbs any bounded
set В of X so that B° 2 л T°, a > 0. B° being a neighbourhood of 0 of
X's, we see that ТУ is a bounded set of X'. By the Proposition and Theo-
rem 2 in Section 1 of this Appendix, we have T =	By the hypothesis
that X is reflexive, we have °(T°) *= (T°)° and so T = (T°)°. Therefore we
have proved that the barrel T is a neighbourhood of 0 of X = (X')'.
Hence X is a barrel space.
By hypothesis, the closed, convex and balanced set X—Conv / U ocB\a
is compact in the weak topology of X. Here we denote by Conv(X)a the
Cv
closure in X of the convex closure (see p. 28) Conv (N) of N. Set Y = иУ К
and let p (%) be the Minkowski functional of K. Then since К is ^-compact in
У, p (%) defines a norm of У. That is, the system {aX} with a > 0 defines a
fundamental system of neighbourhoods of the normed linear space У, and
У is a В-space since X is г^-compact. Hence У is a barrel space. On the
other hand, since X is a bounded set of X, the topology of У defined by the
norm p (x) is stronger than the relative topology of У as a subset of X.
As a barrel of X, T is closed in X. Hence T f\ Y is closed in У with respect
to the topology defined by the norm p (%). Therefore T C\ Y is a barrel
of the В-space Y, and so T Г\ Y is a neighbourhood of 0 of the B-space
У. We have thus proved that TГ\ Y and, a fortiori, T both absorb
К 2 В.
4.	The Eberlein-Shmulyan Theorem
This theorem is very important in view of its applications.
Theorem (Eberlein-Shmulyan). A В-space X is reflexive iff it is
locally sequentially weakly compact; that is, X is reflexive iff every
strongly bounded sequence of X contains a subsequence which converges
weakly to an element of X.
For the proof we need two Lemmas:
Lemma 1. If the strong dual X' of a В-space X is separable, then X
itself is separable.
Lemma 2 (S. Banach). A linear subspace M' of the dual space X' of a
В-space X is weakly* closed iff M' is boundedly weakly* closed; that is,
M' is weakly* closed iff M' contains all weak* accumulation points of
every strdiigly bounded subset of M'.
Lemma 1 is already proved as a Lemma in Chapter V, 2. For Lemma 2,
we have only to prove its “if” part. It reads as follows.
Proof. (E. Hille-R. S. Phillips [1]). We remark that M' is strongly
closed by the hypothesis. Let Xq С M'. Then we can prove that, for each
142
Appendix to Chapter V. Weak Topologies and Duality
constant C satisfying the condition 0 < C < inf ||V— ^j|, there exists
an я0£ X with ||xol| rg 1/C such that
(x0,	= 1 and <x0, x'y ~ 0 for all x' С M'.	(1)
Thus the strongly closed set M' must contain all of its weak* accumula-
tion points.
To prove the existence of %0, we choose an increasing sequence of
numbers {Cft} such that C3 = C and lim C„ — oo. Then there exists a
n—>oo
finite subset cq of the unit sphere S = (x 6 X; 11 x ||	1} such that
II*'= G and sup |O, x'y — <%, x'o) j Cx implies x'£ M'.
V^cr1
If not, there would exists, corresponding to each finite subset er of S, an
xc C Af such that
I -< - 4
|| S C2 and
sup \<x, xay — <x, x'> I g Q.
We order the sets <r by inclusion relation and denote the weak* closure of
the set {xa\ o' or} by N'a. It is clear that N'a enjoys the finite intersection
property. On the other hand, since M' is boundedly weakly* closed,
the Corollary of Theorem 1 in this Appendix, 1, implies that the set
Af', = {x' С M'; ||я'|| C}
is weakly* compact. Hence N'a QM'e for C' — C2 + ||*oll> an(l so there
exists an х'т С П N‘a C M'. Hence we have sup |<x, x[> — <x, x’y I jg C,
a	x$S
and so — xQ || Cp contrary to the hypothesis that 0 < Cx <
inf ||я/ — %q||.
x’tM'
By a similar argument, we successively prove the existence of a se-
quence of finite subsets cq, cr2, . . • of S such that
Ck and
sup I <x, x'y — <x, x'oy I <; Ct
(» = 1, 2...................k - 1)
do not imply x' 6 M'.
Thus, since lim Ci = oo, we see that x' g M' if
i—>oo
j <x, x'y — ^x, x'oy I C for all xE(C/C,)ay (7 = 1,2,...).
Let {ям} be a sequence which successively exhausts the sets (C/Cj)oj
(/=1,2,...). Then lim xn = 0 and so L (xf) = {(xn, x'y} is a bounded
П—M5O
linear transformation of X's into the В-space (c0). We know that the
point {(xn, x'oy) C (c0) lies at a distance > C from the linear subspace
L(M'). Thus, by Corollary of Theorem 3 in Capter IV, 6, there exists a
4. The Eberlein-Shmulyan Theorem
143
continuous linear functional E (c0)' = (Z1) such that
НЫП
oo	oo
_	1/C, — ««Or.. x0> = 1 and
n l	n = l
oo
X /* 0
»=1
for all x E M’.
oo
The element x0 = ocnxn clearly satisfies condition (1).
n=l
Corollary. Let <%', x'o') == F (x) be a linear functional defined on
the dual space X' of a В-space X. If N (F) = N (xq ) = {%' E X'; F (%') = 0}
is weakly* closed, then there exists an element x0 such that
F(%') = « %"> = <x0, x'y
for all x 6 X'.
(2)
Proof. We may assume that N (F)	X'. Otherwise we can take
x0 = 0. Let E X' be such that F (%^) = 1. By (1) of the preceding
Lemma 2, there exists an x0 E X such that
(x0, хоУ ~ 1 and Осе Х'У ~ 0 for a^ x' E N (-F) •	(3)
Hence, for any x' E X', the functional
xf-F(x')x'0 = y'eX'
satisfies F (/) = 0, i.e. у E X (F). Therefore, by (3), we obtain (2).
Proof of the Theorem. “Only if" part. Let be a sequence of X
such that ||%n|| = L The strong closure Xo of the subspace spanned by
{%M} is a separable В-space. Being a В-space, Xo is a barrel space. We
shall show that Xo is reflexive. Any strongly closed, bounded set Bo of
Xo is also a strongly closed, bounded set of X and hence Bo is compact
in the weak topology of X by the reflexivity of X. But, as a strongly
closed linear subspace of X, Xo is closed in the weak topology of X (see
Theorem 3 in Chapter IV, 6). Hence Bo is compact in the weak topology
of Xo. Thus Xo is reflexive by Theorem 2 in the preceding section. We
have thus Xo = ((Xo)')'. By Lemma 1 above, (Xo)' is thus separable. Let
{x'J be strongly dense in (Xo)'. Then the weak topology of Xr0 is defined
by an enumerable sequence of semi-norms pm(x) = \<x, <>| (m =
1,2,...). Hence it is easy to see that the sequence {xrt}, which is compact
in the weak topology of Xo, is sequentially weakly compact in Xo and
in X as well. We have only to choose a subsequence {%„/} of {%„} such
that finite lim x^y exists for m = 1, 2, . . .
n->oo
“If” pcfrt. Let M be a bounded set of X, and assume that every in-
finite sequence of M contains a subsequence which converges weakly to
an element of X. We have to show that the closure M of M in X in the
weak topology of X is weakly compact in X. For, then the barrel space
X is reflexive by Theorem 2 in the preceding section.
144
Appendix to Chapter V. Weak Topologies and Duality
Since Xw g (X's)'w*, we have M = M C\ Xw, where M is the closure
of M in the weak* topology of (X's)'. Let Si be the sphere of X' of radius
r > 0 and centre 0. By the correspondence
M Э m о {<X,	; |]x' || 5g 1} E JJ Zv, where
Л' = {?', |z| sup |« m>|},
M may be identified with a closed subset of the topological product
1^. By Tychonov’s theorem, is compact and hence M is
compact in the weak* topology of (X's)'. Hence we have only to show
MQXW.
Let x'q E (A”')' be an accumulation point of the set M in the weak*
topology of (X')'. To prove that x'q E Xw we have only to show that the
set TV (x'o') = {x' e X'; <x', x"> = 0} is weakly* closed. For, then, by the
above Corollary, there exists an x0E X such that <x', x") = <x0, x'y for
all x' E X'. We shall first show that
for every finite set x'v x'2, . . xn of X', there exists a zE ТИ
such that <xj, %"> = <z, x}> (/ = 1, 2, . . ., n).
The proof is as follows. Since Xq is in the weak* closure of M, there is an
element zm E M such that
I<X. Xj) — <Xj, Xg) I 1/m (/ = 1, 2, . . ., n).
By hypothesis there exists a subsequence of {zm} which converges weakly
to an element z E X and so z E M, since the sequential weak closure of M
is contained in M. We have thus (4).
Now, by Lemma 2, N (х£) is weakly* closed if, for every r > 0, the
set TV(%q) f\ S'T is weakly* closed. Let y'o be in the weak* closure of
N (xq) C\ S'. We have to show that y'o E TV (x^) C\ S{. To this purpose, we
choose an arbitrary e > 0 and construct three sequences {zn} g M,
{%„} С M and {yJJ C TV(xq) A Si as follows: By (4) we can choose
a, EM such that	2i being in the weak closure
of M, there exists an xx E M such that | <хр — (zv | 5g e/4. y'Q being
in the weak* closure of N (x'J) A Si, there exists a y{ E TV (xq ) A Si such
that |<xp yi> — <xx, yo> |	e/4. Repeating the argument and remember-
4. The Eberlein-Shmulyan Theorem
145
ing (4), we obtain {z„} Q M, {xn} С M and {y^} C N (xq) A S, such that
<^1 > Уо> = <Уо. xo> .
<Z„, У'т> = (Ут. *0> = 0
\^n< Ут/ ^n> Ут/ | = ®/4
|<*£. у»>— <*..Уо>| ^e/4
(m = 1, 2, . . ., n — 1),
(m = 0, 1, . . n — 1),
(г = 1, 2, .... n).
Thus we have
|<Уо> *o> — <*i> Уп>| ^/4 + e/4 = e/2 (i = 1,	2, ...,n).	(6)
Since {xn} G M, there exists a subsequence of {%n} which converges weakly
to an element x С M. Without losing the generality, we may assume that
the sequence {xnj itself converges weakly to x £ M. Hence, from (5),
I <% Ут) I £/4- From z^-lim xn = x and Mazur’s theorem 2 in Chapter V, 1,
n>oo
n /	n	\
there exists a convex combination u = ocjxAocj 0, £ <Xj = 1)
\	j=i	/
such that ||% — w||	e/4. Therefore, by (6),
n
|<Уо. *o> — <u. У’п>\
I <yo. *o> — <xi> Уп> I e/2>
and hence
I<Уо> xo> I SJ I<Уо. xo> — <м- У»> | + I<и- У»> — <*> У»> I + I<x> У»> |
< e/2 + ||u — x|| ||y; || + e/4 5g e.
As e was arbitrary, we see that <yo, Xq > — 0 and so	C N (x'o' ). Combined
with the fact that S* is weakly* closed, we finally obtain y'o £N(x'q) A
Remark. As for the weak topologies and duality in В-spaces, there
is an extensive literature. See, e.g., the references in N. Dunford-
J. Schwartz [1]. Sections 1, 2 and 3 of this Appendix are adapted
and modified from N. Bourbaki [1] and A. Grothendieck [1]. It is
remarkable that necessary tools for proving this far reaching theorem of
Eberlein [1]-Shmulyan [1] are found, in one form or other, in the book
of S. Banach [1].
VI. Fourier Transform and Differential Equations
The Fourier transform is one of the most powerful tools in classical
and modern analysis. Its scope has recently been strikingly extended
thanks to the introduction of the notion of generalized functions of
S. L. Sobolev [1] and L. Schwartz [1]. The extension has been applied
successfully to the theory of linear partial differential equations by
L. Ehrenpreis, B. Malgrange and especially by L. Hormander [6].
10 YoHldn, Ftinotlonhl Annlynln
146
VI. Fourier Transform and Differential Equations
1.	The Fourier Transform of Rapidly Decreasing Functions
Definition 1. We denote by S (!?”) the totality of functions / E C°° (Rn)
such that
sup \x^Daf(x) \ < oo /хр = П	(1)
x£Rn	\	3 = 1	/
for every oc — (oclt a2, . . ., an) and Д = (J3lf • • •» fin) with non-negative
integers (Xj and fik. Such functions are called rapidly decreasing (at oo).
Example, exp (— | x |2) and functions / 6 Cq° (Rn) are rapidly decreas-
ing.
Proposition 1. <S(Rn) is a locally convex linear topological space by
the algebraic operation of function sum and multiplication of functions
by complex numbers, and by the topology defined by the system of
semi-norms of the form
p(f) = sup |P(x) Daf(x) |, where P(x) denotes a polynomial. (2)
x£Rn
Proposition 2.	is closed with respect to the application of
linear partial differential operators with polynomial coefficients.
Proposition 3. With respect to the topology of <B(Rn), C^(Rn) is a
dense subset of © (R*).
Proof. Let /С ® (Rn) and take ip E Co° (jRm) such that ip(x) = 1 when
|я|	1. Then, for any e > 0, fe(x) = /(%) ip (ex) E C™(Rn). By applying
Leibniz’ rule of differentiation of the product of functions, we see that
£“(/.«'/«) = D“{f(x) (у>(ех) - 1)}
is a finite linear combination of terms t)f the form
• (с)|,,|{£*г¥'(у)}у=еж> Where |/?| + |y| = |a| with |y| > 0,
and the term Daf(x) • (ip (ex) — 1). Thus it is easy to see that fe(x) tends
to f(x) in the topology of &(Rn) when e 0.
Definition 2. For any /Е (&(Rn), define its Fourier transform f by
f (f) = (2 я)-и/2 / e^f (x) dx,	(3)
Rn
where f = (£b f2, . . x — (хг, x2, . . xn), <g, x) =	^x, and
j=i
dx = dxr dxz .. . dx„. We also define the inverse Fourier transform g of
gee (Rnj by
~g(x) = (2л)-”'2 f e^g^dS.
(4)
Proposition 4. The Fourier transform: f -> f maps 6 (jR*1) linearly
and continuously into <&(Rn). The inverse Fourier transform: g~> g also
maps ® (Rn) linearly and continuously into @ (R*1).
1. The Fourier Transform of Rapidly Decreasing Functions 147
Proof. Differentiating formally under the integral sign, we have
Daf (f) = (2 л)~”/2 J e~i(M (-»)|a| xaf (x) dx.	(5)
The formal differentiation is permitted since the right hand side is, by
(1), uniformly convergent in f. Thus / E C°°(Rn). Similarly, we have, by
integration by parts,
ЮИ^7(Й = (2л)“м/2У e~i{M D?f(x)dx.	(6)
Thus we have
(*)M+N	= (2тг)-л/2/	D?(x“f(x))dx,	(7)
and (7) proves that the mapping / —> f is continuous in the topology of
sm.
Theorem 1 (Fourier’s integral theorem). Fourier's inversion theorem
holds:
7 (x) — (2л)~п/2 f /(£) d£ = /(%), i.e., we have (8)
7 = /, and similarly f = f.	(8')
Therefore it is easy to see that the Fourier transform maps <5(R") onto
® (Rn) linearly and continuously in both directions, and the inverse Fourier
transform gives the inverse mapping of the Fourier transform.
Proof. We have
fg(i)f(i)ei^di = fg(y)f{x + y)dy (J and g € <S(R")).	(9)
In fact, the left hand side is equal to
Jg(f) {(2л) ~nlZf e-^fty) dy} e
= (2тг)_я/2 f {f g(i)	d£} f (y) dy
= fg (y — x) f(y) dy = fg(y) f(x + y) dy.
If we take g(ef) for g(f), e > 0, then
(2л)"л/2 / e~i<y^ g(ef)^f = (2тг)-"/2 £~w J g(z) e~iMe) dz = e~n g (y/e).
Hence, by (9),
Jg(e() ~f (£) е*'<жЛ> di = f g(y) f(x + ey) dy.
We shall take, following F. Riesz, g(x) =e~^'lz and let e | 0. Then
g(0) / 7 (f) di = f(x)fg (y) dy.
This proves (8), since g(0) = 1 and f g(y) dy = (2л)п/2 by the well-
known facts:
(2л)“п/2 f e~Mt/2 dx =	,	(10)
(2 7t)~n}2 f e~ |x|’/2 dx = 1.	(10')
10*
148	VI. Fourier Transform and Differential Equations
Remark. For the sake of completeness, we will give the proof of (10).
We have
(2тг)-1/2 f e“‘V2e-IMf^ = e-“a/2(2?i)"1/2 f e~{t+iuW2dt.
-Л	-2
Let и > 0, and integrate the function e~^2, which is holomorphic in z =
t 4- iu, along the curve consisting of the oriented segments
——--->--------->------------------>-------------------->
—Л, Я, А, A + iu, X 4- iu, —Л 4- iu and — Л 4- iu, — A
in this order. By Cauchy’s integral theorem, the integral vanishes. Thus
л	л
(2пГ'1г f	dt = (2 л)’1'2 f e-'1'2 dt
—А	Д
0
+ (2 л)-1'2 / e-W№ idH
u
u
-I- (2л)-1'2 f
0
The second and third terms on the right tend to 0 as A—> oo, and so, by
(ЮЪ
co	oo	.
(2тгГ1/2 f	= e-“2/2 (2л)“1/2 fe ‘/2 dt e
— 00	—co
We have thus proved (10) for the case n = 1, and it is easy to prove the
case of a general n by reducing it to the case n = 1.
Corollary (Parseval’s relation). We have
/7(f)g(fl« = //WgW*,	(11)
f/(e)g(S^ = I7wtw*.	(12)
(C*4 = (2 л)”'2 J  g and (2n)”'2 (47) = /*g ,	(13)
where the convolution / * g is defined through
(M (%) = f f(x — y) g(y) dy f g(x — y) f(y) dy. (14)
Proof. (11) is obtained from (9) by putting x = 0. (12) is obtained
from (11) by observing that the Fourier transform of g is equal to g . We
next show that
(2л)-”'2/(j * g) (х) dx
= (2л)-”'2 f g(y) <;-«’>{/f(x-у)	dx] dy (15)
Since the product f  g of functions f and g E <S (7?Л) is again a function
of @(2?“), we know that the right hand side of (15) belongs to ®(2?w).
2. The Fourier Transform of Tempered Distributions 149
11 is easy to see that, the convolution f * g of two functions of g (Rn) belongs
also to g(7?“). Thus we have proved the first formula of (13). The second
formula may be proved similarly to (9) by (15).
Theorem 2 (Poisson’s Summation Formula). Let	and
<p f g (7?1) its Fourier transform. Then we have
£ <р(2лп) = S cp(n).	(16)
n = —OO	« =—OO
oo
Proof. Set /(%) = <p(x -j- 2ли). This series is absolutely conver-
« = — 00
gent, € C°° and / (% + 2 л) = /(%), as may be proved by the fact that tp (%)
is rapidly decreasing atoo. In particular, both sides of (16) are convergent.
We have to prove the equality.
The Fourier coefficients ck of /(x) with respect to the complete ortho-
normal system {(2л)-1/2 e~tkx; k 0,±l,±2, • -.} of L2(0, 2 л) are given
by
2л	0O	2л
ck = (2л)-1/2 f f(x) e~lkx dx = £ (2л)-1/2 j fp(x 2лп) e~lkx dx
0	»=—00	0
oo	2л(»+1)
=	(2л)“1/2 f <p{x)e~ikxdx = ^{k).
2m
Thus, by / E La(0, 2л), we have
/(%) = j? <p(x + 2ли) = l.i.m. J:y(k)eikx.
« = —OO	sfoo A —— s
OO
However, since m(x) 6 g(7?H), the series £ (p(k) eikx converges absolu-
fc=—oo
tely. Hence
Д <p{x2лп) = £ у (k) elkx,
м = —oo	A =—oo
and so we obtain (16) by setting x = 0.
Example. We have, by (10),
00	OO	_
(2л)-1/2 / e~tx'e~ixy dx = (2л)“1/2 J e~x^ e~ixy^2t (2if1/a dx
— OO	—oo
= (2Zp1/2e~y'!it, t > 0.
Hence, by (16), we obtain the so-called ^-formula'.
OO	oo
Д	= Д (2t)~112 e~n lit, t> 0.	(17)
«=—OO	» ——oo
2. The Fourier Transform of Tempered Distributions
Definition 1. A linear functional T defined and continuous on g(7?”)
is called a tempered distribution {in 7?”). The totality of tempered distri-
150
VI. Fourier Transform and Differential Equations
butions is denoted by As a dual space of &(Rn), Q>(Rny is a
locally convex linear topological space by the strong dual topology.
Proposition 1. Since (A”) is contained in <5 (A**) as an abstract set,
and since the topology in Ф (Rn) is stronger than the topology in S(Art),
the restriction of a tempered distribution to	is a distribution in
Rn. Two different tempered distributions define, when restricted to
C£° (/?"), two different distributions in R”, because Cx(Rn) is dense in
<S (A") with respect to the topology of <S (R*), and hence a distribution
E (/?**)' which vanishes on Cg° (22м) must vanish on (g(A”). Therefore
ед'£ф(#*)'.	(i)
Example 1.	A distribution in Rn with a compact support surely be-
longs to ©(7?*)'. Therefore
®(7?")' C ®(Rn)'.	(2)
Example 2.	A а-finite, non-negative measure a (dx) which is a-additive
on Baire sets of Rn is called a slowly increasing measure, if, for some non-
negative k,
J (1 + \x\2)~k fi(dx) <oo.	(3)
Rn
Such a measure p defines a tempered distribution by
Тц(<р) = / У (%) ^ (dx), (Rn).	(4)
For, by the condition <p E <S(Z?W), we have <p(x) = 0((l + )%|2)-ft) for
large
Example 3.	As a special case of Example 2, any function /€ Lp (Rn),
p > 1, defines a tempered distribution
Tf(<p) = j<p(x)f(x)dx, <p€&(R”).	(4')
я"
That an f£Lp (Rn) gives rise to a slowly increasing measure u (dx) =
|/(%) | dx may be proved by applying Holder’s inequality to
f (1 + |x |/W | dx.
Rn
Definition 2. A function 0х (Rn) is called slowly increasing (at
co), if, for any differentiation DJ, there exists a non-negative integer N
such that
lim Ixl-" \Dj/(%) | = 0.	(5)
[x|—X»
The totality of slowly increasing functions will be denoted by £)м (Л*).
It is a locally convex linear topological space by the algebraic operations
of function sum and multiplication of functions by complex numbers,
2. The Fourier Transform of Tempered Distributions
151
.иid by the topology defined by the system of semi-norms of the form
P (/) = P^ (/) = sup I h (%) 1У / (%) I, / e (7?”),	(6)
where h{x) is an arbitrary function E ©(7?”) and D] an arbitrary differen-
tiation. As may be seen by applying Leibniz' formula of differentiation
<>! the product of functions, A (%) DJf(x) E <3(7?”) and so />AD;(/) is finite
lor every /6 0^(7?*). Moreover, if />ftr,y(/) = 0 for all h E @(7?w) and 7)J,
i hen /(%) = 0 as may be seen by taking D = I and h£ ®(7?”).
Proposition 2. C™ (Rn) is dense in (7?”) with respect to the topology
Proof. Let f E £>m (7?”), and take yj E C£° (7?”) such that ip (x) = 1 for
| r ]	1. Then fe(x) ~ f(x) ip (ex) E C£°(7?”) for any e > 0. As in Pro-
position 3 in Chapter VI, 1, we easily prove that Д (%) tends to f(x) in the
topology of DM(7?”) when e 0.
Proposition 3. Any function f £	(7?w) defines a tempered distribu-
tion
T, (?) = f f W <f (x) dx, q> e S («”).	(7)
Definition 3. As in the case of a distribution in Rn, we can define the
generalized derivative of a tempered distribution T by
ТУТ® = (- 1)МТ(ОУ), ?€ 6(Л“).	(8)
since the mapping <p (x) -> D^<p (x) of ® (Rn) into © (7?”) is linear and
continuous in the topology of ©(7?”). We can also define a multiplication
by a function /€ (Rn) to a distribution T E ©(7?**)' through
(fT) (<p) — T(М <p€@(7?)\	(9)
since the mapping <p (%) -> / (%) <p (%) of © (7?w) into © (7?”) is linear and
continuous in the topology of 3(7?”).
The Fourier Transform of Tempered Distributions
Definition 4. Since the mapping <p(x) -> (p(x) of ©(7?") onto ©(7?rt)
is linear and continuous in the topology of ©(7?”), we can define the
Fourier transform T of a tempered distribution T as the tempered
distribution T defined through
f(^-T^), <p€©(7?”).	(10)
Example 1.	If / E I? (7?n), then
t, = Tj, where / (x) = (23Г)-'2 / <•<*«> /(f) df,	(11)
Rn
as may be seen by changing the order of integration in =
f /(x'l-q>t.x)dx=(2n)-^ f/W [
K’1	Rn	I
152
VI. Fourier Transform and Differential Equations
Remark. In the above sense, the Fourier transform of a tempered
distribution is a generalization of the ordinary Fourier transform of
functions.
Proposition 4. If we define
/<*)=/(—*).	(12)
then Fourier's integral theorem in the preceding Section 1 is expressed by
/€б(Г).	(13)
Corollary 1 (Fourier’s integral theorem). Fourier's integral theorem
is generalized to tempered distributions as follows:
ft *	V
Z = T, where T((p) ~	(14)
In particular, the Fourier transform T T maps (5(7?*)' linearly onto
<5(2?”)'-
Proof. We have, by definition,
= Z(<p) = Z(^) = t(<p) for all ^®(2?").
Corollary 2. The Fourier transform T T and its inverse are linear
and continuous on 0 (7?n)' onto 0 (Rn) ’ with respect to the weak* topology
of <5(7?”)':
lim ZA (у) = T (9?) for all 9? C © (7?”) implies that
lim (9?) = T (99) for all 99 G ©(7?”).	(15)
Here the inverse of the mapping T—>T is defined by the inverse Fourier
transform T T given by
7^=7^), 9>۩(7?w).	(10')
Example 2.
fa = (2тг)-п/2 Tp Z\ = (2л)п12Тд.	(16)
Proof. ts((p) = Z^ijp) = (0) = (2тг)-я/2 f 1 • <p(y) dy =
я"
(2я)-”'2 П iff}, and TS=TS = TS = (2я)-"'2 Д.
Example 3.
-	(17)
(dTfdXj) ~ixjT,	' ’
FF) = -	.	(18)
2. The Fourier Transform of Tempered Distributions
153
Proof. We have, by (5) in Chapter VI, 1,
(dT/dXj) (99) = (дТ/дх^ (<p) = — T(d<p}c)Xj) = — T (—ix} 93 (%))
= T = (iXjT) (99).
We also have, by (6) in Chapter VI, 1,
(99) = (iXjT} (ф) = T(ixfi} = Т(д(р]дх3) = Т(&р18х3]
= — (8Т1бх3} (99).
Plancherel’s Theorem. If f£L2(Rn), then the Fourier transform T}
of Tf is defined by a function /6 L2(Rn), i.e.,
Tf=Tp with /eL2(R”),	(19)
and
\\/\\ = ( f = f |/«N*y'2= !!/ll-	(20)
\K"	/ \Яя	/
Proof. We have, by Schwarz’ inequality,
W)l = IWI =
f f(x)(p (x) dx
Rn
ll/ll IHI = ll/ll Ikll- (21)
The equality above Ц99Ц = ] |gp |l is proved in (12) of the preceding section
!. Hence, by F. Riesz' representation theorem in the Hilbert space Z2 (R”),
there exists a uniquely determined /£ L2(R”) such that
Tf(<p) = f <p(x) /(*) dx = Tf (9?), that is,
Rn
f f (%) 99 (x) dx — f f (x) <p (x) dx for all (p C (Rn) •
Rn	Rn
Moreover, we have, from (21), |]/||	||/j| since S(R") is dense in L2(Rn)
in the topology of L2 (R”). We have thus ||J [j \\f ]]	\\f ]|. On the other
hand, we have, by (13) and (22),
/ / (x) 99 (%) iix = f / (x) (p(x) dx = f /(— x} 99 (x) dx for all 99 F
А*Л
that is,
„	f\x) = f(—x) = f(x) a.e.	<23)
л	л
Thus ||/ || = ||/]| and so, combined with ||/ || Ц/ |[ < ]| /1|, we obtain
(20).
Definition 5. The above obtained / (%) 6 L2 (Rn) is called the Fourier
transform of the function /(x) £ Z.2(R”).
154
VI. Fourier Transform and Differential Equations
Corollary 1. We have, for any L2(Rn),
f(x) = l.i.m. (2tt)
Л f oo
~»/2 J e^>IMdy,
1*1
(24)
Proof. Put
fh [x] = / (%) or = 0 according as j x | h or | x j > h.
Then lim ||Д — /11 = 0 and so, by (20), lim j/A — /11=0, that is,
f(x) = l.i.m.(%) a.e. But, by (22),
Л—юо
f th (*) ? (x) dx = f /ft (x) ip (x) dx
Rn	Rn
1*1 s*	яп
which is, by changing the order of integration, equal to
J?n
since /*(%) is integrable over |x( S h as may be seen by Schwarz’ ine-
quality. Thus /А (л) = (2я)~я/3 J f {y) dy a<e,, an(] so we obtain
(24).	hlfi*
Corollary 2. The Fourier transform / -> / maps L2 (L’n) onto L2 {Rn)
in a one-one-manner such that
(Ag) = (Л £) br all /,geL2(tf").	(25)
Proof. Like the Fourier transform /->/, the inverse Fourier trans-
form f~+f defined by
f(x) = l.i.m. (2л)^я/2 f f(y) dy	(26)
A~*°°
mapsL2(7?w) into L2{Rn) in such a way that ||/|| = ||/|). Hence we see
that the Fourier transform /->/ maps£2(7?w) onto L2 (Rn) in a one-one
way such that Ц/ || ~ [|/j|. Hence, by the linearity of the Fourier trans-
form and
(%, у) = 4’1 (| |x + у 112 — 11 x — у 112) + 4-1 i
|x jy |[2 „	_;y ||2),
we obtain (25).
Parseval’s Theorem for the Fourier Transform. Let /г(х) and /2(-^)
both belong to L2(!?"), and let their Fourier transforms be [u) and /2 (и),
respectively. Then
f fi {^)f2 («) du = f Л (x) /2 (— x) dx,	(27)
Rn	Rn
2. The Fourier Transform of Tempered Distributions
155
 ind so
f fi («) /2 (u) ? <«- *> du = / A (y) A (x — y)dy.
Rn	.К”
(28)
1'hus, if fi(u) f2(u) as well as both its factors belong to L2(7?n), it is the
I’ourier transform of
(2?r) n/2 f /г(у) f2(x~y) dy.
Rn
(29)
I bis will also be true if /г(ж), /2(%) and (29) all belong to L2(Rn).
Proof. It is easy to see that
Rn
and so, by (25), we obtain (27). Next, since the Fourier transform of the
function of y, f2[x~y), is /2(w)	containing a parameter x, we
obtain (28) by (25). The rest of the Theorem is clear from (28) and f = f.
The Negative Norm. The Sobolev space TF'',2T2) was defined in
Chapter I, 9. Let /(%) G ЖЙ,2(АП). Since /(%) G L2(Rn)> f gives rise to a
slowly increasing measure |/(%) | dx in Rn. Thus we can define the Fourier
transform Tj of the tempered distribution T<. We have, by (17),
it”
DaTf = (?)w П Tf.
By the definition of the space Wk'2 (A”), Dx Tf£L2(Rn) for |a| g k.
1’hus, by Plancherel’s theorem for L2{Rn},
11 Da Ty| [0 = 11 Da Tj\|0, where [| |[0 is the Z,2(/?w)-norm.
I fence we see that (1 +	[2)A/2 Tj£ L2 (Rn), and so it is easy to show that
the norm ||/||fe = / V / |D“ Ty|2 dT\1/2 is equivalent to the norm
\ lafS* Rn	}
Ki + m2)*'2 7}iio = ii/ii;
(30)
in the sense that there exist two positive constants c± and c2 for which we
have
fi	11 /11*/11 f 1c2 whenever / G Ж*’2 (Aw).
We may thus renorm the space W*,2(/?n) ЬуЦ/Ц*; Ж*’2 (7?”) may thus
be defined as the totality of /G L2(Rn} such that ]]/ ]|^ is finite. One ad-
vantage of the new formulation of ЖА’2 (I?”) is that we may also consider the
case of a negative exponent k. Then, as in the case of L2(Rn} pertaining
to the ordinary Lebesgue measure dx, we see that the dual space of the
renormed space Ж*,2(7?и) is the space Ж-Л’2(Т?М) normed by 1|/||L*.
This observation due to L. Schwartz [5] is of an earlier date than the
introduction of the negative norm by P. Lax [2).
156
VI. Fourier Transform and Differential Equations
3.	Convolutions
We define the convolution (Faltung) of two functions f, g of C (Rn), one
of which has a compact support, by (cf. the case in which /, gG @(Й”)
in Chapter VI, 1)
(/* g) W = J — y) g(y) dy = f f(y) g(x — y) dy = (g*/) (x). (1}
Rn	Rn
Suggested by this formula, we define the convolution of a T E ® (Rn)' and
a <p e © (Rn) (or a T G @ (Rn)’ and a <p G ® (2?")) by
(T*(p) (x) = Тм(?(*~ y)),	(2)
where indicates that we apply the distribution T on test functions
of y.
Proposition 1. (T *9?) (x) G	and supp (T * <p) C supp (7) -h
supp (99), that is,
supp(T * (p) C {w G Rn; w = x + y, x G supp(T), у G supp (9?)}.
Moreover, we have
Da(T*(p) = T* (Da<p) = (D*T)	(3)
Proof. Let <p G © (Rn) (or G ©(й”)). If lim xk = x, then, as func-
tions of y, lim <p (xh — y) = cp(x — y) in Ф (Rn) (or in (Rn)). Hence
Тм (<p(x — у)) = (T *99) (x) is continuous in x. The inclusion relation
of the supports is proved by the fact that T[у] (tp (x — y)) = 0 unless the
support of T and that of 99 [x — y) as a function of у meet. Next let fy be
the unit vector of along the xj-axis and consider the expression
ты ((?>(% T Agy — y) — 92 (% — у))/Л).
When h —> 0, the function enclosed by the outer parenthesis converges,
as a function of y, to (x ~y) in © (Rn) (or in @ (I?*1)). Thus we have
proved
£(ТММ = (т*£)(,).
Moreover, we have
„ 5? \ , .	/	8<p(x — y)\	[дт \
V* toj W = Di (-------------У)) = (^-* v) «•
Proposition 2. If 99 and ip are in © (2?“) and T G © (2?n)' (or 99 G (£(!?*),
tp G © (Rn) and T G ® (Я*)'), then
(T -x- 99) * ip - T -x- (9? x- ip).	(4)
3* Convolutions
157
Proof. We approximate the function {(p * ip) (x) by the Riemann sum
Д (x) = hn ^rp{x — kh) ip(kh),
h
where h > 0 and k ranges over points of Rn with integral coordinates.
Then, for every differentiation Da and for every compact set of x,
Dafh{x) - hn £ D*<p(x — kh) ip(kh)
converges, as h ф 0, to (SP^ty) * ip) (x) = (Dx((p * ip)) (x) uniformly in x.
Hence we see that lim Д = <p * ip in Ф (Rn) (or in (£(R”)). Therefore, by
MO
the linearity and the continuity of T, we have
(F* (<p * ip)) (x) = lim (T * /л) (*) = lim
A | О	Ц0
IT У) (T * <p) (x — kh) ip(kh)
ft
= ((F * (p) * M w 
Definition. Let tp £ Ф (Rn) be non-negative, f (p dx = 1 and such that
j?"
supp (<p) Q {x C Rn; x
< 1]. We may, for instance, take
<p(x) = exp (l/(|x |2— 1))/ J exp(l/(|x|2 — l))rfx if\x | < 1,
H<i
_ 0 if \x |	1.
We write y?e(%) for e ” <jp(x/e), e > 0, and call T * the regularization
of Те $(R")' (ore through (pe{x)’s (Cf. Chapter 1,1).
Theorem 1. Let T e ®(R")f (or C @(R”)')- Then lim (T * tpe) = T in
fi ф 0
the weak* topology of ®(RM)' (or of ®(RK)')- this sense, (ge is called
an approximate identity.
For the proof we prepare a
Lemma. For any ip£ ® (RM) (or C	we have lim ip * <pE = ip
in D(R”) (or in S(R”)).
Proof. We first observe that supp(y*^£) £ supp(y>) + supp(<pe) =
supp(y) + £. We have, by (3), D*(ip*<pe) = (D1*ip) * <pE. Hence we
have to show that lim (ip * (pE) (x) = ip (x) uniformly on any compact set
s | 0
of x. But, by f <pe(y') dy — 1,
W — vW = f {v’U — y) — vdy-
Rn
Therefore, by <pe(x) > 0, f <p£{y) dy ~ 1 and the uniform continuity of
j?n
y(x) on any compact interval of x, we obtain the Lemma.
Proof of Theorem 1. We have, by ip(x) = ip(—x),
Г Ы =(T* ip) (0).	(5)
158	VI. Fourier Transform and Differential Equations
Hence we have to prove that lim ((T * <pe) * ip) (0) = (T *	(0). But,
e ф 0
as proved in (4), (Г*^е)	= T* (<pe * y>) and so, by (5),
(T* (qpe* y>)) (0) = T((<pe * Hence we obtain lim ((Г* 99J * w) (0) =
8^0
Т((у»)") = Г (y>) by the Lemma.
We next prove a theorem which concerns with a characterization of
the operation of convolution.
Theorem 2 (L. Schwartz). Let L be a continuous linear mapping on
ф (7Г) into ® (Rn) such that
Lthq> = rhL<p for any h£Rn and 99 G® (7?”),	(6)
where the translation operator ta is defined by
wW = tp(x-h).	(7)
Then there exists a uniquely determined T G ф{7?м)' such that L<p ~
T * 99. Conversely, any T G ®(7?”)' defines a continuous linear map L
on © (7?*) into @(7?M) by L<p = T *99 such that L satisfies (6).
Proof. Since 99 -> <p is a continuous linear map of ©(7?*) onto ®(7?n),
the linear map T : <p - > (L<p) (0) defines a distribution T © (7?*)'. Hence,
by (5), (Ltp) (0) = Г(<р) = (T * 99) (0). If we replace 99 by тл<р and make
use of condition (6), then we obtain (£99) (h) = (T * 99) (Л). The
converse part of Theorem 2 is easily proved by (2), Proposition 1 and (5).
Corollary. Let 7^ g ©(7?rt)' and T2G ®(7?")'. Then the convolution
7\ * T2 may be defined, through the continuous linear map L on
© (7?”) into ® (7?") as follows:
(Л * T2) * <p = L (<p) = Т1*(Т2*9з),9)€®(йи).	(8)
Proof. The map 99 —> T2 * 99 is continuous linear on © (7?w) into
®(7?w), since supp(T2) is compact. Hence the map 99 -> 7\ * (T2 * 99)
is continuous linear on ® (7?”) into @(7?”). It is easily verified that condi-
tion (6) is satisfied for the present L.
Remark. We see, by (4), that the above definition of the convolution
* T2 agrees with the previous one in the case where T2 is defined by
a function G © (Rn)- It is to be noted that we may also define 7\ * T2 by
(T, * Тг) (<p) = (T1W X Тад) (? (x + y)), v e s> (Я*),	(8)'
where	is the tensor product of TT and Tz. See L. Schwartz
И-
Theorem 3. Let Т1С®(7?Л)' and T2£&(Rny, Then we can define
another “convolution” T2 ® through the continuous linear map L;
99 -> T2 * (Tj * 99) on © (7Г) into ® (R")
as follows. (T2 ES 7\) (*99) = £(99), 99 G ®(7?”). Then we can prove 7’3ES
Tj = Tx * T2 so that the convolution is commutative if it is defined
either as 7\ * T2 or as T2 Щ Tv
3. Convolutions
159
Proof. The map <p 7\ * g? is linear continuous on ф (!?”) into
(£(!?*). Hence the map tp -> T2 * (7\ * <p) is linear continuous on
T(J?n) into Thus T2gj J\ is well-defined. Next, we have for any
7’1» <?2 ® (йп)>
(Л* Л) * (<Pi *^2) = Ti * (T2 * (9h * Ф2У) = ?!* ((^2 * <?]) *g?2)
= 7\ * (<p2 * (T2 * tp-J) = (7\ * gw) * (T2 * gpj
by the commutativity of the convolution of functions and Proposition 2,
observing that supp(T2 * gzj is compact because of T2Q^(Rn}'.
Similarly we obtain
(T2S 7\) *	(дрг*^2)) = Г2* ((Л *^2) *9h)
= T2 * (9^ * (7\ * <p2)) = (T2 *	* (TT * g?2).
Thus (Тг * T2) * (9^ * 9?2) = (7'aS Tj) * (<рг *^), and so, by (5)
and the Lemma above, we obtain (Tj*T2) (95) = (T2® 7\) (g>) for all
<P t © (Rn), that is, (7\ * T2) = (T2 ® TJ.
Corollary
Г1*(7’г*7’3) = (Л*^)*7'з	(9)
if all Tj except one have compact support,
ЯЯ(Л % T2) = (LFTJ	(P’T2).	(10)
Proof. We have, by (5) and the definition of Тг * T2,
(Л * (T2 * T,)) (<p) = ((T, * (Ta * Г3)) * ч>) (0)
= (Л * ((т, * Т3) * й) (0)
= (П * (т2 * (тг * й)) (0)
and similarly
((Г, * Т2) * 7-3) (р) = (Г, * (Та * (Г, * у))} (0),
so that (9) holds.
The proof of (10) is as follows. We observe that, by (3),
(D*TJ *<p = T, * (D>) = Dx(T5 *<?) = Dy, (11)
which implies
(D*T) *9) = T* (Dy} = T*((D“Td) *<p) = (T * DaTs} * 9?,
that is, by (5),
*	DaT = (D*TS}*T.	(12)
Hence, by the commutativity (Theorem 3) and the associativity (9),
1Г (T, * 7S) = (D“ 7-,) * (Г, * Г2) = (Jp-T,) * T,)	(D-т,) * T2
--- (/>aГ,} *(T^Tt)= ((ITT,)*T2)* T, = (Й» T2)*Г,.
160	VI. Fourier Transform and Differential Equations
The Fourier Transform of the Convolution. We first prove a theorem
which will be sharpened to the Paley-Wiener theorem in the next sec-
tion.
Theorem 4.	The Fourier transform of a distribution T £ @(7?”)' is
given by the function
= (2лГ”'2Гм(е~‘«>).	(13)
Proof. When e | 0, the regularization TE — T * <pe tends to T in the
weak* topology of (£(7?")' and a fortiori in the weak* topology of <5 (R*)'.
This we see from (T *<pe) (y) =(T* (<pE * fy) (0) = Tw((^e * v>) (— x))
and the Lemma. Thus, by the continuity of the Fourier transform in the
weak* topology of <5(7?")', lim (T *	= T in the weak* topology of
5(7?")'. Now formula (13) is clear for the distribution defined by the
function (T x- <jpe) (x). Hence
(2я)"'2	(f) = (Г *%)м (<г‘«>),
which is, by (5), = (Tw*lf’.*CW)))W = 2'W(i*«''W))-
The last expression tends, as e j. 0, to	uniformly in $ on any
bounded set of $ of the complex «-space. This proves Theorem 4.
Theorem 5.	If we define the convolution of a distribution T £ 5(7?")'
and a function C 5(7?") by (T x-y>) (x) =	(9? (x — y)), then the
linear map L on 5 (7?”) into (£ (7?”) : ср T -X- <p is characterized by the
continuity and the translation invariance = Lxk.
Proof. Similar to the proof of Theorem 2.
Theorem 6. If T C © (7?")' and 99 (, 5 (7?”J, then
(f*J) = (27t)n/2$f	(14)
If T\£ &(Rny and ®(7?M)', then
= (2л)”'2 f,- i\,	<15)
which has a sense since, as proved above in Theorem 4, T2 is given by
a function.
Proof. Let ip£ 5(7?"). Then the Fourier transform of </  ip is, by (13)
of Chapter VI, 1, equal to (2л)-"/2 £	= (2тг)-и/2 Thus
(T *^>) (v) = (T*9?) (у) = ((Г *<p) *') ) (0) = (Г * t'p *?)) (°)
= T((<p * ?)') = T&* V) = Т((2^Ч2 (ч • v,)')
= (2л)”'2 t (4>V) = Wl-ytW,
which proves (14).
4. The Paley-Wiener Theorems. The One-sided Laplace Transform 161
Let yje be the regularization 7 2 Then the Fourier transform of
Л * v, — Ti* (T2 * <p,) = (Tj * Tt) *<p, is, by (14), equal to
(2л)“« Д - ft = (2л)”;2 A  (2 л)”'2 Тг • ft = (2л)”'2 (зТ^Гг)  ft.
Hence we obtain (16), by letting e | 0 and using lim <pt(x') = 1.
e 10
4. The Paley-Wiener Theorems. The One-sided Laplace Transform
The Fourier transform of a function € C^°(flw) is characterized by the
Paley-Wiener Theorem for Functions. An entire holomorphic function
F(C) = F(Ci, C2,   , Cn) of n complex variables + ir^ (/ =
1, 2, . . и) is the Fourier-Laplace transform
F(C) = (2л)"и/2 f e-^f^dx	(1)
of a function / G C£J° (Rn) with supp (/) contained in the sphere [ x | В
oiRn iff there exists for every integer .V a positive constant CN such that
|F(fl|£C„(l + |C|)-''M'”£1.	(2)
Proof. The necessity is clear from
П F(C) = (2л)“”/2 / е~^’х) Dpf(x) dx
j=i	Иfin
which is obtained by partial integration.
The sufficiency. Let us define
f(x) = (2л)-”'2 f г<“'£> F(J) d£.	(3)
Then, as for the case of functions € :S (Rn), we can prove that the Fourier
transform /(£) is equal to F(£) and /6 C°°(.R"). The last assertion is
proved by differentiation:
Z//(%) = (2n)- ”'2 f ?<’•« П F(f) d£,	(4)
Rn J=1
by making use of condition (2). The same condition (2) and Cauchy’s
integral theorem enable us to shift the real domain of integration in (3)
into the complex domain so that we obtain
f(x) = (2лри/2 J F(£ + iyf) d%	(3')
for arbitrary real rj of the form r}=<xxf\x\ with a> 0. We thus obtain,
by taking N = n• -f- 1,
/(%) |	(%n}-n!Z f (IF (£[)"JM-
яп
If | x | > B, we obtain / (%) = 0 by letting a f + oo. Hence supp (/) C
{%€	]%|	Я}-
11 Ycwdda, Functional AiutlyHh
162
VI. Fourier Transform and Differential Equations
The above theorem may be generalized to distributions with compact
support. Thus we have
The Paley-Wiener Theorem for Distributions £(£ (F”)' (L. Schwartz).
An entire holomorphic function F (£) = F(Ci, • • - of w complex vari-
ables Cj = fy +	(/ = L 2, . . ., я) is the Fourier-Laplace transform
of a distribution T £ &(Rn)' iff for some positive constants B, N and C
jF(£)|	C(1 +
(5)
Proof. The necessity is clear from the fact (Theorem 2 in Chapter 1,13)
that, if T C @(F“)', there exist positive constants C, N and В such that
For, we have only to take <p(x) = and apply (13) of the preceding
section.
The sufficiency. F (£) is in ®(й”)' and so it is the Fourier transform
of a distribution T € @(F")'. The Fourier transform of the regularization
T£ = T * <pe is (2T  <pe by (14) of the preceding section. Since the
supp (g?£) is in the sphere j £ of Rn, we have, by the preceding theorem,
|£,(й I g C 	.
Moreover, since T is defined by the function F (£), we see that T • <pe
is defiend by a function (2.я)и/2F(f) -^(f) which, when extended to the
complex я-space analytically, satisfies the estimate of type (5) with
В replaced by В + e. Thus, by the preceding theorem, T£ = T *
has its support in the sphere |л | В -f- e of Rn. Thus, letting e | 0 and
making use of the Lemma in the preceding section, we see that the supp(F)
is in the sphere |x| В of Rn.
Remark. The formulation and the proofs of the Paley-Wiener theorems
given above are adapted from L. Hormander [2j .
The Fourier Transform and the One-sided Laplace Transform. Let
g(t) C L2 (0, oo). Then, for % > 0,
g(t)
as may be seen by Schwarz’ inequality. Hence, by Plancherel’s theorem,
we have, for the Fourier transform
/(x {- ty) = (2л)']/2 /° g(f) e~tx rlty di
° no
= (2л)'1/2 f g(t) di (x > 0),
0
the inequality
/ ]/(% + iy) I2 / Ig (0 I2 F~ix dt g / |g(Z)|2^-	(7)
—oo	0	0
4, The Paley-Wiener Theorems. The One-sided Laplace Transform 163
The function f{x -T iy) is holomorphic in z = x iy in the right half-
plane Re(z) = x > 0, as may be seen by differentiating (6) under the
integral sign observing that g(t) te~l!)a.sa. function of / belongs to L1 (0, oo)
and to Z2 (0,00) when Re(z) — x > 0. We have thus proved
Theorem 1. Let g(i!) 6 L2(0, oo). Then the one-sided Laplace transform
OO
f(z} =--=(2 я)-1'2/ g(l)e~u di (Яф)>0)	(6')
0
belongs to the so-called Hardy-Lebesgue class Zf2(O), that is, (i) f(z) is
holomorphic in the right half-plane Re(z) > 0, (ii) for each fixed x > 0,
f{x + iy) as a function of у belongs to Z.2(—co, oo) in such a way that
This theorem admits a converse, that is, we have
Theorems (Paley-Wiener). Let /(z)g№(0). Then the boundary
function f(iy) E L2(- - oo, oo) of f[x -g iy) exists in the sense that
OO
lim f \f{iy) — f(x-\- iy) |2 dy 0
*40 -6o
(8)
in such a way that the inverse Fourier transform
N
g(t) — (2 zr)~1/2 l.i.m. f f(iy) elty dy	(9)
N—>00 _д?
vanishes for t < 0 and f(z) may be obtained as the one-sided Laplace
transform of g{t).
Proof. By the local weak compactness of L2(—00,00), we see, that
there exists a sequence {.r„} and an f{iy) € Z,2(—oo, oo) such that
xn 0 and weak-lim f(xn -4- iy) = f(iy).
Hr^OQ
For any 5 > 0, there exists a sequence {2Vft} such that
<5
lim = oo and lim f f(x± !'^Tk) I2 dx = 0,
fe-хю	й ->оа
as may be seen from
f (x + i у) j2 dy dx < oo
(for all N
0).
Thus, by Schwarz' inequality,
6
lim J | / (x	i Nk) [ dx = 0.
Jc—>OQ (J.
(10)
11*
164
VI. Fourier Transform and Differential Equations
From this we obtain
/(г) = (2 л)-1
— 00
(W>o).
(И)
The proof may be obtained as follows. By Cauchy’s integral representa-
tion, we obtain
/(г) = (2Л!')-1	(Яф) > 0),	(12)
С
where the path C of integration is composed of the segments
x0 — iNA, x-L — iNk, Xi - iNz, xT + iNk,
-j- iNk> xQ + iNfi, Xq + iNk, я0 — iNk
(x0 < Re(z) < %1( — Nk < Im(z) < Nk)
so that the closed contour C encloses the point z. Hence, by letting k -+ oo
and observing (10), we obtain
OO	oo
/(z)^(2jr)~1 f ~	dt + (2 л)-1 f
'	1 J Z-^ill)	' J (*j + ti) — z
—OO	"OO
We see that the second term on the right tends to 0 as хг oo, because
of (7') and Schwarz' inequality. We set, in the first term on the right,
x0 = xn and letting n—>oo we obtain (11). Similarly, we obtain
0
OO
J l^dt (Кф)
— OO
Let us put, for Re(z) > 0,
Л(x) = 0 (for x < 0); h(%) = e~zx (for x > 0).
We then have
f h (x) eixt dx = J e'xt~zx dx=(z — it)'1,
—oo	0
and so, by Plancherel’s theorem,
У e<t*	о /for x < 0)
l.i.m. (2л)-1 / -—~dt = \ ,x	if Re(z\ > 0.
_£ z	V (for x > 0)
Similarly we have
r e~iia> f— c zx (for x < 0)
l.i.m. (2л)'1 I -—- dt = \	if Re(z) < 0.
лг-юо v }	[ 0 (forx>0)
(13)
(11)
(14')
Therefore, by applying the Parseval theorem (27) of Chapter VI, 2
to (11), we obtain the result that f(z) is the one-sided Laplace transform
4. The Paley-Wiener Theorems. The One-sided Laplace Transform 165
of g(£) given by (9). By applying the Parseval theorem to (13), we see also
that g (t) = 0 for t < 0.
We finally shall prove that (8) is true. By adding (11) and (13), we
see that /(я) admits Poisson’s integral representation
oo
/(«) = /(% + iy) f 7T_J^7-^dt whenever x>0. (15)
—oo
By virtue of
OO
/* dt
J F=yT
—oo
(16)
we have
OO
2----L ds, where f+ (y) = / (z у),
—oo
and so
OO
—oo
00 / oo
2 Г Г ds
00
f
л/ J
—oo
\ / oo
oo
—00
ds idy
' 2
6?s} dy
—oo \—oo
oo
x C ds
я J s2 +
—oo
oo
/ \—00
oo
— 00
—oo
where 0 < /z (/+; s) £ 4 j|/+ |l2 and /z(/+; $) is continuous in s and has
value 0 at 5 = 0.
To prove that the right side tends to zero as % | 0, we take, for any
e > 0, a <5 = <5(e) > 0 such that s) Li e whenever |s[ Li <3. We
decompose the integral
P(t+', s)
S2 + X2
ds
— Л + 12 + ^3 
We havsJ-Zal < e by (16), and Li 4ЛГ1 ||/+||2 • cot-1(c3/^) (/ = 1, 2).
Hence the left integral must tend to zero as x | 0. This proves the Theo-
rem.
Remark 1. For the original Paley-Wiener theorem, see Paley-
Wiener [1]. For the one-sided Laplace transform of tempered distribu-
tions, see L. Schwartz [2].
166	VI. Fourier Transform and Differential Equations *
Remark 2. M. Sato [1] has introduced a happy idea of defining a
“generalized function” as the “boundary value of an analytic function”.
His idea may be explained as follows. Let 93 be the totality of functions
<p(z) which are defined and regular in the upper-half and the lower-half
planes of the complex г-plane, and let 31 be the totality of functions
which are regular in the whole complex z-plane. Then 93 is a ring with
respect to the function sum and function multiplication, and 3t is a sub-
ring of 93. Sato calls the residue class (mod 3i) containing <p(z) as the
“generalized function” <p(x) on the real axis A1 defined through <p(z).
The “generalized derivative d{p(x)[dx of the generalized function ^(x)”
is naturally defined as the residue class (mod 91) containing dxp(z)ldz.
Thus the “delta function д{х)” is the residue class (mod 91) containing
— (Злг)^1^-1. Sato’s theory of “generalized function of many variables”
admits the following very interesting topological interpretation. Let M be
an и-dimensional real analytic manifold and let X be a complexification
of M. Then the и-th relative cohomology group Hr‘(X mod (X —M))
with the germ of regular functions in X as coefficients gives rise to a
notion of the “generalized function on M". That is, the relative cohomo-
logy class is a natural definition of the “generalized function”.
Remark 3. For more detailed treatment of the Fourier transform of
generalized functions, see L. Schwartz [1] and Gelfand-Silov [1].
In the latter book, many interesting classes of basic functions other than
Ф (7?w), S (Z?w) and Cjh (A”) are introduced to define generalized functions;
the Fourier transform of the corresponding generalized functions
are also discussed in Gelfand-Silov [1]. Cf. also A. Friedman [1]
and L.Hormander [6].
5.	Titchmarsh’s Theorem
Theorem (E. C. Titchmarsh). Let /(%) and g(x) be real- or complex-
valued continuous functions, 0 x < oo, such that
</*g)(x) = //(x — y) g(y) dy = fg(x — y)/(y)dy = (g-*/) (x)	(1)
о	0
vanishes identically. Then either one of j (%) or g (x) must vanish identically.
There is a variety of proofs of this important theorem, such as by
Titchmarsh [1] himself and also by Crum and Dufresnoy. The follow-
ing proof is elementary in the sense that it does not appeal to the theory
of functions of a complex variable. It is due to Ryll-Nardzewski [1] and
given in the book by J. Mikusinski [1].
Lemma 1 (Phragmen). If g(u) is continuous for 0 < и T, and
0 t < T, then
oo ,_ ..i-i т	t
lim £ '- f g<„) du = f g(«) du.	(2)
= 1	о	0
5. Titchmarsh’s Theorem
167
Proof. We have J? (-1)*"1 (Л!)”1 ekx(t~u} = 1 - exp (—г((~и)), and,
= I
for fixed x and t, the series on the left converges uniformly in и for
0 < и T. Thus the summation in (2) may be carried out under the
integral sign and we obtain (2) by the Lebesgue-Fatou Lemma.
т
Lemma 2. If f(t) is continuous for 0 < t T and f enl dt
о
for n = 1; 2, ..., where M is a positive constant which is independent
of n, then / (t) must be = 0 for 0 rd t < T.
Proof. We have
M
f(T-u) du'
j А=1 Л'
T
f ekn[T~u} f{T — u) du
0
M (ехр(г~я(г^) - 1).
If t < T, the last expression tends to zero as n —> oo. Hence, by Lemma 1,
t
with g(u) f{T u), we see that f f(T — u) du = 0 for	T.
о
Since / is continuous, it follows that f(t) — 0 for 0 <i t < T.
Corollary 1. If g(x) is continuous for 1 x < X and if there exists
a positive number N such that
Л
xng(x) dx
(к = 1, 2, ...),
then g(x)  0 for 1 -. л "_ A’.
Proof. Putting x = el, X — eT and %g(%) =/(0» we obtain, by
Lemma 2, that /(/) = 0 for 0 t	T. Thus xg(x) = 0 for 1 S x < X,
and so g (x) ~ 0 for I < ?; * .V.
Corollary 2 (Lerch’s theorem). If f(t) is continuous for 0 < t < T and
т
/ dt = 0 (n = 1, 2, . . .), then /(/) = 0 for 0 t T.
0
Proof. Let be any number from the open interval (0, T), and put
1 — tGx, T = tQX, f(t) = g(%). We then obtain
x
t^1 f xng(x)dx = 0 («=1,2,...),
0
1
1
and so j x g(x) dx = , J x g(x) dx J |g(x) dx = .
1	I |o	0
Hence, by Corollary 1, we obtain g(%) = 0 for 1
/(/) = 0 for /0 < t T. Since tQ was an arbitrary point of (0, T), we
must have /(£) = () for 0 < t < T.
The proof of Titchmarsh’s theorem. We shall first prove the Theorem
for the special case when f = g, that is, if /(£) is continuous and (/ * /) (/)
t
f f(t — u) f(u) du = 0 for 0 - t < 2 T, then f(t) = 0 for 0 t T.
168
VI. Fourier Transform and Differential Equations
We have
2T	/ i	\
J ew(2r~0 I J" f(u)f(t—u)du\dt=Q,
0	\0	/
and, by the change of variables и T v, t = 2T — v — w, we obtain
J f en[v+№] f(T— v) f(T — w) dvdw = 0
where d is the triangle v + w 0, v T, w < T in the v — w plane.
Let d' be the triangle v -j- w < 0, ?? >  - 7’, w — T. Then the join
d + d' is the square—T ^v, w <^T. The above equality shows that
the integral of ew!t,+w) f(T — v) f(T — w) over d d' is equal to th^inte-
gral over d'. The integral over d + /Г is the product of two single Ttite-
grals, and in the integral over d' we have	1. Thus
т
/ en« f(T — u} du
2
= // f(T-v) f(T — w) dvdw
A+df
И\НТ~ y) f(T-^)\dvdw 2T2-A2,
A1
where A is the maximum of |/(Z) | for 0 t 2Tt and 2Г2 is the area
of We thus have
and, moreover,
о
/ enuf(T-uj du< TA. Therefore
T
T 0
о
and so, by Lemma 2, f(t) = 0 for 0 £ i g T.
We are now ready to prove Titchmarsh's theorem for the general case.
t
Let f f(t—u) g(u) du = 0 for 0 t < oo. Then we have, for 0 t < oo,
о
t	t	t
f	—	f /(i~u) ug(u) du = f f /(i^u)g(u)du — 0,
о	oo
This may be written as
(/1 * g) w + (/ * gl) (*) = 0 (° t < °°).
where	=tg (t).
Thus
[/*{gi* (/i*g + / *gi)}J (0 = °
[(/*g) * (A *gi)] W + [(/*gi) * (/*gj] ю = 0.
6. Mikusiriski’s Operational Calculus	169
Thus, by (/* g) (f) = 0, we have [(/ * gj * (/*&)] (/) = 0, and so,
by the special case proved above, (/ * gj (/) = 0, that is,
t
f f(t — m) Mg(«) du = 0 (0 <] t < oo).
о
From this we obtain, similarly to above,
t
f f(t — u) u2g(u) du — 0 (0 < 2 < oo) •
о
Repeating the argument, we find that
t
f f(t— w) Mwg(«) du = 0 (0	< oo; n - 1, 2, . . .).
о
Hence, by Lerch’s theorem proved above, we obtain
/ (t — u) g(u) =0 for 0 < и t < oo.
If а и0 exists for which g(«0) =£ 0, then	= 0 for all t uQ,
that is, f{v) =0 for all v 0. Therefore, we have either /(у) = 0 for
all v > 0 or g (<) = 0 for all v > 0.
6.	Mikusiriski’s Operational Calculus
In his “Electromagnetic Theory”, London (1899), the physicist
O. Heaviside inaugurated an operational calculus which he successfully
applied to ordinary linear differential equations connected with electro-
technical problems. In his calculus occured certain operators whose
meaning is not at all obvious. The interpretation of such operators as
given by Heaviside himself is difficult to justify. The interpretation
given by his successors is unclear with regard to its range of validity,
since it is based upon the theory of Laplace transforms. The theory of
convolution quotients due to J. Mi к и si^ ski provides a clear and simple
basis for an operational calculus applicable to ordinary differential equa-
tions with constant coefficients, as well as to certain partial differen-
tial equations with constant coefficients, difference equations and integral
equations.
We &hall give, adapted from K. Yosida-S. Okamoto [40], a simpli-
fied presentation of Mikusihski’s theory by introducing the notion of the
ring in place of his operators {= the convolution quotients). By virtue
of this ring we can derive the operational calculus without appealing
to Titchmarsh's theorem given in Chapter VI, 5.
170
VI. Fourier Transform and Differential Equations
The ring H. We denote by the totality of complex-valued con-
tinuous functions defined on [0, oo). In this section, we shall denote
such a function by {f(t}} or simply by f, while /(t) means the value at
t of this function /. For /, g £ and a, /3 £ C1 (the complex number field),
we define
7	+ g = {/ (0 + g (0} and fg = | // (t - w) g (и)	.
10	J
Then, as proved in Chapter VI,3, we have
(1)
fg = gh f (gty (fg) h and / (g + Л) = fg + fh .
Hence is a commutative ring with respect to the above addition and
multiplication over the coefficients field C1.
We shall denote by h (I in J. Mikusinski [1]) the constant function
{1} £ so that we have
p-i
=	("=1.2,...)	(2)
and
!t	।
ff(u) du\ for	(3)
0 J
i.e., h behaves as an operation of integration. Thus we have the fairly
trivial
Proposition 1. For k £ H = [k] k = hn {n = 1, 2,...) } and /	, the
equation kf --- 0 implies / — 0, where 0 denotes {0} £
Therefore, as in the algebra, we can construct the commutative ring
^ir consisting of fractions of the form //7г, i.e.,
=	(4)
where the equality of two fractions is defined by
| = T Hl W=/'A.
and the addition and multiplication of two fractions are defined through
/ , {'	fk' + f'k	.	/	//'	/m
k + k' kk'	d	k	k' ~ kk'	'
By identifying /С#7 with	the ring can isomorphically
be embedded as a subring of ‘Й’ц.
We next introduce, after Mikusinski, the important notion of the
operation of scalar multiplication. We define, for every a £ C1,
м =-T = {=<}/! e®’,,.	(7)
6. Mikusinski’s Operational Calculus
171
Then we have, for a, /? £ C1, f e and k £ H,
и+ [Д= [а+Д, [а] [Д = [affl ,	(8)
[a]/= «/ = {«/(<)}, И у =	.
Proof, [a] + [/3] = [ a + Д] is clear. We have
г i гZ?i	{«^0 h{<xp} {a fi} R
W и = ~л—тг =	“ —k~ = W. - Wb
if a / (u) du I
r. {/ w _ {«}{Щ} _ u _L = m
LaJ k	hk	hk	A
Hence [a] can be identified with the complex number a, not with {a},
and we see that the effect of the multiplication by [a] is exactly the
scalar multiplication by a. [1] may be identified with the multiplicative
unit I of the ring i.e.,
hn
(«=1.2....).	(9)
We next define
Ln
5 =	(w = 0, 1, 2, . . . ; h° = I) so that sh = hs = I. (10)
Proposition 2. If both / and its derivative f belong to , then
/' = 7-/(0), where /(0) = [/(0)1 ,	(11)
i.e., s behaves as an operation of differentiation.
Proof. Clear from (10) and Newton’s formula
hf = Iff' (u) Д = {/ (i) - f (0)} = f-[f (0)] h .
Corollary. For и-times continuously differentiable function /£^,
/<«> = snf — sn~lf(ty - s’1”2 /л(0) - ... -	(0) ,	(12)
where /<0 (0) = [/<0 (0)] .
Proposition 3. For any tx t C1 and for any positive integer и,
(s - a)n = (s - [a])" = --= “-------------IP"----н
admits a uniquely determined multiplicative inverse in H given by
*	“(JZT= {(и —1)!
Proof. We have, by (11),
(s - a) {eai} - a {ea[} + [1] - a{e“f} = I, i. e., Ij(s - a) = {eat} .
Hence, we have //(s a)a { e™ du} - {t ея(} and so on.
<9
172
VI, Fourier Transform and Differential Equations
Application to the integration of the Cauchy’s problem for linear ordinary
differential equations with constant coefficients £ C1:
+ «мУ41 + ... + аоу = /^(апФО) ,	(14)
У (°) = To- У (°) = П, •  • > УП-1)(О) = Tn-i 
By (12), we rewrite (14) into
(«„s'1 +	, + a0) у = / +	+ fin-2sn-2 + ,.. + /9 о > (14)'
P? = «,+i Уо + СС+2У1 +    + <*nyn^ (v = 0, 1, 2, ,.., n - I) ,
The polynomial ring of polynomials in s with coefficients £ C1 is free
from zero factors, because of the fact that
(<Un + 4-1S"-1 +   •) (W™ + Y^-yS™-' +...)= дпЧт S»+™ + .
Hence we can define rational functions in s:
Fi =-----—--------- - and F2 = —
a„ sn + . . . -J- a0	2 a„ 5" + . . . + a0
and obtain their partial fraction expressions:
(15)
(15)'
of the
j k — 7	; k — 7
where both c?-fe and djk belong to C1 and r/s are distinct roots
algebraic equation an zn H- ап_г zn~i + .. . + a0 = 0 so that 27	= n.
?
By virtue of (13), both 7q and F2 given in (15)' belong to so
that we obtain the solution of (14)' and hence of (14) as well:
W7
0(0} Л Лc* I TV-nyr*'''} {/<')}	<16)
j k = 1
W7
?	= 7
Example 1. Solve the equation
x"(t) - x'(t) - 6x(t) - 2, x(0) - 1, %'(0) = 0 .
Solution. We rewrite the equation
{x"(t) - x'{t} - 6x(t)} = 2/s
by (12), obtaining
s2 x — 5 x(0) — x'(0) — s x -|- x(0) — 6r = 2/s
7. Sobolev's Lemma
173
and so, by substituting the initial condition, we get
(s2 — s — 6) x = s — 1 4- 2/s .
Hence, by (13),
+ 2	11,8	1	4	1
X = Т(Г— 3) (s 4^2)" “~3s + 15s — 3 + 5s + 2
fl 8 _.	4	„.1
.—     Д_ ——	j_ __ p—2t ,
3 1 15 e + 5 e 
Example 2. Solve the system of equations
x'(t} - a x(t) — Py{t) = fie** , y'(t) 4 Д x(t) - <zy(t) = 0 ,
under the initial condition x(0) — 0, y(0) = 1. We here assume that a and
P are real numbers.
Solution. Rewrite the equations by (11) and (13), obtaining
sx — art — Ду — Д/ (s—a), sy — 1 -J- Дх — ay = 0 .
Hence
2/3	(s —a)2 — /32
X ~ (S — a)2 + /4 ' У ~ {$ — a) ((s — a)2 4 /32) ’
from which we get
x = -г------—----o- —-------—гД = 4-	= {2eai sin ДД
г s — a — ijS s — a 4 г ft г L	J 1 r J'
_	2 (s — a)_________1	_	1_____________1___________1
У ~ [s-— a)2 4	(s — a) ~ s — a —-ift	s — a 4 ip $ — a
— {g(a+i/5)t	= COS pt — 1)} .
Remark. Suggested by (2), we can define the fractional power inte-
gration through
= А{/“-1/Г(а)}/7^^11(0<а< 1) .	(17)
Accordingly, we can define the fractional power differentiation through
s“ = h[t-* fr{\ - а)}ДЧ^н(0 < ас 1) ,	(18)
because we obtain
has“ = I	(19)
by the Gamma function formula B(1 — а, a) = Д1 — а) Да).
For further applications of the operational calculus, we refer the
reader J. Mikusinski [1] and also to A. Erdelyi [1].
7.	Sobolev’s Lemma
A generalized function is infinitely differentiable in the sense of
the distribution (see Chapter I, 8). So the differentiability in the gene-
ralized sense has no bearing on the ordinary differentiability. However,
174
VI. Fourier Transform and Differential Equations
we have the following result which is of fundamental importance in
modern treatment of partial differential equations.
Theorem (Sobolev’s Lemma), Let G be a bounded open domain of
Rn. Let a function u(x) belong to Wk (G) for k > n + where <r is an
integer 0. We thus assume that the distributional derivatives of
и (%) of order up to and including k all belong to L2 (G). Then, for any open
subset GT of G such that the closure Gf is a compact subset of G, there
exists a function их(х) £ CCT(GX) such that и (%) = ^(x) a.e. in Gx.
Proof. Let а (ж) be a function £ C“ (Rn) such that
Gx C supp(<x) QG, 0	tx(%)	1 and <x(x) = 1 on Gx.
We define a function v (x) defined on Rn by
v (%) = a (%) и (x) for x £ G; v [%) = 0 for x £ Rn — G.
Then v (%) = w (%) whenever x 6 Gx. Since v (%) is locally integrable over
Rn, it defines a distribution C ® (Rn)’  By the assumption that и £ Wk (G),
the distributional derivatives Dsv(x) £ L2(Rn) when |s [ k. For example,
the distributional derivative
dXj ' ! 8xj ' dx,	8xf
belongs to iZlR*) by the fact that u, du^dXj both belong to L2(G) and
that the infinitely differentiable function л (%) has its support contained
in a compact subset of the open domain G. By the Fourier transformation
v (л') v (y), we obtain
Since, by Plancherel’s theorem, the L2-norm is preserved by the Fourier
transformation, we have (Dsv) (у) 6 L2{7?”) for s k. Thus
(у) У? У2    Упп € L2 (Rn) for |s|^A.	(1)
In particular we have
i (у) ё L2 (R“).
Let q = (gx, g2, . . ., qZ) be a system of non-negative integers. Then
by (1), we can prove that
у(у) yx* yf . . . ye„ is integrable over Rn whenever |?| +	< A• (2)
For the proof, take any positive number C. We have, by Schwarz’ ine-
quality,
/ У1 У2 • • - У«”| dy
|y|Sc
( f lУ1 У2   • dy • f |v(y) I2 dy\li2 < OO,
\hisc	Ы<;с	/
/ | v (у) УI1 У2   - Уяп i dy
hl>c
gf f +	f |?W (1 + Ы2)‘Т dy\m-
\W>c	Ы>с	/
8. Garding’s Inequality
175
The second factor on the right of the last inequality is < oo by (1). The
first factor is finite by
dy — dy\ dy2 . . . dyn = гп~г dr dQni
where dQn is the hypersurface element of the surface (3)
of the unit sphere of Rn with the origin 0 as its centre,
provided
к
2 |^| — 2k + n — 1 < — 1, that is, if k >	+ \q
Now, by Plancherel’s theorem, we have
у (%) = l.i.m. (2 л) rt*2 f v(y) exp(i (y, x)) dy
and so, as in the proof of the completeness of L2(Rn), we can choose a
subsequence {h'} of positive integers h such that
v(x) = lim (2л)~"/2 f v (y) exp (г <y, x>) dy for a.e. x£_Rn.
But, since v{y) is integrable over Rn as proved above, the right side is
equal to
zq(x) = (2л)-к/~ f v{y) exp(i<%, y>) dy,
Л”
that is, у (%) is equal to ух(х) for a.e. x£ Rn. By (2), the differentiation of
tq(%) under the integral sign is justified up to the order cr; and the result
of the differentiation is continuous in x. By taking иг (x) = tq (%) for
xy Glr we have proved the Theorem.
Remark. For the original proof, see S. L. Sobolev [1], [2] and
L. Kantorovitch-G. Akilov [1].*
8.	Garding’s Inequality
Consider a quadratic integral form defined for C°° function u(x) =
w-(%!, x2, . . ., x„) with compact support in a bounded domain G of Rn:
В \u, w]
where the complex-valued coefficients csl are continuous on the closure
Ga of G, and (и, y)0 denotes the scalar product in £2(G).
Then we have
Theorem (L. Garding [1]). A sufficient condition for the existence of
positive constants с, C so that the inequality
||« £ c Re B + C ||w ||o	(2)
holds for all м-t Cj°(G), is that, for some positive constant c0,
Re A1 f > co | £ |2”* forall x^G and all real vectors
IT(<!=«’ '	(3)
f- (*U8..........
* Sen also Supplrrurnlnry Noles» p. 1GG.
176
VI. Fourier Transform and Differential Equations
Remark. The inequality (2) is called Garding’s ineqitality. If condi-
tion (3) is satisfied, then the differential operator
£ = 2 (-1)1'1	(4)
is said to be strongly elliptic in G, assuming that cs/ is on Ga.
Proof. We first prove that, for every £ > 0, there is a constant C
such that for every (G) function u,

IIм!& + c(e) ll<-
To this purpose, we consider и to belong to C£°(7?M), defining its value as
0 outside G. By the Fourier transformation we have

«	j2
0 = Uo= f П^И(у) dy,
Rn 3=1 ’
in virtue of Plancherel's theorem. Thus (o) is a consequence from the fact
that
2 nyj“) [с+у n = £ Sj, \i\ = .s tj)
tends to zero uniformly with respect to у = (ylr y2>   , Уn) as C f oo.
Suppose that the coefficients cst are constant and vanish unless
| s | = | £ | = m. By the Fourier transformation и (%) -> « (£) and Plan-
cherel's theorem, we have, by (3),
Re В [и, u\=Ref У |w(£) |2 d£
s,t
where сх > 0 is a constant which is independent of u. Hence, by (5), we see
that (2) is true for our special case.
We next consider the case of variable coefficients cst. Suppose, first,
that the support of « is sufficiently small and contained, say, in a small
sphere about the origin. By the preceding case, we have, with a constant
Co > 0 which is independent of w,
Cq |(u Re В [и, и] + Re £ f (cst(ty— cst{x}) Dsu • Dludx
-Re	+ C(e)
If the support of « is so small that c5( has very small oscillation there,
we see that the second term on the right may be bounded by 2^x c'o 11 w
The third term on the right is bounded by constant times ||« ||w • ]| и
Hence we find that, constants denoting positive constants,
2“% I|«i&	ReB\u, u] + constant  ||«(|w_i -h C (e) ||w |g.
9. Friedrichs’ Theorem
177
Thus, by
2 H 
£ |« |2	t-1 j/3 |2 which is valid for every
£ > 0,	(6)
we obtain |j« Si constant  Re В [и, w] + constant 	+ C (e\ -
||z(j|o, and so, by (5), we obtain (2).
Next we consider the general case. Construct a partition of unity in G:
N
1 - 21 oj$} coy E C%> (G) and co,- (x) > 0 in G,
such that the support of each coy may be taken as small as we please.
Then, by Leibniz' rule of differentiation of the product of functions,
Schwarz’ inequality and the estimate of the case obtained above, we have
Re В [и, и] = Re £ f cstDsuDtudx = Re 2* 2. ( cojc^D^uD^dx
S,t	S.i J J
— Re f S 2 csi и) Р((со,- и) dx + 0(|fw||m ||«||TO-i)
j; constant (|]wy«|£ — ||ш7-и||и-1) + 0(|j« j|ro * liMIU-i)
J
Sr constant + 0(||«||OT • ||«||w_1).
Thus we obtain (2) by (5). We remark that the constants c,C in (2)
depend upon c0, csi and the domain G.
9.	Friedrichs’ Theorem
Let
L= ,2? D\t(x]Dl	(1)
be strongly elliptic with real G°° coefficients cst (x} in a bounded open domain
of Rn. For a given locally square integrable function /(,r) in G, a locally
square integrable function « (%) in G is called a weak solution of
Lu = f,	(2)
if we have
(».L4=».A.e=„<? (-Dfc^x)D>,	(3)
for every <p € G£° (G), Here (/, g)0 denotes the scalar product of the Hilbert
space L2(G). Thus a weak solution «of (2) is a solution in the sense of the
distribution. Concerning the differentiability of the weak solution u, we
have the following fundamental result:
Theorem (K. Friedrichs [1]). Any weak solution и of (2) has square
integrable (distributional) derivatives of order up to (2 m + p} in the
domain Gx £ G where j has square integrable (distributional) derivatives
of order up to p. In other words, any weak solution и of (2) belongs to
IT7' l2m(G1) whenever / belongs to W^fGjJ.
12 YoMdfi* Functional AnatyHh
178
VI. Fourier Transform and Differential Equations
Corollary. If p = oo, then, by Sobolev’s lemma, there exists a func-
tion u0(x)€ C00^) such that u(x) = w0(%) for a.e. % 6 Gr Thus, after a
correction on a set of measure zero, any weak solution u(x) of (2) is C°°
in the domain £ G where /(%) is C°°; the corrected solution is hence a
genuine solution of the differential equation (2) in the domain where /(x)
is C00.
Remark. When L ~ Л, the Laplacian, the above Corollary is Weyl’s
Lemma (see Chapter II, 7). There is extensive literature concerning
the extensions of Weyl’s Lemma to general elliptic operator L;
such extensions are sometimes called the Weyl-Schwartz theorem.
Among an abundant literature, we refer to the papers by P. Lax [2],
L. Nirenberg [1] and L. Nirenberg [2]. The proof below is due
to the present author (unpublished). A similar proof was given by L. Bers
[1]. It is to be noted that a non-differentiable, locally integrable function
/ (%) is a distribution solution of the hyperbolic equation
dx dy
as may be seen from
^pdy\dx &(х,у)еС?(КУ).
Proof of the Theorem. We shall be concerned only with real-valued
functions. Replacing, if necessary,!, by I + <xL with a certain constant
tx 0, we may assume that the strongly elliptic operator L itself satisfies
Garding's inequality
(^, L*(p)0	(3 [|<p
0),
(4)
I (99, L*v>)o| У H?’Dm IMI™ (/ > 0), whenever <p, Cf(G}.
The latter inequality is proved by partial integration. We assume here
that each of the derivatives of the coefficients csi (ж) up to order m is
bounded in G, so that the constants d and у are independent of the
test functions ip, (p£C™(G).
Suppose that GT is a periodic parallelogram
0	2 л (/ = 1, 2, . . ., n),	(5)
and that the coefficients of L and / periodic with the period 2 л in each
Xj. Under such assumptions, we are to deal with functions 9?(%) defined
on a compact space without boundary, the «-dimensional torus Gx given
by (5), and the distribution £ C00(G1)' associated with the space of test
functions 99 6 C°° (G-J consisting of C°° functions 99 (%) = 99 (x1( x2, . . ., x„)
periodic with period 2 л in each of the variables Xj. It is to be noted that,
since GT is without boundary, we need not restrict the supports of our
test functions <p(x).
9. Friedrichs’ Theorem
179
The condition PF9 (GJ thus means that the Fourier coefficients vk
<>f у (%) defined by the Fourier expansion
v (%) —' £ vk exp {i k  x)
k
n X (6)
k = (Ax, . . ., kn), X = (xx, . . xn), and Л • % kjXj)
/
satisfy
(n \
|A|2=^^.	(7)
;=1 /
For, by partial integration, the Fourier coefficients of the distributional
derivative Dsv satisfy
(Dsv(%), ехр(гЛ • x))0 = (— 1)|js| (v (x), Ds ехр(1Л - x))0
= (— Jls| П kJ* vk, s = (slf s2, . . ., s„),
7 = 1
and so, by Parseval’s relation for the Fourier coefficients of Dqv 6 L2(GJ,
we obtain (7).
It is convenient to introduce the space PR7(G\) with integer q | 0, by
saying that a sequence {wk] k — (&х, k2,  	£„)} of complex numbers
with wk=w_k belongs to Wq (G^) if (7) holds. This space Wq (GJ is normed by
|2 (1 + |&|2)9)1/2. By virtue of the Parseval relation
with respect to the complete orthonormal system {(2л)-^2 ехр(гЛ  x)} of
L2 (GJ, we see that, when q 0, the norm l|y||? = f У f ]Dsv (x) |2 dx\1!2
\lds? Gl	/
is equivalent to the norm || {yj ||f, where у (x) ~ £ vk exp (г k  x).
The above proof of (7) shows that, if / £ Wp(GJ, then DsfG
and q>fE B^fGJ for q> e 0х {Gv). Hence
if /€ ^(Gj), then, for any differential operator N of
order q with C00 (GJ coefficients, N/£ ^“^(GJ.
(8)
To prove the Theorem for our periodic case, we first show that we may
assume that the weak solution L2 (GJ = И70 (GJ of (2) belongs to
PPm(GJ. This is justified as follows. Let
и (%)	£ uk exp (i k  x), v (x)	+ Н2Г"ехр (ik  x).
Л	A
Then it is easy to see that у (x) 6 W2m (GJ and that у is a weak solution
of (Z — у = и, where Д denotes the Laplacian <92/6x2. Hence у is
7 = 1
a weak solution of the strongly elliptic equation of order 4ш:
If we can show that this weak solution v 6 IF2'" (GJ actually belongs to
iP4m4^(GJ, then, by (8), и =: (Z -d)wy belongs to IF4wi+^~2w(GJ =
180
VI. Fourier Transform and Differential Equations
Therefore, without losing the generality, we may assume
that the weak solution и of (2) belongs to IF”1 (GJ, where m is half of
the order 2 m of L.
Next, by Garding’s inequality (4) forL and that for {I —4)m, we may
apply the Lax-Milgram theorem (Chapter III, 7) to the following effect.
The bilinear forms on C°° (GJ:
(fP. vO' = (<P. and {(p, ip)" = (<p, (I — J)”»o	(9)
can both be extended to be continuous bilinear forms on	such
that there exist one-one bicontinuous linear mappings T', T" on IF^fGJ
onto Wm (Gj) satisfying the conditions
(T'<P> ip)' = (ip, ip)m, (T'f(p, ip)" = (pp, ip)m for <p, ip E IF” (GJ.
Therefore, there exists a one-one bicontinuous linear map Tm = T" (T')~l
on	onto IF" (GJ such that
(Z- VO' = VO" whenever y>, IF"(GJ.	(10)
We can show that
for any /	1, Tm maps JF"H(GJ onto IF”+j(GJ in
a one-one and bicontinuous way.
In fact, we have
(9,.£*(^-^)G)o = U„?>.(/-zl)"+7v>)o for «р.уёС^С,).
On the other hand, there exists, by the Lax-Milgram theorem as applied
to the strongly elliptic operators (Z— JJZ and (Z—a one-one
bicontinuous linear map Tm+j of IF"+J(GJ onto JF”+j(GJ such that
(<p,L*(Z~d)^)0 = (Tm+y<P,(Z-Zir+>V;)0 for у EC00 (GJ.
Therefore, the function w — (Tm+j — Tm) <p is, for any (p E C°°(GJ, a
weak solution of (Z —d)m+j w — 0. But, such a w(x) is identically zero.
For, the Fourier coefficient of w (x) satisfies
0 = ((Z — d)w+j w(x), exp(ik  x))0 = (w(x), (Z — Zl)m+j exp(ik - x))0
= (1 + |A|2)”+j(»(x).exp(!A.x))0=(l + |A|2)"+’X.
and so wk = 0 for all k. Thus (Tm+}- — Tm) is 0 on C°°(GJ. The space
C00^) is dense in lF"+j(GJ £ IF” (GJ, since trigonometric polynomials
£ wk exp[ik • x) are dense in the space IF”+j(GJ. Hence Tm+}- = Tm
on JF”+J(GJ.
We are now ready to prove the differentiability theorem for our
periodic case. We have, for чр E C°°(GJ,
(/. Wo = («>	= (w> V7)' = {Tmu, ip)'1 = .(Tmu, {I ~A)m ip\.
9. Friedrichs’ Theorem
181
Hence, for
Tmu £ ck ехр(х’Л  Jt), ip(x} yk exp (i k • x),
k	fe
we obtain, by Parseval’s relation,
(T„«, (I-J)”»o=	+ |A|2)“	= 2fo.
A	«
By the arbitrariness of ip£ C00^^, we have cft(l -f- |&|2)w = Д, and so,
by 10* (GJ, we must have TmuG Wp+Zm (G-^. Hence, by (11),
п С- ИТ+2"’(GJ .It is to be noted that the above conclusion «4 W7>+2m(GJ
is true even in the case 0 p S: (1 —w), i.e., the case {fh} C JV^G)
with 0 p > (1 — m). For, by /> + m + 1, we can apply (11).
We finally shall prove, for the general non-periodic case, our differen-
liability theorem. The following argument is due to P. Lax [2].
We want to prove, for the general non-periodic case, the differentia-
bility theorem in a vicinity of a point л° of G. Let J (x) £ C£° (G) be identi-
cally one in a vicinity of x°. Denote /3 м by u' is a weak solution of
Lu' = pf + Nu,
(12)
where N is a differential operator of order at most equal to (2m — 1)
whose coefficients are, together with p(x), zero outside some vicinity V
of x°, and the operator 2V is to be applied in the sense of the distribution.
We denote the distribution pf + Nu by
Let the periodic parallelogram GT contain V, and imagine the coeffi-
cients of L so altered inside G2 but outside F that they become periodic
without losing their differentiability and ellipticity properties. Denote
the so altered L by L'. Thus u' is a weak solution in G of
L' u' = where f = pf + Nu.	(13)
We can thus apply the result obtained above for the periodic case to
our weak solution u!. We may assume that the weak solution и belongs
to H/W(GJ. Thus, since N is of order (2m— 1) and with coefficients
vanishing ontside V, /' = pf + Nu must satisfy, by (8),
/'e’ Wfi' (GJ with pf = min(^>, m — (2m — 1)) == min(^>, 1 —m) > 1 — m.
Therefore, the weak solution uf of (13) must satisfy
u' € Wp" (GJ with p" = min (J + 2m, 1 — m + 2 m)
= min (p + 2 m, m + 1).
Hence, in a certain vicinity of x°, и has square integrable distributional
derivatives up to order p" which is > (m 4- l).Thus /' = pf + Nu
has, in a vicinity o^f x°, square integrable distributional derivatives of
order up to
p'" - - min (/>, p" — (2m — 1)) min (p, 2 — m).
182
VI. Fourier Transform and Differential Equations
Thus again applying the result already obtained, we see that u' has, in a
certain vicinity of %0, square integrable distributional derivatives up to
order
= min (ft 4- 2m, 2 — m + 2 m) = min (/> + 2m, m 4 2).
Repeating the process, we see that w has, in a vicinity of x°, square
integrable distributional derivatives up to order p 4- 2m.
10.	The Malgrange-Ehrenpreis Theorem
There is a striking difference between ordinary differential equations
and partial differential equations. A classical result of Peano states
that the ordinary differential equation dyfdx ~ f(x, y) has a solution
under a single condition of the continuity of the function /. This result
has also been extended to equations of higher order or to systems of
equations. However, for partial differential equations, the situation is
entirely different. H. Lewy [1] constructed in 1967 the equation
which has no solution at all if / is not analytic, even if / is C°°. Lewy’s
example led L. Hormander [3) to develop a systematic method of
constructing linear partial differential equations without solutions. Thus
it is important to single out classes of linear partial differential equations
with solutions.
Let P(f) be a polynomial in	gn, and let P(D} be the linear
differential operator obtained by replacing fy by Dy — idf&Xj. P (D) may
be written as
P(D} = 2? СЛ where, for a = (a, <x2, . . a„) ,
a
Da = Dp IP1 . . . D“".
*	12	n
Definition 1. By a fundamental solution of P(D) we mean a distribu-
tion E in Rn such that
P(D) E = 6 = T6.
The importance of the notion of the fundamental solution is due to the
fact that
«=£*/, where /G С§°(Л”),
gives a solution of the equation
P(D) w = /.
In fact, the differentiation rule (10) in ChapterVI,3 implies that
P(D) и = (P(D) E) * f = <5 * / = /.
10. The Malgrangc-Ehrenpreis Theorem
183
Example. Let P(D) be the Laplacian A = in Rn with n > 3.
7=1 j
Then the distribution

E = T,t where g(x) - ----:-c- \x I2 ” and S,, - the area of the
surface of the unit sphere of Rn,
is a fundamental solution of Zl.
Proof. We have, in the polar coordinates, dx = |%|” 1
so the function g(x) is locally integrable in Rn. Hence
rf x dSn,
and
А T|r|i-n (9?) = lim f j x |2 " • Дер dx, ер^'ДЬ (7?’1).
e ! 0
e
Let us take two positive numbers e and R (> e) such that the supp(y>)
is contained in the interior of the sphere x | R. Consider the domain G:
Ix j	R of Rn and apply Green’s integral theorem, obtaining
f	I x I2-  <p1 dx =
G	SG
= e and |%I = R,
where S = 8G is the boundary surfaces given by \x
and v denotes the outwards normal to S. Since ep vanishes around |x | = R,
we have, remembering that A [%	= 0 for x 0 and that —— =
.it the points of the inner boundary surface l%j = e,
1“” ep dS.

2—w	j с I
Q ] ]	“j
n
When e j 0, the expression |x | = £ (%j/l% |)  dep/dx, is bounded
and the area of the boundary surface |%| = e is Sme"-1. Consequently
the first term on the right tends to zero as e 0. By the continuity of <p
and a similar argument to above, the second term on the right tends, as
e j, 0, to (2 — n) Sn • 99(0). Thus T„ is a fundamental solution of A.
The existence of a fundamental solution for every linear partial diffe-
rential equation with constant coefficients was proved independently by
B. Malgrange [l]andL. Ehrenpreis [1] around 1954—55. The exposition
of the result given below is due to L. Hormander [4].
Definition 2. Set
p(f) = (x |p(”WY/2
[)a ™ . __ -
where PW(^)=^P(^),
(1)
184
VI. Fourier Transform and Differential Equations
We say that a differential operator Q(D) with constant coefficients is
weaker than P(D) if
Q(g) CP($), ^Rn, C being a positive constant. (2)
Theorem 1. If Q is a bounded domain of Rn and /£ £2(12), then there
exists a solution м of P (D) и = / in Q such that Q (D) L? (Q) for all Q
weaker than P. Here the differential operators P (E>) and Q (D) are to be
applied in the sense of the theory of distributions.
The proof is based on
Theorem 2. For every e > 0, there is a fundamental solution E of
P(D) such that, with a constant C independent of w,
|(£*и) (0)[£Csup /	d(,	(3)
Rn
Here и is the Fourier-Laplace transform of u:
«(С) = (2тг)-^2 f e-'W) u(x) dx, £ = £ + i?],
Rn
and the finiteness of the right side of (3) is assured by the Paley-Wiener
theorem in Chapter VI, 4.
Deduction of Theorem 1 from Theorem 2. Replace и in (3) by
Q (D) и * v, where и and v are in C£° (Rn). Then, by (10) in Chapter VI, 3,
|((?(£») E * и * v) (0)| = |(E*^(P)w*d(0)|^ CN(Q(D) i4*v),
where V(w) = sup f |«(£ + iq)\/P(g)  d$.
|i]|gs Я"
The Fourier-Laplace transform of Q (D) и * v is, by (17) in Chapter VI, 2
and (15) in Chapter VI, 3, equal to (2л)”/2 Q(£) w (£) v (£). Since, by
Taylor’s formula,
1 K
Q (£ +	~i (— T[Y Dlx Q (£), where (—	= П > (4)
a	/ - 1
we have, by (2),
| (?(f + iri)\IPtf) C' for |^| Те and
where the constant C may depend on e. Thus
N (Q u-x-v) (2 л)”/2 C  sup J" |«(£ + i rj) v (£ + i r>} j d£.
Ids* кп
By Parseval’s theorem for the Fourier transform, we obtain, denoting
by j) || the E2(R“)-norm,
/ [w (£ + t ?]} |2	= f | u (x) |2 e2 (x,tl'} dx < 11 и (x)	112 when | rj I	e,
RP	Rn
and a similar estimate for v. Thus, by Schwarz' inequality,
N (Q (D) и * v)	C" 11 и {x) e^xl (| 11 v (x) e£^ | [ whenever и and v 6 CJJ° (RM),
C" denoting a constant which may depend on e.
10. The Malgrange-Ehrenpreis Theorem
185
Therefore we have proved
/ (2(D) E * u) {x} v{-x) dx (CC")||w^l || |REW||(wJi^C^(7?n)).
кя	'	(5)
We shall write L2E {Rn} for the Hilbert space of functions w (%) normed by
/ f |де(%) |2 dx\1/z = ||w(x)ri^ ||.
Since C^°(£rt) is dense in LE(Rn) and, as may be proved easily, LLe(RM)
is the conjugate space of L2(Rn), we obtain, by dividing (5) by |[y (ж)^*11
and taking the supremum over v E C^° (/?”),
||ДО)£*«) (x)	(CC") ||гфИх1||, u€C^(Rn).
Hence the mapping
ii -» 2 (D) E * «•	(6)
can be extended by continuity from (Rn) to LE (Rn), so that it becomes
a continuous linear mapping on Ll(Rn) into L^_e(Rn). Thus, to prove
Theorem 1, we have to take ~ / in Q and = 0 in Rn —Q and define
и as equal to и = E * /r
For the proof of Theorem 2, we prepare three Lemmas.
Lemma 1 (Malgeange). If /(z) is a holomorphic function of a complex
variable z for I z | S 1 and p (z) is a polynomial in which the coefficient
of the highest order term is A, then
И /(0)1 < (2Л)~1 / |/(?8) />(?») I M.	(7)
Proof. Let zj’s. be the zeros of p (z) in the unit circle ] z) < 1 and put
j
Then q (z) is regular in the unit circle and | p (~) |= | q (г) | for | z | = 1. Hence
we have
Л	Л
(2 л)”1 f |/(?e) p(eie) | d$ = (2 л)-1 f |/(rie) q(eie) | dO
-’Tt	—Л
a (2 я)-1
f f (?“) q (?») M
—n
= 1/(0) 9(0)1-
Thus Lemma 1 is proved since ^(Oj/ri | is equal to the product of the
absolute values of zeros of p (г) outside the unit circle.
Lemma 2. With the notations in Lemma 1, we have, if the degree of
p(z) is m,
|/(0)/>“’(0)|<	(2я)-> / !/(?“)/.(?») I d».	(8)
1	— л
186
VI. Fourier Transform and Differential Equations
Proof. We may assume that the degree of p (2) is m and that
m
£ w = ~ •
k
Applying the preceding Lemma 1 to the polynomial ]J (г — zj) and the
i
holomorphic function f(z) * fl (г — Zj)f we obtain
J=:A+1
«°>X/’
£(2я)-1 / [/(?’)/.(?’) | df).
—ft
A similar inequality will hold for any product of (m — k) of the numbers
Zj on the left hand side. Since (0) is the sum of w!/(w — k)! such terms,
multiplied by (— we have proved the inequality (8).
Lemma 3. Let F(£) = F (Ci, C2< •  • - C«) be holomorphic for |C| —
»	\l/2
S < oo and P(C) = P(CP C2-  • •• £»)' a polynomial of
m. Let 0(f) = 0(£p C2, . . ., f„) be a non-negative integrable function
with compact support depending only on |f2 j, . . ., ]f„ ]. Then we
have
/	»? f	f*
j moww, (9)
Klcoo	'•	|f|<TO
where tZf is the Lebesgue measure d^di^ . . . d^ndr)n {fft = £k -j- and
m is regarded as multi-index {m, m, . . m).
Proof. Let f(z) be an entire holomorphic function and apply (8) to the
functions j{rz} and 'p(rz), where r > 0. Then we obtain
1/(0) />'*’ (0) | • r* fi (2»)"1 /	^ге“'>1M-
Let y(r) be a non-negative, integrable function with compact support.
Multiplying the above inequality by 2л-гу(г) and integrating with
respect to r, we obtain
[/(0)/>w(0) I/ PI* V(\t I) di fi J |/(/) />(«) I V(|«|) dt (10)
where dt = rdrdf) and the integration is extended over the whole complex
i-plane. Lemma 3 is obtained by applying (10) successively to the va-
riables fa, . . ., С», one at a time.
Proof of Theorem 2. Put P(D) и = v, where «€C§°(7?ft). Then
P(t) w(C) = £(£). Apply Lemma 3 with F(£) = и -j- £), with P(£)
replaced by P(£ -j- £) and with |0(£) | = 1 when |t | e and = 0 other-
wise. Since P(£)	|D*P(^) we obtain, from (9),
I «(i)jp(i)|sc, / |i(f + i}P(i + г)рс = с, f ii(« + f)|«.
|CI^	|C|^
10. The Malgrange-Ehrenpreis Theorem
187
Hence, by Fourier's integral theorem,
l«(o)।g(2я)-”'2/।#gc; f (ji;« +
HIS-
^c;f( f Rf + «' + •>/')|/p(f)
On the other hand, we have
P(f+ f')/P(f)^C2 when’ |f'|^E,
because

D^Ptf),
so that \DaP($ + £') |/P(f) is bounded when If' | e. Therefore we have
l«(o)| g c;c2/f J |S(f + c + ir)'>\iP{t+ f‘}d?dV'\dt
— C3 1ЫГ’ W11<.TC
(11)
IMI' = / (J|5(f+ *»I/P(f) ayv (uec?(«„)).
C3 denoting a constant depending only on e.
By the way, the finiteness of ||wj|' is a consequence of the Paley-Wiener
theorem in Chapter VI, 4. Consider the space C£°(/?”) which is the
completion of C£° (Rn) with respect to the norm 11 v 1Then, by the Hahn-
Banach extension theorem, the linear functional L :
v = P(D) u^u(0) (where и € C?(Rn))
can be extended to a continuous linear functional L defined on (/?*).
As in the case of the space Lx{Rn}, we see that there exists a Baire func-
tion A(f T it]) bounded a.e. with respect to the measure P^^d^dq
such that the extended linear functional L is represented as
£(y) = f (J u(f + iv}) A(f 4- i7])/P(g)  d£) dy. (12)
BIS15
When vA(%) С C^°(P”) tends, as Л->оо, to 0 in the topology of ® (/?”),
uA(%) е-х,ъ) also tends to zero in the topology of Ф(/?”), uniformly with
respect to Y] for | t} | iC e. Hence, as in Chapter VI, 1, we see easily that
vk (f H- irj), as function of f, tends to zero in the topology of @(7?n),
uniformly in тэ for |??j e. Therefore, by (12), L defines a distribution
Tt$(Kn)'. Thus, by (5) in Chapter VI, 3.
L(y) = (T*i] (0) = (f * v) (0).	(13)
We have thus proved Theorem 2 by taking E = T. (3) is clear from (11).
188	VI. Fourier Transform and Differential Equations
11. Differential Operators with Uniform Strength
The existence theorem of the preceding section may be extended to a
linear differential operator
P{xtD} = £aJx}Da	(1)
a
whose coefficients are continuous in an open bounded domain Q of
R*.
Definition. P (x, D) is said to be of uniform strength in Q, if
sup P(x,£)/P(y,£) < oo,	(2)
where P (x, g) is defined by | PM (x, g) |2W2 considering x as para-
meters.
Examples. The differential operator P(x, D) = Dsast(x) D*
with real, bounded C00 coefficients ast (x) = at(%) in Q is strongly elliptic
in D (see Chapter VI, 9) it there exists a positive constant 6 such that
In such a case, P\x, D) satisfies the condition (2). Next let P{x, D) be
strongly elliptic in an open bounded domain Q of R’: Then

(4)
is said to be parabolic in the product space Qx{xn; 0 < x„}. It is easy
to see that the operator (4) is of uniform strength in the above product
space.
Theorem (Hormander [5]). Let P(x, D) be of uniform strength in an
open bounded domain Q of Rn. For any point x°, there exists an open
subdomain of Q such that x° E and the equation P (x, D) и = f has,
for every /Е L2^), a distribution solution Т2(£?х) for which, more-
over, Q (D) if f L2 (I?x) for every Q (£>) weaker than P (%, D) for any fixed
x-L\.
Proof. Write P(%°, D) ~ P0(R)- The set of all the differential opera-
tors with constant coefficients weaker than Po (D) is a finite dimensional
linear space. For, the degree of such operators Q (D) cannot exceed that
of Р0(Р). Thus there exist PX(Z)), P2(D), . . ., PN(D) which form a basis
for the differential operators weaker than PQ{D). Hence we can write
P(x,D)—P0(D) + J&^P^D), ^(x°)-0,	(5)
j=i
with uniquely determined bj(x) which are continuous in Q.
12. The Hypoellipticity (Hormander’s Theorem)	189
By the result of the preceding section, there exists a bounded linear
operator T on L2{Q^ into Е2(Рг) such that
P0(D) Tf = f for all f € L2 (^),	(6)
and that the operators Pj(D) T are all bounded as operators on T2^)
into L2(Q). Here!?! is any open subdomain of Q. We have only to take
Tf as the restriction to Q1 of E * /, where Д = / in and = 0 in
Rn -
The equation P(x, D) и = / is equivalent to
P0(^H+	=	(7)
j=i
We shall seek a solution of the form и = Tv. Substituting this in (7), we
obtain, by (6),
n
«+ £bj{z)Pi(D)Tv = f.	(8)
3 = 1
Let the sum of the norms of the bounded linear operators Pj(E>) T on
L2{Qr) into L2(Q-f) be denoted by C. Since bj{x) is continuous and
bj{xQ) — 0, we may choose Qr Э x° so small that
C [ bj (x) | < 1/N whenever x £	(/ = 1, 2, . . ., w).
We may assume that the above inequalities hold whenever x belongs
to the compact closure of £?r Thus the norm of the operator
N
£ bj(x) Pj(D} T is less than 1, and so the equation (8) may be solved
7 = 1
by Neumann's series (Theorem 2 in Chapter II, 1):
»=(/+ lbjPj(D)TV1f = Af,
\	j = i	/
where A is a bouned linear operator on L2 (йг) into L2 (X2J. Hence
и = TAf is the required solution of P(x, D)u = f.
12. The Hypoellipticity (Hormander’s Theorem)
We have defined in Chapter II, 7 the notion of hypoellipticity of
P(P) and proved Hormander’s theorem to the effect that, if P (D) is
hypoelliptic, then there exists, for any large positive constant Clt a
positive constant C2 such that, for all solutions £ = £ -j- w? of the alge-
braic equation P (£) = 0,
H<c2if ЫсСр	(1)
To prove conversely that (1) implies the hypoellipticity of P(Z)),
we prepare the
Lemma (Hormander [1]), (1) implies that
XJP'-’ral2/.X [P<“>(f)|2^0 as fes-, IfHoo. (2)
'Д 1 "	| I Hh * *
190
VI. Fourier Transform and Differential Equations
Proof. We first show that, for any real vector 0 € Rn, we have
Ptf + 0)1 P (£} -> 1 as Rn, |£| -> oo.	(3)
We may assume that the coordinates are so chosen that 0 = (1,0,0,..., 0).
We have, by (1),
P(£ +	0 when |^| < Cj and |£| > C2.
Then the inequality	holds if |£| Q + C2 and P (C) = 0.
For, setting £'==£' + itf, we have either |?/| Cx or else |£'| < C2 so
that |£ — £'| Cr Giving fixed values to, £2, f3, we can write
m
A=1
where (tk, £rt) is a zero of P. Hence we have |tfc — £x | C*i if
l£| > CT + C2. Thus
P(£+6>)_ Л^+.l-^ Л/ 1 \
satisfies
P(|+ ei ~ 11 £ mCr1 (1 + Cf1)—1 if |f|SC, + Cs.
As we may take Ct arbitrarily large by taking C2 sufficiently large, we
have proved (3).
We have, by Taylor’s formula,
p(f + r,i =2’тгр<"’®
tx
and so
A	A
1=1	Л	J^l
where are arbitrary real vectors and arbitrary complex numbers.
k
The coefficients £	Й m> can be given arbitrary values by
<=i
a convenient choice of k, ti and ??(*). If otherwise, there would exist con-
stants Сж, |a| < m, not all equal to zero and such that JC Ca^“ — 0 for
every rj. Thus	“
Pw(g) = ^tiP(^+ j/i)) with real vectors .
1 = 1
Since the principal part on the right must cancel out when |oc| 7^ 0, we
k
must have £ = 0. Hence, by (3), we obtain (2).
Corollary. Suppose that P1(^) and P2(g) satisfy (2). Then P(f) =
Pi(^) • P2(^) satisfies (2). Moreover, if Qj(D) is weaker than Pj(D)
(/ = 1, 2), it follows that Q2 (-D) is weaker than P(P).
12. The Hypoellipticity (Hormander's Theorem)	191
Proof. By applying Leibniz’ formula of differentiation of product of
functions, we see that (£) is a linear combination of products of
derivatives of Pr($) and P2(£) ^he order sum (a). Hence (2) holds
for P(£). The latter part of the Corollary may be proved similarly.
We are now ready to prove
Theorem (Hormander [1]). P(D) is hypoelliptic iff the condition (2)
is satisfied.
Proof. The "only if" part is already proved (Chapter II, 7 and what
was proved above). We shall prove the "if" part.
Let Q be an open subdomain of Rn. A distribution w 6 ®(£?)' is said
to belong to	if, for any y>0 G Cq° (.Qj, the Fourier transform w0
of u0 = <f>Qu satisfies (see Chapter VI, 2)
/(14- |£|2)* |«оЮ I2 that is- if м0 = %«е^2(йл). (5)
By virtue of Sobolev’s lemma in Chapter VI, 7, the "if" part is
proved by the following proposition:
Let P(£) satisfy (2). If a distribution «Е Ф(Р)' satisfies
P (D) и G Hioc (&) with a positive s, then и belongs to (6)
For, if P (£>) и G C°° in Q, then P (D) и G Htoc (12) for every positive s
because of Leibniz’ formula of differentiation.
The following proof of (6) is based upon two Lemmas:
Lemma 1. Let /£ W*>2 (Rn) and y> G C“ (Rn), s |0. Then y/G lVB>2(Z?n).
Lemma 2. Let P(£) satisfy (2). Then there exists a positive constant
fj, such that P(<x) (£)	|/| P (£) |	0 as £ G Rn, |£ | -ч* oc> for every л VO.
The proof of Lemma 1 will be given later, and the proof of Lemma 2
will not be given here (for the latter, we refer the reader to L. Hor-
mander [6] or to A. Friedman [1]).
Next, letA^ and/20 be arbitrary open subdomains of/2 such that their
closures I2i and -OS are compact and 12^ C 120, 12q CI2. By Schwartz’
Theorem in Chapter III, 11, the distribution «t ®(12)' is, when conside-
red as a distribution G ®(120)', a distributional derivative of the form
Dlv of a function. v(x) G L2(12o). Let C^°(12o) be such that <p(x) = 1
in^j. Then « = u0 = D^v as distributions G ©(L^)'. <pv beingG L2(Rn),
we see that there exists a (possibly) negative integer k such that
P(a) (D) w0 =	(£>) Dl(pv G (Rn) for every a.	(7)
Hence, by Lemma*l and the generalized Leibniz formula (see Chapter 1,8)
7-471)	= q\P(D) «о +	Ра<Р!  PH(P) «0,	(8)
192
VI. Fourier Transform and Differential Equations
we see by P (D) w0E Hfoc(£?) that, whenever <рг С C^° (I3X),
P (H) yh wo £ Ж*1’ (/?”) with k-y = min (s, k),	(9)
Thus the Fourier transform of u^x) ~ <рг{х) w0(%) satisfies
/|Р(Я«,(£)|2(1 + |£|2Л<*£<оо	(10)
Rn
and hence, by Lemma 2,
f [Pw(f) wx(£) I2 (1 +	< oo, that is,
*” (11)
P*{D) wx £ И^+/‘>2 (£«) for every a 0.
Let be any open subdomain of such that its closure is compact
and contained in Qx. Then, for any £ C^° (£}%), we prove, by (8) and (11)
as above, that
P(D) <p2ui^ Ж*”2(7?я) with k2 = min(s, kr + ^u) and hence
P^ (D) e Wzftl+z‘,2(i?w) for every a VO.
Repeating the argument a finite number of times, we see that, for any
open subdomain Q' of Q such that its closure is compact and contained
Р(я) (D) <p и 6 Ws’2 (Rn) for all a 7^ 0 whenever q> C Cq° (Q'}.
Thus, P(a) (£) = constant =}= 0 gives <pu G Ws>2 (Rn).
Proof of Lemma 1. The Fourier transform of ipf is
(2л;)_я/2 J V (^) / (ъ " >/') dy (see Theorem 6 in Chapter VI, 3)
and thus we have to show that, for $ g 0,
oo.
яп
Rn
By Schwarz’ inequality, this can be estimated from above by
= f Iy’to) \<fy Г f f (i + If l2)5^to)I  |7(^ —	.
Rn	Я"
We then make use of the inequality
(12)
2V
(13)
which may be proved by
1 _L [ Л8
1 +	v 1 > ‘ v 1	i + Hp •
By (13), the right side of (12) is estimated by / [y'(^) j dr]-times
Rn
1. Dual Operators
193
This integral converges since /£	and (??) E ©(7?”).
We have thus proved our Theorem.
Further Researches
1. A linear partial differential operator P(x, D) with C°°(Q) coeffi-
cients is said to be formally hypoelliptic in L? C Rn if the following two
conditions are satisfied: i) P(x°, D) is hypoelliptic for every fixed x°EP
and ii) P(x°, g) = 0 (?(%', £)) as £ E Rn, |£ | oa for every fixed x° and
x'(-Q. L. Hormander [5] and B. Malgrange [2] have proved that,
for such an operator P(x, D), any distribution solution и E ® (£?)' of the
equation P(x D) и = / is Cx after correction on a set of measure zero
in the open subdomain C Q where / is C°°. The proof above for the con-
stant coefficients case may be modified so as to apply for the formally
hypoelliptic case, see. e.g., J. Peetre [1].
2. It was proved essentially by I. G. Petrowsky [1] that all distri-
bution solutions w t X (Rn) of P (D) и = 0 are analytic functions in Rn
iff the homogeneous part Pm (£) of P (£) of the highest degree m does not
vanish for £ E Rn- If this condition is satisfied then P (D) is said to be
(analytically) elliptic. It is proved that in such a case the degree m is
even and P(D) is hypoelliptic. It is to be noted that the hypoellipticity
of an analytically elliptic operator P (D) can also be proved by Friedrichs’
Theorem in Chapter VI, 9. For, by the non-vanishing of Pw(£), we easily
see, by the Fourier transformation, that P(D) or —P(D) is strongly
elliptic. For the proof of Petrowsky’s Theorem, see, e.g., L. Hormander
[6], F. Treves [1] and С. B. Morrey-L. Nirenberg [1].*
VII. Dual Operators
1. Dual Operators
The notion of the transposed matrix may be extended to the notion of
dual operator through
Theorem 1. Let X, Y be locally convex linear topological spaces and
X', У' their strong dual spaces, respectively. Let T be a Unear operator on
D (T) QX into У. Consider the points {x', y'} of the product space X's x Y'
satisfying the condition
(Tx, у’У = (%, x'~) for all x£D(T).	(1)
Then x' is determined uniquely by y' iff D (T) is dense in X.
Proof. By the linearity of the problem, we have to consider the condi-
tion:
<x, x'y -= 0 for all %E implies x' = 0.
* Sec also Siip|ijriiietil.;ii у Noles, ]>. 4G6.
13 VohUIu, FuiuilJoiml Antd
194
VII. Dual Operators
Thus the "if" part is clear from the continuity of the linear functional
x'. Assume that D (T)a ^4 X. Then there exists, by the Hahn-Banach
theorem, an %q 7^ 0 such that (x,	= 0 for all x E D IT}; consequently,
the "only if" part is proved.
Definition 1. A linear operator T' such that T'y' ~ x’ is defined
through (1) iff = X. T’ is called the dual or conjugate operator of
T; its domain D (T') is the totality of those у’ E V' such that there exists
x’ E X' satisfying (1), and T'y' = x’. Hence T’ is a linear operator defined
on D (T'} С V' into X' such that
(Tx, у'у ~ (x, T'yfor all x£D(T} and all y'£/)(T').	(2)
Theorem 2. If D (T} = X and T is continuous, then T' is a conti-
nuous linear operator defined on У' into X{.
Proof. For any у' E Y's, (Tx, y'y is a continuous linear functional of
% - X and so there exists an x' E X' with T'y' — x', Let She a bounded
set of X. Then, by the continuity of T, the image T  В = {Tx; x E B}
is a bounded set of У. Thus, by the defining relation x, y'y = (%, %'>,
the convergence to 0 of y' in the bounded convergence topology of Y'
(given in Chapter IV, 7) implies the convergence of x' in the bounded
convergence topology of X'. Thus T' is a continuous linear operator on
У' into X'.
J	0
Example 1. Let X = У be и-dimensional euclidean spaces normed by
the (Z2)-norm. For any continuous linear operator T £ L(X, X), set
Tx = ys where x = (x1( x2, . . ., xn} and у = (уъ y2, . . ., ул).
Then y{ — XJ tijXj (i = 1, 2, . . ., n) and so, for z = (zlf z2, . . ., zM),
(Tx, zy =	<y, zy - 2? y-z-	- XJ (XJ tijXA z<	-- J?	(XJ TzA	.
3	i \ J 7	 j	\ i /
л
Thus Tr z = до is given by Wj =	(/ = 1, 2, . .	n). This proves
t=i
that the matrix corresponding to T' is the transposed matrix of the matrix
corresponding to T.
Example 2. Let X = У be the real Hilbert space (Z2), and let
Tn E L (X, X) be defined by
Then from
<Trt(%b x2, . . (zt, z2, ...)> =	4-	+ xn+2z$ + • • •,
we obtain	"T1
К (^i, z2, . . .) = (0, 0, . . ., 0, zlt z2, . . .).
/ 00	\ 1/2
Since jj Tn{xt, x2, . . .) |] = ( XJ x„\	->0asw—>oo and 11 T'n(z11z2,. . .) ||
= || Oh-	we have
2, Adjoint Operators
195
Proposition 1. The mapping T -> T' of L(X, Y) into L(Y'S) X's) is
not, in general, continuous in the simple convergence topology of opera-
tors, that is, lim Tnx = Tx for all x £ X does not necessarily imply
lim T'ny' = T'y' for all y’ £ У', in the strong dual topology of X\.
n~>0Q
Theorem 2'. Let T be a bounded linear operator on a normed linear
space X into a normed linear space Y. Then the dual operator T’ is a
bounded linear operator on У' into X"s such that
(3)
Proof. From the defining relation (Tx, y'> = (x, x'y, we obtain
is T>y’ II — l!%/II = SUP I(x> Х'У
~ sup
<74, y'> | ^ ||y'|
• sup | Tx
IWlsi
and so || T’ |[	|| T ||. The reverse inequality is proved as follows. For
any %0EX there exists an /0£ Y' such that |j/0|| = 1 and /0(Tx0) ~
<74>/o> = II TxQ ||. Thus /о = Т'Д satisfies (x0,	=
|740|| and so
r||.
Theorem 3. i) If T and S are £L(X, У), then (aT q- fiS)' =
(осT' q~ flS'). ii) Let linear operators T, S be such that D(T), D(S),
R (T) and R (S) are all contained in X. If S is 6 L (X, X) and D (T)a = X,
then
(ST)' = T'S'.	(4)
If, moreover, D(TS)a = X, then
(TS)’ 2 ST, i.e., (TS)' is an extension of S'F. (5)
Proof, i) is clear, ii) D(ST) = D(T) is dense in X, and so (ST)'
exists. If у E D ((S T)'), then, for any x £ D (T) — D (S T), (Tx, S' y) —
(ST x, y> = <x, (ST)' y). This shows that S'y£D(T') and T'S'у =
(S T)'y, that is, (ST)' С T'S'. Let,conversely,у (T'S'),i.e.,S'yED (T').
Then, for any x £ D(T) = D(ST), (STx, y) = <74, S'y> = <x, T'S'y).
This shows that у 7) ((ST)') and (STYy^T'S’y, that is, TeS/Q(ST)\
We have thus proved (4).
To prove (5), let yeD(S'T') =D(T'). Then, for any ^D(TS),
(TSx, y) = <Sx, T'y) = (x, S'T’yy. This shows that y£D((TS)')
and (TS)'y = S'T'y, that is, S'Г Q (TS)'.
»	2. Adjoint Operators
The notion of transposed conjugate matrix may be extended to the
notion of adjoint operator through
13*
196
VII. Dual Operators
Definition 1. Let X, Y be Hilbert spaces, and T a linear operator
defined on D(T) С X into У. Let D(T)a = X and let T' be the dual
operator of T. Thus (Tx, y'> = (x, T’y'y for x^D(T), y'^D{T'). If
we denote by Jx the one-to-one norm-preserving conjugate linear corre-
spondence X' 3/ yyC X (defined in Corollary 1 in Chapter III, 6), then
(Tx, y'> = y'(Tx} = [Tx, JYy'), (x, Ту'у = (Гу') (%) = (%, JxT'y').
We have thus
I
{Tx, JYy') = (x, JxT'y'), that is, [Tx, y) = (x, JXT' J^y).
In the special case when У = X, we write
T* = J XT’JX'
and call T* the adjoint operator of T.
Remark. If X is the complex Hilbert space (Z2), we see, as in the Exam-
ple in the preceding section, that the matrix corresponding to T* is the
transposed conjugate matrix of the matrix corresponding to T.
As in the case of dual operators, we can prove
Theorem 1, T* exists iff D (T)a X. It is defined as follows: у f X
is in the domain of D (T*) iff there exists а у* С X such that
(Tx, y) = (x, y*) holds for all x^D(T},	(1)
and we define T* у = у*.
We can rewrite the above theorem in terms of the graph G(X) of the
linear operator A (the graph was introduced in Chapter II, 6):
Theorem 2.	We introduce a continuous linear operator V on XxX
into X X X by
V{x,y] = {~y, x}.	(2)
Then (I7 G (Т)У is the graph of a linear operator iff D(T}a —- X, and, in
fact, we have
G(T*) = (LG(T))±.	(3)
Proof. The condition {—Tx, x} | {y, y*} is equivalent to (Tx, y) =
(x, y*). Thus Theorem 2 is proved by Theorem 1.
Corollary. T* is a closed linear operator, since the orthogonal comple-
ment of a linear subspace is a closed linear subspace.
Theorem 3.	Let T be a linear operator on D (T) S X into X such that
D(Ty = X. Then T admits a closed linear extension iff T** =(7"*)*
exists, i.e., iff D(T*y = X.
Proof. Sufficiency. We have T** 2 T by definition, and T** = (T*)*
is closed by the above Corollary.
Necessity. Let S be a closed extension of T. Then G(S) contains
G (Ту as a closed linear subspace, and so G (T)a is the graph of a linear
operator. But G(T}a = G(T)11 = (G(7')±)± by the continuity of the
3. Symmetric Operators and Self-adjoint Operators	197
scalar product, and, moreover, by FG(7'*) = GfT)1-, we obtain
(l/G(7'*))± — G(7')-L-L. Therefore, (FGfjC*))-1- is the graph of a linear
operator. Thus by Theorem 2 D(T*)a = X and T** exists.
Corollary. Under the condition that D (Т)д = X, T is closed linear iff
у _ y*+
Proof. The sufficiency is clear. Necessity is proved by observing the
formula G(T)a = G(T**) obtained above. For, G(T) = G^T)11 implies
that T = T**.
Theorem 4.	An everywhere defined closed linear operator T is a con-
tinuous linear operator.
Proof. Clear from the closed graph theorem.
Theorem 5.	If T is a bounded linear operator, then T* is also bounded
linear and
|T|| = || T* ||.	(4)
Proof. Similar to the case of dual operators.
3. Symmetric Operators and Self-adjoint Operators
A Hermitian matrix is a matrix which is equal to its transposed
conjugate matrix. It is known that such a matrix can be transformed
into a diagonal matrix by a suitable (complex) rotation of the vector
space on which the matrix operates as a linear operator. The notion of the
Hermitian matrices is extended to the notion of self-adjoint operators
in a Hilbert space.
Definition 1. Let X be a Hilbert space. A linear operator on D (T) Q X
into X is called symmetric if T* 2 T, i.e., if T* is an extension of T. Note
that the condition of the existence of T* implies that D(T')a = X.
Proposition 1. If T is symmetric, then T** is also symmetric.
Proof. Since T is symmetric, we have D (T*) 2 & (?) and D (T)a =X.
Hence Z>(T*)“ = X and so T** = (Г*)* exists. T** is surely an exten-
sion of T and so Z>(T**) 2 D(T). Thus, again by D(T)a = X, we have
£)(y**)“ = X and so T*** = (y**)* exists. We have, from T* 2 T,
T** £ y* an(£ hence T*** 2 T** which proves that T** is symmetric.
Corollary. A symmetric operator T has a closed symmetric extension
y** = (y*)*
Definition 2. A linear operator T is called self-adjoint if T = T*.
Proposition 2. A self-adjoint operator is closed. An everywhere defined
symmetric operator is bounded and self-adjoint.
Proof. Being the adjoint of itself, a self-adjoint operator is closed.
The last assertion is proved by the fact that an everywhere defined closed
operator is bounded (closed graph theorem).
Example 1 (integral operator of the Hilbert-Schmidt type). Let
•oo a < b • ’ ow and consider L2(a, b). Let K(s, t) be a complex-
198
VII. Dual Operators
valued measurable function for a s, t ,b such that
b b
f f |K($, Z)|2dsdZ <00.
a a
For any x(t) € L2(a, b), we define the operator К by
b
(K • x) (s) = f K(sft) x(t) dt.
A
We have, by Schwarz’ inequality and the Fubini-Tonelli theorem ,
b	b b	b
J | (K • x) (s) |2 ds 5^ J J | К (s, t] |2 dt ds f | % (Z) |2 dt.
a	a a	a
Hence К is a bounded linear operator on 7.2 (a, b) into L2 (a, b) such that
/ b b	\ 1/2
||К || I J" J |K(s, Z) |2 ds dt j . It is easy to see that the operator K*
\д 4	/
Ъ ______
is defined by (Z<* y) (Z) = f K(t, s) y(s) ds. Hence К is self-adjoint iff
К (s, t) = К (t, s) for a.e. s, t.
Example 2 (the coordinate operator in quantum mechanics). Let
X = L2(— oo, oo). Let D = {%(/); x[f) and t • x(t) both g L2(— oo, oo)}.
Then the operator 7 defined by Tx(t) - t  %(Z) on D is self-adjoint.
Proof. It is clear that Da = X, since the linear combinations of
defining functions of finite intervals are strongly dense in L2(—oo.oo).
Let у D (T*) and set T*y = y*. Then, for all x £ D ~ DIT),
f tx(t) y(t) dt = J" x(t'} y*{i} dt.
—OO	— oo
If we take for x(Z) the defining function of the interval [a, Zo], we have
^0	___ ^0	____ _____
f t  y(t) dt ~ f y* (Z) dt, and hence, by differentiation, tQ  у (Zo) = y* (Zoj
a	a
for a.e. Zo. Thus у £ D and T*y(Z) = t • y(Z). Conversely, it is clear that
у C D implies that у € D (T'*) and T*y (Z) = t - у (Z).
Example3 (the momentum operator in quantum mechanics). Let
X = Z,2(—oo, oo). Let D be the totality of x(t} C L2(— oo, oo) such that
x (Z) is absolutely continuous on every finite interval with the derivative
x'(t) C L2 (—00,00). Then the operator T defined by Tx(t') = i-1xf(t)
on D is self-adjoint.
Proof. Let a continuous function xn (i) be defined by
xn{t) = 1 for t£ (a, Zo],
xn (Z) = 0 for Z a — n-1 and for Z > Zo + Л"1,
x„(Z) is a linear function on [a — w-1, a] and on [Zo, Zo 4-	.
3. Symmetric Operators and Self-adjoint Operators	199
Then the linear combinations of functions of the form xn {t) with different
values of oc, tQ and n are dense in £2(— oo, oo). Thus D is dense in X.
Let у £ D (T*) and T*y = y*. Then for any x£ D,
f f-1 x' {t'j y(t) dt = J" x (/) у* (/) dt.
—oo	—co
If we take x„ (£) for x{t), we obtain
n J" i-1y(tf)^ —w У i~ly{t}dt~ У xn{t} у* {t) dt,
Л— n-1	—00
_________________________________________________ _______ ___________
and so, by letting n -> oo, we obtain i-1 (y (a) — у (£0)) = f y* (i) dt for
a
a.e. oc and tQ. It is clear, by Schwarz’ inequality, that y* (/) is integrable
over any finite interval. Thus y(Z0) is absolutely continuous in t0 over
any finite interval, and so we have t-1y'(f0) = y*(£0) for a-e- Hence
yQD and T*y{t) = i~1y'(£). Let, conversely, y^D. Then, by partial
integration,
ь	____ _________________ ь ___________________
f i~Yx! (t) у {t) dt = i-1 [x(t) у (£)]£ + У x (t)	(t)) dt.
a	a
By the integrability of x (t) у (t) over (— oo, oo), we see that
lim | [x (t) у (£) ]£ | = 0, and so У i~xx' (tf) у {t) dt — f x (tf) (t-1y' {t)) dt.
oorbjoo	—oq	—00
Thus ye D(T*) and T*y{t) = i~~ly'{t}.
Theorem 1. If a self-adjoint operator T admits the inverse T-1, then
T-1 is also self-adjoint.
Proof. T = T* is equivalent to (VG(T)')1- —G(T). We have also
G(T^) = VG(—T). Hence, by (—T)* = — T* = — T, (FG(— T))3- =
G (— T) and so
(FGfT-1))1 = G{-T)± = (HG(- T))~L± = VG{~T) = С(Г~3),
that is, (GT-1)* = T-1.
We have used, in the above proof, the fact that (FG(—T))a = VG{— T)
in virtue of the closedness of (— T).
Corollary. A symmetric operator T in a Hilbert space X is self-ad-
joint if D{T) = X or if R(T) = X.
Proof. The case D (T) = X was proved already. We shall prove the
case R (T) = X. Tx = 0 implies 0 = {Tx, y) = (%, Ty) for all
D(T), and so, by R(T) = X, we must have x = 0. Therefore the
inverse T-1 exists which is surely symmetric with T. D (T"1) = R (T) = X,
and so the everywhere defined symmetric operator T"1 must be self-ad-
joint. Hence T ~ (T"1)'1 is self-adjoint by Theorem 1.
200
VII. Dual Operators
We can construct self-adjoint operators from a closed linear operator.
More precisely, we have
Theorem 2 (J. von Neumann [5]). For any closed linear operator T
in a Hilbert space X such that D (T)a = X, the operators T* T and 1 T*
are self-adjoint, and (I + T*T) and (7 4- 7'7'*) both admit bounded
linear inverses.
Proof. We know that, in the product space XxX, G(T) and VG(T*)
are closed linear subspaces orthogonal to each other and spanning the
whole product space XxX. Hence, for any h 6 X, we have the uniquely
determined decomposition
{h, 0} = {x, Tx] + {—T*y,y} with x G D (T), у G T) (T*).	(1)
Thus h = x — T*y, 0 = Tx + y. Therefore
x G D (7'* T) and x + T* Tx = h.	(2)
Because of the uniqueness of decomposition (1), x is uniquely deter-
mined by h, and so the everywhere defined inverse (7 + T* 7')-1 exists.
For any h, k G X, let
x = (7 + T*T)~ih, y= (I + T*^-^.
Then x and у G D(T*T) and, by the closedness of T, (7'*)* = T. Hence
(A, (7 T T*T)-4) = ((7 + T*T) x, у) = (x, у) 4 (T*Tx, y)
= (X У) + (Tx, Ту) = (x, y) + (%, T* Ty)
= (x, (I + T*T) y) = ((7 + T'*7')~1A, k),
which proves that the operator (7 + 7'*7')“1 is self-adjoint. As an every-
where defined self-adjoint operator, (I 4- 7'*7')-1 is a bounded operator.
By Theorem 1, its inverse (7 + T*T) and hence T*T are self-adjoint.
Since T is closed, we have (Г*)* — T, and so, by what was proved
above, TT* = (7'*)*7'* is self-adjoint and (7 4- 7'7'*) has a bounded
linear inverse.
We next give an example of a non-self-adjoint, symmetric operator:
Example 4. Let X = L2(0, 1). Let D be the totality of absolutely
continuous functions %(^)G72(0, 1) such that x(0) = a(1) = 0 and
x(l)gL2(0, 1). Then the operator Tx defined by Trx{t) = i~'xx(t) on
D = 7)(ТХ) is symmetric but not self-adjoint.
Proof. We shall prove that T* = T2, where T2 is defined by:
T2x(t) = i~rx' (t) on D[T2) = {x(t) G L2(0, 1); x(t) is
absolutely continuous such that x (t) G 7.2 (0, 1)}.
Since D = D(T^ is dense in L2(0, 1), the operator T* is defined.
Let у G T)(T*) and set T*y = y*. Then, for any x G D = D {T^,
1	____ i ______________
f i~Ax' (t) у (г!) dt = f x(t) y* (t) dt.
о	0
3. Symmetric Operators and Self-adjoint Operators
201
By partial integration, we obtain, remembering я(0) = %(1) — 0,
i _________ 1	______ t
f %ф у*ф dt = — f %'(/) У*ф dt, where Y*(£) = J y*(s) ds.
oo	о
i
Hence, by %(1) ~ f x'(t) dt = 0, we have, for any constant c,
о
i ___________________________
J xf ф (У* (i) — ir1 у (Z) — c) dt — 0 for all x £ D (7\).
о
i
On the other hand, for any яф 6	1), the function Z (/) = f z(t) dt~
о
i
t J гф dt surely belongs to Z)(7\). Hence, taking Z(f) for the above
о
x ф, we obtain
z
z(t) dt}-(Y*(t)
— ъ~г y(t) — c) dt — 0.
i
If we take the constant tin such a way that J (У*ф— i“1y(/) —cjdt — 0,
о
then
i ______ _____________ _
J зф (У* ф — i~Y y{t) — c) dt = 0,
о
and so, by the arbitrariness of z6£2(0? 1), we must have У*ф ~
t
f у*ф dt= г"1 уф + c. Hence у e Т)(Г2) and T2y = y*. This proves
° . *
that T* C T2. It is also clear, by partial integration, that T2 £ T\ and
—	mi
so T2 .
Theorem 3. If Я is a bounded self-adjoint operator, then
sup
(Hx, %) [.
(3)
Proof. Set sup ] (Hx, x) | = y. Then, by | (Hx, x) |	||Я% || ||x||,
ihllsi
у |[H1|. For any real number A, we have
| (H(y ± Az), у ± Az) | = | (Hy, y) ± 22 Re(Hy, z) -h z)
Iy ± 2г112.
Hence
142 Re(Hy\ z)
y(|[y + Az ||2 +
y —2^]|2) = 2y (||y ||2 4- 22 IHI2)-
By taking 2 — |]yj|/||^||> we obtain \Re(Hy, z) | у
by substituting хегв for z, we obtain [ (Hy, z") [ у ||y |[
у |j ||z||. Hence,
z 11, and so
(Hy, Hy) - ||Яу ||2 < у ||y || ||Яу||, i.e., ||Я|| у.
202
VII. Dual Operators
4. Unitary Operators. The Cayley Transform
A symmetric operator is not necessarily a bounded operator. Various
investigations of a symmetric operator H may be made through the
continuous operator (H — iT) (// + z'7) 1 called the Cayley transform
of H. We shall begin with the notion of isometric operators.
Definition 1. A bounded linear operator T on a Hilbert space X into
X is called (bounded) isometric if T leaves the scalar product invariant:
(Tx,Ty) = (x, y) for all x, у £ X.	(1)
If, in particular, R(T) — X, then a (bounded) isometric operator T
is called a unitary operator.
Proposition 1. For a bounded Unear operator T, condition (1) is
equivalent to the condition of the isometry
||T%|| = ||% || for all x 6 X.	(2)
Proof. It is clear that (1) implies (2). We have, by (2),
±Re(Tx, Ту) = ||T(% + y)||2— ||T(%-y)|J2
= 11 % 4- у 112 — [ | x — у 112 = 4 Re (x, y).
By taking iy in place of y, we also obtain Hm(Tx, Ту) — klm(x,y),
and so (2) implies (1).
Proposition 2. A bounded linear operator on a Hilbert space X into X
is unitary iff T* = T"1.
Proof. If T is unitary, then T~r surely exists in virtue of condition (2),
and D (7' ') = R (T) = A. Moreover, by (1), 7'*T — I and so T* = Y1.
Conversely, the condition T* = 7' 1 imphes T*T = I which is the con-
dition of the invariance of the scalar product. Moreover, T* = 7' 1
implies that R(T) = D(T-1) = D(T*) = X and hence we see that T
must be unitary.
Example 1. Let X = Z2(—oo, oo). Then, for any real number a, the
operator T defined by Tx(t) = x(i + a) on L2(—oo,oo) is unitary.
Example 2. The Fourier transform on L2 (A") onto L2 (К”) is unitary,
since it leaves the scalar product (/, g) = f f(x) g(x) dx invariant.
Definition 2. Let X be a Hilbert space. A linear operator T defined
on D(T) Q X into X such that D(T)a = X is called normal if
(3)
Self-adjoint operators and unitary operators are normal.
The Cayley Transform
Theorem 1 (J. von Neumann [1]). Let H be a closed symmetric
operator in a Hilbert space X. Then the continuous (but not necessarily
4. Unitary Operators. The Cayley Transform
203
everywhere defined) inverse (H + H)^1 exists, and the operator
UH = (Н — И) (H + rl)-1 with the domain D (UH) = D((H +	(4)
is closed isometric (||Z7H%|| — ||%||), an<^	exists.
We have, moreover,
H = i (I + ин) (I - UH)-1.	(5)
Thus, in particular, D(H) = R(I — UH) is dense in X.
Definition 3. UH is called the Cayley transform of H.
Proof of Theorem 1. We have, for any x € D (H),
{(H ±	%, (H i %) = (Hx, Hx) ± (Ux, ix) ± (ix,Hx) + (x, x).
The symmetry condition for H implies (Hx, ix) = — i(Hx, x) =
— i(x, Hx) = — (ix, Hx) and so
|| (Hi H) % ||2 = ||H%||2 + Ik IP-	(6)
Hence (H + iI) x = 0 implies x ~ 0 and so the inverse (H 4- ii) 1
exists. Since || (H T H) *||	|k|	, the inverse (H + H)~l is continuous.
By (6), it is clear that ]| UHy || = |y [|, i.e., UH is isometric.
UH is closed. For, let (H + ii) xn = yn~> у and (H — ii) xn~ zn-> z
as n->oo. Then we have, by (6), ||y„ — ym ||2 = || H(xn — xm) ||2 +
||%« — xm ||a, and so (xn — xm) -> 0, H(xn — xm) 0 as n, m -> oo. Since
H is closed, we must have x = s-lim xnC D(H) and s-lim Hxn = Hx.
я—юо	rt—>oo
Thus (H + il) xn-> у = (H + ii) x, (H — ii) xn -> z ~ (H — ii) x and
so UHy = z. This proves that UH is closed.
From у = (H -|- ii) x and UHy = (H — ii) x, we obtain
2^(1 — UH) у = ix and 2-1(T -f- UH) у — Hx. Thus (I— UH) у = 0
implies x = 0 and so (I U^) у = 2Hx — 0 which implies у =
2~г((1 — UH) у + (I -f- UH) y) = 0. Therefore the inverse (I — UM)~l
exists. By the same calculation as above, we obtain
Hx = 2-Ц1 + UH) у == i(I + UH) (I — UH)-i x, that is,
H = i(I + UH)(I-~ UH)-1.
Theorem 2 (J. von Neumann [1]). Let U be a closed isometric opera-
tor such that R (I — U)a = X. Then there exists a uniquely determined
closed symmetric operator H whose Cayley transform is U.
Proof. We first show that the inverse (I — U)-1 exists. Suppose that
(I — U) у - 0. For any z = (I ~ U) w C. R (I — U), we have, by the
isometric property of U, (y, w) = (Uy, Ute>) as in Section 1. Hence
(y, z) = (y, w) — (y^Uw) = (Uy, Uw) — (y, Uw) = (Uy ~y, Uw) — 0.
Hence, by the condition R(I — L7)a = X,y must be = 0. Thus (I— U)~r
exists. Put H i(I I- U) (I - U)-1. Then D(H) D((I- U)^1) =
204
VII. Dual Operators
R (I — U) is dense in X. We first prove that H is symmetric. Let
x, у £ D (H) - R (I — U) and put x = (I — U) и, у = (7 - 17) w. Then
(Gw, Uw) = (w, w) implies that
(Hx, y) = (i(I 4- U) u, {1 — U) w) = i((Uu, w) — («, Gw))
= ((/ - G) w, i (I 4- G) w) = (x, Hy).
The proof of UH = G is obtained as follows. For x = (I — U) u, we have
Hx ~i(I T G) w and so (H 4- il) x = 2 гм, (H — il) x = 2 гUu. Thus
Z)(GH) ‘ {2гм; u E D(G)} ~ D(G), and GH(27w) - 2г • Gw = G(2гм).
Hence UH = U.
To complete the proof of Theorem 2, we show that H is a closed
operator. In fact, H is the operator which maps (I — G) и onto i {I + G) u.
If (I — G) un and i(I 4- G) un both converge as n -> oo, then un and
Gm„ both cohverge as n oo. Hence by the closure property of G, we
must have
wn^w, (I — G) мп-> (Z-G) м,г(/ + G) i(I + G) u.
This proves that H is a closed operator.
For the structure of the adjoint operator of a symmetric operator,
we have
Theorems (J. von Neumann [1]). Let H be a closed symmetric
operator in a Hilbert space X. For the Cayley transform G H =
(H — il) (H 4- il)-1 of H, we set
Xi=D(U„)\ Xh = R(Uh^.	(7)
Then we have
X% = {xf X; H*x = ix}, XH ~ {xfX; H*x = — ix}, (8)
and the element x of D (H*) is uniquely expressed as
x = x0 4- хг 4- x2, where xQ E D(H), xx E Xfi, x2 € Хц so that
H*x ~ Нх^ 4- ixx + (— ix2).	(9)
Proof. гЕШуГ = D((H + г/)"1)± implies (x, (H 4- H) y) = 0
for all у £ G(H). Hence (%, Hy) = — (%, iy) = (ix, y) and so %G D(H*),
H*x = ix. The last condition implies (%, (H 4- H) y) = 0 for all у G D (H),
i.e. x£D(\H 4- г’/)-1)-1- =D(GW)±. This proves the first half of (8);
the latter half may be proved similarly.
Since UH is a closed isometric operator, we see that D(UH) and
R(UH) are closed linear subspaces of X. Hence any element x£X is
uniquely decomposed as the sum of an element of D(UH) and an element
of DIJJh)1. If we apply this orthogonal decomposition to the element
(H* 4- il) x, we obtain
(H* 4- H) x = (H 4- г/) x0 4- where x0 E D (H), x' 6 D (GH)J .
4. Unitary Operators. The Cayley Transform	205
Hut we have (H + il) %0 = (H* 4- il) x0 by x0£ D(H) and H £ H*.
We have alsoH*x' = ix' by x' £	and (8). Thus
x' = (Я* + il) x„ x, = (2»)-1 x'£ DIUh)1- ,
and so
(H* + il) x = (H* 4- il) xG + (H* + il) хг where
х^ЩН), ^(UJ1
1'herefore (x — x0 — х2) 6 R (Uh)1 by H* (x — x0~ xj = — i(x — z0 — xj
and (8). This proves (9). The uniqueness of the representation (9) is
proved as follows. Let 0 = xQ 4- хг 4- x2 with x0 € ЩН), xl Q DtUn)1-,
t2C RIJJx)1-. Then, by H*xQ = Нхй, H*xr = ix±, H*x2 = — ix2,
0 = (H* + il) 0 = (H* + il) (x0 +	+ x2) ~ (H 4- il) x0 + 2ixY.
But by the uniqueness of the orthogonal decomposition of X as the sum
of D(UH) and DlU^)1, we obtain (Zf + H) xQ = 0, 2ixt = 0. Since the
inverse (H + Н)~г exists, we must have x0 = 0 and so x2 — 0 — x0 —
%1 = o — 0 — 0 = 0.
Corollary. A closed symmetric operator H in a Hilbert space X is self-
adjoint iff its Cayley transform UH is unitary.
Proof. The condition D (H) = D (Я*) is equivalent to the condition
Z) (Uh)1 — R (Uh)1 = {0}. The last condition in turn is equivalent to the
condition that UH is unitary, i.e. the condition that UH maps X onto
X one-one and isometrically.
5. The Closed Range Theorem
The closed range theorem of S. Banach [1] reads as follows.
Theorem. Let X and У be В-spaces, and T a closed linear operator
defined in X into У such that D(T)a — X. Then the following proposi-
tions are all equivalent;
R(T) is closed in У,	(1)
R (Tr) is closed in X!,	(2)
B(T) = N(T')X ={yC У;<у,у*> = 0 for ail	(3)
R(T‘) = W(T)± = {x*E X'; <%,%*> = 0 for all xOVT)}. (4)
Proof. The proof of this theorem requires five steps.
The first step. The proof of the equivalence (1)	(2) is reduced to
the equivalence (1) ** (2) for the special case when T is a continuous
linear operator such that D(T) = X.
The graph G = G (T) of T is a closed linear subspace of X X У, and so
G is a В-space by the norm 11 {x, y} 11 = 11 x 11 4- 11 у 11 of X X У. Consider a
continuous linear operator S on G into У:
S {x, Tx} = Tx.
206
VII. Dual Operators
Then the dual operator S' of S is a continuous linear operator on Y' into
G', and we have
<{*, тx}, S'y*> = <S{x, Tx}, _v*> = <7X, y*>
= <{x, Tx}, {0, ?•}>,	D(T), y* e Y-.
Thus the functional S' • y* — {0, у*} G (XX У)' = X'xY' vanishes at
every point of G. But, the condition <{x, Tx}, {л*, y*}> = 0, x G D(T),
is equivalent to the condition <A,	Tx, y*>, xf D(T), that is,
to the condition — T'y* = x*. Hence
s' • У* = {0, У*} + tf} = {-T'y*, y* + y*}, y* G Y'.
By the arbitrariness of у*, we see that R (S') = R(—T') X Y' — R(T') X Y'.
Therefore R (S') is closed in X' X Y' iff R (T') is closed in X', and, since
R(S) =R(T), R(S) is closed in Y iff R(T) is closed in Y. Hence we
have only to prove the equivalence (1)	(2) in the special case of a
bounded linear operator S, instead of the original T.
The second step. Let X and У be В-spaces, and T a bounded linear
operator on X into У. Then (1) -> (2).
We consider T as a bounded linear operator Тг on X into the B-space
Ух = R(T)a = J?(T). We have to prove that (2) is true. T^y*, у* € У{,
is defined by
<Г1А >*> = <Tx. y*> = <x, Г1У*>, X eD(T,) = D(T) = x.
By the Hahn-Banach theorem, the functional y* can be extended to a
у* G У' in such a way that <Tx, y*> = <Tx, y*>, x G D (T) = X. Hence
T[y* = T'y* and so R(T^) = R(T'). Thus it suffices to assume that
R(T) = Y. Then, by the open mapping theorem in Chapter II, 5, there
exists a c > 0 such that for each у G У, there exists an x G X with
Tx — y, ||x|| c |[у 11. Thus, for each y* in D (T'), we have
I <У> У*> | = | <Tx, y*> | = | <%, T'y*>
< 11 x II • T' y* | < c | у I  11T' y* |.
Hence	-ill / । _ I / i и x ।
|y*|| = sup |<y, y*> | S с ЦТ'у* ||
and so (T')-1 exists and is continuous. Moreover, (T')-1 is a closed linear
operator as the inverse of a bounded linear operator. Hence we see that
the domain D {(T')~r) = R (Tr) must be closed in X’.
The third step. Let X and У be B~spaces, and T a bounded linear
operator on X into У. Then (2)	(1).
As in the second step, we consider T as a bounded linear operator
on X into Ух = В(Т)Я. Then Т'г has the inverse, since T[y* = 0
implies
<T,X, yb = <Tx, = <%, т;У1*> = 0, x& D(T,} =D^= X,
5. The Closed Range Theorem.
207
and so, since R (Tj) = 2? (Г) is dense in У* = R (T)a, y* must be 0. There-
fore, the condition that R (T') = R (T'j) (proved above) is closed, implies
that T[ is a continuous linear operator on the В-space (R(T)ay = Y2
onto the В-space R (7^) in a one-one way. Hence, by the open mapping
theorem, (7^)-1 is continuous.
We then prove that R{T) is closed. To this purpose, it suffices to
derive a contradiction from the condition
there exists a positive constant e such that the image
{T±x; ||% || < e} is not dense in all the spheres ||у ||
(n = 1, 2, . . .) of Yr = R(T)a = R{Ttf.
For, if otherwise, the proof of the open mapping theorem shows that
R(Tf) = R(T) = Yv Thus we assume that there exists a sequence
with
s-lim yn 0, y„e ||%
n—ЮО
(n = l,2, ...).
Since {Т-^х; ||%
c}a is a closed convex, balanced set of the В-space YT
there exists, by Mazur’s theorem in Chapter IV, 6, a continuous linear
functional fn on the В-space Ух such that
/nW > sup |/„(7» | (« = 1, 2, . . .).
1Idi £e
Hence 11 T\Jn 11 < e-1 | [ fn 11 11 yn ] |, and so, by s-lim yn = 0, T\ does not have
>00
a continuous inverse. This is a contradiction, and so R (T) must be closed.
The fourth step. We prove (1) -> (3). First, it is clear, from
<Tx, y*> = (x, T'y*), x(D (T), y*((D (Г),
that R (Г) g N(T')-"-. We show that (1) implies 2V(T')1- C R(T). Assume
that there exists ay0€ W{Г')1 with y0€ В (T). Then, by the Hahn-Banach
theorem, there exists a y* £ Y' such that <у0, у*) =£ 0 and (Tx, y*> = 0
for all %E В(Г). The latter condition implies (x, T'y*) = 0, x£ D(T),
and hence T'y* = 0, i.e., y0 E Л7(В',)±. This is a contradiction and so we
must have 2У(Г)Х CB(B).
The implication (3)	(1) is clear, since NIT')1 is closed by virtue
of the continuity in у of (y, y*).
The fifth step. We prove (2) -> (4). The inclusion R(T") С Л^В)1 is
clear as in the case of (3). We show that (2) implies that N (В)1 C B(B').
To this purpose, let x* E TV (B)1, and define, for у = Tx, the functional
/г(у) of у through Д (y) = (x, x*). It is a one-valued function of y, since
Tx = T x' implies (% — %') E-V (B) and so, by %* E TV (B)1-, <(%—%'),%*> = 0.
Thus /г(у) is a linear functional of y. (2) implies (1), and so, by the open
mapping theorem applied to the operator S in the first step, we may
choose the solution % of the equation у — ?’% in such a way that s-lim
208
VII. Dual Operators
у = 0 implies s-lim x = 0. Hence Д (у) = <%, %*> is a continuous linear
functional on Ух = R(T). Let У' be an extension of Then
f(Tx) = fi(Tx} = <*> %*>-
This proves that T'f = x*. Hence N(T")^ Q R(T').
That (4) implies (2) is clear, since (x, x*> is a continuous linear func-
tional of x.
Corollary 1. Let X and У be В-spaces, and T a closed linear operator
on D (T) С X into У such that D (T)a = X. Then
R (У) = У iff Tr has a continuous inverse,	(5)
R{T') = X' iff T has a continuous inverse.	(6)
Proof. Suppose that R(T) — У. Then, from (Tx, y*>	=	<x,	T'y*>,
x£ D(T)	and T'y* = 0, we obtain y* = 0," that is, T' must	have	the
inverse (T')~L Since, by R(T) = У and (2), R(T') is closed, the closed
graph theorem implies that (T')-1 is continuous. Next let T‘ admit a
continuous inverse. Then 2V(T') ={0} and also (2) holds since T' is
closed. Thus, by (3), R(T) = У.
Suppose that R[T')=Xr. Then, from (Tx, y*> = <x, Tfy*yt
y* £ D(Tf] and Tx = 0, we obtain x = 0, i.e., T must have the inverse
T~l. Since, by R(T‘) = X' and (1), R(T) is closed, the closed graph
theorem implies that T-1 must be continuous. Next let T admit a con-
tinuous inverse. Then N(T) = {0} and also (1) holds since T is closed.
Thus, by (4), R(T‘) = X1.
Corollary 2. Let X be a Hilbert space with a scalar product (m, v), and
T a closed linear operator with dense domain D (7j £ X and range
R(T) C X. Suppose that there exists a positive constant c such that
Re ITu, w) c |[w |j2 for all u£D{T).
Then R(T*) = X.
Proof. By Schwarz’ inequality, we have
(?)
| T и 11  11 и 11 Re (Tu, и) > с 1| и | j2 for all и £ D (T).
Hence || Tu || c | u\(, u^_ D (T), and so T admits a continuous inverse.
Thus, by the preceding Corollary, R (T') = X. HenceR (Г*) = R(T') = X.
Remark. A linear operator T on D(T) С X into X is called accretive
(the terminology is due to K. Friedrichs and T. Kato) if
Re(Tu, w) 0 for all w £ T) (T).	(8)
T is called dissipative (the terminology is due to R. S. Phillips) if — T
is accretive.
References for Chapter VII
For a general account concerning Hilbert spaces, see M. H. Stone
[1], N. I. Achieser-I. M. Glasman [1] and N. Dunford-J. Schwartz
[5]. The closed range theorem is proved essentially in S. Banach [1].
1. The Resolvent and Spectrum
209
VIII. Resolvent and Spectrum
Let T be a linear operator whose domain D (T) and range R (T) both
lie in the same complex linear topological space X. We consider the linear
operator

where Л is a complex number and I the identity operator. The distribution
of the values of Я for which Tx has an inverse and the properties of the
inverse when it exists, are called the spectral theory for the operator T.
We shall thus discuss the general theory of the inverse of
1. The Resolvent and Spectrum
Definition. If Яо is such that the range R{Tx) is dense in X and
has a continuous inverse (Яо/ —T)-1, we say that AG is in the resolvent
set q(T) of T, and we denote this inverse (Яо/ — 7')-1 by А(Яд; T) and
call it the resolvent (at Яд) of T. All complex numbers A not in q (T) form
a set a(T) called the spectrum of T. The spectrum o(T) is decomposed
into disjoint sets Pa(7'), CO(T) and Ra(T) with the following properties:
Pe(7') is the totality of complex numbers A for which 7\ does not have
an inverse ;Ра(Т) is called the point spectrum of T.
Cg(T) is the totality of complex numbers A for which T? has a discon-
tinuous inverse with domain dense in A ; Cg(T) is called the con-
tinuous spectrum of T.
Rg(T) is the totality of complex numbers A for which Тл has an inverse
whose domain is not dense in X; Rg(T) is called the residual
spectrum of T.
From these definitions and the linearity of T we have the
Proposition. A necessary and sufficient condition for Яо 6 Pg (T)
is that the equation Tx = Aox has a solution x 0. In this case Яд is
called an eigenvalue of T, and % the corresponding eigenvector. The null
space N [Ao I — T) oil\ is called the eigenspace of T corresponding to the
eigenvalue Яд of T. It consists of the vector 0 and the totality of eigen-
vectors corresponding to Яд. The dimension of the eigenspace correspond-
ing to Яд is called the multiplicity of the eigenvalue Яд.
Theorem. Let X be a complex В -space, and T a closed linear operator
with its domain D (T) and range R (T) both in X. Then, for any Яо € q (T),
the resolvent (Яо1 — T)-1 is an everywhere defined continuous linear
operator,
Proof. Since Яд is in the resolvent set q (Г), R (A0I ~T) =D (JAqI — T)^1)
is dense in X in such a way that there exists a positive constant c for
which
l(V
7) x 11 > c 11 % 11 whenever x{D (Г).
14 Yoalda, b'ltHnlJotin! Armlyuln
210
VIII, Resolvent and Spectrum
We have to show that — T) = X, But, if s-lim ~T) xn = у
rt—>00
exists, then, by the above inequality, s-lim xK = x exists, and so, by the
n—ЮО
closure property of T, we must have (2qZ — T) x = y. Hence, by the
assumption that R(^I — T)a = X, we must have 7?(2oZ — T) = X.
Example 1.	If the space X is of finite dimension, then any bounded
linear operator T is represented by a matrix (^ ). It is known that the
eigenvalues of T are obtained as the roots of the algebraic equation,
the so-called secular or characteristic equation of the matrix (^):
det (Л<5^ — Zo) = 0,	(1)
where det(X) denotes the determinant of the matrix A.
Example 2.	Let X = L2(—00,00) and let T be defined by
7". x(t) — tx(t),
that is, D(T) = {%(/); x(t) and tx(t) £ L2(—00,00)} and Tx(t) — tx(t)
for x(t) £ D (T). Then every real number 2q is in Ca(T).
Proof. The condition (^Z — T) x ~ 0 implies — t) x(t) = 0 a.e.,
and so x(t) =0 a.e. Thus (^Z — T)"1 exists. The domain D((2gZ — T)-1)
comprises those у (/) 6 Z2 (— 00,00) which vanish identically in the neigh-
bourhood of t — the neighbourhood may vary with y(rf). Hence
D((A^I — Z)-1) is dense in L2(— 00,00). It is easy to see that the operator
{XqI — T)~l is not bounded on the totality of such y(t)’s.
Example 3.	Let X be the Hilbert space (Za), and let TQ be defined by
^2’ • • ) = (0» fl» $2>   ) 
Then 0 is in the residual spectrum of T, since 7?(7()) is not dense in X.
Example 4.	Let H be a self-adjoint operator in a Hilbert space X.
Then the resolvent set q(H) of H comprises all the complex numbers 2
with Im (2) 7^ 0, and the resolvent 7? (2; H) is a bounded linear operator
with the estimate
||7?(2; H) |1 l/|Im (2) |.	(2)
Moreover,
Im((2Z — H) x, %) = Im(2) ||%||2, x^D(H).	(3)
Proof. If % £ D(H), then (Hx, x) is real since (Hx, x) = (x, Hx) =
(Hx, x). Therefore we have (3), and so, by Schwarz' inequality,
II № — H) % || - ||x|| Sr |((2Z — H) x, x)\ S: |Im(2) |  ||x||2	(4)
which implies that
\\(AI — H) x|| I Im (2) I • 1|% 1|, x^D(H).	(0)
Hence the inverse (2Z — TZ)-1 exists if Im(2) =£ 0. Moreover, the range
7?(2Z— H) is dense in X if Im (2) 7^ 0. If otherwise, there would exist a
у 7^ 0 orthogonal to 7?(2Z — H), i.e., ((2Z — H) x, y) — 0 for all x£ D(H)
and so (%, (2Z — Z7) y) — 0 for all x £ D (H). Since the domain D (H) of a
2. The Resolvent Equation and Spectral Radius	211
self-adjoint operator H is dense in X, we must have (Л/ — H] у — 0,
that is, Hy ~Ay, contrary to the reality of the value (Hy, y).
Therefore, by the above Theorem, we see that, for any complex
number Л with Im(A) =£ 0, the resolvent 2?(Л; H) is a bounded linear
operator with the estimate (2).
2.	The Resolvent Equation and Spectral Radius
Theorem 1.	Let T be a closed linear operator with domain and range
both in a complex В-space X. Then the resolvent set q(T) is an open
set of the complex plane. In each component (the maximal connected
sets) of q(T), R(A; T) is a holomorphic function of A.
Proof. By the Theorem of the preceding section, R(A; T) for A C q (T)
is an everywhere defined continuous operator. Let A$ € Q (T) and consider
(1)
The series is convergent in the operator norm whenever |A0 — Я| •
] 7? (^; T) || < 1, and within this circle of the complex plane, the series
defines a holomorphic function of A. Multiplication by (Al — T) =
(A — A^ I + (AqI — T) on the left or right gives I so that the series
S(A) actually represents the resolvent R(A; T). Thus we have proved
that a circular neighbourhood of belongs to q(T) and R(A;T) is
holomorphic in this neighbourhood.
Theorem 2.	If A and /г both belong to q(T), and if В (A; T) and
R (/a; T) are everywhere defined continuous operators, then the resolvent
equation holds:
R(A-T)-R(^T} = (^A)R(A-,T) R(fa-T).	(2)
Proof. We have
R(A;T)=R(A;T) (fil-T) R(ja; T)
= R(A; T){(fi~A)I + (Al — T)} R(/a', T)
= fc-A) R(A-, T) R(fi; T) + R(fi; T).
Theorem 3.	If T is a bounded Enear operator on a complex В-space X
into X, then the following limit exists:
Hm Ill'll1'” = r„(T).	(3)
It is called the spectral radius of T, and we have
rAT) =S||T||.	(4)
If |Л| > rCT(7X then the resolvent 7?(A; T) exists and is given by the
series
• R(A;T) - ^A-n,T~l	(5)
which converges in (he norm of operators.
11*
212
VIII, Resolvent and Spectrum
Proof. Set r = inf |j Tn We have to show that lim
«^1	ft-*co
For any e > 0, choose m such that ||Tm|)1/w r + g. For arbitrary n,
write n = pm + q where 0 q (m — 1). Then, by j| A В11 S 11A || •
||B [|, we obtain
Since	1 and	0 as n —> oo, we must have lim J| Tn ||1/”
П—ИЗО
r + £. Since g was arbitrary, we have proved lim j | T** ] |1/w r.
Since ||T”||	|[T]|W, we have hm ||Гп||1/я^ ||T||. The series (5)
is convergent in the norm of operators when |Л| > го(Т). For, if |A|
ra(T) 4- e with e > 0, then, by (3), (рГ”Т”|| 5g (ra(T) 4- g)~n •
(Л>(Г) + 2~1e)w for large n. Multiplication by (Л/ — T) on the left or
right of this series gives I so that the series actually represents the resol-
vent R (Л; T).
Corollary. The resolvent set q (T) is not empty when 7" is a bounded
linear operator.
Theorem 4.	For a bounded linear operator T £ L(X, X), we have
= sup |A |.
лесг(Г)
(6)
Proof. By Theorem 3, we know that ra(T) Sr sup [Л |. Hence we
fam
have only to show that ra(T) 4= sup |Я
Aia(T)
By Theorem 1, T) is holomorphic in Л when [Я| > sup |A|.
AW)
Thus it admits a uniquely determined Laurent expansion in positive and
non-positive powers of A convergent in the operator norm for |Я| >
sup By Theorem 3, this Laurent series must coincide with
Л€а(Г)
JU'*?’’1"1. Hence lim	= 0 if |Я|> sup |2|, and so, for
any e>0, we must have S p + sup
proves that	'	л€<г(Т)
ra m — lim IT™ I 1/n < sup
A€tr(T)
Corollary. The series 2rnTn~1 diverges if 111 < ra(T).
Proof. Let r be the smallest number 0 such that the series
Л||” for large n. This
14
oo
converges in the operator norm for ЯI > r. The existence
«=1
of such an r is proved as for ordinary power series in Z”1. Then, for
я*+со
we must haye lim I IT** l1/M < r. This proves that ra[T) r.
ft—H5O
Л€сг(Т)
3. The Mean Ergodic Theorem
213
3.	The Mean Ergodic Theorem
For a particular class of continuous linear operators, the mean ergodic
theorem gives a method for obtaining the eigenspace corresponding to the
eigenvalue 1. In this section, we shall state and prove the mean ergodic
theorem from the view point of the spectral theory, as was formulated
previously by the present author. The historical sketch of the ergodic
theory in connection with statistical mechanics will be given in
Chapter XIII.
Theorem 1.	Let X be a locally convex linear topological space, and T
a continuous linear operator on X into X. We assume that
the family of operators {7'n; n = 1, 2, . . .} is equi-
continuous in the sense that, for any continuous semi-
norm q on X, there exists a continuous semi-norm q' on
X such that sup qlT^x) q' (x) for all x £ X.	(1)
Then the closure R(I — T)a of the range R(I — T) satisfies
R (I — ТУ =
Я
 x e X; lim Tnx = 0, Tn = n 1 X
«—►OQ	W=1
and so, in particular,
R(I~ T)aC\N(I — T) ={0}.
(2)
(3)
Proof. We have —T)=n —T”+1). Hence, by (1),
w 6 R {I —T) implies that lim Tnw = 0. Next let z E R (7 — T)a. Then,
n—>oo
for any continuous semi-norm q' on X and 0, there exists &	(I—T)
such that q' (z — w) < e. Thus, by (1), we have q(Tn (z — ж))
;г-1 X q (Tm {z — w\)	q' (z — w) <Z £. Hence q (Tn z) < q (Tn w) +
m = l
q(Tn(z— w)) q(Tnw) + e, and so lim l'nz = 0. This proves that
R (I — T)a Q к 6 X; lim Tnx = Ol.
rt—>co	J
Let, conversely, lim Tnx = 0. Then, for any continuous semi-norm
л—>oo
q on X and s > 0, there exists an n such that q (x — (x — T„xy =
q(Tnx) < £. But, by
x~Tnx = n~1 X (I — Tm)x
1H = 1
= n-1 x (I-T) (I + T + T2 + •  . + Г*-1) x,
m=l
(% — Tnx) E R (I — *). Hence x must belong to R (I — T)a.
Theorem 2	(the mean ergodic theorem). Let condition (1) be
satisfied. Let, for a given x С- X, there exist a subsequence {n'} of {w}
214
VIII. Resolvent and Spectrum
such that
weak-lim Tn-x = x0 exists.
(4)
Then Txq = %0 and lim Tnx ~ x0.
Proof. We have TTn — Tn = n~x (T”+1 — T), and so, by (1),
lim [TTnx - Tnx) = 0. Thus, for any / £ X', lim <TTn,x, /> =
Л—ЮО	ft—HX
lim (Tn, x, T'fy exists and = lim <Z'„, x, /> = <x0,/>. Therefore
ft^-ХЭО	П—>00
<x0, h ~ (Txo> f> and so, by the arbitrariness of /6 X’, we must have
T%0 - x0.
We have thus T™x = T™x0 T 7™ (x — xQ) = x0T™ (x — x0) and
so Tnx = x0 + Tn [x — x0). But, (x — z0) — weak-lim (x — TnrX) and,
Я--ЮО
as proved above, (x — Tn>x) R(J -7"). Therefore, by Theorem 11 in
Chapter V, 1, (%—x0)£R(7— T)a. Thus, by Theorem 1, lim Tn{x —x0) = 0
and so we have proved that lim Tnx = x0.
Corollary. Let condition (1) be satisfied, and X be locally sequen-
tially weakly compact. Then, for any x E X, lim Tnx — x0 exists, and
n—изо
the operator 70 defined by Tox = xQ is a continuous linear operator such
that
* __ ^2 __
0 — J 0 —
TT0 — TQT,
(o)
7?(T0)=W-T),	(6)
^To) =/?(/-zy = To).	(7)
Moreover, we have the direct sum decomposition
X — R(I -ту ® N(I — T),	(8)
i.e., any x £ X is represented uniquely as the sum of an element £R{1 — T)a
and an element € TV (Z — T).
Proof. The linearity of To is clear. The continuity of TQ is proved by
the equi-continuity of {Tn} implied by (1). Next, since Tx0 = x0, we have
TT0 = TQ and so TnTQ = TQf TnT0 = To which implies that Т%~Т0.
On the other hand, Tn—TnT п~л {T—Z^+1) an(j (i) imply that
TQ = TQT. The equality (6) is proved as follows. Let Tx = x, then
T^x = x, Tnx = x and so Tox = x, that is, x £ R(T0). Let conversely,
xE A(T0). Then, by Tfi = To> we have TQx = x and so, by TT0 = To,
Tx = TTQx — Tox = x. Therefore, the eigenspace of T corresponding
to the eigenvalue 1 of T is precisely the range R (Z'o). Hence (6) is proved.
Moreover, we have, by Theorem 1, W(To) = R (Z — T)a. But, by Tq = Тй,
we have R(I — To) C N(T0), and if xE^T(Z0'), then x = x — TQxf
R(7- To). Thus N (To) = R (7 - To).
4. Ergodic Theorems of the Hille Type Concerning Pseudo-resolvents 215
Therefore, by I = (I — To) + To and (6) and (7), we obtain (8).
Remark. The eigenspace Д'"{Al — T) of T belonging to the eigenvalue
A with A
{TfA)w x.
n
= 1 may be obtained as R (T(A)), where T(A) x = lim и-1 £
n—+CQ	in = 1
The Mean Ergodic Theorem of J. von Neumann. Let (5, S3, m) be a
measure space, and P an equi-measure transformation of S, that is, P
is a one-one mapping of S onto S such that P  В € 53 iff В is 6 S3 and
m{P  B) = m{B). Consider the linear operator T on L2(S, 53, m) onto
itself defined by
{Tx) (s) = x (Ps), x^L2 (5, 53, m).	(9)
By the equi-measurable property of P, we easily see that the operator T
is unitary and so the equi-continuity condition (1) is surely satisfied by
;| Tn j| = 1 {n = 1, 2, . . .). Therefore, by the sequential weak compactness
of the Hilbert space L2{S, 53, m), we obtain the mean ergodic theorem of
J. von Neumann:
n
Bor any x С- J2 {S, 53, m), s-lim и-1	T™ x f= x0 C L2 {S, 53, m)
tt—>oo
exists and T xQ = %0.
(10)
Remark. Theorem 1 and Theorem 2 are adapted from K. Yosida [3].
Cf. also S. Kakutani [1] and F. Riesz [4]. Neumann’s mean ergodic
theorem was published in J. von Neumann [3].
4. Ergodic Theorems of the Hille Type Concerning Pseudo-resolvents
The notion of resolvent is generalized to that of pseudo-resolvent by
E. Hille. We can prove ergodic theorems for pseudo-resolvents by a
similar idea to that used in the proof of the mean ergodic theorems in the
preceding section. See K. Yosida [4]. Cf. T. Kato [1]. These ergodic
theorems may be considered as extensions of the abelian ergodic theo-
rems of E. Hille given in E. Hille-R. S. Phillips [1], p. 502.
We shall begin with the definition of the pseudo-resolvent.
Definition. Let X be a locally convex complex linear topological
space, and L {X, X) the algebra of all continuous linear operators defined
on X into X. A pseudo-resolvent /л is a function defined on a subset
/) {]) of the complex Z-plane with values in L {X, X) such that
Л “ Д = (p — Л) (the resolvent equation).	(1)
Proposition. AH J,, AfD(J), have a common null space, which we
denote by N{J), and a common range which we denote by R{J)- Simi-
larly, all (/ 'Af^),A( /)(/), have a common null space, which we denote
216	VIII. Resolvent and Spectrum
by N(I — /), and a common range which we denote by R{I — J). More-
over, we have the commutativity:
W))-	(2)
Proof. By interchanging л and fi in (1), we obtain
k = — /*) Л/л = — G« ~ Л) 7Дд,
and hence (2) is true. The first part of the Proposition is clear from (1)
and (2). The second part is also clear from
V - АЛ) = (I -- (Л -д) Л) (I-fiJJ	(1')
which is a variant of (1).
Theorem 1. A pseudo-resolvent /д is a resolvent of a linear operator A
iff N (/) = {0}; and then 7? (/) coincides with the domain D (A) of A.
Proof. The “only if” part is clear. Suppose N (J) = {0}. Then, for
any /. C D (J), the inverse exists. We have
и - j,1 =	- j? (>.. fie d(j)).	(3)
For, by (1) and (2),
ЛЛ(Л/ - Ji1 -fil + Л1) = (A -//) JJ, - JJMi1 - J?)
= (A-rtZJ,,-(/,,-n) = o.
We thus put
A = (Л/ - K1).	(4)
Then J, = (II - .4)-1 for Лей (/).
Lemma 1. We assume that there exists a sequence */.„} of numbers
6 D (J) such that
lim Лп = 0 and the family of operators {Л„7лп} is equi-continuous. (5)
п—ЮС
Then we have
R(I-J)’=(xeX-, lim Л,7Лях = 01,	(6)
\	ft—ЮО	J
and hence
»(Z-J)n«(Z-J)- = {0}.	(7)
Proof. We have, by (1),
АЛ(А-дЛ) = (i-fiifi-^UJi-W-iir'fiJ,..	(8)
Hence, by (5), the condition x C- R (I - и J=. R{I — J) implies that
lim Л/Дп% - 0. Let у 6 R (I — /)*• Then, for any continuous semi-norm
ft—*-oo
q on X and e > 0, there exists an x € 7? (Z — /) such that q (y— %) < a.
By (5), we have, for any continuous semi-norm q' on X,
/(Л7лп(у — *)) q{y — *) (n = 1,2,...)
with a suitable continuous semi-norm q on X. Therefore, by А„7лпУ =
Л7лЛ + 47л„(У — x), we must have lim Л7л„У = 0.
ft—КЮ
1. Ergodic Theorems of the Hille Type Concerning Pseudo-resolvents 217
Let, conversely, lim kjx x = 0. Then, for any continuous semi-
norm q on X and e > 0, there exists а Лл such that q(x — {x — kjxnx)) <
Hence x must belong to 7? (I — kJka = R (I — J)a-
Lemma 1'. We assume that there exists a sequence {/.„} of numbers
C D (J) such that
lim |Z„ | = oo and the family of operators {Я„7лп} is equi-continuous. (5')
л-*со
Then we have
; lim
W)^ Wr = {0}.	(7')
R(J)a = tx£X
and hence
= <6')
Proof. We have, by (1),
А Л7я = т Я Л Л —1V» +	•
Hence, by (5'), the condition x E 2? (JJ) = R (J) implies that
lim kJxnx = x. Let у E R{J)a- Then, for any continuous semi-norm q
>co
on X and s > 0, there exists an x£ R(J} such that q(y— x) < e. By
(S'), we have, for any continuous semi-norm q' on X,
q QnLJy-xy^qly — x} (« = 1,2, ...)
with a suitable continuous semi-norm q on X. Therefore, by (5') and
kJ^V ~y — (kJxnx - x) + (x - у) + А„7Ля(у - %),
we must have lim kJ^У = У-
Я—>00
Let, conversely, lim kJ лnx — x- Then, for any continuous semi-
ft—H5Q
norm q on X and e > 0, there exists a к such that q(x — kJ^x) < e.
Hence x must belong to R (Jk* = R (7)“-
Theorem 2. Let (5) be satisfied. Let, for a given х& X, there exist a
subsequence {«'} of {«} such that
weak-limA„' Jin,x ~ xk exists.	(9)
Then xh = lim kj*nx and xh E X (I — 7), xp = (x — xA) € 7? (Z — 7)“.
я—>oo
Proof. Setting /z = к hr (V) and letting n’ oo, we see, by (5),
that (I — 77л) x = (Z — Л7л) (x ~ хл)> that is, (Z — Л7л) хь = 0. Hence
xk£N (I — J) and so
kJxnx ' xh d~ kJxn{X xh)"	(Ю)
Therefore we have only to prove that lim kJxn(x — xh) = 0, or, by
Lemma 1, (x — xk)£R(I—J)a. But (x — kJinx) 7?(Z — J), and so,
by Theorem 11 in Chapter V, 1, we must have (% ~ xh) £ R (I — J)a.
Corollary 1. Let*(5) be satisfied, and let X be locally sequentially
weakly compact. Then
X = W(Z-7) ®R{I-jy.	(11)
2 18	V111. Resolvent and Spectrum
Proof. For any x e’ X, let xh = lim Л x and xp = (% — xA) be the
n-->OQ	F
components of x in N (I J) and R(I— /)a, respectively.
Theorems'. Let (5') be satisfied. Let, for a given x£ X, there exist
a subsequence {и'} of {и} such that
weak-lim Aw- J\n,x = xh> exists.	(9')
n—ХЭО
Then xh> = lim x and xK E R (J)a, xp. = (x — xk,} £NIJ).
n—хзо
Proof. Setting /r = in (8) and letting n' oo, we see, by (5'), that
2— xv) = 0» that is, (x — xhi) C N(J). Hence
— ^nJxnXh’-	(Ю )
Therefore we have only to prove that lim xh- = xh>, or, by
л—хэо n
Lemma 1', xv £ R(J)a. But J}^,xv 6 R(J), and so, by Theorem 11 in
Chapter V, 1, we must have xh- £ R(J)a-
Corollary 1'. Let (5') be satisfied, and let X be locally sequentially
weakly compact. Then
X = N(J) ® R(J)a.	(11')
Proof. For any x£ X, let xh- = lim 2.nJ^x and xP> = (% — xh>) be
the components of X in R(J)a and N(J), respectively.
Remark. As a Corollary we obtain: In a reflexive 5-space X, a pseudo-
resolvent Jx satisfying (5') is the resolvent of a closed linear operator A
with dense domain iff R (/)“ = X. This result is due to T. Kato, loc.
cit. The proof is easy, since, by Eberlein’s theorem, a i?-space X is
locally sequentially weakly compact iff X is reflexive.
5. The Mean Value of an Almost Periodic Function
As an application of the mean ergodic theorem we shall give an
existence proof of the mean value of an almost periodic function.
Definition 1. A set G of elements g, h, ... is called a group if in G a
product (in general non-commutative) gh of any pair (g, Л) of elements
£ G is defined satisfying the following conditions:
gh^G,	(1)
(gA)A = g(^&) (the associativity),	(2)
there exists a unique element e in G such that eg =
ge = g for all gfc G; e is called the identity element
of the group G,	(3)
for every element g£ G, there exists a uniquely deter-
mined element in G, which is denoted by g'1, such
that gg^1 = g-Lg = e; the element g-1 is called the
inverse element of g.	(4)
5. The Mean Value of an Almost Periodic Function
219
* learly, g is the inverse of g-1 so that (g^1)-1 = g. A group G is said to be
/ ommutative if gh = hg for all g, h £ G.
Example. The totality of complex matrices of order n with determi-
nants equal to unity is a group with respect to the matrix multiplication;
11 is called the complex unimodular group of order n. ‘The identity of this
croup is the identity matrix and the inverse element of the matrix a is
I lie inverse matrix ar1. The real unimodular group is defined analogously,
these groups are non-commutative when n 2.
Definition 2 (J. von Neumann [4]). Given an abstract group G. A
complex-valued function /(g) defined on G is called almost periodic on G
if the following condition is satisfied:
the set of functions {gs(f, h); s£ G}, where /s(g, h) =
f(gsh), defined on the direct product GxG'S totally
bounded with respect to the topology of m orm con-
vergence on G x G.	(5)
Example. Let G be the set A'1 of all real numbers in which the group
multiplication is defined as the addition of real numbers; this group R1
is called the additive group of real numbers. The function /(g) = e1™,
where a is a real number and i = [/— 1, is almost periodic on 7?1. This
we easily see from the addition theorem f(gsh) = el'xs elxk and the
fact that {?a/; t G E1} is totally bounded as a set of complex numbers
of absolute value 1.
Proposition 1. Suppose /(g) is an almost periodic function on G. If
we define, following A. Weil,
dis(s, u) = sup |/(gsA)— f(guh} |,	(6)
then
dis (s, w) = dis(as&, aub).	(7)
Proof. Clear from the definition of the group.
Corollary 1. The set E of all elements s which satisfies dis(s, e) = 0
constitutes an invariant subgroup in G, that is, we have
if ev e2G E, then G E and aejOT1 G E for every a G G. (8)
Proof. Let dis(ep e) = 0, dis(e2, e) = 0. Then, by (7) and the triangle
inequality, we obtain
dis(exe2, e) disfe^, e1e) + dis(tqe, e") = 0 -j- 0 = 0.
Similarly we have disfa^a-1, e) = disfa^a-1, й^д-1) = 0 from
dis (e-p e) = 0.
Corollary 2. If we write s = и (mod E) when sir1 G E, then s~ и
(mod E) is equivalent to dis(s, w) = 0.
Proof. Clear from dis (хм l, c) - - dis(s, eu) = dis(s, m).
220
VIII. Resolvent and Spectrum
Corollary 3. The concept s == w (mod E) has all the general properties
of equivalence, namely
s = s (mod E),	(9)
s = и (mod E), then и = s (mod E),	(10)
if sx ~ - s2 (mod E) and s2 ~ s3 (mod E), then .q = s3
(mod E).	(11)
Proof. Clear from Corollary 2 and the triangle inequality for the
dis(s, «).
Hence, as in the case of the factor space in a linear space, we can
define the factor group or residue class group G/E as follows: we shall
denote the set of all elements C G equivalent (mod E) to a fixed element
xf G by $xt the residue class (mod E) containing x\ then the set of all
residue classes £x constitutes a group G/E by the notion of the product
(12)
To justify this definition (12) of the product, we have to show that
if = x2 (mod E), уг = y2 (mod E), then x1y1 = x2y2 (mod E). (13)
This is clear, since we have by (7) and Corollary 2,
dis^yp x3y2) disfsqyp x2yr) + dis(x2yp л2у2)
= dis (Xp x2) + dis (yp y2) = 0 + 0 = 0.
Since the function f(x) takes the same constant value on the residue
class we may consider /(x) as a function E (£J defined on the residue
class group G/E.
Corollary 4. The residue class group is a metric space by the distance
dis (£,, £y) = dis (x, y).	(14)
Proof, x ? Xj (mod E) and у = уг (mod E) imply
dis (x, y) dis (x, Xj) + dis (xv yx) + dis (yp y) = 0 + dis (Xp yx) + 0
and dis(#p yj < dis (ж, у) to the effect that dis(x, y) = dis(xp yx). Thus
(14) defines a distance in G/E.
Corollary 5. The group G/E is a topological group with respect to the
distance dis(^, £y), that is, the operation of multiplication is con-
tinuous as a mapping from the product space (G/E) X (G/E) onto G/Er
and the operation is continuous as a mapping from GfE onto G/E.
Proof. We have, by (7),
dis(s«, s'и') < dis(s«, s'u) -j- dis (s’ u, s'u') — dis(s, s') + dis(w, w')
and
dis(.$‘ l, и~г) = dis(ss~1w, su^u) == dis (и, $) = dis(s, u).
We have thus proved the following
5. The Mean Value of an Almost Periodic Function
221
Theorem 1 (A. Weil). The topological group G/E, metrized by (14),
is totally bounded, and the function /(x) gives rise to a function F($x)
(= /(%)) which is uniformly continuous on this group G/E.
Proof. The uniform continuity of the function F is clear from
l-F&r) ~ F(^) | = !/(*)— /(y) | dis(x, y) = dis (^, fy).
The almost periodicity of the function /(x) implies, by (7) and (14), that
the metric space G/E is totally bounded.
By the above theorem, the theory of almost periodic functions is
reduced to the study of a uniformly continuous function / (g), defined on a
totally bounded topological group G, metrized by a metric dis (gT, gz)
satisfying condition (7). By virtue of this fact, we shall give a proof for
the existence of the mean value of an almost periodic function.
Since G is totally bounded, there exists, for any s > 0, a finite
system of points glf g2, . . ., gn such that min dis (g, gj e for any gC G.
Hence, collecting these finite systems of points corresponding to e =
1, 2-1, 3-1, . . ., respectively, we see that there exists a countable
system {gj} of points 6 G such that {gj} is dense in G. We take a sequence
OO
of positive numbers ocj such that = 1- Let C (G) be the set of all
uniformly continuous complex-valued functions h (g) defined on G. C (G)
is a В-space by the operation of the function sum and the norm |[A|| =
sup \h(g) [. We define an operator T defined on C (G) into C (G) by
OO
(Т-Л) (g) = .^«^(gjg).	(15)
3 = 1
By the uniform continuity of A(g) on G, there exists, for any e > 0, a
d > 0 such that dis(g, g')	(5 implies |A(g) — h(g') |	£. Thus, by (7),
|^(gyg) — h(gjg') [	£ (/ = 1, 2, . . .) whenever dis(g, g') d. Hence it is
OO
easy to see, by tx, > 0 and = 1, that T is a bounded linear opera-
tor on C (G) into C(G). By the same reasoning we see that the set of
functions hn (g) defined by
hn(g) = и-1 (TmK) (g), which is of the form
m=l
OO	oo
Zpjhigjg) with &>0,^ft=l,
is equi-bounded and equi-continuous with respect to n. Hence, by the
Ascoli-Arzela theorem, the sequence {A„ (g)} contains a subsequence which
is uniformly convergent on G.
Therefore, by thejnean ergodic theorem, there exists an h* (g) £ C (G)
such that
lim sup |A„(g) — &*(g) | = 0 and Th* = h*.	(17)
222
VIII. Resolvent and Spectrum
Proposition 2. A*(g) is identically equal to a constant.
Proof. We may assume, without losing the generality, that A(g) and
A* (g) are real-valued. Suppose that there exist a point g0 6 G and a posi-
tive constant 5 such that
A* (g0) = ft ~~ % where Д = sup A* (g).
By the continuity of A*(g), there exists a positive number e such that
dis(g', g") S. e implies [ A* (g') — A* (g") | d; in particular, we have
A* (g")	/? — <5 whenever dis (g0, g") e. Since the sequence {gj is dense
in G, there exists, for any e > 0, an index n such that, for any g€ G, we
have min dis(g, g?) <= £ Hence, by (7), we have, for any g6 G,
min dis(g0,gyg) e.
Let the minimum be attained at j = f0. Then
OO
A* (g) = (TA*) (g) = S «,A* (g,g) g	tfi—&) + (1 -«,.) ? = p» < Д,
contrary to the assumption that g was an arbitrary point of G. Therefore
A*(g) is identically equal to a constant.
Definition 3. We shall call the constant value A* (g) the left mean
value of A(g) and denote its value by M*(A(g)):
4(*te)) = bm 2 P"”*) to-	<18)
n—КЮ	W = 1
Theorem 2 (J. von Neumann). We have
Mj(«*to) = «M'(A(g)).	(i)
M‘t 01 to + Л2 to) = 0i (g)) + m‘s (g)), (ii)
M‘(l) = l,	(in)
if h{g) 0 on G, then Af|(A(g)) > 0;
if, moreover, A(g) ф 0, then Ag(A(g)) > 0,	(iv)
|M'(A(g))| g M'(|A(g)|),	(V)
= M'(A(gj),	(vi)
M‘s 0 (g «)) = M‘r. 0 to).	(™)
M'(A(ag))=M'(A(g)).	(vii'J
^(*te'1» = Af'.(*to)-	<™i)
Proof. By definition (18), it is clear that (i), (ii), (iii), the first part of
(iv), (v) and (vi) are true. The truth of (vii) is proved by Proposition 2.
(vii') is proved as follows.
5. The Mean Value of an Almost Periodic Function
223
Starting with the linear operator Tr defined by
И
{T'h} (g) = ^^h(gg}),
we can also define a right mean Mg{h(g)') which, as a functional of h[g),
satisfies (i), (ii), (iii), the first part of (iv), (v), (vi) and (vii'). We thus
have to prove that the left mean M!g(h (g)) coincides with the right mean
Mg(A(g)). By its definition of the left mean, there exist, for any £ > 0,
a sequence of elements {kj} £ G and a sequence of positive numbers (P
OO
with = 1 such that
OO	j
sup	<ge.	(19)
g	i
Similarly, there exist a sequence of elements {s;} £ G and a sequence of
OO
positive numbers with 2? y, = 1 such that
sup 2? yj h (g sy) — Mrg (h (g))
g
(20)

We have, from (19) and (vii)
sup 12? y^h(fygSi) — Mlg(h(g))| ^ £,
g ;	I
and, similarly from (20) and (vii'),
sup	— M^(A(g))

g
Hence we must have Mg(A(g)) = Mrg(h(g)').
We next remark that a linear functional Mg (h (g)) defined on C (G) is
uniquely determined by the properties (i), (ii), (iii), the first part of (iv),
(v), (vi) and (vii) (or (vii') as well). In fact, we have, by (20),
OO
Mg(h(g)) ~	£7ih(sst) Mrs(h(?)) + e for real-valued h(g).
1 “1
Hence, for real-valued h(g), Mg(h(gy) must coincide with the right mean
Mrg (h (g)) and hence with the left mean Mlg (h (g)) as well. Therefore we
see that we must have Mg = M'g = Mlg. Being equal to the right mean,
Mg must satisfy (vii'). Moreover, since M\(h{g~1}') satisfies, as a linear
functional, (i), (ii), (iii), the first part of (iv), (v), (vi) and (vii'), we must
have M'= M'(Afe)) = M‘s (Л(й).
Finally we shall prove the last part of (iv). Suppose A(g0) > 0. For
any e > 0, there exists, by the total boundedness of G, a finite system
of elements slf s2>  • sn such that
min sup, | h (gSi) —h (gs)
£
224
VIII. Resolvent and Spectrum
for all s E G. This we see by the uniform continuity of h (g) and the fact
that dis(gsi( gs) — disfo, s). Hence, for e = A(g0)/2, we obtain, for any
s € G, a suffix i, 1 У i < n, such that
Thus, by the non-negativity of the function h{g}, we obtain
Sh(goSi*1(go)/2> 0 for all G.
Therefore, by taking the right mean of both sides, we have
(n	\
XAfeoVM = n- Afe0)/2 > 0.
1 = 1	/
Remark. The happy idea of introducing the distance (1) is shown in
A. Weil [1]. The application of the mean ergodic theorem to the exi-
stence proof of the mean value is due to the present author. See also
W. Maak [1].
6. The Resolvent of a Dual Operator
Lemma 1. Let X and У be complex В-spaces. Let T be a linear
operator with D (T)a — X and 7? (T) С У. Then (T')^1 exists iff R (Т)л ~ Y.
Proof. If T'y* = 0, then
<%, T'y*> == <Tx, y*> = 0 for all x£D(T),
and hence y* (R (T)a) = 0. Thus R (T)a = У implies y* = 0 and so T'
has an inverse. On the other hand, if y06 R(T)a, then, the Hahn-Banach
theorem asserts that there exists a continuous linear functional у* C Y'
such that y* (y0) = 1 and y*(R(T)“) = 0. Hence (Tx, y*> = 0 for all
x£ D(T), and so y* 6 D(T') and T'y* — 0, whereas y* (y0)	0, i.e.,
у* 0. Therefore, the condition R{T)a У implies that T' cannot
have an inverse.
Theorem 1 (R. S. Phillips [2]). Let T be a linear operator with an
inverse and such that D (T)a = X and R (T)a = У, where X and У are
В-spaces. Then
(7^ = 0.	(1)
Т~г is bounded on У iff T is closed and {T'y1 is bounded on X'.
Proof. ( L ]y exists because D (7м) = R (T) is dense in У. (T')-1
exists by Lemma 1. We have to show the equality (1). If у C R(T) and
y* 6 D (T'), then
<y, y*> = (TT^y, y*y = (T~xy, T'y*).
Hence R(T') g DftT-1)') and (TM)' (T'y*) = y* for all y* 6 В(T'). Thus
(T’1)' is an extension of (T')M. Next, if xf D(T)r then
(x, x*) = (T~XT x, x*) = (Tx, (T-1)' x*) for all x* € D{{T^').
7. Dunford’s Integral
225
Hence	CD(T') and Г (Г-1)' x* = x* for all x* 6 B((7'~1)').
Thus (T-1)' is a contraction of (L')-1. Therefore we have proved (1).
If, in addition, 7' 1 is bounded on Y, then (T-1)' is also bounded.
Conversely, if (B')-1 is bounded on X', then, for all x£R(T) and
%* £ X', we have, by (1),
<T-4, %*> j = | <%, (T-1)' x*> | = | <x, (Г)-1 x*>
S 11(^41  Ill’ll  INI-
Since Г 1 is closed and R (T)a = Y, 7’-1 must be bounded.
Lemma 2. Let T be a linear operator with D [TV = X and R (T) C Y,
where X and Y are В-spaces. If R (T') is weakly* dense in X'. then T has
an inverse.
Proof. Suppose that there exists an x0 -/ 0 such that Tx0 = 0. Then
<%0, B'y*> = <7%, y*> = 0 for all y*£D(T').
This shows that the weak* closure of R (Г') is a proper linear subspace
of X', contrary to the hypothesis.
Theorem 2 (R. S. Phillips [2]). Let X be a complex В-space, and T a
closed linear operator with D (T)a = X and R (Г) £ X. Then
q(T)=q(T') and R(A; T)f = R(A; Г) for XEq(T). (2)
Proof. If AC q(T)> then, by Theorem 1, At. q(T') and R(A; T)' =
R(A; T'). On the other hand, if A C then Lemma 2 shows that
(AZ—T) has an inverse (AZ — T')-1 which is closed with (AZ—7').
Lemma 1 then shows that Z)((AZ — 7’)-1) = R(XI — T) is strongly dense
in У. Hence, by Theorem 1, AC Q(T)-
7. Dunford’s Integral
Let X be a complex В-space and T a bounded linear operator £ L {X, X).
We shall define a function / (T) of T by Cauchy’s type integral
/(T) = (2^1)1 //(Л) Я(Л; T) M.
c
To this purpose, we denote by F (T) the family of all complex-valued
functions f (A) which are holomorphic in some neighbourhood of the spec-
trum or(T) of T; the neighbourhood need not be connected, and can
depend on the function / (A). Let /С B(B), and let an open setU 2
of the complex plane be contained in the domain of holomorphy of /,
and suppose further that the boundary QU of U consists of a finite num-
ber of rectifiable Jordan curves, oriented in positive sense. Then the
bounded linear operator f(T) will be defined by
=	ffWR(X,T)M,	(1)
du
15 Yosldft, Funotlouid Апп1ун1н
226
VIII. Resolvent and Spectrum
and the integral on the right may be called a Dunford’s integral. By
Cauchy’s integral theorem, the value f('T") depends only on the function f
and the operator Tf but not on the choice of the domain U.
The following operational calculus holds;
Theorem (N. Dunford). If / and g are in F(T), and a and p> are
complex numbers, then
«/Zpgis inF(T) anda/(T) +	= (<xf + @g)(T),
/ • g is m F(T) and /(T) • g(T) = (/ • g) (T),
cc
if f has the Taylor expansion /(Я) = £ ocnAn con-
№0
vergent in a neighbourhood U of ct(T), then
OO
f(T) = £ (in the operator norm topology),
let fnG F (T) (w = 1, 2, , . be holomorphic in a fixed
neighbourhood U of ст (T). If /rt(Z) converges to /(Л)
uniformly on U, then fn(T) converges to f{T) in the
operator norm topology,
if /€ F(T), then fe F(T') and /(Г) = /(?')'.
(2)
(3)
(4)
(5)
(6)
Proof. (2) is clear. Proof of (3). Let Ux and be open neighbourhoods
of ct(T) whose boundaries диг and dU2 consist of a finite number of
rectifiable Jordan curves, and assume that Ux dUt Q U2 and that
U% T 8U2 is contained in the domain of holomorphy of f and g. Then,
by virtue of the resolvent equation and Cauchy's integral theorem, we
obtain
= (2ni)~1 f f^)R(A;TyU27i^ J (ft - Z)'1 g (jz) dfA dk
dU,	(	BU,	J
dUt	[	J '
= (2«0-* f fW gW T) db = (h g) (T).
3Ul
Proof of (4). By hypothesis, U must contain a circle {Я; |Z p (T) + e}
e > 0, in its interior, where ro(T) is the spectral radius of T (Theorem 4
00
in Chapter VIII, 2). Hence the power series /(Z) = £ converges
n—0
uniformly on the circle C = {Z; |Z | r0 (T) -be}, for some c > 0. Hence,
7. Dunford’s Integral
227
by Cauchy’s integral theorem and Laurent’s expansion 7?(Я; Г) =
j? Я"”^”1 of Я(Я; T) (Chapter VIII, 2, (о)),
n=l
_ / oo	\	oo
/(T) = (2л;0"1 f £	}	(Ziii}-1 У ak J 7') dl
ac \fe=o	/	ft=o	5C
(5) is proved by (1), and (6) is also proved by (1) and formula (2) of
the preceding section.
Corollary 1 (Spectral Mapping Theorem). If / is in 7?(Z), then
/(a(T)) = a(/(T)).
Proof. Let Я 4 oJT), and define the function g by g(u) = (/(A) — /(«))/
(Л -jU). By the Theorem, /(Я) I — /(7J = (Я/— 7J g(7J. Hence, if
(/(Я) I — /(7J) had a bounded inverse B, then g(T) В would be the boun-
ded inverse of (Я/ — T). Thus Я€сг(Т) implies that /(Я) £ cr(/(7')). Let,
conversely, Я 4 tr(f(T}), and assume that ЯЁ f(o(T}). Then the function
g(/z) = (/(fi)—Я)^1 must belong to F(7'), and so, by the preceding
Theorem, g(T) (/(7J — Я7) = I which contradicts the assumption
Яёо(/(Т)).
Corollary 2. If f£F(T), g^F(f(T}) and h (Я) = g(/W), then h is
in F(T} and h(T} = g(/(T)).
Proof. That h C F(T} follows from Corollary 1. Let 1)г be an open
neighbourhood of	whose boundary (^consists of a finite number
of rectifiable Jordan curves such that U± + 8Уг is contained in the
domain of holomorphy of g. Let U2 be a neighbourhood of a (T} whose
boundary &U2 consists of a finite number of rectifiable Jordan curves
such that U2 4- 8U2 is contained in the domain of holomorphy of / and
/(I72 T oU2} C Uv Then we have, for Я 4 dUlt
К(Л;/(T)) = J
SUt
since the right hand operator S satisfies, by (3), the equation
(Я7 — f(T}) S = 5(Я/ — fi'F}) = -7. Hence, by Cauchy's integral theo-
rem,
s(/(O) = (a^)-1 f s(W; /(?)) <a
= (-4^)-i f f	/(?))-> Rf^T) dp <U
8U2
Remark. The introduction of the operational calculus based on a
formula like (1) goes back to the investigations by H. Poincare on
15*
228
VIII. Resolvent and Spectrum.
continuous groups (1899). The exposition of the operational calculus in
this section is adapted from N. Dunford-J. Schwartz [1]. In the next
chapter on semi-groups, we shall frequently make use of Dunford’s inte-
gral for a closed unbounded operator T.
8. The Isolated Singularities of a Resolvent
Let Z{) be an isolated singular point of the resolvent R (A; T) of a closed
linear operator T on a complex 5-space X into X. Then R (Л; T) can be
expanded into Laurent series
Я(Л;Т)= J (Л-ад-л., x. = (2^r*/(Л—Л>)—
"- “ c (1)
where C is a circumference of sufficiently small radius: | Я - Aq | = e
such that the circle |Л — AjiSe does not contain other singularities
than A = Aq, and the integration is performed counter-clockwise. By
virtue of the resolvent equation we obtain
Theorem 1. H’s are mutually commutative bounded linear operators
and
TAk x = AkTx for x^ D(7’) (k — 0, ± 1, i 2, . . .),
AkAm - 0 for k 0, ms - I,
л, = (~1)”лг+1 (»s i).	(2)
- A_pA_q = 1) 
Proof. The boundedness and the mutual commutativity of H's
and the commutativity of H's with T are clear from the integral
representation of H's.
We substitute the expansion of 7?(Л; T) in the resolvent equation
R(A; T) — R(p;T) = (p — A) R(A; T) R(p; T), and obtain
J? a	& A>)*   у A A (A A \k (u
The coefficient of An on the left is
(Л - ад»-’ + (Л - - ад-2 (u - ад + • •  + (д - ад""1. « a 1.
-  {(я - адя и*- ад-1 + (я—ад”+1 & - ад-2 +  • 
+ (л-ад-10*-ад’}> «<о.
Непсе the terms containing (Л — А^ (/г — Д,)ж with k 0 and w <^ — 1
are missing so that we must have the orthogonality HAHW = 0 (k 0,
m S — 1). Hence
R+(A;T) =	and R~(A;T) = J? Нп(Л-Л0Г
n^—oo
8. The Isolated Singularities of a Resolvent
229
must both satisfy the resolvent equation. Substituting the expansion
of R+ (Я; T) in the resolvent equation
Я+(Л; T) - Я+(Л; I) = —Я) Л+(Я; T) £+(//; T),
we obtain, setting (Л — Л,,) = h, (/z — Яд) — k,
J A (V - A*) = (k - А) (Д At v) (Д A, A’).
Hence, dividing both sides by [k — h), we obtain
- J + hp~z k + - - • +	= J hp kq ApA,
so that we have — Л^+?+1 = ApAq (ft, q Sr 0). Thus, in particular,
A = -< 4 = -л,а1 = (-1)!4..............A, =- (-1)“ Л5+1
Similarly, from the resolvent equation for R~iZ; T), we obtain,
setting (Я — 4)“x = h> ~	= k>
j? AAhp-y + hp~2 k +  • - + &*-1) = J h^lkq-1A_pA_,l,
j*=i p	*
so that we have	= A_pA_g (p, q> 1). In particular, we have
А_г Allt A_2 = A_t А-t, . , ., A_„ = А_г А_„ (и 1).
Theorem 2.	We have
ля.-=(т-адл+1
(T - Ло/) A^n = Л_(п+1) = (T - A^ (n 1),	(3)
Proof. By the integral representation of /ft) we see that the range
R (Ak) is in the domain of T, so that we can multiply Ak by T on the left.
Thus our Theorem is proved by the identity
I = (H-T) J Д»(А-ад*
A==—00
= {(;-ад1 + (4/-т)} J л*(Л-А>)‘-
Л=—oo
Theorem 3.	If is a pole of T) of order m, then is an eigen-
value of T. We have
= W((V— T)H) and R^I-A-J	T)n') for n^m, (4)
so that, in particular,
X N{^I-T}n) ф R((^I-T)n) for	(5)
230
VIII. Resolvent and Spectrum
Proof. Since A_± is a bounded linear operator satisfying A%t = A_lf
it is easy to see that
N(A^)^R{I-~A^}.	(6)
We put A^ =	= R (I — A^), and put also
X2 = R(A_1),	and Rn = Rn(^I-Tr). (7)
Let x G N„ where n 1. Then we see, by (T — Я0/)м An_t = (T — Z0T) Ao
= Л_!-1, that 0 =	-- (Т-Ло/рА^х =
(T — Лц1) Аох -- А_хх - х so that x = t X2. Thus Nn with n У 1
belongs to X2. Let, conversely, x-.'- X2. Then we have x = А_гу and so
x=-.4,.1/L.j}1 - .-Li.rby/Lj = Aix; consequently, we have (T -Л{}Г)п
x = A_{n+1]xby (T — ЛоZ)wЛ_г = Л_(п+1). Since Л_(л+1) = 0 for n m
by hypothesis, it follows that X2 g NH for n 2g m and so
7Vn = X2 if w ;> w.	(8)
Because (T — Ло1) A_m = A_(w+1) = 0 and A_m у 0, the number Ло
is an eigenvalue of T.
We see that Xt = X(A_j) = R(I -g Rn by (T- Vf An^ =
Л_х -1. If n > m, then x£ Rn Л Nn implies x = 0. For, if x = (Л§1 -7T у
and (Лц1 — T)nx — 0, then, by (8), у £ TV2w = Nn and therefore x = 0.
Next suppose x£ Rn with n > m, and write x — xx + x2 where xx =
(/ —Л_х) x€ Xx, xz = A_tx^ X2. Then, since Xj g Rn, x2 = x — хг 6 Rn.
But x2 £ X2 = N„ by (8), and so x2 £ R,t C\ Nn, x2 = 0. This proves that
x = xz £ Хг Therefore we have proved that Rn = Xj if n 2g m.
Theorem 4.	If, in particular, T is a bounded linear operator such that
X2 = R (Л-j) is a finite dimensional linear subspace of X, then Ло is a pole
of T?(A; T).
Proof. Let Xj, %2, . . ., xk be a base of the linear space X2. Since
Xj, Tхг, T^xlt . . TkXj are linearly dependent vectors of X2, there
exists a non-zero polynomial Рг(Л) such that Px(T}Xi = 0. Similarly
there exist non-zero polynomials P2 (Л), . . ., Pk (Л) such that Pj (T) x^ = 0
й
(/ = 2, 3, . . ., k). Then for the polynomial Р(Л) = П Р^(Л), we must
have P(T)%y = 0 (/ = 1, . . ., k) and hence P(T)x = 0 for every x 6 X2.
Let
Р{Л)=х П	0)
be the factorization of Р(Л). Then we can prove that [T — Л^ТУ’х ~ 0
for every x£X2. Assume the contrary, and let x0C X2 be such that
(T — ЛоГ^х^^ 0- Then, by P(T)%0= 0, we see that there exist at
least one Л] (/ У 0) and a polynomial Q (Л) such that
(T’-V)e<7')(r-A1/C^=o
8. The Isolated Singularities of a Resolvent	231
and у = Q(T) (T—0. Thus у C X2 is an eigenvector of T
corresponding to the eigenvalue Hence (2.1— T)y = (Л — Л^у and
so, multiplying both sides by T), we obtain у = (Л — Лу) R(2.; T)y
which implies
У = А_1У = (ЯлрГ1 f R(2; T)yd2= / (Л-Л^у d2 = 0,
c	c
by taking the circumference C with Ло as centre sufficiently small. This
is a contradiction, and so there must exist a positive integer m such
that (T-V)”^=0. Thus, by	and (T-Vf^-i =
Л_[я+1), we see that Л_(и_н1) = 0 for n m.
Comments and References
Section 6 is adapted from R. S. Phillips [2]. Section 8 is adapted
from M, Nagumo [1] and A. Taylor [1], Parts of these sections can
easily be extended to the case of a locally convex linear topological space.
See, e.g,, section 13 of the following chapter.
IX. Analytical Theory of Semi-groups
The analytical theory of semi-groups of bounded linear operators in a
ЛЗ-space deals with the exponential functions in infinite dimensional
function spaces. It is concerned with the problem of determining the
most general bounded linear operator valued function T(t), t S: 0, which
satisfies the equations
T(t + s) = T(t)  T(s), T(0)=I.
The problem was investigated by E. Hille [2] and K, Yosida [5] inde-
pendently of each other around 1948. They introduced the notion of the
infinitesimal generator A of T (/) defined by
A = s-limt-\T(t) -I),
i 0
and discussed the generation of T (t) in terms of A and obtained a cha-
racterization of the infinitesimal generator A in terms of the spectral
property of A.
The basic result of the semi-group theory may be considered as a
natural generalization of the theorem of M. H. Stone [2] on one-para-
meter group of unitary operators in a Hilbert space, which will be ex-
plained in a later section. Applications of the theory to stochastic processes
and to the integration of the equations of evolution, which include diffusion
equations, wave equations and Schrodinger equations, will be discussed
in Chapter XIV.*
In this chapter, we shall develop the theory of semi-groups of con-
tinuous linear operators in locally convex linear topological spaces rather
than in Banach spaces.
* S(*c also Siip|ilriiiriihiry* Niilrs, p. 468. >
232
IX* Analytical Theory of Semi-groups
1. The Semi-group of Class (Co)
Proposition (E. Hille). Let X be a В-space, and Tt, t 4 0, a one-
parameter family of bounded linear operators £ L (X, X) satisfying the
semi-group property
= for t, s > 0.	(1)
If p{t) = log |[TJ| is bounded from above on the interval (0, a) for each
positive a, then
limг! 1 log |[T(|| = inft ЧойЦТ^
Moo	/>0
(2)
Proof. We have p{t 4“ s) p(t) 4- p{s) from	= || TtTs || 4
(|Tf[| • [|TS||. Let p = in4-1 • p(p). p is either finite or -oo. Suppose
that p is finite. We choose, for any e > 0, a number a > 0 in such a way
that р{а} 4 (p 4- e) a. Let t > a and n be an integer such that na
t < (w 4~ 1) « Then
P W P (na> P(l — na)	nap (a)	p{t — na)
— “F —	+ t = T ' Г
- wa . .	.
— ~ (P 4- s) 4-
P(i — na)
By hypothesis, p(t— na) is bounded from above as Zoo. Thus,
letting t -> oo in the above inequality, we obtain lim p (t) = p. The
/->oo
case p = — oo may be treated similarly.
Definition 1. If {Tt; t 4 0} g L(X, X) satisfy the conditions
TtTs^Tt+s (for i,s^0),	(Г)
T0 = I,	(3)
s-lim Ttx = Tltx for each 4) ' - 0 and each x Q X,	(4)
then {TJ is called a semi-group of class (Co).
In virtue of the Proposition, we see that a semi-group {Tt} of class
(Co) satisfies the condition
11 Tt | [ <:_ M (for 0 < t <Z oo),	(5)
with constants M > 0 and p <_ oo.
The proof is easy. We have only to show that 11 Tt 11 is, for any interval
(0, a) with oo > a > 0, bounded on (0, a). Assume the contrary and let there
exist a sequence {^} £ (0, a) such that II Tt 11 > n and lim t* =	< a.
я~*оо
By the resonance theorem, |f Ttnx || must be unbounded at least for one
X, which surely contradicts the strong continuity condition (4).
Remark. By multiplying e~&, we may assume that a semi-group
{T,} of class (Co) is equi-bounded;
(for 0^/<oo).	(6)
1. The Semi-group of Class (Co)
233
If, in particular, M is < 1, i.e., if
(for 0 <: t < oo),	(7)
then the semi-group {TJ is called a contraction semi-group of class (Co).
As for the strong continuity condition (4), we have the following
Theorem. Let a family {Tg, t 0} of operators 6 L{X, X) satisfy (1')
and (3). Then condition (4) is equivalent to the condition
w-lim Ttx = x for every x £ X.
Цо
(8)
Proof. Suppose that (8) is satisfied. Let xQ be any fixed element of
X. We shall show that s-lim Ttx0 = for each Ц 0. Consider the
function x{t) = TtxQ. For each £0	0, x(t) is weakly right continuous at
t0, because	= w-Iim Th 1\ xQ = Tt x0. We next prove that
| Tj | is bounded in a vicinity of i = 0. For, otherwise, there would exist
a sequence {Ц} E such that tn | 0 and lim II Ttnx0=oo, contrary
n—>oo
to the resonance theorem implied by the weak right continuity of x (t) =
Ttx0. Thus by (I'), we see that TtxQ = %(/) is bounded on any compact
interval of t. Moreover, х(р] is weakly measurable. For, a right continuous
real-valued function ftp is Lebesgue measurable, as may be proved from
the fact that, for any a, the set {if; /(/) < a} is representable as the union
of intervals of positive length. Next let {tn} be the totality of positive
rational numbers, and consider finite linear combinations £ faxty)
where are rational numbers (if X is a complex linear space, we take
fy = а+ ibj with rational coefficients Oy and 6y). These elements form a
countable set M = {%„} such that {x (t); t 0} is contained in the strong
closure of M. If otherwise, there would, exist a number t' such that x(t')
does not belong to Ma. But, being a closed linear subspace of X, Ma is
weakly closed by Theorem 11 in Chapter V, 1; consequently, the condi-
tion x(t') £ Ma is contradictory to the weak right continuity of x(t), i.e.,
to x (if7) = w-lim x (Ц).
We may thus apply Pettis’ theorem in Chapter V, 4 to the effect that
x(t) is strongly measurable, and so by the boundedness of ||х(/) j| on any
compact interval of t, we may define the Bochner integral J x(t) dt
and we have
f X (t) dt
a
p
f 11x W 11	for 0 л < /5 < oo. By virtue
A
£	P
of the strong continuity in s of the integral f x (t + s) dt — J x(t) dt,
a	a+$
which is implied by the boundedness of x(t) on any compact interval
of t, N. Dunford [3] proved that %(f) is strongly continuous in t > 0.
We shall follow his proof.
234
IX. Analytical Theory of Semi-groups
Let OL л <	— with e > 0. Since x(g) — T^xQ —
Tr/ x0 = Tri x (£ — rft, we have
(Д —«) %(£} f x(£) drj = f Tn x^ — v}} dy,
and so, by sup ||M|[ < oo which is implied by (1') and (3), combined
«^*7^0
with the boundedness of [j7\|| near t ~ 0, we obtain
(3
(0-«) {x(£± s) — %(£)} = f Tv{x(g ± с — rf} — x($~ tfftdri,
&
(M oc} ||x(£± e) — %(£)|| sup |[TJ| • f ^x(r± £)~x(r)^dr.
«S*!^	£~P
The right hand side tends to zero as г | 0, as may be seen by approximat-
ing x (t) by finitely-valued functions.
We have thus proved that x(t} is strongly continuous in t > 0. To
prove the strong continuity of z(i) at t = 0, we proceed as follows. For
any positive rational number tn> we have Ttx{tn} ~ TtTinxQ ~ ?Mnx0 =
x{t + in). Hence, by the strong continuity of x{t} for / >0 proved
above, we have s-lim Ttx{tn} = x(in). Since each xm£ M is a finite linear
combination of x(^)’s, we have s-lim Tfxw = xm (m — 1, 2, . . On the
other hand, we have, for any [0, 1],
j|M) ~MI H2'**
-Mil

MlM™ —Ml + IM~ Ml + S^P ||TJ| • ||%0 — Mi'
___	0£tgl
Hence lim ||x(£) — Ml + sup | [ Tt j |) j | xw — %oj|, and so s-lim x (t}
/to	og/gi	do
= x0 by inf ||x0 —Ml = °-
(1)
(2)
(3)
2. The Equi-continuous Semi-group of Class (Co) in Locally
Convex Spaces. Examples of Semi-groups
Suggested by the preceding section, we shall pass to a more general
class of semi-groups.
Definition. Let X be a locally convex linear topological space, and
[Tf; t > 0} be a one-parameter family of continuous linear operators
€ L (X, X} such that
TtTs^ Tt+s, To = I,
lim Ttx ~Tt x for any t0 2c 0 and x^X,
i^c„	°
the family of mappings (TJ is equi-continuoiis in t,
i.e., for any continuous semi-norm fi on X, there
exists a continuous semi-norm q on X such that
p{Ttx} g q{x) for all t 0 and all x б X.
Such a family {TJ is called an equi-continuous semi-group of class (Co).
2. The Equi-continuous Semi-group of Class (Co)
235
The semi-groups {T(} satisfying conditions (1'), (3), (4) and (6) of
Section 1 are example of such equi-continuous semi-groups of class (Co).
We shall give concrete examples.
Example 1. Let C [0, oo] be the space of bounded uniformly con-
tinuous real-valued (or complex-valued) functions on the interval [0, oo),
and define Tt, t 0, on C [0, oo] into C [0, oo] by
(7» (s) = x(i + s).
Condition (1) is trivially satisfied. (2) follows from the uniform con-
tinuity of x(s). Finally ||T(|j	1 and so {Tt} is a contraction semi-
group of class (Co). In this example, we could replace C [0, oo] by
С [— OC', oo] or by Lp (— oo, oo).
Example 2.	Consider the space C [— oo, oo]. Let
Nt(u) = (27if)-1/2	— oo < w < oo, t > 0,
which is the Gaussian probability density. Define Tit t Sr 0, on C [—00,00]
into C [— 00, ex?] by
(Tt%) (s) = _f Vf(s — u) x(u) du, for t > 0,
— OO
= x(s), for t = 0.
OO
Each Tt is continuous, since, by / IV/(s — u) du = 1,
—co
OO
11 Tt x I [	[ I x 11 J Nt(s — u) du = J j x I j.
— OO
Тй — 1 by the definition, and the semi-group, property TtTs = Ti+s
is a consequence from the well-known formula concerning the Gaussian
probability distribution:
- - - 1 -= _U _ * F	.
y2n{t 4- f)	у2л*
This formula may be proved by applying the Fourier transformation
on both sides, remembering (10) and (13) in Chapter VI, 1. To prove the
00
strong continuity in t of Tt, we observe that x (s) = J Nt (s — u) x (s) du.
— CO
Thus
(Ttx) (s) — x(s) =; f Nt (s — u) (% (m) — x (s)) du,
— OO
which is, by the change of variable (s w)/|/7 — z, equal to
f (z) (x (s — j/i z) — x (s)) dz.
— OU
IX. Analytical Theory of Semi-groups
By the uniform continuity of x(s) on (—00,00), there exists, for any
f> 0, a number <5 = <5 (s) > 0 such that |x^J — x(s2) j e whenever
jsj — s2| d. Splitting the last integral, we obtain
j {Ttx) ($)— x(s)	f Nx(z) |x(s—]/i-z)—x(s)|dz-|-	{ • •} dz
e / Л\ (z) dz 4- 2 11 x | j J" (z) dz
|УГй| g<5	[УТг|><5
s E + 2 j IX ] I	J Nj (z) dz.
| VT><5
The second term on the right tends to 0 as 0, since the integral
00
J ^(z) dz converges. Thus lim sup | (Ttx) (s) — x(s) | = 0 and hence
—oo	HO s
s- lim Ttx = x; consequently, by the Theorem in the preceding section,
we have proved (2).
In the above example, we can replace C[— oo, oo] by Z?(—00,00).
Consider, for example, L1 (-—00, 00). In this case, we have
00
]| Tt x 11 < ffN((s — u) |x(«) j ds du ||x ||,
—oc
by Fubini's theorem. As for the strong continuity, we have, as above,
|j Ttx — x || = flf N1(z) (x(s — j// • z) — x(s)) dz|ds
—00
00
—00
—00
Hence, by the Lebesgue-Fatou lemma, we obtain
__	00	/___ 00
lim I j Tt x — x jI < f N.{z\\ lim f
x (5 — J/T z) — x (s) I ds j dz = 0,
because of the continuity in the mean of the Lebesgue integral, which may
be proved by approximating x(s) by finitely-valued functions.
Example 3.	Consider C[—00,00]. Let Л > 0, и > 0. Define Tt,
i 0, on C [—00,00] into C [—00,00] by
(^)*
A!
x (s — kp).
3. The Infinitesimal Generator of an Equi-continuous Semi-group 237
We have
rr it .n	V’	V	l \
m=0	’ [ fc = O
=2 £ к ± ЦГ	*(» - м
^-0 r w-0	vr 1
°o
= ^ i('"+'1 Z *i (Яи + Л')’ x <s = (Г-+-*) (s) 
Thus it is easy to verify that Tt is a contraction semi-group of class (Co).
3. The Infinitesimal Generator of an Equi-continuous
Semi-group of Class (Co)
Let {7'z; t 0} be an equi-continuous semi-group of class (Co)
defined on a locally convex linear topological space X which we assume
to be sequentially complete. We define the infinitesimal generator A of Tt
by
Ax = lim Л-1(ТА — I) x,	(1)
л;о
i.e., A is the linear operator whose domain is the set D {A} =
{% С X; lim Л-1(ТА — I) % exists in X}, and, for x£D(A), Ax =
h | 0
lim h~1(Tk - -1} x. D(A) is non-empty; it contains at least the vector 0.
Л ф 0
Actually D(A) is larger. We can prove
Theorem 1. Я(Л) is dense in X.
Proof. Let <p,,(s) -=o’l!, n > 0. Consider the linear operator
which is the Laplace transform of Tt multiplied by n:
oo
C^x = f <pn(s)Tsxds for x£X,	(2)
о
the integral being defined in the sense of Riemann. The ordinary proce-
dure of defining the Riemann integral of numerical functions can be
extended to a function with values in a locally convex, sequentially
complete space X, using continuous semi-norms p on X in place of the
absolute value of a number. The convergence of the improper integral is a
consequence of the equi-continuity of Tt, the inequality
P (SPn (s) Tsx)=ne ~ns p (Tsx)
and the sequential completeness of X.
We see, by
P(CVnx) f ne~™ p(7\x) ds ^supptT.x),
О	зйО
238
IX. Analytical Theory of Semi-groups
that the operator С^п is a continuous linear operator C L (X, X). We shall
show that
R(CVn) for each « > 0,	(3)
and
lim Cax = x for each xfX.	(4)
n^c Vn	 '
OO
Then U R{C^ will be dense in X, and a fortiori D(A) will be dense in
X. To prove (3), we start with the formula
oo	oo
—/) C^x = Л'1 f <pn(s) Tj.T.xds — h-1 f yw(s) Tsxds.
о	о
OO	00
The change of the order: Th J •  = f Th  -  is justified, using the
о о
linearity and the continuity of Th. Thus
oo	oo
(Л ~ /) C^x = A-1 J <pn(s) T^xds — A’1 f <pn(s) Tsxds
о	0
= -^-- n f eT™T«xda~±n J e^T.xds
я
я
”Л
фп
0
By the continuity of <pH(s) Tsx in s, the second term on the right tends
to — <prt(0) TQ x — — nx as A | 0. Similarly, the first term on the right
tends to nC^x as h | 0. Hence we have
ACVnx ^n{C^l}x, x£X.	(5)
OO
We next prove (4). We have, by J ne~ns ds — lt
о
00
С^пх ~x = n f e~ns (Tsx — %) ds}
0
CO	6	OQ
p (Сфпл — x) < n f p (Tsx — x} ds = nf 4- n f =	+ T2, say,
0	0 d
where <5 > 0 is a positive number. For any e > 0, we can choose, by the
continuity in s of Tsx, a 6 > 0 such that p (TEx — x) < £ for 0 X s dL
Then	d	oo
1\ < en f e~ns ds en f e~ns ds = e.
о	0
For a fixed d > 0, we have, by the equi-boundedness of {Tsx} in s 0,
I2 5^ n f e~ns (j> (Tsx) + p {xp) ds^> 0 as n | o©.
<5
Hence we have proved (4).
3. The Infinitesimal Generator of an Equi-continuous Semi-group 239
Definition. For % £ X, we define
Dt Ttx = lim Л-1 (Tt+„ - Л) x	(6)
A—>0
if the right hand limit exists.
Theorem 2. If x € D (A), then x 6 D (DtTt) and
D.Ttx AT.x- TtAx,r>^i.	(7)
Thus, in particular, the operator A is commutative with Tt.
Proof. If	then we have, since Tt is continuous linear,
Mo
40
40
40
Thus, ifxCD(^), then Ttx£D(A) and TtAx=ATtx = lim/g1
(Tt+h —Tt)x. We have thus proved that the right derivative of Ttx
exists for each xE D(A'). We shall show that the left derivative also
exists and is equal to the right derivative.
For this purpose, take any /0£ XA Then, for a fixed %£ D(A), the
numerical function fo(Ttx) = <Ttx, is continuous in t 0 and has
right derivative dAf^Tixydt which is equal to /о(ЛТ(х) = /0(Т(Лг) by
what we have proved above. Hence d+ fQ(Ttx)[di is continuous in t. We
shall prove below a well-known Lemma: if one of the Dini derivatives
D+/(^), />+/(*), and
of a continuous real-valued function f(t) is finite and continuous, then
f(t) is differentiable and the derivative is, of course, continuous and equals
Thus /о (Ttx) is differentiable in t and
/0(Г,х-х) = /0(7» ~/0(T„x) = /W,x)/i>  ds = f f„(T,Ax)ds
0	0
Since /0 £ Xf was arbitrary, we must have
t
Ttx — x = f TsAxds for each
о
Since 7\A x is continuous in s, it follows that Т>х is differentiable in t in
О	J	Й
the topology of X and
£ + A
DtTtx = lim hr1 f TsAxds = TtAx.
Thus we have proved (7).
Proof of the Lemma. We first prove that the condition: D+/(t)	0
[or a < t ft implies that /(ft) —/(u)	0. Assume the contrary, and
IX. Analytical Theory of Semi-groups
let /(b) — /(a) < — e(b ~ a) with some e > 0. Then, for g(t) = f(t) —
/ (a) 4- e (tf — a), we have ZH g («) = LAf (a) 4- e > 0, and so, by g (a) = 0,
we must have g(tG) > 0 for some t0 > a near a. By the continuity of g(t)
and g (b) < 0, there must exist a tfT with a < tf0 <	< b such that
g(tfj) = 0 and g(tf) < 0 f°r < t < b. We have then ZWgftfj) 14 0 which
surely contradicts the fact D+g(tfj) = D+f(t1) + 8> 0.
By applying a similar argument to /(t) — oct and to/itf — /(tf), we prove
the following: if one of the Dini derivatives Df (tf) satisfies
oc < Df (if) S4 /? on any interval tf2],
then oc 54 (/(tf2) — /(^i))/(^2 —	/? Hence, the suprema (and the infima)
on [tfp t2~] of the four Dini derivatives of a continuous real-valued func-
tion / (tf) are the same. Thus, in particular, if one of the Dini derivatives of
a continuous real-valued function /(tf) is continuous on [tfp tf2], then the
four Dini derivatives of /(tf) must coincide on [tfp tf2].
4.	The Resolvent of the Infinitesimal Generator A
Theorem 1.	If n > 0, then the operator (nl — A) admits an inverse
R(«; A) = (nl -A)-*GL(X, X), and
2?(«;A)x = f e~ns Tsxds for x£X.	(1)
о
In other words, positive real numbers belong to the resolvent set q(A)
of A.
Proof. We first show that (иТ - Л)'1 exists. Suppose that there
exists anro^0 such that (nl—A) x0 = 0, that is, Ax0 — nx0. Let /0 be a
continuous linear functional С X' such that /0(x0) = 1, and set tp(tf) =
/0(7/x0). Since л*0 € D (A), <p (tf) is differentiable by Theorem 2 in the prece-
ding section and
d<p(i)/dt f0(DtTtxQ) = /0(TtA%0) = f0(Ttnx0) = n<p(t).
If we solve this differential equation under the initial condition 9?(0) =
/0(х0) = 1, we get ^(tf)=ew(. But, gj(tf) = f$(Ttx^} is bounded in tf,
because of the equi-boundedness of TtxQ in tf 0 and the continuity of
the functional /0. This is a contradiction, and so the inverse (nl — A)’1
must exist.
Since, by (5) of the preceding section, A C9.nx ~ W(C?„ — f) x, we have
(nl — A) C^x — nx for all x 6 X. Thus (nl —A) maps 7?(CVn) £ D(A)
onto X in a one-one way. Hence, a fortiori, (nl — A) must mapD(A)
onto X in a one-one way, in virtue of the existence of (nI - - A)-1. There-
fore, we must have K(CVn) = D(A) and (nl — A)'1 = п~гСуп.
4. The Resolvent of the Infinitesimal Generator A
241
Corollary 1. The right half plane of the complex Я-plane is in the
resolvent set о (A) of A, and we have
R(X; A) %	(Я7 — A)-1 x = f er^Tixdt for 2?с(Я)> 0 and xf A’. (2)
о
Proof. For real fixed t,	0}, constitutes an equi-continuous
semi-group of class (Co). It is easy to see that the infinitesimal generator
of this semi-group is equal to (A — ixl). Thus, for any u> 0, the resolvent
R{a + it; A) = ((<r -f- it) I — A)-1 exists and
R((o + tr) I — A} x — J	Tsxds for all xyX. (2')
о
Corollary 2.
D(A) = R((tJ~A'rr) = R(R(A; A)) when йе(Я) > 0,	(3)
АЯ(Я; А)х = 7?(Я;А) Ax = (ЯЯ(Я; A) - Z) x for x^D(A), (4)
АК(Я; А)х = (Я7?(Я;А)—Z)x for x£X,	(5)
lim Я7?(Я; A) x = x for x£X.	(6)
Л f'Oe
Proof. Clear from R(A; A) = (Л1 — A)-*1 and (4) of the preceding
section.
Corollary 3. The infinitesimal generator A is a closed operator in the
following sense (Cf. Chapter II, 6):
if xK C D (A) and lim xA = x С X, lim A xh - у £ X, then
Й—КЮ	k—HX>
x£D{A) and Ax = y.
Proof. Put (Z — A) xh — xA. Then lim xA = x — у and so, by the
A—
continuity of (Z — A)-1, lim xA = lim (Z — A)-1 xA = (Z — A)’'1 (x — y)
A—>oo	h^OQ
that is, x = (I — A)-1 (x — y), (Z — A)x — x — y. This proves that
у — Ax.
Theorem 2.	The family of operators
{(Я2?(Я; A))”}	(7)
is equi-continuous in Я > 0 and in n = 0, 1, 2, . . .
Proof. From the resolvent equation (Chapter VIII, 2, (2))
Я(^; A) — 2?(Я; А) = (ЯR&-, А) 7?(Я; A),
we obtain
lim (/z —Я)”1^^; А)—7?(Я;л))х = АК(Я;А) х/с?Я = —7?(Я;A)2x,x£X.
To derive the above formula, we have to. appeal to (2) in order to show
that lim R(ju', A) у = 7?(Я; А) у, у E X.
10 Yosldfli Functional Anaiyaia
242	I X. Analytical Theory of Semi-groups
Therefore, R (Л; A) x is infinitely differentiable with respect to A when
Re (A) > 0 and
d”R(A‘, A) x/dAn ~ (~l)Bn! Я(Л; A)n+1 x (« = 0,1,2,...). (8)
On the other hand, we have, by differentiating (2) и-times with respect
d*R(A;A) xjdA* ~ f	Ttxdt.	(9)
0
Here the differentiation under the integral sign is justified since {7^%}
is equi-bounded in t and J tn dt~ (^!)/Лл+1 when Re{ty > 0, Hence
о
an+i 00
(ЛК(2; A))w+1 x = —у- J e~uinTtxdt for x(~X and Re (A) > 0, (10)
n' о
and so, for any continuous semi-norm ф on X and A > 0, n > 0,
£((27?(Л; А))«+1я) f e~^tndt  sup/>(7» = sup/>(7».	(11)
0	4Й0
This proves Theorem 2 by the equi-continuity of {TJ in t.
5.	Examples of Infinitesimal Generators
We first define, for n > 0,
J„ = (f~h-M)-1 = nR{n; A),	(1)
so that
AJ„^n(J„-J).	(2)
Example 1.	(Ttx) (s) ~ x(t + s) on C [0, oo].
Writing yn(s) — (Jnx) (s) = n f е~** x{t + s) dt = n f e~nd-^x^j di,
0	s
we obtain y^(s) == — и e~n(s-sbc(s) + и2 J e~n((~s) x(t) dt — — nx(s) +
n yft(s). Comparing this with the general formula (2):
(А/Пх)(з) = и ((/„-/) x)(s),
we obtain Ayn(s) = >£($). Since R(Jn) ~ R(R(к; A)) ~ D(A), we have
A у (s) = y' (s) for every у D (A).
Conversely, let у (s) and y' (s) both belong to C [0, co]. We will then show
that у C £)(A) and Ay(s) — y' (s). To this purpose, define #($) by
y' (s) — ny (s) = — hx[s) .
Setting (Jnx)(s) = yn(s), we obtain, as shown above,
y^(s) — nyn(sj = — wx(s).
5. Examples of Infinitesimal Generators
243
Hence w(s) = y(s) — yn (s) satisfies wr (s) = nw(s) and so w(s) = Ce*”.
But, w must belong to C [0, oo] and this is possible only if C = 0. Hence
У ($) = У» (s) € D (A) and A у (s) = y' (s).
Therefore, the domain D (A) of A is precisely the set of functions у £
C [0, oo] whose first derivative also € C [0, oo], and for such a function
у we have А у = у'. We have thus characterized the differential operator
d/ds as the infinitesimal generator of the semi-group associated with the
operation of translation by t on the function space C [0,oo],
Example 2.	We shall give a characterization of the second derivative
d2/dsz as the infinitesimal generator of the semi-group associated with
the integral operator by Gaussian kernel. The space is C [--oo,oo] and
(Ttx)(s) = (2л7) 1/2 f	dv if t > 0, = x(s) if t = 0.
—oo
We have
OO
= CM) CO = f x(v)
— 00
J п(2л1)~1/2 e~^s~vM2tdi
.0
dv
oo
—oo
2 ]/я f (2 л)-1/2	da
о
dv
(by substituting t — cr2/^).
Assuming, for the moment, the formula
/	da =	c = /Й |s-u|//2 >0,	(3)
о
we get
y„ (s) = f x (и) (n/2)1^2 l4 s-"l dv.
—OO
x (v) being continuous and bounded, we can differentiate twice and obtain
yrt'(s) = « f x(v) e~^2n dv — n f x(v)	dv,
5	“OO
yM"(s)=wj—x(s)—%($)+j/2 j/w f %(v)	(«-«») dv
I	5
b
4- |/1T \fn f x(y)	dv 
—oo
= — 2 я x(s) 4- 2w y„(s).
Comparing this with the general formula (2):
= M7»(s) =«((/« — /) %)(s) = n(yn(s) — %(s)),
we find that dyn(s) ~ y”(s)l2. Since R(Jn) = 2?(7?(я; Л)) = D(A), we
16*
244
IX. Analytical Theory of Semi-groups
have proved that
A у (s) = y"(s)/2 for every y£D(A).
Conversely, let у (s) and y” (s) both belong to C [— oo, 00]. Define x (s) by
y"(s) — 2ny (s') — — 2wx(s).
Setting yn(s) = (Jnx')(s), we obtain, as shown above,
y"($) — 2ny„(s} = — 2nx(s).
Hence w(s) = у (s) — yw(s) satisfies w" (s) — 2nw(s) = 0, and so w{s) ~
(\	4- C26~^2ws. This function cannot be bounded unless both Ct
and Cz are zero. Thus у (s) = yn(s), and so у £ Р(Л), (A y) (s) = y" (s)/2.
1 d2
Therefore, the differential operator -g- is characterized as the
infinitesimal generator of the semi-group associated with the integral
transform by the Gaussian kernel on the function space C[—00, сю].
The proof of (3). We start with the well-known formula
J* e~x* dx = |6r/2.
0
Setting x = a — cfa, we obtain
Setting a = eft in the last integral, we obtain
Exercise. Show that the infinitesimal generator A of the semi-group
{7]} on C [—00,00] given by
OO
^)(s) = e~u £
A = 0
is the difference operator A ;
(Л x) (s') — Л {x(s — fj,) — x ($)}.

6.	The Exponential of a Continuous Linear Operator
whose Powers are Equi-continuous
Proposition. Let X be a locally convex, sequentially complete, linear
topological space. Let В be a continuous linear operator £ L(X, X) such
6.	The Exponential of a Continuous Linear Operator
245
that {Bk; A = 1, 2, . . .} is equi-continuous. Then, for each x £ X, the
series
oo

(1)
converges.
Proof. For any continuous semi-norm ф on X, there exists, by the
equi-continuity of {BA}, a continuous semi-norm q on X such that
p(Bkx} q{x} for all k 0 and x £ X. Hence
ф ( J (£B)ft x/kl\ £ ikp{Bkx)[k\ <. q{x} - £ £*/Л 1.
\A==n	/ k=n	k—n
f n	>
Therefore | £ (tB)kx/kl is a Cauchy sequence in the sequentially
complete space X. The limit of this sequence will be denoted by (1).
CO
Corollary 1. The mapping x-+ £ (tBjh xfkl defines a continuous
A—0
linear operator which we shall denote by exp (ZB).
Proof. By the equi-continuity of {Bft}, we can prove that Bn —
fl
£ (tB)kjk\ (л ~ 0,1, 2,. . .) are equi-continuous when t ranges over
Л«0
any compact interval. In fact, we have
n	H
£(Bwx) g v tkp(Bkx)/kl q(x) 	t  ?(x).
Л®0	я—v
Hence the limit exp(/B) satisfies
P (exp (/B) x) exp (Z) • q (%) (2^0).	(2)
Corollary 2. Let В and C be two continuous linear operators € L {X, X)
such that {BA} and {C*} are equi-continuous. If, moreover, ВС = CB,
then we have
exp(/B) exp(£C) = exp(/(B -f- C)).
Proof. We have
A
(3)
k
p((B + C)kx)^ rkCsp(B^Cs x)^ J^CS ?{CS %) 2й sup ?(Csx)-
s=0	s=0	q<j
Hence {2-* (B -4- C)A} is equi-continuous, and we can define exp (J (В + C)).
By making use of the commutativity BC ~CB, we rearrange the series
00
fe=O
/00	\ j oo	\
so that we obtain I £ {tB)k/k\ 1 i (tC)k x/k\\ as in the case of
\A “0	/ \Л	j
oo
numerical series	+ c))*/A!.
Corollary 3. For every x £ X,
lim h~r (exp (AB) — Г) x — Bx,
(4)
246
IX. Analytical Theory of Semi-groups
and so, by making use of the semi-group property
exp((i + A)B) = exp{/B) ехр(ЛВ)	(5)
proved above, we obtain
Dt exp(tB) x = exp(£B) Bx = В exp(£B) x.	(6)
Proof. For any continuous semi-norm on X, we obtain, as above
(A'1 (exp (AB) — I) x-Bx) £ h*-1 p(B*x)/kl g q(x) h^/kl
which surely tends to 0 as h | 0.
7.	The Representation and the Characterization of Equi-continuous
Semi-groups of Class (Co) in Terms of the Corresponding
Infinitesimal Generators
We shall prove the following fundamental
Theorem. Let X be a locally convex, sequentially complete, linear
topological space. Suppose A is a linear operator with domain В(Л)
dense in X and range 2?(Л) in X such that the resolvent Bfn; A) =
(n l - - A )-1 E L (X, X) exists for n ~ 1, 2, . . . Then A is the infinitesimal
generator of a uniquely determined equi-continuous semi-group of class
(Co) iff the operators {(7 - n1 are equi-continuous in n — 1, 2, . . .
and m ~ 0, 1, . . .
Proof. The "only if" part is already proved. We shall prove the "if"
part.
Setting
= (1)
we shall prove
lim Jnx = x for every x 6 X.
Я—ЮС
(2)
In fact, we have, for D[A), A Jnx = Jn^x = nUn — I) x and so
Jnx — x = п~г}п(Аx) tends to 0 as и ->oc, in virtue of the equi-
boundedness of {/„(Ax)} in n = 1,2,.... Since D(A) is dense in X and
{Jn} is equi-continuous in и, it follows that lim J„x = x for every
n-kOC
хе X.
Put
= exp (M J„) = - exp (tn (/„ — 7)) = exp (— nt) exp (nt/„), f J> 0. (3)
Since {/*} is equi-bounded in n and k, the exp (tn J„) can be defined, and
we have, as in (2) of the preceding section,
/>(exp(wt/„) x) У. (nf)k (A!)-1 p(Jk  x) expM - q(x).
A=0
7. The Representation and the Characterization
247
Consequently, the operators {Tjn)} are equi-continuous in t 0 and
« = 1, 2, ... in such a way that
/>(T<”x)g?(x).	(4)
We next remark that	for n, m > 0. Thus Jn is commu-
tative with Hence, by DtT^ x = A x =	/„x, proved
in the preceding section,'
plT^x-T^x) =p( f D,(T^'sT^x)ds)
\« _	/	(5)
= p(f TH 71"' (Л /. - A J„) x *).
Hence, if x 6 D (Л), there exists a continuous semi-norm q on X such that
P (T^x - Tj-> x) S f q ((Л J. - A J„) x) ds = A - J„ Л) x).
0
Therefore, by (2), we have proved that lim рП\п}х—T^x) = 0
юс
uniformly in t when t varies on every compact interval. Since D (Л) is dense
in the sequentially complete space X, and since the operators are
equi-continuous in and in w, we see that lim T^x = Ttx
exists for every x-'X and t > 0 uniformly in t on every compact interval
of t. Thus the operators {TJ are equi-continuous in t > 0, and from the
uniform convergence in t, Ttx is continuous in t 0.
We next prove that Tt satisfies the semi-group property TtTs = Tt+s.
Since = T^T™, we have
p t,Tl+.x  T.T.x} £ p (T,^x - 71” x) + p (T&x - T^T^x)
+ p (71” 71” x - 71” 7» + )> (71” 7> - T, T, x)
=S P {T,+,x - T&x) + q{T^x - T,x)
+ p ((71” - Г.) T,x) -> 0 as n —> <X>.
Thus P(Tt+sx — TtTsx) = 0 for every continuous semi-norm p on A’,
which proves that Tt+s = TtTs.
Let A be the infinitesimal generator of this equi-continuous semi-
group {Tt} of class (Co). We have to show that A = A. Let x^ D(A).
Then lim J„x = TtAx uniformly in t on every compact interval
«—►CO
of t. For, we have, by (4),
P(T,Ax-T?>AM ^P^Ax-T^Ax) +p(T<?Ax-T<?AJnx)
 P'S!’, - 7-j”) A x) + ?(A x - J,Ax)
248
IX. Лn;ily t ie;d Theory of Semi-groups
which tends to 0 as n -> oo, si ore lim J nA x = A x. Hence
И ЬПО
i
Ttx~x == lim	x) lim f 7^4 Jnxds
iwcc	n—к» q
r
t
A Jnx\ ds = f TsAxds
о
t
so that lim tr1 (Ttx — x) = lim t~} f TsAxds exists and equals A x. We
40	go 0J	ч
have thus proved that «еР(Л) implies х££>(4) and Ax = dx, i.e.
A is an extension of A. A being the infinitesimal generator of the semi-
group Tti we know that, for n > 0, (nl — A) maps D(4) onto X in
one-one way. But, by hypothesis, (nl - A} also maps Z)(4) onto X in
one-one way. Therefore the extension A of A must coincide with A.
Finally, the uniqueness of the semi-group Tt is proved as follows.
Let Tt be an equi-continuous semi-group of class (Co) whose infinitesimal
generator is precisely A. We construct the semi-group T<tn). Since A is
commutative with Tt, we see that A Jn and are commutative with
Tt. Then, for x£ D{A), we obtain, as in (5),
(t	\
f	x) ds)
x (S)
= P (f	(A - A J.} xdsj .
Thus, by virtue of lim A Jnx — Ax for all x£D(A), we prove lim
«—►OO	П—o-OO
Tln) x = Ttx for all xr'.Y similarly to the above proof of the existence of
Um T^x, x£X.
Therefore, Ttx = ftx for all x£X, that is, Tt = Tt.
Remark. The above proof shows that, if A is the infinitesimal genera-
tor of an equi-continuous semi-group Tt of class (Co), then
Ttx = lim exp(/4 (I — n^d)-1) x, x£X,	(7)
n->oo
and the convergence in (7) is uniform in t on every compact interval of I.
This is the re-presentation theorem for semi-groups.
Corollary 1. If X is a Z?-space, then the condition of the Theorem
reads: D(A)a = X and the resolvent (I — и-14)-1 exists such that
||(£-л~14Г”‘|!^с («=1,2,...; w=l,2, ...)	(8)
with a positive constant C which is independent of n and m. In particular,
for the case of a contraction semi-group, the condition reads: D(A)a = X
7. The Representation and the Characterization
249
and the resolvent (I — м-1Л)-1 exists such that
g 1 (w = 1, 2, . . .)•	(9)
Remark. The above result (9) was obtained independently by E. Hil-
le [2] and by K. Yosida [5]. The result was extended by W. Feller
[1], R. S. Phillips [3] and I. Miyadera [1] and the extension is
given in the form (8). It is to be noted that in condition (8) and (9), we
may replace (и — 1, 2, . . .) by (for all sufficiently large n). The exten-
sion of the semi-group theory to locally convex linear topological spaces,
as given in the present book, is suggested by L. Schwartz [3].
Corollary 2. Let X be a В-space, and {Tt; t > 0} be a family of
bounded linear operators L (X, X) such that
Г,Г, = Т,+,(О^0),Т, = Л	(10)
s-lim Ttx = x for all x£ X,	(11)
Ц0
11Г, II < M(F for all t > 0, where M '> 0 and в > 0 are
П «II-	-	t'-
independent of t.
Then (A — ft I) is the infinitesimal generator of the equi-continuous semi-
group St — e~&Tt of class (Co), where A is the operator defined through
A x — s-lim Is {Tt — I) x. Thus, by Corollary 1, we see that a closed
linear operator A with В(Х)Л = X and R(T) £ X is the infinitesimal
generator of a semi-group Tt satisfying (10), (11) and (12) iff the resolvent
(7 — n~l(A — exists such that
||(I —И-1(Л -	\ <M (for m = 1, 2, . . . and all large n).	(13)
This condition may be rewritten as
(7 — «-1ri)-m|j < M(1 —	"* (for m = 1, 2, . . . and all large n.
(13')
In particular, for those semi-groups Tt satisfying (10), (11) and
for all /SO,	(14)
condition (13') may be replaced by
| j (I - n^A)-11|^(1- «"W1 (for all large n).	(13")
An application of the representation theorem to the proof of Weierstrass’
polynomial approximation theorem. Consider the semi-group Tt defined by
(Ttx) (s) = x(/ 4-s) on C [0, oo]. The representation theorem gives
oo pm
(Ttx) (s) = x(i 4- s) = s-lim exp(tAJ„x) (s) = s-lim ~y (A Jn)mx(s),
»—К£>ж=0"*-
and the above s-lim is uniform in t on any compact interval of Л From
tt— ЮО
this result we can derive Weierstrass’ polynomial approximation theo-
250
IX. Analytical Theory of Semi-groups
rem. Let 2(5) be a continuous function on the closed interval [0, 1]. Let
x(s) € C [0,00] be such that x(s) ~ z(s) for s £ [0, 1]. Put s = 0 in the
above representation of x (t + s). Then we obtain
(Ttx) (0) = x(t) = s-lim J tm(AJn)Mx (0)/w! in C[0,1],
n->00
uniformly in t on [0, 1]. Hence z (#) is a uniform limit on [0,1] of a sequence
of polynomials in t.
8.	Contraction Semi-groups and Dissipative Operators
G. Lumer and R. S. Phillips have discussed contraction semi-groups
by virtue of the notion of semi-scalar product. The infinitesimal generator
of such a semi-group is dissipative in their terminology.
Proposition. (Lumer). To each pair {r, y} of a complex (or real)
normed linear space X, we can associate a complex (or real) number
[x, y] such that
[x + y, 2] = [x, z] 4- [y, z],	[/U, y] = A[x, y],	[x, x] = 11 x 112,	(1)
IMI • ||y||-
[%, у J is called a semi-scalar product of the vectors x and y.
Proof, According to Corollary 2 of Theorem 1 in Chapter IV, 6, there
exists, for each х0€ X, at least one (and let us choose exactly one)
bounded linear functional	such that ||Л,|| = and
<%Ae> = ll*o! I2- Then dearly
[x, y] = (x, /y>	(2)
defines a semi-scalar product.
Definition. Let a complex (or real) В-space X be endowed with a
semi-scalar product [x, у]. A linear operator A with domain D(A) and
range R (A) both in X is called dissipative (with respect to [x, y]) if
Re [Ax, z]^0 whenever	(3)
Example. Let X be a Hilbert space. Then a symmetric operator A such
that (A x, x) < 0 is surely dissipative with respect to the semi-scalar
product [%, y] = (x, y), where (x, y) is the ordinary scalar product.
Theorem (Phillips and Lumer). Let A be a linear operator with do-
main D(A) and range R(A) both in a complex (or real) В-space X such
that D (A)® = X. Then A generates a contraction semi-group of class (Co)
in X iff A is dissipative (with respect to any semi-scalar product [x, y])
andR(Z-A) = X.
Proof. The “if” part. Let A be dissipative and л > 0. Then the inverse
(Л/— A)-1 exists and |j (Л/ — A)-1y|)	2г1 j|y||wheny E D((a.I — A)"1).
9. Equi-continuous Groups of Class (Co). Stone’s Theorem
251
For, if у = Ax — Ax, then
Z||%||2 = A{xt %] Re{A{x, x] — [Ax, %]) = x] ||y ||  |Jx||, (4)
since A is dissipative. By hypothesis, R{I — Л) = X so that A = 1 is in
the resolvent set q(A) of A, and we have ||2?(1; Л)||	1 by (4). If
|Z — 1 [ < 1, then the resolvent R(A', A) exists and is given by
2?(Л;Л) = R(1; Л) (I + (Л - 1) R (1; A))-1
= R(1;A)- J((l-Z)2?(l; Л))л,
л=0
(see Theorem 1 in Chapter VIII, 2). Moreover, (4) implies that
||7?(А; Л) ||	A-*1 for A > 0 with |Z — 11 < 1. Hence, again by
Rfa A) = R(A; A) (I + & - Я) R(A; A))~\
which is valid for /j, > 0 with |/z — A j ||R(Z; Л) |] < 1, we prove the exi-
stence of R(/i; A) and ||Z?(/G Л) ](	/г1. Repeating the process, we see
that R (Я; A) exists for all 2 > 0 and satisfies the estimate | j R (A; A) | J Z '1.
As D (Л) is dense by hypothesis, it now follows, from Corollary 1 of the
preceding section, that A generates a contraction semi-group of class (Co).
The “only if” part. Suppose {Tf; t 2? 0} is a contraction semi-group
of class (Co). Then
Re\J\x — %, x] = Re[Ttx, %] — ||^||a = IMM[ ' IM! ~ I Ml* =
Thus, for x£D[A}, the domain of the infinitesimal generator A of T, we
have Re[Ax,x] = lim Re[Rl [T,x x, x~\} g 0. Hence A is dissipative.
t ;o
Moreover, we know that R(I — A) = D(R(1; Л)) = X, since A is the
infinitesimal generator of a contraction semi-group of class (Co).
Corollary. If Л is a densely defined closed linear operator such that
D(A) and R(A) are both in a В-space X and if A and its dual operator
A' are both dissipative, then A generates a contraction semi-group of
class (Co).
Proof. It sufficies to show that R{/ — A) = X. But, since (Z — A)”1
is closed with A and continuous, R(I — Л) is a closed linear subspace of X.
Thus R [I — Л) X implies the existence of a non-trivial x £ X' such
that	ф - A %), V> = 0 for all x e D (A).
Hence x' — A'x' = 0, contrary to the dissipativity of A' and x' =4 0,
Remark. For further details concerning dissipative operators, see
G. Lumer-R. S. Phillips [1]. See also T. Kato [6].
9.	Equi-continuous Groups of Class (Co). Stone’s Theorem
Definition. An equi-continuous semi-group {Tt} of class (Co) is called
an equi-continuous group of class (Co), if there exists an equi-continuous
semi-group of class (Co) with the condition:
252
IX. An.tlyliv .il 'theory of Semi-groups
If we define 5, by ,S( i\ for /.	0 and = Tt for t 0, then
the family of operators «S',, nu < i, < oo, has the group property
' Z, x < oo), So = I.	(1)
Theorem. Let X be a locally convex, sequentially complete linear
topological space. Suppose A is a linear operator with domain D (A) dense
in X and range R (A) in X. Then A is the infinitesimal generator of an
equi-continuous group of class (Co) of operators St £ L (X, X) iff the
operators (J— n~1A)~m are everywhere defined and equi-continuous in
n = dz 1, i 2, • - • and in m = 1, 2, . . .
Proof. The “only if" part. Let Tt — St for t 0 and 7\ = S„( for
t 0. Let A and A be the infinitesimal generator of Tt and Tt, respec-
tively. We have to show that A = — A. If x £ T>(A), then, by putting
xh= h~1(Th~ I) x and making use of the equi-continuity of Th,
Р(1\хк—Ах) ^p(Thxh — ThAx) + p((Th — I) Ax)
?(xa~Ax) + p ((Tk — I) Ax)
where p and c/arc continuous semi-norms on X such that, for a given/», we
can choose a q satisfying the above inequality for all h 7? 0 and all x<- D (A)
simultaneously. Thus lim L\xh = Ax, and so x QD (A) implies Ax —
lim TXh-1 (Th -1)) x = lim A-1 (I — TA x = - A x,
л;о xx»	д^0
Hence —A
is an
extension of A. In the same way, we can prove that A is an extension of
—A. Therefore A = - A.
The “if” part. Define, for t -T 0,
Ttx = lim T^x = lim exp(ZA (I плА)г) x,
n—К»	n—>00
Ttx = lim T(t^x = lim exp(£A (I — n^1A)~l) x, where A - - A .
>00	ГН-КЮ
Then Tt and Tt are both equi-continuous semi-groups of class (Co). We
have
p(TtTtx —T^f^x) p(TtTtx - T^Ttx) + p{T^Ttx~ T^t^x)
g p((Tt - T^) Ttx) + q(ftx - f^x)
by the equi-continuity of {T;n!} in n and Z >- 0. Thus we have
lim T^T^x = Tt Ttx. We have incidentally proved the equi-con-
tinuity of TtTt by the equi-continuity of T^T^. On the other hand,
we have T^	by the commutativity of (I — и1!)-1
and (/-ж-М)-1, Thus (T^T^) (T^T(”}) ^T^_sT^st and so
9. Equi-continuous Groups of Class (Co). Stone’s Theorem 253
(TtTt) with > 0 enjoys the semi-group property (T^T,) (Ts7\) =
Tt+sTt+s. Hence {TtTt} is an equi-continuous semi-group of class (Co).
If x^D(A) = 0(A), then
lim к-ЧТьТь — I) x = lim Thh^1(fh — 7) x + lim h~1(Th — I) x
л;о	но	но
~ Ax У Ax = 0,
so that the infinitesimal generator A1 of {Г4Т(} is 0 at every x{:_ D(A).
Since {I — A J is the inverse of a continuous linear operator (Z - - AJ"’ E
L (X, X), we see that Аг must be closed. Hence Ax must be = 0, in virtue
of the fact that Аг vanishes on a dense subset Z>(A) = D(A) of X.
Therefore (7j,7\) x = lim exp(if  0  (I — h'1 • 0)-1) x = x, that is,
Tt Tt = I. We have thus proved the group property of S(, — oo < t < oo,
where St = Tt and S_t = Tt for t > 0.
Corollary 1. For the case of a 2?-space X, the condition of the Theorem
reads: D(A)a = X and the resolvent (I — n-1A)-1 exists such that
|j (7 —	[j < M (for m -- 1, 2, . . . and all large \n |,	0). (2)
For the group St satisfying Ц5, || M(^ > 0) for all t, — oo< t < oo,
the condition reads: 7)(A)“ = X and the resolvent (7 —	! exists
such that
||(7 —w~1A)-'nj| M (1 — | n-11 /9)-w (for m =
and all large | n |, n 0).
For the particular case |JSJ| <• for all#, — oo < t < oo, the condi-
tion reads: £>(A)e = X and the resolvent (7 -	j”1 exists such that
||(7 — n^A)-1 [] SZ (1 — |n-1|/?)-1 (for all large |я|, я§0). (4)
Proof. As the proof of Corollary 1 and Corollary 2 in Chapter IX, 7.
Corollary 2 (Stone's theorem). Let Ut, —oo < t < oo, be a group of
unitary operators of class (Co) in a Hilbert space X. Then the infinitesimal
generator A of Ut is 1 times a self-adjoint operator H.
Proof. We have (Ut x, y) = (x, U^y) = (x, V_ty) and so, by diffe-
rentiation,
(Ax, y) = (x, —A y) whenever x and у C D (A).
Thus — г A - //is symmetric. A being the infinitesimal generator of Uit
(I — n-1A)-1 = (7 — п'ЧЩ1 must be a bounded linear operator such
that || (7 - Xj 11|	1 for я = ± 1, i 2, . . . Hence, taking the case
n = i 1, we see that the Cayley transform of H is unitary. This proves
that H is self-adjoint.
254
IX. Analytical Theory of Semi-groups
Remark. If A is of the form А |/ Л  H, where H is a self-adjoint
operator in a Hilbert space A', then condition (4) of Corollary 1 is surely
satisfied, as may be proved by the theory of Cayley's transform. There-
fore, A is the infinitesimal generator of a group of contraction operators
Ut in X. It is easy to see that such Ut is unitary. For, a contraction
operator Ut on a Hilbert space X into AT must be unitary, if by 1
is also a contraction operator on X into X.
10.	Holomorphic Semi-groups
We shall introduce an important class of semi-groups, namely, semi-
groups Tt which can, as functions of the parameter t, be continued holo-
morphically into a sector of the complex plane containing the positive
/-axis. We first prove
Lemina. Let X be a locally convex, sequentially complete, linear
topological space. Let {Tt;i 0} £ L (X, X) be an equi-continuous semi-
group of class (Co). Suppose that, for all t > 0, TtX QD (A), the domain
of the infinitesimal generator A of Tt. Then, for any x £ X, Ttx is in-
finitely differentiable in t > 0 and we have
for all />0,	(1)
where Tt = DtT(, Л" = DtT'...... T<n) =
Proof. If t > /0 > 0, then T\x = A Ttx = T^A T(ox by the com-
mutativity of A and Ts, s 0. Thus T't X £ Tt~laX £ 15(A) when
t > 0, and so T” x exists for all / > 0 and x 6 X. Since A is a closed linear
operator, we have
T”x = Dt(ATt)x = A • lim - n (Tt+1/n -Tt}x = A(A Tt) x
n-KOO
— A T//3A Тц2х = (Tt/2)2 x.
Repeating the same argument, we obtain (1).
Let A be a locally convex, sequentially complete, complex linear
topological space. Let {Tt; t 0} £ L (X, X} be an equi-continuous semi-
group of class (Co). For such a semi-group, we consider the following three
conditions:
(I)	For all t > 0, T(x£ D(A], and there exists a positive constant
C such that the family of operators {(C7TJ)*} is equi-continuous in
n by 0 and t, 0 < / id 1.
(II)	Tt admits a weakly holomorphic extension Tx given locally by
T\x = £ (Я — t)nT^x/nl for |arg2[<tan T(Ce 1),	(2)
and
the family of operators {е~л7\} is equi-continuous in /. for
|arg Л| < tan’ 1 (2 ACe-1) with some positive constant k. (3)
10. Holomorphic Semi-groups
255
(III)	Let A be the infinitesimal generator of Tt. Then there exists
a positive constant CT such that the family of operators {(^ЯВ (Я; Л))”}
is equi-continuous in н > 0 and in Я with Re (A) > 1 + s, e > 0.
We can prove
Theorem. The three conditions (I), (II) and (III) are mutually equi-
valent.
Proof. The implication (I)	(П). Let p be any continuous semi-
norm on X. Then, by hypothesis, there exists a continuous semi-norm q
on X such that	x) C~nq(x), 0 C for 1 Si 0, w 0
and xC X. Hence, by (1), we obtain, for any t > 0,
Л"	'F(n) I	<r-	И-Zl”	И” 1 . /7 *	ГТ' V
P ((^ *0 Х1П !)	= tn	w'[ Cn \\ n Xj
— ( C~le) ' ?(x)> whenever 0 < tjn 1.
Thus the right side of (2) is convergent for |argA| < tan"1(C«-1), and
so, by the sequential completeness of the space X, TAx is well defined
and weakly holomorphic in Я for |arg Я [ < tan~1(Ce"1). That is, for any
x £ X and /О", the numerical function f(Ttx) of t, t> 0, admits a
holomorphic extension f(TAx) for |argA[ < tan"1 (Ce-1); consequently,
by the Hahn-Banach theorem, we see that TAx is an extension of Ttx for
|argA| < tan_1(Ce_1). Next put St = е~*Т{. Then SJ = e~(T't— e~(Tt
and so, by 0	1 (0 < t) and (I), we easily see that the family
of operators {(2"*C7SJ)”} is equi-continuous in t > 0 and и Si 0, in virtue
of the equi-continuity of {TJ in t > 0. The equi-continuous semi-group
St of class (Co) satisfies the condition that t > 0 implies StX £ D (Л — /)
= D(A), where (A  - /) is the infinitesimal generator of S(. Therefore,
by the same reasoning applied above to Tt, we can prove that the weakly
holomorphic extension e~*TA of St — e~lTt satisfies the estimate (3).
By the way, we can prove the following
Corollary (due to E. Hille). If, in particular, X is a complex B-space
and lim IItT(t\I < e"1 then X = D(Л).
GO 11	1	___
Proof. For a fixed t > 0, we have lim 11 (£/«) 7'(/„ [ j < e-1, and so the
series
i (л - o* n”x/»i = Д!4- S (^ n.)“ -
strongly converges in some sector
{Я; |Я — t\ft < 1 + 8 with some <5 > 0}
of the complex Я-plane. This sector surely contains Я = 0 in its interior.
256
IX. Analytical Theory of Semi-groups
The implication (II) (III). Wc have, by (10) in Chapter.IX, 4,
(Л/е(Л; Л))”+1.% = —, f c~uinTtxdt for Re(ty > 0, x&X. (4)
Hence, putting St^e^Tt, we obtain
((o	1 + ix) R(a + 1 + j't; A))”+1 x
j fT^+^rStxdt, (j> 0.
Let т < 0. Since the integrand is weakly holomorphic, we can deform,
by the estimate (3) and Cauchy’s integral theorem, the path of integra-
tion: 0 < t < oq to the ray: re*6 (0 LI r < oo) contained in the sector
0 < arg Я < tan 1(2ACc J) of the complex Я-plane. We thus obtain
((<? + 1 + ix) R(a + 1 -f- ix; A))n+1 x
OO
=	f e-(a+^ernein&S^xei9drt
221	J	re	r
0
and so, by (3),
p({[a + 1 -h ix) R[a + 1 + it; A))n+l x)
sup p(Sreif>x} l(ff t 1 +	J° e(~^6+^in0)rr»dr
0<r<oo	п-	о
|ст + 1 -I- гт|п+1	, .	.
— ? W JTsinT — CT cos |"+i ’ where q is a continuous semi-norm on X.
A similar estimate is also obtained for the case x > 0. Hence, combined
with (7) in Section 4, we have proved (III).
The implication (III) -r (I). For any continuous semi-norm /> on Xt
there exists a continuous semi-norm q on X such that
/>((С1ЯЯ?(Я; A))n%) q(x) whenever Я?е(Я) St 1 -j- s, e> 0, and 0.
Hence, if Re^) 1 + £, we have
S	(« = 0, 1. 2.. . .).
Thus, if )Я — Яо 1/Сх jЛ, j < 1, the resolvent R (Я, A) exists and is given by
Я(Я;А)х= J? (Яо-Я)” Й(ЯО; Af+1 x such that
tt=0
P(R(2; Л) x) £ (1 - Cf1	л) XY
Therefore, by (HI), there exists an angle в0 with тт/2 < 0O < л such that
7? (Л; A) exists and satisfies the estimate
/>(7?(Я; A) x) д-| q' (x) with a continuous semi-norm qf on X (5)
10. Holomorphic Semi-groups
257
in the sectors л/2 < arg A < 0O and —0o < arg Я < — л/2 and also for
Re (A) Я 0, when [Л| is sufficiently large.
Hence the integral
Ttx = (2ТЙ)”1 f euR(A; A) x dA (f>0, x£X) (6)
c,
= oo and, for some e > 0,
converges if we take the path of integration C2 = Я(ст), — oo < о < oo,
in such a way that lim IЯ (cr)
|a|foo 1
л/2 + s arg Я (a) < 0O and —0Q arg Я (or) — л/2 — e
when ст | -j- oo and aj-oo, respectively; for |o| not large, Я (ст) lies
in the right half plane of the complex Я-plane.
We shall show that Tt coincides with the semi-group Tt itself. We
first show that lim Ttx = x for all x £ D (Я). Let %0 be any element
€ £)(Я), and choose any complex number Я^ to the right of the contour
C2 of integration, and denote (Я^Я — A) x0 = y0. Then, by the resolvent
equation,
J-.Xo = T, «(J,; A) y0 = (2Я»)-1 f S‘R(X,A') K(V. A) y9<U
= (2лг)~1 /^(Я0-Я)-17?(Я;А)у0Л
ca
- (2ТПГ1 f ^(Яо-Я)-1 7?(Я0; A) yodA.
c,
The second integral on the right is equal to zero, as may be seen by
shifting the path of integration to the left. Hence
Г,х0 = (2 л»)-1 J ^(Ло — Л)-1 Л) y0 = (V - A) x0.
c,
Because of the estimate (5), the passage to the limit 0 under the
integral sign is justified, and so
lim ttx0 = (2тп)-1 f (Яо — Я)-1 Л (Я; Я) yodA, у0 = (AqI — А) х0.
*1°	с,
То evaluate the right hand integral, we make a closed contour out of the
original path of integration C2 by adjoining the arc of the circle |Я | = r
which is to the right of the path C2, and throwing away that portion
of the original path C2 which lies outside the circle [Я | = r. The value of
the integral along the new arc and the discarded arc tends to zero as
rfoo, in virtue of (5). Hence the value of the integral is equal to the
residue inside the new closed contour, that is, the value R (Лц; A) y0 = x0.
We have thus proved lim Ttx0 = x0 when x0 С 7)(Я).
We next show that T'x = ATtx for t>Q and x£X. We have
7?(Я; Я) X -  D(A) and A R{A; Я) = AR (Я; Я) — I, so that, by the
17 Yosldu, Functhmid Лиа1ун1н
258
IX. Analytical Theory of Semi-groups
convergence factor the integral (2лг) 1 f e** AR(A; A)x dA has a
л
sense. This integral is equal to A Ttx, as may be seen by approximating
the integral (6) by Riemann sums and using the fact that A is closed'.
lim x„ = x and lim Axn = у imply x 6 D (Л) and A x = y. Therefore
WOO	rt—*OQ
A Ttx = (2лгГх J e“AR(A; A) xdA, £>0.
c,
On the other hand, by differentiating (6) under the integral sign, we
obtain
t\x = (Злг'Г1 /e?AR(A', A) xdA, t > 0.	(7)
c,
The difference of these two integrals is (2л»)“1 J e^xdA, and the value
c.
of the last integral is zero, as may be seen by shifting the path of integra-
tion to the left.
Thus we have proved that x(t) = Ttx0, %0€-О(Л), satisfies i)
limx(Z) = x0, ii) dx(t)]dt = Ax (I) for t > 0, and iii) {%(/)} is by (6) of
exponential growth when t j co. On the other hand, since %0C О(Л)
and since {LJ is equi-continuous in t 0, we see that x(t) = Т\хС) also
satisfies limx(i) = x0, dxtyjdi = A x(t) for t > 0, and {%(£)} is bounded
when t 0. Let us put x(t) - x(t) - y(Z). Then limy(Z) = 0, dy(t}ldt =
A у (t) for t > 0 and {y (/)} is of exponential growth when t f oo. Hence we
may consider the Laplace transform
00
L{A;y) = f е~*у{1) dt for large positive Re (A),
о
We have
fl	fl	p
f e-^y'fydt — f e~^Ay(t}dt~A f e~uy(t)dt, O^a</J<oo,
л	ex	a
by approximating the integral by Riemann sums and using the fact that
A is closed. By partial integration, we obtain
fl	fl
f е~*‘у'ty) dt = e“^y(j8) — е~Лау{ос) + A f e~A‘y(t) dt,
Л	a
which tends to AL (A; y) as a | 0, ft j oo. For, y(0) = 0 and {y(/?)} is
of exponential growth as p' j co, Thus again, by using the closure pro-
perty of A, we obtain
AL (Л; y) == AL (A; y) for large positive Re (A).
Since the inverse (Al — А)~г exists for Re (Я) > 0, we must have L (A; y) = 0
when Re (A) is large positive. Thus, for any continuous linear functional
11. Fractional Powers of Closed Operators
259
/ С X', we have
oo
J	dt = 0 when Re(A) is large positive,
о
We set Я о 4 гт and put
g^t) =e	or = о
according as t > 0 or
t < 0.
Then, the above equality shows that the Fourier transform
OO
(2л;)-1 f e~iTt ga(t) dt vanishes identically in r, — oo < т < oo, so
—co
that, by Fourier’s integral theorem, ga(t) =* 0 identically. Thus / (y (i)') = 0
and so we must have у (i) = 0 identically, in virtue of the Hahn-Banach
theorem.
Therefore Ttx = Ttx for all t > 0 and %£ D (Л). D (A) being dense in
X and Tt, Tt both belonging to L {X, X),we easily conclude that Ttx = Ttx
for all я £ X and t > 0. Hence, by defining 70 = I, we have Tt = Tt
for all t S; 0. Hence, by (7), T'.x = (2лг)-1 J e-uAR(A\ A) xdA, t > 0,
and so, by (1) arid (5), we obtain
(7?/л)" x = T^x = (2n i)1 f e^AnR(A; A) xdA, t > 0.
ct
Hence
(tT'y. X = (2тп)-1 J enU(tA)n R (A; A) xdA, t>0.
Therefore, by (III),
p ((tT'J* x) (2тг)-1 q(x) J| en^\ tn |Я |n-1 d |Л
c.
If 0 < t 1, then the last integral is majorized by rig where Сг is a
certain positive constant. This we see, by splitting the integration path
C2 into the sum of that in the right half-plane 7?е(Л) > 0 and those in
the left half-plane Re (A) < 0, and remembering the integral representa-
tion of the T'-function.
References. The result of the present section is due to K. Yosida [6].
See also E. Hille [3] and E. Hille-R. S. Phillips [1].
11.	Fractional Powers of Closed Operators
Let X be a В-space, and {Tt; t Sr 0} C L (X, X) an equi-continuous
semi-group of class (Co). We introduce
2Vi f
а— юо
ж < 1),
r ~ 0 (when A < Э),
(1)
17*
260
IX. Analytical Theory of Semi-groups
where the branch of z'4 is so taken that /и1 (Y) > 0 for Re(z) > 0. This
branch is a one-valued function in the z-plane cut along the negative
real axis. The convergence of the integral (1) is apparent in virtue of the
convergence factor e~tza. Following S. Bochner [2] and R. S. Phillips
[5], we can show that the operators
Ttietx = ftx = f //>a(s) Tsxds (/> 0),
0
= x(* = 0),	(2)
constitute an equi-continuous semi-group of class (Co). Moreover, we can
show that {7’J is a holomorphic semi-group (K. Yosida [8] and V. Bala-
krishnan [1]). The infinitesimal generator A = Ал of Tt is connected
with the infinitesimal generator A of Tt by
Aax^-(-A)“x for х€Т>(Л),	(3)
where the fractional power (— А)л of (— A) is given by
(~A)ax =^5-^ Г (2.1-Ay1 (~Ax) dA for x^D(A) (4)
я о
and also by
(—Л)* x = Г^хУ1 f A~“-1 (Л - I) xd2. for x^D(A).	(5)
о
Formula (4) and (5) were obtained by V. Balakrishnan. For the
resolvent of Ал, we have the following formula due to T. Kato :
-	oo	tx
(f,i -Л.Г' =51J {rl - Л)-1 ——У--------------------dr. (6)
71 0	ц —-	cos octi + r
In this way, we see the abundance of holomorphic semi-groups among
the class of equi-continuous semi-groups of class (Co).
To prove the above result, we have to investigate the properties of
the function yjfy in a series of propositions.
Proposition 1. We have
OO
«-'- = /	(<>0,a>0).	(7)
0
Proof. It is easy to see, by the convergence factor that the
function ft>a (A) is of exponential growth in A. By Cauchy's integral theo-
rem, integral (1) is independent of a > 0. Let a > a == Re(z) > 0.
Then, by Cauchy’s theorem of residue,
OO	. cr+toO
0	(7—400
Л—оа
Г*4 dz
4=0
— a
11. Fractional Powers of Closed Operators
261
Proposition 2. We have
/(>ДЛ)^0 for all Л>0.	(8)
Proof. If we set a* = g(a), e~tx == k(x), then
(-1)”-1. g<"> (a) J; 0 (n = 1, 2, . . .), g(«) 2: 0 and
(—l)n> (x) О (я = 0, 1, . . .), when a St 0 and x > 0.
Hence k(a) = h(g(a)) = е~^№ satisfies
(-1)* *>>(«) - (-1) A'(*) (-I)""1 gW(«)
u>)
0, j>Q > 2, > 0,..., p„ St 0 with />0 A = я> and v arbitrary)
> 0, (я = 0, 1,...).
(9)
That is, the function k (a} = 0-te“ is completely monotone in « > 0.
We next prove the Post-Widder inversion formula
/1,.(Л) = 1ш1^)-(у)’+1Л|“>(у), Л>0	(10)
*-*oo ™1	\ л /	\ л /
so that, by (9), ftia(ty > 0. The proof of (10) is obtained as follows. We
find, differentiating (7) «-times,
. 00
k‘~! (t) = (-1)’ f Л-*/иВ
\ Л /	0
Substituting this into the right side of (10), we get
OO
.. «’“+1 1 r rs /. s \“|*
0
Since
lim nnlV2^n e*n\ = 1 (Stirling's formula),
Л—КХ) r
we have to prove that
= J™ f n?,!2 Ц-  exp (1 —	4,«(s) ds, > 0. (11)
Let rj be a fixed positive number such that r/ < z^. We decompose the
last integral into three parts,
oo	ос
f — f + J + f = Ji + + Is-
о о K—n	.
Since x  exp(l x) increases monotonically in [0, 1] from 0 to 1, we see,
262
IX. Analytical Theory of Semi-groups
by the boundedness of /^(s) in v, that lim Д = 0. Next, since
n—>00
x • exp(l — x) decreases monotonically in [1, 00] from 1 to 0, we have
4 ’exP(/“	)<P<1'
and so, since ftia(s) is of exponential growth as s f oo,
IЛI S n3'2 у-" J (^)- exp (- n£) I (s) |	0
n —> 00.
as
By the continuity of Дя($) in s, we have, for any positive number e,
(A)) — £ <	(s) _< ttfX (Aq) + c whenever ?/S x :<	— 77,
if we take 77 > 0 sufficiently small. Thus
where
(A,« (Яо) £) 7o =; /2 = (ft,» (Л)) + £) To •
(12)
(13)

The whole preceding argument is true for the particular case of the
completely monotone function
k(a) = a'1 = f e Ы(1л.
о
In this case, (м/Яо) = (—1)” nl (Я0/и)'‘+1, Substituting this in (10), we
find that (10) holds for /<1Л(Я) = 1. Since (10) and (11) are equivalent,
(11) must also hold for ft л (A) = 1. Thus, since lim Л = 0 and lim Д = 0
*	ИЗО
for a general we obtain
1 = lim l/z?r Л01/0.
IWOO r
Therefore, by (12), we get (11) and the equivalent formula (10) is proved.
Proposition 3.
00
=	(14)
/>+,„« - 7 /..«(A -я) fs,M dp.	(15)
0
Proof. Since the function Да(Л) is non-negative, we have, by the
Lebesgue-Fatou lemma and (7),
Г lim (e-^ft,* (Я)) dA < lim e~ta* = 1.
О л|0	*	дфО
Thus А<Л(Я) is integrable with respect to A over (0, co) and so, again by
11. Fractional Powers of Closed Operators	263
the Lebesgue-Fatou lemma and (7),we obtain (14). We next have, by (7),
u - д) (д) 44«
= p-	(A _ м) d (A _ M) . f	до
=	= «-<'+•>' = f e-“ ft+s^ (Я) Л, a > 0.
Hence, by inverting the Laplace transform as in the preceding section,
we get (15).
Proposition 4. We have
/ £/(,* (Я)/# • Л = 0, i > 0.	(16)
о
Proof. By deforming the path of integration in (1) to the union of
two paths re“,e (—oo < — r < 0) and re'6 (0 < r < oo), where tc/2
0 < tt, we obtain
1 ?
/(„(s) = — j exp (sr  cos 6 — cos oc &)
X sin (sr  sin в — tr* sin ocO + 0) dr.	(17)
Similarly, by deforming the path of integration in
&+too
f	dz
400
to the union of two paths, re~'6 (— oo < — r < 0) and re'6 (0 < r < oo),
we obtain
/ co
ft,* (*) = 8ftia
X sin(
If we take
then

(s)fdt = д j exp (sr  cos 0 — tir cos «0)
зг • sin 0 — tr* sin ocG + oc6 + 0) r*dr.	(18)
0 =	= тг/(1 -J- ос),
((sr + ira) cos 0Я)  sin ((sr — tr^) sin 0Л) radr.	(19)
ft,As)
Thus we see, by the factor ra (0 < oc < 1), that /('a(s) is integrable with
respect to s over (0, oo). Hence, by differentiating (14) with respect to t,
we get (16).
We are now able to prove
Theorem 1.	{Tt} is a holomorphic semi-group.
Proof. That {7\} enjoys the semi-group property TtTs = Tt+s
{t, s > 0) is clear from (2) and (15). We have, by (2) and (17) with 0 = 6a,
I OO	°°
Ttx = — J Tsxds f exp((sr + ir“) cos 0Я)
’ X sin ((sr — ir") sin 0Я + 0Я) dr,	(20)
264
LX. Analytical Theory oi Semi-groups
which gives by the change of variables
s = г и.г11л,	(21)
00	OU
Ttx = — J Tvtiix  xdv / exp((wv + «*) cos 6a)
я 0	0
X sin ((му — «“) sin 0a + 0„) du.	(20')
The second integral on the right is exactly л • Д * (w), and so, by the
equi-boundedness of {|| Ttx ||} in t 0, we see, by (14), that
sup [|T(x|| f /iiCt(v) dv — sup ЦТ^хЦ. (22)
f'frO	о ’	c>o
By passing to the limit £ j, 0 in (20'), which is justified by the integra-
bility of /iFa(w) over [0,oo), we obtain, by (14),
OO
s-lim Ttx = f dv  x = x.
^0	о
Hence {'/’/} is an equi-continuous semi-group of class (Co) such that (22)
holds.
By the integrability of fji£l(s)	over [0, oo) and the equi-
continuity of {Tt}, we obtain, by differentiating (2) with respect to t
under the integral sign,
OO
r.xds
0
— — f Tsxds f exp(($r 4- fr*) cos 0Я) • sin((sr — £гя) sin 03 rxdr,
л о о
(23)
which is, by the change of variables (21),
OO
= / (ЛД/«  x) • f1>cAy) dv  Г1.
о
Thus, by the integrability of (v) over [0, oo) and the equi-continuity
of {Tt} in t > 0, we see that
lim 11T*t\| < oc,
qo 1	11
that is, {Tt} is a holomorphic semi-group.
Theorem 2.	The infinitesimal generator ЛЛ of Ti is connected with
the infinitesimal generator A of T by (3), where (—А)л is defined by (4)
and also by (5). We have, moreover, (6).
11. Fractional Powers of Closed Operators
265
Proof. By (16) and (23), we obtain
oo	oo
T'tx = — f (Ts - - Z) xds<J exp ((sr + ir*) cos
о	0
X sin ((sr — tra) sin 0a) radr.
(24)
If x 6 D (A), then s-lim s-1 (Ts -1) x ~ Ax and 11 (Ts — I) • x I [ is boun-
s|0
ded in s 0. Thus, we obtain, letting t ф 0 in (24),
OO	DO
s-lim T'tx = - f (Ts — I) x ds f exp(sr  cos &л)  sin (sr • sin 0a) r* dr
ЦО	я 0	0
oo
= (-Г (-«1Г1 f
0
because, by the Г-function formulae
P(z) = cs J e'cr r1"1 dr (Re (z) > 0, Re (c) > 0)	(25)
о
and
Г(г) Г(1 — z) = тг/sin nz,	(26)
we obtain, by (« + 1)	= я,
1 00	00	'
— f exp (sr - cos 0Л) • sin (sr • sin 0л) radr — Im • f e~r^~set r*dr 
n о	[o
= (лг)-1 Im((- ^Л'Гл~’) Г(1 + a) - s-'x’1 л-1 sin («л) Г(1 + «)
_ c-«-i___Г(—«IV1 s~“-1
Г (— л)Г(1 + a) ~
Thus, by T\x = A^TfX (when t > 0), the continuity at t = 0 of Ttx and
the closure property of the infinitesimal generator Ал, we obtain
OO
A* % = (—Г(—a))"1 f 5""а“г(Т5 —Г) xds when x£D(A),
о
oo
Hence, by (25)^ (26) and (tl — ^)-1 = /	we obtain
о
OO f 00
e-s‘Fdt\(I — Ts) xds
f ta ((tl — Ap1 - - t~xT) xdi
oo
Sinn'xл J ta~l (il - A)-1 A xdi for x € D (A).
266	IX. Analytical Theory of Semi-groups
Finally we have, by taking 6 = л in (17) and (2),
oo
GuZ-Л)-1 = J e~fitTldt
0
OO OO	OO
= я'1 f dr f e~srTsds f exp (— pt — tr* cos осп} • sin(/r“ sindt
0	0	0
oo
= л'1 f (rZ —Л)”1
0
exp(—pt — if* cos o) ‘ sin^r* sin ал} dn dr
 oo	<%
sin ал г t jr л \ — i	j
-—— / [rl — A) 1 ——————r dr.
"W j *	<	2 л ct	। 2л
л о	/* — 2r fi COS a.71 4- r
Remark. Formula (2) was devised by S. Bochner [2] without de-
tailed proof. Cf. R. S. Phillips [5]. That Tt is a holomorphic semi-group
was proved by K. Yosida [8], V. Balakrishnan [1] and T. Kato [2].
Formulae (4) and (5) are due to V. Balakrishnan [1], who, by virtue of
(4), defined the fractional power (—A)a of a closed linear operator A
satisfying the condition:
the resolvent R{X; A) = {XI - .4)’1 exists for Re{X) > 0
and sup |Re(A) |  |[2?(Л; Л) || < oo.	(27)
Р«(Л)>0
He also proved that (—Лр enjoys properties to be demanded for the
fractional power. In fact we have
Theorem 3. Let a closed linear operator A satisfy condition (27). Then,
by (4), a linear operator (—is defined, and we have
(-A)a (—Л)д x - (— A)a+fi x if x^D{A2) and 0<л,/3
with a + fi < 1,	(28)
s-lim (—Л)л x — — Ax if x£D(A)>	(29)
afl
s-lim (—Л)“ x~. x if s-lim XR{X; A) x = 0,	(30)
and, if A is the infinitesimal generator of an equi-continuous semi-group
Tt of class (Co),
(A^ = A^, where Aa is the operator Aa defined through
(31)
Kato’s formula (6).
Remark. The last formula (31) is due to J. Watanabe [1].
Proof. ||ra“1 {rl — X)-1 (— A x) || is, by (27), of order O^-2) when
r f oo, and, by (rZ—Л)"1 (—Ax) — x— r(rl — А)~гх and (27), it is
of orderOfr®^1) when r | 0. Thus the right side of (4) is convergent.
11. Fractional Powers of Closed Operators
267
It is clear that x£ D(A2) implies (— A }px C D (A). For, by approximat-
ing the integral by Riemann sums and making use of the closure property
of A, we have (—A)^x C D(A). We can thus define (—A)“(—A)&x.
o oo oo
(—A)* (--A/%_<3in^Sm/-- f f A^~1fia-1R(A;A)R(ji;A)AzxdAdfi
can be rewritten, decomposing the domain of integration into those for
which A > fi and A < fi, as follows.
• a 1
Sin <X7t Sin рЛ Г
л	0
(«A1 + c/*-1) da f A*^"1 Я(Аа; А) R(A; A) A2xdA.
о
By the resolvent equation R(A't A) — R(/r, A) = (ji — A) R(A‘, A) R(/i; A)
and R(A; A) (—A) = I -AR(X; A) valid on D(A), we obtain
R(Aa; A)R(A; A) A2x = (1 — ^{-oRfAcr; A) + R(A; A)} (-Ax),
and hence
(—A)a (—A)^ x
=	s-lim f (</ 1 j-
Л л f f 1 J
0
oo
X f (-aR(Acr; A) + R(A; A)) (-Ax) dA
Q
sin ал sin /?л Г tZ”1 +	1 — a-£* — a
л n J	1 — a
0
X	A)(-A%) Л.
The coefficient ( ) above is evaluated as л-1 sin л (a 4- /?), as may be
seen by expanding (1 — o’)1 into powers of a. We have thus proved (28).
00
To prove (29), we shall make use of J 1“-1 (1 4- л)”1^/ = л/этссл.
о
Thus
oo	1
(-A)“ X - (—A) X = S-^ J A*’1^; A) -jprj) (-Ax) Л.
We split the integral into two parts, one from 0 to C and the other from
C to oo. For a fixed C, the first part goes to zero as a f 1, since
R(A; A) (—Ax) = x —A) x is bounded in A > 0. The second part
is, in norm,
(вдЛ}-щ)ах .
We haves-lim AR(Я; A) x = x by x — AR{A’> A) x ~ R(A\ A) (—Ax) and
Л foo
(27). Hence s-lim of the second part is arbitrarily near 0 if we take C
«ii
sufficiently large. This prove ’ (29).
sin ал „j
268
IX. Analytical Theory of Semi-groups
To prove (30), we split the integral into two parts, one from 0 to C and
the other from C to oo. Because of (27), the second part tends to zero
as tx j 0. By R(X; A) (— A x) = x — лЛ’ (Л; A) x and the hypothesis
s-limA2?(Л; Л) x — 0, the first is, for sufficiently small C, arbitrarily
near the value (ал)"1 sin octi  Cxx, which tends to x when a | 0. This
proves (30).
We shall prove (31). By virtue of the representation (6), we obtain
oo co	.
0*1 - И-W 1 = / / (2'2 (7277.7^ - TZ/Tisa)
x trFa
This double integral is absolutely convergent in norm, and so we may
interchange the order of integration. Hence we obtain (31), because the
inner integral is
= (2 л i) -2 f—[ ------—dz
J fz - / V-
where the path of integration C runs from oo etn to 0 and from 0 to
oo
An example of the fractional power. Их-- 1/2, then we have, by
taking 0 = я in (17),
Z.i/2 (s) = J e~sr sin (tr1!i) dr == л~1 ]/nt (23]/s]~Z  e~tIf4s. (32)
о
Thus, if we take the semi-group {TJ associated with the Gaussian kernel*.
л	00
(Ttx) («) =
then
—oo
oo
oo
^—2	dv
4л s4
—oo
0
oo
—oo
that is, the semi-group {Tt} is associated with the Poisson kernel. In this
case, the infinitesimal generator A of is given by the differential opera-
tor d^/ds2, while the infinitesimal generator A of Tt is given by the singular
integral operator
xi \ t 1 f x—-y) —x(s) J
(4w»)(s) = s-lnn- J --------------------dv,
—OO
and not by the differential operator djds. For another example, see
K. Yosida [30].
12. The Convergence of Semi-groups. The Trotter-Kato Theorem 269
12. The Convergence of Semi-groups. The Trotter-Kato Theorem
We shall denote by exp(M) the semi-group of class (Co) with the
infinitesimal generator A. Concerning the convergence of semi-groups,
we have
Theorem 1. Let X be a locally convex, sequentially complete complex
linear space. Let {exp Q L (X, X) be a sequence of equi-continuous
semi-groups of class (Co) such that the family of operators {ехр(/Л„)} is
equi-continuous in t > 0 and in n = 1, 2, . . . Thus we assume that, for
any continuous semi-norm p {x) on X, there exists a continuous semi-norm
q(x) on X such that
p (exp (Mn) %) q (%) for all t 0, x G X and w = 1, 2, . . . (1)
Suppose that, for some with Re (л0) > 0,
lim R (Ao; Лл) x = J (Ло) x exists for all x G X in such
a way that the range 7? (J (Aq)) is dense in X.
Then J (Яо) is the resolvent of the infinitesimal generator A of an equi-
continuous semi-group exp(M) of class (Co) in X and
lim exp(/Л„) x = ехр(/Л) x for every x£X.	(3)
WOO
Moreover, the convergence in (3) is uniform in t on every compact interval
of t.
For the proof, we prepare
Lemma. Let T. = exp(L-l) be an equi-continuous semi-group of class
(Co) in X. Then, for any continuous semi-norm p(x) on X, there exists a
continuous semi-norm q{x} on X such that
p(Ttx- (I — tiT1 АГп x) (2«)“Ч2 q(A2x} [n = 1, 2, . . .)
(4)
whenever x G D (Л2).
Proof. Set T(t,n) = (I	n. Then we know (Chapter IX, 7)
that {T(t, n)} is equi-continuous in t 0 and n = 1, 2, . . . Moreover,
we have (Chapter IX,4), for any x G £>(Л),
DtT(t,n) =	Ax = A (I -n-HA)-^1 x,
DtTx = TtA x - A Ttx.
Thus, by the commutativity of Tt and T(t, n),
I
Ttx — T(t, n) x = f [DsT(t — s, n) Tsx]ds
t ° <5>
/	/	ft ____ c	\
= f T(t  s,n)Tx\Ax — (l-------/ Axjds, x£ D(A).
270
JX. Analytical Theory of Semi-groups
Hence, if x^D(A2), we have, by (/ -ж 1 T)_C4 x - - ш (I—
(f — wr^A^^x,
/Г	C — f “1
p(Ttx~T(t.n)x)^ J p T(t~stn}Ts(l~n-^t-s)Ay^ — A2x\dsf
0 L	«J
and so, by a continuous semi-norm qix} on X which is independent of x
and n,
p (Ttx — 7'(Z, n) x) (2n)~1t2q(A2x).
Corollary. For any x £ D{A2), s > 0 and t 2' 0,
p(Ttx — (7 — sX)“Ws] x) 4(zk) + ~ q(A2x), where
Ы
qr (x} is a continuous semi-norm on X which is indepen-
dent of x, i and s, and [7/s] is the largest integer ж Z/s.
Proof. We have, for t = ns,
P(T^x-(I- sAYn x) 2-1 stq^x}.
If t = ns ~i~ и with 0 u < s and n = [i/s],
р^х—Т^х} —p( fT'axdo\<L f p(TaAx) da £. sqv(Ax).
\ns	/ П$
Proof of Theorem 1. By (1) and by (11) in Chapter IX, 4, we see that
{(Re (z) 7?(Я; A„)*”)} is equi-continuous in Re (A) > 0, in n = 1, 2,. . . and
in ж = 0, 1, 2, . . . From this and by (2), we can prove that J(Я^) ==
(AqI — /1)‘ 1 with some A and
lim A’(z; An) x = R(A; A) x whenever 7?е(Л) > 0, and
the convergence is uniform with respect to A on every com- 
CO
pact subset of the right half-plane Re (A) ’> 0.
To this purpose, we observe that
R(A; An) x= J (Яо-ЯГ Я(Я0; A)w+1 *	| A - Яо \\Re (Ло) < 1)
and that the series is, by the equi-continuity of {(2?e (Яо) 7? (Я^>; 4*))ж} in
n = 1, 2, . . . and wz — 0, 1, 2, .. ., uniformly convergent for [Л — Aq\}
Re(/^ > 1 —sandw = 1, 2, . . , Here e is a fixed positive number. There-
fore, for any d > 0, there exists an m0 and a continuous semi-norm q(x)
on X such that, for |Я — \/Re (Aq) 1 — e,
p(R(A-,An)x-R(A;An.)x-)^	|^-Я|ж.
m-=0
p(R(AQ; An)^+1x~R^; An^ x) + 2<5 q(x) for ah x£X.
12. The Convergence of Semi-groups. The Trotter-Kato Theorem 271
Hence, by (2), we see that lim А (Л; Л„) % — J (Я) x exists, uniformly for
rt—ИХ
[Я — Яо \[Re (Яо) ‘Al — s. In this way, extending the convergence domain
of the sequence {7?(Я; Ял)}, we see that
lim R (Я; A) x = J (Я) x exists and the convergence is uniform
on any compact set of Я in the right half-plane Ае(Я) > 0.
Thus J (Я) is a pseudo-resolvent, because J(Я) satisfies the resolvent
equation with R (Я; A„) (n = 1, 2, . . .). However, by R (J (Яо))л = X and
the ergodic theorem for pseudo-resolvents in Chapter VIII, 4, we see that
J (Я) is the resolvent of a closed linear operator A in such a way that
7(Я) = Л(Я; A) and D(A) = R (Я(Я; Л)) is dense in X.
Thus we see that exp (M) is an equi-continuous semi-group of class
(Co) in X. We have to show that (3) is true. But we have, by (6),
/>((exp(M.) -	(Z-
. S	 Л„)-2я) + 2-1isy(A2(I — A„)~2x),
for any x E X, s > 0 and t > 0.
The operators
and A2(I~ A„Y* = (An[I — An)~^
axe equi-continuous in n = 1, 2, . . . On the other hand, by (7),
lim' {I - s(I - X„)-2 x = (I - sA) (I - A)~2 x
tt—ЮО
uniformly in s and t, if s > 0 is bounded away from 0 and oo, and if t
runs over a compact interval of [0, oo). Moreover, we have, by (6),
ф ((exp (M) - (/ - s Л)~№]) (I - A}~2x]	s?1 (A (1 - A)~2x)
+ 2~iisq (A2 (I — A)~zx)
for every x С X, s > 0 and t 0. Thus, by taking s > 0 sufficiently
small, we see that
lim exp(M„J у ~ exp(M) у for any у £ R(l; A)2 • X,
n—XX>
and the convergence is uniform in i on every compact interval of t.
1?(1;Л)2 • X being dense in X, we see, by the equi-continuity of exp
(M) and exp(A4„) in t 0 and in n = 1, 2, . . ., that (3) holds.
Theorem 2. Let a sequence {exp (t A „)} of equi-continuous semi-groups
of class (Co) in X be such that {exp (£.4„)} is equi-continuous in t A: 0 and
in и = 1, 2, . . . If, for each x g X,
lim exp(Mw) x = exp(M) x
№—*O0
IX. Analytical Theory of Semi-groups
272
uniformly with respect to t on every compact interval of t, then
lim A„) x = (Я; /1) x for each Xz X and /?в(Л) > 0
and the convergence is uniform on every compact set of A
in the right half-plane Re (A) > 0.
Proof* We have
R(A; A) x — R(A; A„) x = f e~u (exp(£A) — exp (tA„)) xdt.
о
Hence, splitting the integral into two parts, one from 0 to C and the
other from C to oo, we obtain the result.
Remark, For the case of a Banach space X, Theorem 1 was first
proved by H. F. Trotter [1], In his paper, the proof that J (A) is the
resolvent R (Л; Л) is somewhat unclear. This was pointed out by T. Kato.
The proof given above is adapted from Kato's modification of Trotter’s
proof. For perturbation of semi-groups, see E. Hille-R. S. Phillips [1],
T. Kato [9] and K. Yosida [31].
13, Dual Semi-groups. Phillips’ Theorem
Let X be a locally convex sequentially complete linear topological
space, and	Z.L(X, X) an equi-continuous semi-group of
class (Co). Then the family {T*; t <- 0} of operators E L (A", A?') where (♦)
denotes the dual operators in this section, satisfies the semi-group
property: T*T* = T*+s> T* = I* — the identity in X's (see Theorem 3
in Chapter VII, 1). However, it is not of class (Co) in general. For, the
mapping T* does not necessarily conserve the continuity in i (see
Proposition 1 in Chapter VII, .1). But we can show that {T*} is equi-
continuous in t 0. For, we can prove
Proposition 1. If {$,; t 0} g L (X, X) is equi-continuous in t Y 0,
then {S*; t 2; 0} g L (Xf, A') is also equi-continuous in t > 0.
Proof. For any bounded set В of X, the set Uq S(  В is by hypothesis
a bounded set of X. Let U' and V' be the polar sets of В and St - В:
U' = {x'eX'; sup | gl], Г =	X'- sup |<Sr 6,x7> j 1}.
b£B	1>(8,Гг0
Then (see Chapter IV, 7), U' and V are neighbourhoods of 0 of Xj. From
|<St • b, x'y [ = | < b, S* V>| g 1 (when Ь^В, x’ € Г)
we see that S* • V' £ U' for all i 2_ 0. This proves that {St*} is equi-
continuous in t -P 0.
Let A be the infinitesimal generator of the semi-group Tt. Then
D(A)“=X, 2?(Л)^Х, and, for A > 0, the resolvent (AI-AY^C
L (X, X) exists in such a way that
{Лт(Л/ — Лр”’} is equi-continuous in A > 0 and in m = 0, 1 . . .	(1)
We can prove (cf. Theorem 2 in Chapter VIII, 6)
13. Dual Semi-groups. Phillips' Theorem
273
Proposition 2. For Л > 0, the resolvent (AZ* — Я*)-1 exists and
(AZ* — A*)-1 = ((AZ-A)"1)*.
(2)
Proof. We have (AZ — A)* = AZ* — A*. Since (AZ - A)-1 € L [X, X),
the operator ((AZ -A > T* {-Z (X', X'j exists. We shall prove that
(AZ* — A*)-1 exists and equals ((AZ — A)1)*. Suppose there exists an
x' E X' such that (AZ* — A*) x' = 0. Then 0 = <x, (AZ* — A*) x'> =
<(AZ — A) x, x'y for all X Z)(A). But, since R(AZ — A) = X, we must
have x' = 0. Hence the inverse (AZ* — A*)-1 must exist. We have, for
x£X, х'£О(А*),
<%, %'> = <(AZ - A) (AZ - A)-1 x, x'y = <(AZ - A)-1 x, (AZ* - A*) x'y.
Thus Z)(((AZ-A)-1)*)2R (AZ*-A*) and((AZ-A)"1;*.(AZ* A*)x'-T
for every x' EZ)(A*). This proves that ((AZ — A)-1)* 2 (AZ* —A*)-1.
On the other hand, if x E D(A) and x' E Z)(((AZ — A)'1)*), then
<x, x'y = <(AZ —A)-1 (AZ —A) x.x'y = <(AZ-A) x, ((AZ - A)-1)* x'y.
This proves that Z>(A*) = Z>((AZ — A)*) 2 7?(((AZ — A)-1)*) and
(AZ — A}* • ((AZ — A)-1)* x' = x' for every x E Z)(((AZ — A)”1)*), that
is, ((AZ — A)*1)* C (AZ* — .4*1' We have thus proved (2).
We are now ready to prove
Theorem. Let X be a locally convex sequentially complete linear
topological space such that its strong dual space X' is also sequentially
complete. Let {TJ C L(X, X), i 0, be an equi-continuous semi-group
of class (Co) with the infinitesimal generator A. Let us denote by X+ the
closureD (A*)a of the domain D (A*) in the strong topology of X’. Let Tf
be the restriction of T* to X' . Then Т/ € L (X+, X‘“) and {7'2 ; t X 0}
is an equi-continuous semi-group of class (Co) such that its infinitesimal
generator A+ is the largest restriction of A* with domain and range in X+.
Remark. The above Theorem was proved by R. S. Phillips [2j for
the special case of a ZLspace X. The extension given above is due to
H. Komatsu [4].
Proof of the Theorem. We have the resolvent equation Z?(A;A) —
R(p; A) = (/z — A) R(A; A) R(u; A) and the equi-continuity of
{AWR(A; A)w} in A > 0 and in m = 0, 1, 2, .. . Thus, by Proposition 1
and 2,
(AZ* - A*)-1 - (/zZ* - A*)-1 = & - A) (AZ*-A*)-1 (/zZ* —A*)-1 (3)
{AW(AZ* — A *)"”*} is equi-continuous in A > 0 and in m = 0,1,2,... (4)
Thus, if we denote by J (A) the restriction to X+ of (AZ* — A*) we have
7W-7(^) = (^-A)7(A)7W,
(3')
{Am / (A)"*} is equi-continuous in A > 0 and in m — 0, 1, . . .	(4')
18 Yotida, Funuthmftl, Aimlynik
274
X. Compact Operators
Since D (A *) is dense in X * and since (4') holds, we see, as in Chapter IX, 7,
that Jim X J (X) x ~ x for л i. A''. '1'hus we have R(J (Л))® = A'+ and
so, by (7') in Chapter VIII, 4, that N(J(X)') = {0}. Thus the pseudo-
resolvent J (Я) must be the resolvent of a closed linear operator A+ in X+.
Hence, by the sequential completeness of X+ and (4'), A+ is the infinite-
simal generator of an equi-continuous semi-group of class (Co) of operators
Tp £ L (X+, X+). For any x£ X and y' € X+, we have
{(1 — тгЧАГтх, /> = <x,
and so, by the result of the preceding section, we obtain by letting
m -> oc the equality (Ttx, y') = <x, F(+ y'y. Hence T* y' = Tp y', that
is, Tp is the restriction to X+ of 7*.
We finally show that A+ is the largest restriction of A* with domain
and range in X+. It is clear, by the above derivation of the operator A+,
that A+ is a restriction of A*, Suppose that x' £ D (A*) and that x' С X+,
A* x' € X+. Then (XI* - A*) x1 E X+and hence (Л7* - A+)-l(A7* — A*)
x' =%'. Thus, applying (XI* — A+) from the left on both sides, we obtain
A*x' = A+x'. This proves that A+ is the largest restriction of A* with
domain as well as range in X+.
X. Compact Operators
Let X and У be complex В-spaces, and let S be the unit sphere in X.
An operator T € L (X, Y) is said to be compact or completely continuous
if the image T • S is relatively compact in У. For a compact operator Tg
L (X, X), the eigenvalue problem can be treated fairly completely, in
the sense that the classical theory of Fredholm concerning linear integral
equations may be extended to the linear functional equation Tx--Xx-- у
with a complex parameter X. This result is known as the Riesz-Schauder
theory. F. Riesz [2] and J. Schauder [1].
1. Compact Sets in B-spaces
A compact set in a linear topological space must be bounded. The
converse is, however, not true in general; we know (Chapter III, 2) that
the closed unit sphere of a normed linear space X is strongly compact
iff X is of finite dimension. Let S be a compact metric space and C (S]
the В-space of real- or complex-valued continuous functions r(s) on S,
normed by ||x|| = sup ]x(s) J. We know (Chapter III, 3) that a subset
sCS
{хл($)} of C (S) is strongly relatively compact in C (S) iff is equi-
bounded and equi-continuous in a. For the case of the space IP (S, S3, w),
1 == P < 00 > we have
1. Compact Sets in B-spaces
275
Theorem (Frechet-Kolmogorov). Let S be the real line, 35 the
cr-ring of Baire subsets В of S and m (B) -- J dx the ordinary Lebesgue
в
measure of B. Then a subset К of LP(S, 33, m), 1 p < oo, is strongly
pre-compact iff it satisfies the conditions:
sup ||x|| = sup / f |x(s)	< oo,	(1)
ж£Х	лк \s	/
lim f |x(i + s) — x (s) |^ ds = 0 uniformly in x £ К,	(2)
s
lim f |x(s) p ds = 0 uniformly in x^K.	(3)
“too |s|<.a
Proof. Let К be strongly relatively compact. Then К is bounded and
so (1) is true. Let e > 0 be given. Then there exists a finite number
of functions € Lp : Д, /2, • • -, fn such that, for each f£.K, there is a /
with | / — fj 11 e. Otherwise, we would have an infinite sequence {/J £ К
with | f.j — Д-j j > e for j i, contrary to the relative compactness of
K. We then find, by the definition of the Lebesgue integral, finitely-
valued functions glf g2, • • • > such that ЦД- — gj|| e (/= 1, 2,..., w).
Since each finitely-valued function g}- (x) vanishes outside some sufficiently
large interval, we have, for large a,
(OO —a	\1/P	/ OO —л	\ l//>
f + f i/COl^s SU + / |/(s)~gj<s)psl
a — OO	/	\a	“OO	/
(co —a	\ Ijp	/ OO —a	\AIP
f + f |g;(s)^s ^Il/-gfll+ / + f UAS)^S| •
a —OO	/	\<x —oo	/
This proves (3) by ||/ — gy|| [[/— /> | [ + ||/; — gy|[ ё 2а.
The proof of (2) is based on the fact that, for the defining function
C/(s) of a finite interval 7, lim f |C/(s -f- t)— CI(s)lfpds = 0 (see
— oo
Chapter 0, 3). Thus (2) holds for finitely-valued functions gy(s)
(/ = 1, 2, . . ., n). Hence we have, for any j^K,
lim/ J |/(s + t) — f (s) I* (fs'j ^lim/ / |/(s + i)~/,($-H)
*-*O \-co	/	\—oo	/
(oo	\l//>	__ f oo	\l/£
f + gj (s + 0 +Jhn / |gj(s4-^-gy(s)l^^s
—oo	/	"° oo	/
(oo	\i//>	/ oo	\up
f |gj(5)—Zf(s)	+( f |A(S)—+ O + s +
by taking fj in such a way that || / — 41|	£• This proves (2).
We next prove the converse part of the Theorem. We define the trans-
lation operator Tt by (Ttf) (s) =/(£-)-$). Condition (2) says that
18*
276	X. Compact Operators
s-lim Ttf — f uniformly in f£K. We next define the mean value
+0
a
(Maf) (s) = (2a)'1 f (Ttf) (s) dt. Then, by Holder’s inequality and
—a
the Fubini-Tonelli theorem,
i/fi
t	\llp
S(2«r’ / f +	*
\—oo —a	/
((2a)-1 fdt f |/(s-H)- /($)|^sV if lg/><oo.
\	—a — OO	/
Thus we have
so that s-lim Maf = f uniformly in ftK. Therefore, we have to prove
the relative compactness of the set {MJ; ftK} for a sufficiently small
fixed a > 0.
We will show that, for a fixed a > 0, the set of functions {(Maf) (s);
/€ K} is equi-bounded and equi-continuous. In fact, we have, as above,
I M./) (Si) — (AT./) (Sb) I •= (2a)-i / |/(s, + Z) -/(s2 + t) | dt
—a
< (2s)-1 / l/fe + t) -/(s, +
\	—л	/
Thus, by (2), we have proved the equi-continuity of the set of functions
{(MJ) (s); ft K] for a fixed a > 0. The equi-boundedness of the set may
be proved similarly. Thus, by the Ascoli-Arzela theorem, there exists, for
any positive a > 0, a finite number of functions MJlf Maf2, . . Mafn
with /j- G К (j = 1, 2, . . ., w) such that, for any ftK, there exists some f
for which sup [ (Maf) (s) — (Мя/у) (s) | g s. Therefore
ЛГ.4-1К < / I (MJ) (s) - (MJ,) (s) I* ds
— a
|s[>a
(4)
The second term on the right is, by Minkowski’s inequality, smaller than
The term 11MJ — / j| is small for sufficiently small я > 0, and, by virtue
of (3), f |f (s) — (s) I* ds and J" J/y(s) —	(s) \p ds are both
ls!>oc	l5l><v
2. Compact Operators and Nuclear Operators
277
small for sufficiently large cc > 0, if a > 0 is bounded. Also the first term
on the right of (4) is 2 oc eP for an appropriate choice of ;. These esti-
mates are valid uniformly with respect to Thus we have proved
the relative compactness in of the set {Maf; K} for sufficiently
small a > 0.*
2. Compact Operators and Nuclear Operators
Definition 1. Let X and У be В-spaces, and let S be the unit sphere of
X. An operator T f L (X, У) is said to be compact or completely continuous
if the image T • S is relatively compact in У.
Example 1. Let К (x, y) be a real- or complex-valued continuous
function defined for -	<.1:4’ < Л < 00. Then the integral opera-
tor К defined by
ь
= f K(x,y)f(y)dy	(1)
a
is compact as an operator € L (C [a, ft], C [a, ft]).
Proof. Clearly К maps C [a, ft] into C [a, ft]. Set sup |X(x, у) | = M.
x,y
Then || К • /1| (ft — a) M ||/|| so that К • S is equi-bounded. By
Schwarz’ inequality, we have
ь	ь
I W) (*1) - (^7) (*2) II 2 * * s f IK(xi> y)~K(x2’ y) \2dy- J |/(y) I2 dy,
a	a
and hence К  S is equi-continuous, that is,
lim sup J (K f) (xj — (Kf) (x2) I — 0 uniformly in f£S.
Therefore, by the Ascoli-Arzela theorem (Chapter III, 3), the set К  S is
relatively compact in C [a, ft].
Example 2. Let К (%, у) be a real- or complex-valued SB-measurable
function on a measure space (S, S3, m) such that
// IA(x, y) |2 m(dx) m(dy) <00.	(2)
s s
Then the integral operator К defined by the kernel К (x, y):
(Kf) (x) = fK(x, y) f(y) m(dy), ft L*(S) = L*(S, SB, m),	(3)
s
is compact as an operator C L(L2(S), L2(S)). The kernel K(x, y) satis-
fying (2) is said to be of the Hilbert-Schmidt type.
Proof. Take any sequence {/„} from the unit sphere of L2(S). We have
to show that the sequence {K  fn} is relatively compact in £2(S). Since a
Hilbert space L2 (S) is locally sequentially weakly compact, we may assume
that {/w} converges weakly to an element /6 L2 (S); otherwise, we choose a
suitable subsequence. By (2) and the Fubini-Tonelli theorem,
* See also Supplcmrnhn v Noles, p. !(>«
278
X. Compact Operators
f |K{x, y) |2 m(dy) < oc for m-a.e. x. Hence, for such an x,
s	_____
lim {Kfn) (*) = lim f K{x, y) /„(y) m{dy) = lim (/„(.), K{x,.)) =
Л-КЮ	WOO g
(/(.), К{x,.)) = f К{xf y) f (y) m{dy). On lhe other hand, we have, by
s
Schwarz’ inequality,
\(Kfn) {x) |2g f\K{x,y) \*m{dy)-J |/M(y) \2m{dy)^ f \K{x, y) \*m{dy)
s	s	s
for w-a.e, x.	(4)
Hence, by the Lebesgue-Fatou theorem, lim f \{Kfn) {x)\2m{dx) =
s
f | (Kf) {x) \2m{dx). This result, if combined with zc-Iim К  fn — К  /,
5	n—изо
implies s-lim К • fn = К - / by Theorem 8 in Chapter V, 1. But, as proved
woo
above in (4), we have
J” [ {Kh) {x) |a m {dx) < ff |K{x, у) [2 ж {dy) m{dx) • f (Л (y) |2 m{dy),
s	s s	s
and so
(b)
11К [ | < ff | К (x, y) |2 m {dx) m {dy)'^'2 •
Hence, from да-lim /„ = /, we obtain да-lim К • /„ = К • /, because, for
n—MX	woo
any g € £2(S), lim (K  fn, g) = lim (/„, K*g) = (/, /<*g) = {K-f, g).
Theorem, (i) A Unear combination of compact operators is compact,
(ii) The product of a compact operator with a bounded Unear operator
is compact; thus the set of compact operators E L {X, X) constitutes a
closed two-sided ideal of the algebra L {X, X) of operators, (iii) Let a
sequence {Tn} of compact operators 6 L {X, Y) converge to an operator
T in the sense of the uniform operator topology, i.e., lim IIT — Tn = 0.
Then T is also compact.
Proof, (i) and (ii) are clear from the definition of compact operators-
The closedness, in the sense of the uniform operator topology, of the ideal
of compact operators in the algebra L {X, X) is imphed by (iii).
We shall prove (iii). Let {xA} be a sequence from the closed unit sphere
S of X. By the compact property of each T„, we can choose, by the dia-
gonal method, a subsequence {xA.} such that s-lim 7n%y exists for every
Л-+ОО
fixed n. We have
TxAJ| ||Txr— Tnxv || + \\Tnxhr— TnXft-|| + ||Tn^— T^||
and so lim ||T • xh> - T 	2 j|T- 7\|j. Hence {TxA-} is a
AFA—MX
Cauchy sequence in the В-space У.
2. Compact Operators and Nuclear Operators
279
Nuclear Operator. As an application of the Theorem, we shall consider
the nuclear operator introduced by A. Grothendieck [2].
Definition 2. Let X, Y be 2?-spaces and T G L(X, Y). If there exist a
sequence {/^} G X't a sequence {yM} £ Y and a sequence {cft} of numbers
such that
sup
n
co, |crt| < oo and
ft
T  x = s-lim £ cn (x, fny yn in У for every % £ X,
m—изо
(6)
then T is called a nuclear operator on X into У.
Remark. The existence of the s-lim in (6) is clear, since
к
I m
. g <x- t'i> Уз
m
’ 114' 11' 11 Уз 11 = constant •	| Cj |  11 x 11.
J =»
The nuclear condition says that the s-lim is equal to T • x for every
x £ X.
Proposition. A nuclear operator T is compact.
Proof. Define the operator Tn by
ft
Tnx = G <x, /J> yj.
3~1
(П
Since the range R (Tn) is of finite dimension, Tn is compact as may be
proved by the Bolzano-Weierstrass theorem. Moreover, by (6) and
| Tx — Tnx
oo
oo
constant | [ •

= 0 and so T must be compact.
we have lim IT
n—>oo
An Example of the Nuclear Operator. Let G be a bounded open do-
main of Rn, and consider the Hilbert space Hq(G). Suppose (k — /) > n.
Then the mapping T
WS(G)3?>->9>eff0(G)
is a nuclear operator G L(H$(G), Hjq(G)).
(8)
Proof. We may assume that the bounded domain G is contained in the
interior of the parallelogram P:
0 <i Xj 2 л (/ = 1, 2,. . ., n).
We recall that H%(G) is the completion of Hq(G) = Cg(G) with respect
to the norm ||дз|[А = / £ f |Ds<p(x) |2 dxV^2 (see Chapter I, 10). We
extend the functions GHq(G) to be periodic with period 2л in each
variable xs by defining the function values as 0 in P - G. The functions
/Дх) = (2л)~п/2 exp(r^  x), where /5 = (&, 02, - - ., /?„)
ft
is an «-tuple of integers and ft  x — (}sxs,
S=1
(9)
280
X. ( oinpjf t ()pri a tors
form a complete orthonormal syshtm of /?(P) = Hq(P). Thus, denoting
by Ds the distributional derivative, we have, for |s| < k, the Fourier
expansion in L2(P) of functions /)xtp(x) where <p £ #o(Q :
Ds<p(x) = £ (D>, /Д, ffll where /₽)0 = J y(*) fp W dx- (10)
?	p
We have, by
(^V. Wo = (—I)1*1 ((P.	= П w- {</>, ff)„
and the Parseval relation
X](WWol2 =	IWII (|s|S*),
p	p
the inequality
j(p, (1 + \0 |2)A/2 Л)о I2 constant У I (Ds<p, ffi)012 g constant ||<p ||t
Therefore the functional fp £ H^(Gy defined by
<f.fe> = {4>. (1 + PI2)*'2Wo
satisfies sup ||/^|| < oo. Moreover,
’	yf = (1 + p |2)->ra f,
satisfies sup 11 115- < oo by Dsfo — Ц /д. We also have
J£jc;j[ < oo, where = (1 +
fi
because, for positive integers fis,
Therefore, we have proved the (Fourier) expansion
<P =	yy$.
p
Remark. If there exists a complete orthonormal system {9^} of eigen-
functions of a given bounded linear operator K£ L(LZ(S}, LZ(S)) such
that Ktpj = Aj- <.pj (7 = 1, 2, . . .), then, from the Fourier expansion
/=	/e£2(s),
we obtain
3. The Rellich-Gardmg Theorem
281
We have Лу = (Xg?y,g>y), and so, if the eigenvalues are all > 0 and
oo	oo
2 Лу < сю, then the operator К is nuclear. The condition | (K<p,,	|
< oo is surely satisfied, if the operator К is defined by a kernel
K(x, y) = J K2(z, x) Кг(г, у) m(dz), where the kernels
<	s
KT (x, y) and K2 (x, y) are of the Hilbert-Schmidt type.
For,
oo	oo	/ oo	oo	\l/2
2? I (A* X19?y, I = 2; I (A^y, К2щ) I 2 II ] |2. 2? II A2 112
J = 1	\J —1	J = 1	/
and, by Parseval’s relation, we have
00
x||/W2 =
J =1
oo
2'
s
s
00
f f K}(z,y)<pj(y)m(dy)
S j=1 s
2 m (dz)
Ai (z, y) |a tn (dy) m (dz)
oo
OO
and similary for 2 ||A2<py||2. A bounded Unear operator К in a sepa-
co
rable Hilbert space A is said to be of the trace class if 2 | (A^y,	f < oo
j'=1
for arbitrary complete orthonormal systems {<py} and {yy} of X. For a
general account concerning the trace class and the nuclear operator, see
R. Schatten [1], and I. M. Gelfand-N. Y. Vilenkin [3].
3. The Rellich-Gardmg Theorem
Theorem (Garding [1]). Let G be a bounded open domain of R*. If an
operator T C L(Hq(G), Hq(G)) satisfies, for j < k,
||7V||* С ЦуЦу for all (p€ Hq(G), where C is a constant, (1)
then Г is compact as an operator £ L(Hq(G), Hq(G)).
Proof. By the definition of the space Hq(G) (see Chapter I, 10), it
would be sufficient to show the following: Let a sequence {yv.} £. Hq (G) =
Cq(G) be such that ||^ ||A A 1 (v = 1, 2, . . .). Then the sequence {Tyy}
contains a subsequence strongly convergent in Hq(G). The Fourier
transform <pv(g) — (2л:)-”'’’2 f <pt,(x) exp(— ix£)dx satisfies, by Schwarz’
G
inequahty,
|£Д£) |2	(2?c)_* f dx f \<pv(x) |2 dx	(2 л)-” f dx,
G G	G
and hence is equi-bounded in £ C Rn and in v. We may assume,
by the boundedness of ||(p^ ||0, that a subsequence {y-',/} is weakly conver-
282
X. Compact Operators
gent in L2(G') =	Siner, for each g, the function exp(—ixgj
belongs to LZ(G), we know that the sequence of bounded functions
<Ps(£) =	(2a)~nl2 exp( - tx£))o converges at every £. Thus, by (1)
and the Parseval relation for the l''ourier transform (Chapter VI, 2),
II — Ttfy ||J = ||7'(y,. — px) 111 g C2||y,. — <p„.||^
= C2 S ||Р>, -ft.) ||? = С2	||(A(?,. -^ ))||§
pls;
n
пе,‘^АО-ч>А^ г<4
+ C2C, I IfIй'Iq>,® -y„,(f)I2di,
i«i>r
where C\ is a positive constant.
The first term on the right converges, for fixedr, to Oasv' and/Z->oo.
This we see by the Lebesgue-Fatou Lemma. The second term on the right
is, for r > 1.
£С2С,^“ / |f|“|?>,.(fl-^(flM
iei>'
<- c2 c, r’-2'- J If I2* 1$, (fl -	(f) I2 di
g C2C,C2 г*’2* S || (D>Z-r>!%-) IIS
|s|S*
= C2C1C2r2>-2‘	||D*(^-^.)||S
(s|gA
= С2 C, C2 r2’~2‘ 11	112 <: 4 С2 C, C2 r2’ -2i
with a constant C2.
The last term converges, by /< k, to 0 as r>rx. Therefore,
lim ЦТ?’,,- — T(p= °-
v,fi—изо
4. Schauder's Theorem
Theorem (Schauder). An operator T C L(X, У) is compact iff its
dual operator Tf is compact.
Proof. Let S, S' be the closed unit sphere in X, V', respectively. Let
T C L{X, V) be compact. Let {y'} be an arbitrary sequence in S'. The
functions .Fj'(y) = <y, yj> are equi-continuous in the sense that
|Fj(y) — F)(z)| = |<y — z,y;>| g ||y —г||.
Moreover, {Ру(у)} is equi-bounded in j on any bounded set of y, since
|-Pj (y) I = ||У (I- Therefore, by the Ascoli-Arzeli theorem, as applied to the
5. The Riesz-Schauder Theory
283
functions {7q(y)} defined on the compact set (T • S)a, we see that some
subsequence (y)} ' converges uniformly in у G (T  S)a. Hence
(Tx, у£> = (x, T'y'j) converges uniformly in x(S, and so {T'  yj}
converges in the strong topology of X'. This proves that T' is compact.
Conversely, let T' be compact. Then, by what we have proved above,
T" is compact. Hence, if S” is the closed unit sphere in X", (T"  S") is
relatively compact. We know that У is isometrically embedded in Y" (see
Theorem 2 in Chapter IV, 8), Hence T • S С T" • S" and so T ♦ S is relati-
vely compact in the strong topology of Y" and so in the strong topology
of Y. Therefore, T is compact.
&. The Riesz-Schauder Theory
We prepare
Lemma (F. Riesz [2]). Let У be a compact operator G L {X, X}, where
X is a В-space. Then, for any complex number	0, the range R (Яр I— V)
is strongly closed.
Proof. We may assume that z0 = 1. Let be a sequence of X such
that yn = (I - У) xn converges strongly to y. If {x„} is bounded, then by,
the compactness of the operator V, there exists a subsequence {%„-} such
that {Vx„} converges strongly. Since xn> = yn' + Vxn>, {%„'} converges to
some %, and so у = (/ — V) x.
We next assume that {||%w ||} is unbounded. Set T = (I — V) and put
= dis(#„, N(T)), where N (T) = {x; Tx = 0}. Take awn(N(T) such
that <xn
in the case when {«„} is bounded, we can prove, as above, that у G R (T) =
R(I — V). Suppose that lim = oo. Since zn — (xn — w„)/ \xn — wn I
satisfies |fzell = 1 and s-lim Tzn = 0, we can prove, as above, that
rt—
there exists a subsequence {z„-} such that s-lim zn> = w0, s-lim Tzn< = 0.
Я—НЮ	я—изо
Hence гг-’oG А1’ (Г). But, if we put zn — w0 — un, then, in
(1 + и-1) ая. Then T (хл — wn) = Tx„, and so
Я
xn —	—	||*n — “Ц =
the second and the third terms on the left belong to N (T) so that we must
have \un I - |xn — I -> <%„. This is a contradiction, since s-lim un = 0,
1	Я—ЮО
— wn I (1 + я-1) ocn and lim «„ = oo.
я—юо
We are now able to prove the Riesz-Schauder theory; for convenience
sake, we shall state the theory in a series of three theorems.
Theorem 1.	Let V be a compact operator G L (X, X). If 0 is not
an eigenvalue of V, then /Ч) is in the resolvent set of V.
Proof. By the preceding Lemma and the hypothesis, the operator
'1\ =• (Л07 — У) gives a one-one map of X onto the set which is
strongly closed in X. Hence, by the Corollary of the open mapping theo-
rem in Chapter II, Б, 7'^ has a continuous inverse. We have to show that
284
X. Coinpact Operators
R(T^) = X. If not, the topological image = T^X of X is a proper
closed subspace of X. Hence, if we set X2 == Xlt Xs = T, X2,...,
then VnJ1 is a proper closed subspace of Xn (w = 0, 1, 2, . . .; Xo =- X).
By F. Riesz’ theorem in Chapter III, 2, there exists a sequence {yM} such
that y„€ Xn> ||y„|| = 1 and dis(yn, Xn+1)	1/2. Thus, if n > m,
1 (VyM — Vyn) - ym + {—yn — {T^ym - - 7\уй)До} = ym — у
with some у€ЯГл+1.
Hence |j Vyn — Fyw|| > |Я0|/2, contrary to the compactness of the
operator V.
Theorem 2.	Let V be a compact operator £L{X,X). Then, (i) its
spectrum consists of an at most countable set of points of the complex plane
which has no point of accumulation except possibly Л = 0; (ii) every
non-zero number in the spectrum of V is an eigenvalue of V of finite
multiplicity; (iii) a non-zero number is an eigenvalue of V iff it is an
eigenvalue of V'.
Proof. By Theorem 1, a non-zero number in the spectrum of V is an
eigenvalue of V. The same is also true of V', since, by Schauder’s theo-
rem, V' is compact when V is. But the resolvent sets are the same for V
and V (see Chapter VIII, 6). Hence (iii) is proved. Since the eigenvectors
belonging respectively to different eigenvalues of V are linearly indepen-
dent, the proofs of (i) and (ii) are completed if we derive a contradiction
from the following situation:
| There exists a sequence {xn} of linearly independent vectors
| such that V xn
= Лпх (и = 1, 2, . . .) and
lim - Z # 0.
ft—ЮО
To derive a contradiction, we consider the closed subspace Xn spanned by
хг, x2, . . ., xn. By F. Riesz' theorem in Chapter III, 2, there exists a
sequence {y„} such that y„£Xn/ ||y„|| ~ 1 and dis(y,„	1/2
(n = 2, 3, . . .). If n > m, then
V Vyn — Я"1 Vym == yn + (-уж ~ Я-1 TAnyn + Я"1^yn~z,
where z £ Хп_л.
ft	ft
For, if yn =	then we have yn - V У» = xj “
J-l	J—1
ft
and similarly	Therefore ПЯ»1^ —
J ““
Л"1 Vym |j 1/2. This contradicts the compactness of V combined with
the hypothesis lim 7--0.
ft—wo
Theorem 3.	Let yk 0 be an eigenvalue of a compact operator V C
L(X, X). Then is also an eigenvalue of V by the preceding theorem.
We can prove: (i) the multiplicities for the eigenvalue are the same for V
5. The Riesz-Schauder Theory
285
and Vf. (ii) The equation (A^I — V) x ~ у admits a solution x iff у €
N(^r — V')1, that is, iff V' f = A^/ implies <y, /> = 0. (iii) The equa-
tion (А^Г — V f} — g admits a solution / iff g£ N (AqI — 7)1, that is,
iff Vx == AqX implies <%, g) ~ 0.
Proof. Since the eigenvalue Aq 0 is an isolated singularity of the
resolvent R(A; V) = (А/ — F)-1, we can expand R(A; 7) in Laurent
series
OO
Я(А;У) =	(Л—Л-0ГЛ„.
л=—oo
We are particularly interested in the residue A_r = (2ni) 1 ] R (Л; 7) dA.
As was proved in Chapter VIII, 8, Л_х is an idempotent, i.e., Aij = A_j.
If we set (Л/ - У)"1 = A"1 J + 7A,'then from (Al - V) (A 1Z + 7A) = I
we obtain Vx ~ V (A-1 7A + A~aJ), and so Vx is compact when V is.
Hence, by
: (2л1)’1 / R(A; 7) dA =	1	/ A^dA-I
I A—-Afl| = e
f VxdA = (2л0 1 f VxdA.
|Л~Л,|=«	|Л—^[=s
Thus, by the Theorem in Chapter X, 2, is a compact operator.
Therefore, by Л_1Х = Л_А (Л_2 X) and the compactness of A_n the
unit sphere of the normed linear space А X is relatively compact. Hence,
by F. Riesz' theorem in Chapter III, 2, the range 7?(Л_!) is of finite
dimension. On the other hand, Vx~AqX, x 0, implies that
(А/ — 7)-1я — (A — Л))	by (AZ — 7)x = (A — Aq)x, and so A_^x =
(2лг)‘1 f (A — Ao)-1 dA • x = x. Therefore, the eigenvalue equa-
ls—
tion Vx = A^x is equivalent to Vx = A^x, x C R(A.л). In the same way,
we can prove that the eigenvalue equation V / = A^/ is equivalent to
7'/ = Ао/, /€ R(A'_^ But 7?(Л_1) and 7?(ЛСг) are of the same dimen-
sion. For, Л1А/ = g satisfies ALxg = AL^A^/) = g, and this is equi-
valent to <*, g) — <Л_1г, g> for all x С X and so the functional g may be
considered as a functional defined on the finite dimensional space R (^-i)-
Now, by the well-known theorem in matrix theory, the eigenvalue
equation Vx = AqX (in RfA^)) and its transposed equation 77 — A^f
(in R (ALa)) both have the same number of linearly independent solu-
tions. We have thus proved (i). The propositions (ii) and (iii) are already
proved by the Lemma and the closed range theorem (Chapter VII, 5).
Extension of the Riesz-Schauder Theory. Let a power 7” of 7£ L (X, X)
be compact for some positive integer n. Then, by the spectral mapping
theorem in Chapter VIII, 7, <r(V”) = <j(V)” and, by the compactness of
Vя, a (7**) is either a finite set or else a countable set accumulating only
at 0. Therefore, a(V) is either a finite set or a countable set accumulating
286
X. Compact Operators
only at 0. V* being compact,
(2лг) ' f R(2;Vn)d2
M~A,|=s
is,	for any йф # 0 of <r(F") and sufficiently small e > 0, of finite-dimen-
sional range. Hence Zy is a pole of R(Z; Vn) (see Chapter VIII, 8). But
(Z* I — F”) = (Z ~ V) (Л”'1/ + Л*~2 V + - - + F^1) and hence
(Zw А — F”)-1 (Z*-1A + -  • + F”-1) = (1/ - F)'1,
which proves that any Zq 0 of cr (Fn) is a pole of А (А; V) and so is an
eigenvalue of V. These facts enable us to extend the Riesz-Schauder theory
to operators V for which some power Vn is compact. This extension is
highly important in view of its application to concrete problems of
integral equations, such as the Dirichlet problem pertaining to potentials.
See, e.g., O. D. Kellogg [1]. It can be proved that the Riesz-Schauder
theory for >.() - L is valid also for an operator V L {X, X) if there exist a
positive integer m and a compact operator K( L (X, X) such that
||K— F*”||< 1. See K. Yosida [9]. It is to be noted here that, if
Ar1(.s\ Z) and K2(s, t) are bounded measurable for 0 F s, t 1, then the
integral operator T defined by
1
%(s) -> (Tx) (s) = (КгК2х) (s), where (KjX) (s) = f Kj(s, t) x(t) dt,
о
is compact as an operator L(IX(Q, 1), 7.J(0, 1)). See K. Yosida-Y. Mi-
mura-S. Kakutani [10].
6. Dirichlet’s Problem
Let G be a bounded open domain of Rn, and
L = S Ds csi(x) Dl
a strongly elliptic differential operator with real C°°(Ge) coefficients
cst(x) = cts(x). We shall deal only with real-valued functions. Let /£ L2(G)
and Ui £ H™(G) be given. Consider a distribution solution w0£ L2(G) of
Lu — f such that (m0 — иг) £ H™ (G).	(1)
The latter condition (u0  -£ Hq (G) means that each of the distribu-
tional derivatives
(DJ«0 — for |/j <; m	(2)
is the L2(G)-limit of a sequence {D’tphj}, where <p^j£ Co°(G) (see Chap-
ter 1,10 ). Thus it gives roughly the boundary conditions:
D^Uq = D^ur on the boundary dG of G for \j| < m. (3)
In such a sense, (1) will be called a Dirichlet's problem for the operator
6. Dirichlet’s Problem
287
L. We shall follow the treatment of the problem as formulated and solved
by L. Garding [1].
We first solve
« 4- xLu = /, (« — «J C Hq (G),	(4)
where the positive constant a is so chosen that Garding’s inequality
(<p + xL*<p, <p)0 Sr <5 holds whenever Co°(G) .	(5)
Here /-* JE (—	/У Cj( (x) £>s and <5 is a positive con-
stant. The existence of such an x is guaranteed if the coefficients cst(%)
are continuous on the closure Ga of G. We also have, by те-times partial
differentiation, the inequality
|(g? + xL*(p, y>)oj у ||(р||ж • |]v»||„ whenever <p, y»€ C^G),	(6)
where у is another positive constant independent of <p and tp.
We have, for £ Hm (G) and <y € C£° (G) >
«Д, = 2. ((—1)H+I'I	и,)0 - £ (—!)>*'
J,I	s,t
by partial differentiation. By Schwarz’ inequality, we obtain, remember-
ing that the coefficients c., are bounded on G“,
|«Jo[ t? £ ||DV||0 ||Z)4|lo (sup IcsJx) I = tJ.
The right hand side is smaller than constant times [ jgo |jw.
Thus the linear functional
F(<f>) — (<p + xL*(p, «Jo, <p C Cg° (G),
can be extended to a bounded linear functional defined on Hq (G) which
is the completion of Cq° (G) with respect to the norm ||g? ||m. Similarly, we
see, from
|(<p,/)o| IWIo • Jl/llo Hdk ‘ ll/llo,
that the linear functional (pp, /)0 of (p € C£° (G) can be extended to a
bounded linear functional of <p 6 Hq (G). Hence, by F. Riesz’ represen-
tation theorem as applied to the Hilbert space (G), there exists an
/' = f (/, «J c (G) such that
(<p, /)o — (? + xL*tp, «Jo = (<p, f)m whenever <p € Cg°(G).
Hence, by the Milgram-Lax theorem in Chapter III, 7, applied to the
Hilbert space Hq[G}> we have
{<p, fio - (? + otL*cp, «Jo = (?, f)m = B(<p, Sf), Sf e ^(G), (7)
where
B(y>, y) = (95 + xL*(p, ^)o for 99 £ Cg° (G), у € H^ (G).	(8)
Thus
(У» /)o = (9s + ocL*(p, ur + 5/')0 whenever <p C C^° (G),
and so «0 = «j -| S/' is the desired solution C L2(G) of (4).
288
X. Com pad Operators
We shall next discuss the original equation (1). If u^Q L2(G) satisfies
(1), then m2 = uQ — wx E (G) satisfies
(m0, L*<j5)0 = (мъ L*<p)0 + (u2t L*<p)0 = (/, <p}0, cp e Cg°(G).
We obtain, by partial integration as above,
| («i, i*9?)o |	« constant times ||9?||m,
|(0)оЫ|/||оН<=01/1ИЫк
We may thus apply F. Riesz' representation theorem in H™(G) to the
linear functional (/, 95)0 — (и1г L*tp)0 of 99 G Cq°(G). Hence there exists a
uniquely determined v C H™ (G) such that
(Л p)o ' (ui> = (f, <p)m whenever <pGCg°(G).
By the Milgram-Lax theorem, applied to (v, <p)m,-we obtain an v€ H™ (G)
such that
(w,	99) whenever <p £ C^(G), v 6 H™ (G).
Thus the Dirichlet problem (1) is equivalent to the problem: For a given
Sx v C H™ (G), find a solution w2 6 H™ (G) of
(«2.^)0 = B(S1v,<p), <p€Cg°(G).	(Г)
Now, for a given и 6 L2 (G) = Hq (G),
(u,<p}o\	||и||о  Iloilo ||«||o • IHk
so that, by F. Riesz’ representation theorem in the Hilbert space H™ (G),
there exists a uniquely determined ur = Tи 6 Hq (G) such that, whenever
?€C?°(G)>
(m, 99)0 = (u'3 <p)m and IIм'Ik — IHlo-
Hence, by the Milgram-Lax theorem, we obtain
(«,93)0= (и',<p)m~ В (Sxu’,tp) = В(ЗгТи3<р)} |[S17'w]|m^<5-1 jj«!|0. (9)
Therefore, by (1'), we have, whenever <p £ C£°(G),
B(^2>	= (из- + <xL*<p)Q = (u2, 9?)0 4- a(«2, L*<p)Q
= B(S1Tu2,(p) + aB{S1v, 99),
that is,
В (u2 — Sj Tu2 — a Sx v, 99) = 0.
Because of the positivity В (<p,	> 0 of В, we must have
w2 —5iTm2 = a Sxv.	(1")
The right hand term a Sx v€	(G) is a known function. By 11 Sx Tи [ jm
Й-1 ||w||o, we see that the operator SjT defined on Hq (G) into H%(G) is
compact (the Rellich-Garding theorem in Chapter X, 3). Therefore, we
Appendix to Chapter X. The Nuclear Space of A. Grothendieck 289
may apply the Riesz-Schauder theory to the effect that one of the
following alternatives holds:
'Either the homogeneous equation и — S^Lu^Q has a non-trivial
< solution и C Hq (G), or the inhomogeneous equation и — S1Tu = w has,
for every given w G H™ (G), a uniquely determined solution «6 H™ (G).
The first alternative corresponds to the case (u, tp + ex L*q))Q = (м, дз)0,
that is, to the case Lu = 0. Hence, returning to the original equation (1),
we have
Theorem. One of the following alternatives holds: Either i) the homo-
geneous equation Lu = 0 has a non-trivial solution и G or ii) for
any /tI2(G) and any u1^Hm(G), there exists a uniquely determined
solution u0 G L2(G) of Lu = /, и — ux£ H™ (G).
Appendix to Chapter X. The Nuclear Space of
A. Grothendieck
The nuclear operator defined in Chapter X, 2 may be extended to
locally convex spaces as follows.
Proposition 1. Let X be a locally convex linear topological space, and
Y a В-space. Suppose that there exist an equi-continuous sequence
{fj} of continuous linear functionals on X, a bounded sequence {y,} of
elements € У and a sequence of non-negative numbers {cj with Д' с,- < oo,
7=1
Then
T  x = s-lim £ с,- <%, /'> yi
«->007=1 J
(1)
defines a continuous linear operator on X into У.
Proof. By the equi-continuity of {/y}, there exists a continuous semi-
norm p on X such that sup | <% /'> | p (x) for x G X. Hence, for w > n,
£ С/ <x, y.
sup|[yy|| • д;
This proves that the right hand side of (1) exists and defines a continuous
linear operator T on X into the В-space Y.
Definition 1. An operator T of the form (1) is said to be a nuclear
operator on X into the В-space У.
Corollary. A nuclear operator T is a compact operator in the sense
that it maps a neighbourhood of О of. X into a relatively compact set
of Y.
19 YoHidn, ii'iinetkmid AiiiUjkIh
290 Appendix to Chapter X. flir Nuclear Space of A. Grothendieck
Proof* We define
Л,  x У с - (x, у .
> -1
Tn is compact, since the image by Tn of the set V = {x; p (л) < 1} of
X is relatively compact in У. On the other hand, we have
and so Tnx converges to Tx strongly and uniformly on V. Hence the
operator T is compact.
As was proved in Chapter X, 2, we have a typical example of a nuclear
operator:
Example. Let К be a compact subset of R. Then, for (k — ;) > n, the
identity mapping T of (K) into H\y (K) is a nuclear operator.
Proposition 2. Let X be a locally convex linear topological space, and
V a convex balanced neighbourhood of 0 of X. Let pv(x) = ini Л
be the Minkowski functional of V. pv is a continuous semi-norm on X.
Set
Nv — {x£ X; py(x) = 0} = {x С X; Zx € V for all Z>0}.
Then Nv is a closed linear subspace of X, and the quotient space Xv =
X/Ny is a normed linear space by the norm
11 % 11 у — py {x\, where x is the residue class mod Nv
.	(2)
containing the element x.
Proof. Let (x — хг) £Nr. Then pv (xx) pv (x) + pv (xL — x) = pv (x),
and similarly ^>г(х) < Pv{x-l)- Thus pv{x) = if x and x± are in the
same residue class mod Nv. We have ||x]jy S; 0 and ||6||F = 0. If
(|x||F = 0, then x£x implies x£Nv and so x = 0. The triangle ine-
quality is proved by ||x + у[jF = pv(x + y) <i pv(x) + pv{y) =
| x||k + ||y]jv- We have also |jax||F = pv(ax) = |a|pv(x) = |a | ||x||F.
Corollary. By the equivalence
(3)
we can define the canonical mapping
XFt->XFi (when C Li)
by associating the residue class xVt (mod Ny^ containing x to the residue
class xVi (mod .Vcontaining x. The mapping thus obtained is continuous,
since
We are now ready to give the notion of a nuclear space, introduced in
analysis by A. Grothendieck [2].
Appendix to Chapter X. The Nuclear Space of A. Grothendieck 291
Definition 2. A locally convex linear topological space X is said to
be a nuclear space, if, for any convex balanced neighbourhood V of 0,
there exists another convex balanced neighbourhood U С V of 0 such
that the canonical mapping
T: Xv -> Xv	(4)
is nuclear. Here Xy is the completion of the normed linear space Xy.
Example 1.	Let RA be the topological product of real number field
R in such a way that RA is the totality of real-valued finite functions x (a)
defined on A and topologized by the system of semi-norms
= \x(a) |, aG A.	(5)
Then Ra is a nuclear space.
Proof. Ny is the totality of functions x(a) C RA such that, for some
finite set {«y £ A ; / = 1, 2, . . ., n], x(ay) = 0 (/ = 1, 2, . . ., и). Hence
Xy = RA/Ny is equivalent to the space of functions xv(a) such that
Xy(a) = 0 for a aj (/=1,2,..., n) and normed by
||xy(a) ||v = SUP |*(й;) •
We take Nv the totality of functions % (a) 6 RA such that x(as) = 0 for
oc&A' where A' is any finite set of integers containing 1, 2, . . ., n.
Thus, for U С V, the canonical mapping Xv = RA/NU^ RA/NV = Xv
is nuclear. For the mapping is a continuous linear mapping with a finite-
dimensional range.
Example 2.	A nuclear В-space X must be of finite dimension.
Proof. Since X = Xv for any convex balanced neighbourhood У of 0
of a 5-space, the compactness of the identity mapping X -> X implies
that X is of finite dimension by F. Riesz’ theorem in Chapter III, 2.
Example 3.	Let К be a compact subset of Rn. Then the space ®к(7?л)
introduced in Chapter I, 1 is a nuclear space.
Proof. As in Chapter I, 1, let
pK,k (/) = sup Ds f (x)
be one of the semi-norms which define the topology of	Let
La = {/C j 'pK,k(f) ~ 1}- Then NVk is {0}, and XVf. = X/NVk =
(R")/Ayft precisely the space (Rn) normed by pKfk. If (k—/) > n,
then it is easy to prove, as in the example following the Corollary of
Definition 1 above, that the canonical mapping of XVk into XVj is a nuclear
transformation. Hence ©x(.R") is a nuclear space.
Theorem 1. A locally convex linear topological space X is nuclear,
iff, for any convex balanced neighbourhood V of 0, the canonical mapping
X Xy is nuclear.
19*
292 Appendix to Chapter X. The Nuclear Space of A. Grothendieck
Proof. Necessity. Let U <, Г be a convex balanced neighbourhood
of 0 of X such that the canonical mapping Xv~> Xv is nuclear. The
canonical mapping T : X > Xr ‘s product of the canonical mapping
X > Xy and the canonical nuclear transformation Xv >XV. Hence T
must be a nuclear transformation.
Sufficiency. Let the canonical mapping T: X -> Xy be given by a
nuclear transformation
OQ
Tx = Д ty <x, /}> yy.
j=i
For any oc > 0, the set {% € X; | (x,	| < oc for j = 1, 2, . . .} is a convex
balanced neighbourhood U.x of 0 of X, because of the equi-continuity of
{//} £ %' Moreover,
[ | Tx j | у —-
£ 4	/i> Vj
7
X oc sup 11 yy jД Cj whenever x € Ua.
v i	J
Let oc be so small that the right hand side is < 1. Then |j Tx|]r < 1 and
Ua £ V. Each /' may be considered as belonging to the dual space X'(^,
and so
Tx = Tz = Д' Cj (x, fj) yy whenever (x — z) £ 2Vy
Thus the canonical mapping Xv Xv is given by a nuclear transforma-
tion
xUa^ Д 9<5Уя,/;>уу.
Theorem 2. Let a locally convex linear topological space X be nuclear.
Then, for any convex balanced neighbourhood V of 0 of X, there exists a
convex balanced neighbourhood W £ V of 0 of X such that Xw is a Hilbert
space.
Proof. The nuclear canonical mapping Xv Xy (U £ F) defined by
T Xu = X Су /y> yy
is factored as the product of the two mappings
<x\ Xv-+ and ffi.	Xу, where oc is given by
-> {c}/2 <xv, /у» and is given by {£,} Д c}/2 £уУу.
The continuity of oc is clear from
£ ky/2 <%u> /J> j2 (sup	!|u?  Д
J	\ j	/	1
and that of /2 is proved by
IX *S x • X If,r s sup цу,<. lift} if  x С,- •
i J	I У 7	J	/	J
Let U2 be the inverse image in (22) by Д of the unit sphere of Xy. Then U2
is a neighbourhood of 0 of (Z2) and so contains a sphere S of centre 0 of (Z2).
Appendix to Chapter X. The Nuclear Space of A. Grothendieck 293
Let W be the inverse image in X of -S’ by the continuous mapping a
defined as the product of continuous canonical mapping X -> Xv and the
continuous mapping a : Xv (Z2). Then clearly W Q V and, for any
XW(~~ Xw,
xw — inf z - inf
1	x/A6lV3>0 axlKS.ZX)
A — ] <xx [t
(the radius of S).
Since || ||p is the norm in the Hilbert space (l*),Xw is a pre-Hilbert space.
Corollary. Let X be a locally convex nuclear space. Then, for any
convex balanced neighbourhood V of 0 of X, there exist convex balanced
neighbourhoods WT and W2 of 0 of X with the properties:
A	A
H'2 S £ b", Xw, and XWt are Hilbert spaces and the canonical
mappings X -> A^, XWt XWi, XW1 —> Xv are all nuclear.
Therefore, a nuclear space X has a fundamental system {F„} of neigh-
bourhoods of 0 such that the spaces Xv^ are Hilbert spaces.
Further Properties of Nuclear Spaces. It can be proved that:
1.	A linear subspace and a factor space of a nuclear space are also
nuclear.
2.	The topological vector product of a family of nuclear spaces and
the inductive limit of a sequence of nuclear spaces are also nuclear.
3.	The strong dual of the inductive limit of a sequence of nuclear
spaces, each of which is an .F-space, is also nuclear.
For the proof, see the book by Grothendieck [1] referred to above,
p. 47. As a consequence of 2., the space ®(R”), which is the inductive
limit of the sequence {®кг(Ал); r = 1, 2, . . .} (here Kr is the sphere
|x| r of Rn), is nuclear. Hence, by 3., the space ® (/?“)' is also nuclear.
The spaces @(RW), @(Rn)', and S(Rn)' are also nuclear spaces.
The importance of the notion of the nuclear space has recently been
stressed by R. A. Minlos [1]. He has proved the following generalization
of Kolmogorov s extension theorem of measures;
Let X be a nuclear space whose topology is defined through a countable
system of convex balanced neighbourhoods of 0. Let X' be the strong dual
space of X. A cylinder set of Xе is defined as a set of the form
Z' = {/'€X'; ^ <<%,/'><	(* =
Suppose there is given a set function /z0, defined and 0 for all cylinder
sets. Let /z() be о additive for those cylinder sets Z’ with fixed xlf , xn.
Then, under a compatibility condition and a continuity condition, there
exists a uniquely determined extension of which is tr-additive and
0 for all sets of the smallest u-additive family of sets of X' containing all
the cylinder sets of X'.
For a detailed proof and applications of this result, see I. M. Gelfand-
N. Y. Vilenkin [31.*
* Sec also Stlpph'inriil л у Nolc:;, p. 166.
294	XI. Normed Rings and Spectral Representation
XI. Normed Rings and Spectral Representation
A linear space A over a scalar field (F) is said to be an algebra or
a ring over (F), if to each pair of elements x, у 6 A a unique product
xy G A is defined with the properties:
[xy) 2 = x(yz) (associativity),
x(y + 2) = xy 4- xz (distributivity),
«/3 (xy) = (ocx) (fiy).
(1)
If there exists a unit element e such that ex = xe = x for every x£ A,
then A is said to be an algebra with a unit. A unit e of A, if it exists, is
uniquely determined. For, if e' be another unit of A, then we must have
ее' = e — e'. If the multiplication is commutative, i.e., xy = yx for
every pair x, у 6 A, then A is called a commutative algebra. Let A be an
algebra with a unit e. If, for an A, there exists an x' 6 A such that
xx' = x'x = e, then x' is called an inverse of x. An inverse x' of x, if it
exists, is uniquely determined. For, if x” be another inverse of x, then
we must have
x"(xx') = x” e - x" - (x'x) x' = ex' = x'.
Thus we shall denote by x-1 the inverse of x if x has an inverse.
An algebra is called a Banach algebra, or in short a B-algebra if it is a
В-space and satisfies
1И1IMI-	(2)
The inequality
\\x„yn — xy II |K(y„ —y)H + ||(**_ x) у ||
IWi ll(r« — у) II + !1Сгя^х)П 1И
shows that xy is a continuous function of both variables together.
Example 1.	Let X be a В-space. Then L (X, X) is a B-algebra with a
unit by the operator sum T : - S and operator product TS; the identity
operator I is the unit of this algebra L (X, X), and the operator norm
|| T || is the norm of the element T of this algebra L(X, X).
Example 2.	Let S be a compact topological space. Then C (S) is a
B-algebra by (%x 4- x2) (s) = хг(х) + %2(s), (ocx) (s) — <xx(s), (хгх2) (s) =
X1 (s) %2 (S) an<^ 11 X 11 = SUP I X (S) |-
ses
Example 3.	Let В be the totality of continuous functions %(s),
0<^5= 1, which are representable as absolutely convergent Fourier
series:
OO	OQ
x(s) = Xi Cne2mns with X |cw(<oo.	(3)
n=—0O	M--OO
1. Maximal Ideals of a Normed Ring
295
Then it is easy to see that В is a commutative B-algebra with a unit by
the ordinary function sum and function multiplication, normed by
OO
I '
(5)
n=—oo
In the last two examples, the unit is given by the function e(s) = 1
and ||e|| = 1. In the following sections, we shall be concerned with the
commutative B-algebra with the unit e such that
||e || = 1
Such an algebra is called a normed ring.
A Historical Sketch. The notion of Banach algebras was introduced
in analysis by M. Nagumo [1]. He proved that Cauchy’s complex func-
tion theory can be extended to functions with values in such an algebra,
and applied it to the investigation of the resolvent of a bounded linear
operator around an isolated singular point. The result is an abstract
treatment of those given in our Chapter VIII, 8. K. Yosida [11] proved
that a connected group embedded in a B-algebra is a Lie group iff the
group is locally compact. This result is an extension of a result due to
J. von Neumann [6] concerning matrix groups. Cf. E. Hille-R. S. Phil-
lips [1], in which the result of K. Yosida [11] is reproduced.
The ideal theory of normed rings was initiated by I. M. Gelfand [2].
He has shown that such a ring can be represented as the ring of continuous
functions defined on the space of maximal ideals of the ring. By virtue of
this representation, we can give an integration free treatment of the
spectral resolution of bounded normal operators in a Hilbert space; see
K. Yosida [12]. This result will be exposed in the following sections. The
Gelfand representation may also be applied to a new proof of the Tauberian
theorem of N. Wienee [2]. We shall expose this application in the last
section of this chapter. For further details about B-algebras, see N. A. Nai-
mark [1], С. E. Rickart [1] and I. M. Gelfand-D. A. Raikov-
Г. R firrrw r«ll *
ell = L
1. Maximal Ideals of a Normed Ring
We shall be concerned with a commutative B-algebra В with a unit e
such that
Definition 1. A subset J of В is called an ideal of В if %, у C J implies
that (л% -|- /5y) € / and zx£ J for every z£ В. В itself and {0} are ideals
of B. Ideals other than В and {0} are called non-trivial ideals. A non-
trivial ideal J is said to be a maximal ideal if there exists no non-trivial
ideal containing J as a proper subset.
Proposition 1. Each non-trivial ideal /{( of В is contained in a maximal
ideal J.
296
XL Normed Rings and Spectral Representation
Proof. Let [/0] be the set of all non-trivial ideals containing Jo. We
order the ideals of [Jo] by inclusion relation, that is, we denote «< J2
if is a subset of J2. Suppose that {/a} is a linearly ordered subset of
[70] and put J? = U L. We shall show that J is an upper bound

of {/«}. For, if x, у G Jp, then there exist ideals and JSi such that
% € Д and у € Д. Since {Ja} is linearly ordered, Д < Д (or Д > / J
and so x and у both belong to ; consequently (x — у) £ /Л1 g Jp and
zx E JC for any z E B. This proves that J $ is an ideal. Since the unit
element e is not contained in any fa,e is not. contained in J3 = U J*.

Thus Jp is a non-trivial ideal containing every Ja. Therefore, by Zorn’s
Lemma, there exists at least one maximal ideal which contains /0.
Corollary. An element x of В has the inverse x-1 £ В such that
x"1 x = xx"1 = e iff x is contained in no maximal ideal.
Proof. If x~! E В exists, then any ideal J '}x must contain e = xx1
so that J must coincide with В itself. Let conversely, x be contained in
no maximal ideal. Then the ideal xB = {xb; b^B} # {0} must coincide
with В itself, since, otherwise, there exists at least one maximal ideal
containing xB 3 x = xe. It follows that xB = B, and so there must exist
an element b E В such that xb == e. By the commutativity of B, we have
xb = bx -- es that is, b = x“L
Proposition 2. A maximal ideal J is a closed linear subspace of B.
Proof. By the continuity of the algebraic operations (addition, multi-
plication and scalar multiplication) in B, the strong closure Ja is also an
ideal containing J. Suppose Ja =4 J. Then Ja B, because of the maxi-
mality of the ideal J. Thus e E Jat and so there exists an x E J such that
||e— %|| < 1. x has the inverse x-1E В which is given by Neumann’s
SerieS	в + (e - x) + (<? - x)2 +  - •
For, by |[ (e — x)" ||	|| e — x ||“, the series converges to an element E В
which is the inverse of x, as may be seen by multiplying the series by
x = e — (e — x). Hence e ~ x"1 x E J and so J cannot be a maximal ideal.
Proposition 3. For any ideal J of B, we write
x = у (mod J) or x <-*> у (mod J) or in short x	(1)
Then x у is an equivalence relation, that is, we have
x ~ x (reflexivity),
 x у implies у x (symmetry),
x у and у >—• z implies x^z (transitivity).
We denote by x the set {у; (у — x) E J}> it is called the class (mod J)
containing x. Then the classes (F+ y), ax and (xy) are determined inde-
pendently of the choice of elements x and у from the classes x and y,
respectively.
1. Maximal Ideals of a Normed Ring
297
Proof We have to show that x <-*> x', y^y' implies that (x 4- y)
{x' + /), ocx ax' and xy x'y'. These are clear from the condition
that J is an ideal. For instance, we have xy — x'y' = (x — x’} у 4-
(У ~У'К1 by (x — x')ej and (y — y') E /.
Corollary. The set of classes x (mod J) thus constitutes an algebra by
x + у = x + y, ax = ax, x у = xy.	(2)
Definition 2. The above obtained algebra is called the residue class
algebra of В (mod J) and is denoted by BjJ. Thus the mapping x  > x
of the algebra В onto В = В}J is a homomorphism, that is, relation (2)
holds.
Proposition 4. Let / be a maximal ideal of B. Then В = В/J is a
field, that is, each non-zero element x E В has an inverse x-1 £ В such
that x 1 x — x x-1 = ё.
Proof. Suppose the inverse x-1 does not exist. Then the set xB =
{x b’, b £ B} is an ideal of B. It is non-trivial since it does not contain e,
but does contain x^ 0. The inverse image of an ideal by the homomor-
phism is an ideal. Therefore, В contains a non-trivial ideal containing J
as a proper subset, contrary to the maximal!ty of the ideal J.
We are now able to prove
Theorem. Let В be a normed ring over the field of complex numbers,
and J a maximal ideal of B. Then the residue class algebra В = Bjf is
isomorphic to the complex number field, in the sense that each x 6 В is
represented uniquely as x = where f is a complex number.
Proof. We shall prove that В — B[J is a normed ring by the norm
11x|| = inf |jx||.	(3)
If this is proved, then Bj/ is a normed field and so, by the Gelfand-
Mazur theorem in Chapter V, 3, В = BjJ is isomorphic to the complex
number field.
Now we have ||ax]| = [л
|x ||, and |[x -j- У |
= inf_ J[x T у [| <
inf ||x|| 4- inf ||y f| = ||x|| + ||y||; ||xy [|	||x|| |[y || is proved simi-
larly. If ||x|| = 0, then there exists a sequence {x„} Qx such that
s-lim xw — 0. Hence, for any x E x, (x — x„) E J and so s-lim (x — xft) = x
ft—MX	tt—>CO
which proves that x E ,/‘г - J, that is, x = 0. Hence
x|| =0 is equi-
valent to x = 0. We have | e 11	|p|| = l. If||e||< 1, then there exists
an element x E J such that \e — x 11 < 1. As in the proof of Proposition 2,
the inverse x-1 exists, which is contradictory to the Corollary of Propo-
sition 1. Thus we must have ||ё || = 1. Finally, since В is a В-space and J
is a closed linear subspace by Proposition 2, the factor space В = BjJ
is complete with respect to the norm (3) (see Chapter I, 11). We have
thus proved the Theorem.
298
XI. Normed Rings and Spectral Representation
Corollary. We shall denote by x {J) the number £ in the representation
a - . Thus, for each x€ B, we obtain a complex-valued function x(J)
defined on the set {J} of all the maximal ideals of B. Then we have
(* + y) (7) = x(7) + у(/), (at) (7) = «x(7),
(*У) (7) = *(7) y(7)> and e(7)el.	(4)
We have, moreover,
sup i*(7)| (Mb	(5)
Л{Л
and
sup I % (7) I ~ 0 implies x = 0 iff f] J — {0}.	(6)
J€{J1	Ml
Proof. The mapping x - > x  - x (/) e of the algebra В onto the residue
class algebra В = BjJ is a homomorphism, that is, relation (2) holds.
Hence we have (4). Inequality (5) is proved by
|£| ' |£| [|e|| - ||%|| - inf (|%|| < ||%||.
Property (6) is clear, since x(J) — 0 identically on {7} iff x £ J.
Definition 3. The representation
*^M7)	(7)
of the normed ring B, by the ring of functions x (J) defined on the set {7}
of all the maximal ideals J of BJ is called the G elf and-r epresentation of B.
2. The Radical. The Semi-simplicity
Definition 1. Let В be a normed ring over the complex number field,
and {7} the totality of the maximal ideals J of B. Then the ideal
J is called the radical of the ring В. В is said to be semi-simple if its
radical R = Г) J reduces to the zero ideal |0L
JGJ}	1 -1
Theorem 1.	For any x C B, lim ||%* ||1/n exists and we have
lim ||%w ]|1/я = sup | % (7) | -	(1)
je{J)
Proof. Set a = sup | x (7) |. Then, by | [ xn 11 > | xn (J) | = | x (J) |n, we have
|%”||^a”, and so lim ||%"|p"^a. We have thus to prove
lim ||%та(|1/п < a.
Let |/?| > a. Then, for any J € {7}, x(J) — 0 =/= 0, i.e., (% — fie) G 7-
Hence the inverse (fie — x)' 1 exists. Setting = Л, we see that the
inverse (/?e — x)^1 = X(e— Лх)^1 exists whenever |Л| < a-1. Moreover,
as in Theorem 1 in Chapter VIII, 2, we see that Л(е — лх)'1 is, for
2. The Radical. The Semi-simplicity
299
tx-1, holomorphic in Л. Hence we have the Taylor expansion
Л(е — Лх) 1 = Л (e + Лху 4- Л2х2 + •  • + Лп хп + ••).
That хп = хп may be seen by Neumann’s series
OO
(е — ЛхГ1 =	Лпхп
n-0
which is valid for ||Ях|| < 1. By the convergence of the above Taylor
series, we see that
Thus 11%” 11 = |Л[ w || < |Я | ” for large n when |Я [ < tx \ and so
lim 11 xn | lln |Л ~1 when |Л|-1 > tx, that is, lim | x”|1/n AJ tx.
n—KJO	Я--ХЭО
Corollary. The radical R П J of В coincides with the totality
of the generalized nilpotent elements x С В which are defined by
lim | xn |1/K = 0.
(2)
Definition 2. A complex number Л is said to belong to the spectrum
oi x& B, if the inverse (x — Ле)-1 does not exist in B.
If Л belongs to the spectrum of x, then there exists a maximal ideal
J such that (% —Ле) Conversely, if (x — Ле) belongs to a maximal
ideal J, then the inverse (x — Ле)-1 does not exist. Hence we obtain
Theorem 2.	The spectrum of x £ В coincides with the totality of
values taken by the function x (/) on the space {J} consisting of all the
maximal ideals J of В.
Application of Tychonov’s theorem. We define, for any /()C {/},
a fundamental system of neighbourhoods of JQ by
{7е{7};|М7)-М7о)|<<ч (i = 1,2.(3)
where c, > 0, n and х^ С В are arbitrary. Then {/} becomes a topological
space and each x (/), x^B, becomes a continuous function on {J}. We
have only to verify that if JQ =£ Jlf then there exist a neighbourhood
VQ of Jo and a neighbourhood V\ of with empty intersection. This may
be done as follows. Let xn 6 Jn and xn Ё Д so that (7n) = 0 and (JR ~
«#0. Then v„ = {je{j}; |x„(7)|< |«|/2} and 71 = {7e{7};
|ж0(J) — x0(J1) | < |tx |/2} have an empty intersection.
Theorem 3.	The space {/}, topologized as above, is a compact space.
Proof. We attach, to each x£ B, the compact set
Kx = {z; |z|^ ||%||}
of the complex z-plane. Then the topological product
300
XI. Normed Rings and Spectral Representation
is a compact space, by virtue of Tychonov’s theorem. See Chapter 0. To
any maximal ideal /0 G {/}, we assign the point
if
{J} is in one-one correspondence with a subset Sx of S by the above
correspondence /0-> s (/0). Moreover, the topology of {/} is the same as
the relative topology of as a subset of S. Hence, if we can show that Sx
is a closed subset of the compact space S', then its topological image {/}
is compact.
To prove that Sx is a closed set. we consider an accumulation point
co = П Ax € S the set Sx in S. We shall show that the mapping
x£B
x —>	is a homomorphism of the algebra В into the complex number field
(K\. Then Jo = {x; Ax = 0} becomes, as may be seen from the isomor-
phism of B[JQ with (K), a maximal ideal of В and (x— t Jo, that
is, x (Jo) = Ax. This proves that the point <y = fl Ax = fl x (/0) belongs
x£S x^B
to S-p
We thus have to show that
^’X I " Ayj ^(XX ” " f ^'XV AxAy, Ag — 1 •
We shall, for instance, prove that Лж+у = Ax + Ay. Since m = JI Ax is
an accumulation point of Sv there exists, for any e > 0, a maximal ideal
J such that
—*(/)]< fi. |ЛУ —У(/)[<£, (x + у) (J)\ < £-
By (x + y) (/) = x (/) -J- у (/) and the arbitrariness of e > 0, we easily
see that Ax+y = Ax + Ay is true.
We can now state the fundamental facts about the Gelfand represen-
tation v > x (/) of the normed ring В in the form of
Theorem 4.	A normed ring В over the complex number field is repre-
sented homomorphically by the ring of functions x(Jj on the compact
space {/} of all the maximal ideals J of B. The radical R of В consists of
those and only those elements which are represented by functions iden-
tically equal zero on {/}. The representation x -> x (/) is isomorphic iff
the ring В is semi-simple.
Application of the Stone-Weierstrass Theorem (of Chapter 0). The
above obtained ring of functions is dense in the space of all complex-
valued continuous functions on {/} with uniform convergence topology
if the ring В is symmetric (or involutive) in the following sense:
For any x £ B, there exists an x* С В such that x* (J) = x (J) on {/}. (4)
Examples of Gelfand Representations
Example 1.	Let В = C (S) where S is a compact topological space, and
/о a maximal ideal of C(S). Then there exists a point s0C S such that
2. The Radical. The Semi-simplicity
301
x(s0) = 0 for all x £ Jo- Otherwise, for any C S, there exists an xa £ Jo
such that xa(sa) #= 0. xa(s) being a continuous function, there exists a
neighbourhood Va of зл such that xa(s) 0 in Va. Since S is compact,
there exists a finite system, say, УЛ1,
Hence the function
Т’я, . . ., such that U Ул = S.
J «1
n_______
X(s) = .J, Xxi (s) xai (s) £ Jo
*=1
cj < oo
%(s) ==
does not vanish over S, and the inverse x~J x-1(s) = %(s)-1, of x£ Jo
exists, contrary to the maximality of the ideal /0. Thus we see that To is
contained in the maximal ideal J' = {%t B ; x(s0) = 0}. By the maxi-
mality of /о, we must have JQ — ]’. In this way we see that the space
{/} of the maximal ideals J of В is in one-one correspondence with the
points s of S.
Example 2.	Let В be the totality of functions x(s), 0 s 1, which
can be represented by absolutely convergent Fourier series:
CO	00
v r	у
n~ — CQ	П—~	OO
В is a normed ring by (x + y) (s) = x(s) -f- у (s), (xy) (s) — x(s) y(s)
and ||x || = Д? | с?. |. Let JQ be a maximal ideal of B. Set eZnts = xr Then
xf1 = and so, by jxJTo) | Н^Ц = 1, ^p1 (Л) I = К(ЛН
11xf111 = 1, we see that |x1(/0) | = 1. Hence there exists a point s0,
0 < s0 < 1, such that x,(Jn) = e2mSo. Thus xn = e2msn = x” satisfies
oo
xn(J0) = ez™°n, and so x(JQ) = £ cne2mStn = x (sQ). In this way, we
n=—OO
see that for any maximal ideal /0 of B, there exists a point s0, 0 4= s0 1,
such that the homomorphism x-+x(J0) is given by x(J0) = x(s0), for all
% С В. It is also clear that the mapping x -> % (s0) gives a homomorphism
of the algebra В into the complex number field. Therefore we see that
the maximal ideal space of В coincides with {e2,IW; 0 < s 1}.
Corollary (N. Wiener’s theorem). If an absolutely convergent Fourier
OG
series x(s) = cwe2"WM does not vanish on [0, 1], then the function
n ——co
l/x(s) is also representable as an absolutely convergent Fourier series.
For, x does not belong to any of the maximal ideals of the normed ring
of the above Example 2.
Example 3.	We take Br = C [0, 1], and define, for x, у £ B^
(x + y) (s) = x (5) 4- у (s), (ar) (s) = ixx(s), (xy) (s) =
s
f x (s — Z) у (/) dt and 11 x | ] — sup \x (s) |.
0	s€[0,I]
302
XI. Nurmed Kings and Spectral Representation
Bj is a commutative B-algebra without unit. Adjoining formally a unit e
by the rule ex = xe = x, ||e|| = L, the set В = {z = 2e + x; x £ BJ
becomes a normed ring by the operations
{(^i e -j- x J -p (Л2 e -j- x2) = (Л^ -I- 22)	(^1 “F ^2) » d- %)
=/хЛе + ax, (Лге + xj (Л2 e -f- %2) = ЛгЛ2е + Л1х2 + Л2х1 + xrx2
and ||Ле + %|| = p[ + |p||.
We have, by induction,
r2	q”-1
|V(s)| s Mzs, 1^(5)I ' Ms~ . .p»(s)| <
where M = sup p(s) | == 11 x ||. Thus every x^Br is a generalized
s€S
nilpotent element of B, due to the fact that lim (n!)1/w = oo.
n~ 00
3. The Spectral Resolution of Bounded Normal Operators
Let X be a Hilbert space, and let a system M of bounded normal
operators G L (X, X) satisfy the conditions:
T, S 6 M implies TS = ST (commutativity),	(1)
T £ M implies T* £ M,	(2)
A system M consisting of a bounded normal operator T^L (X, X) and its
adjoint T* surely satisfies (1) and (2).
Let M' be the totality of operators G L (X, X) which commute with
every TGM, andletB = Af" = (Af')/be the totality of operators ££ (X,X)
commutative with every operator S G M'.
Proposition 1. Every element of В is a normal operator. В is a normed
ring over the complex number field by the operator sum, the operator
product, the unit I (the identity operator) and the operator norm [| T ||.
Proof. M Q M' by (1), and so Г2Г'. Hence M”' = (М'У 2 M”
and so В = M" is a commutative ring. The identity operator I belongs
to В and is the unit of this algebra B. By (2), we easily see that every
operator £ В is normal. Since the multiplication TS and the adjoint
formation T -> Г* in the algebra В are continuous with respect to the
norm of the operator, it is easy to see that the ring В is complete with
respect to the operator norm.
Theorem 1.	By the Gelfand representation
Вэт^тцу	(3)
the ring В is represented isomorphically by the algebra C ({/}) of all
continuous functions T{/) on the compact space {/} of all the maximal
3. The Spectral Resolution of Bounded Normal Operators 303
ideals J of В in such a way that
|r|| = sup IT(J)
(4)
T (/) is real-valued on {J} iff T is self-adjoint,	(5)
T(/)	0 on {J} iff 7' is self-adjoint and positive,
that is, (Tx, x) 0 for all x£ X.	(6)
Proof. We first show that, for any bounded normal operator T,


By the normality of T, we see that H = TT* = T*T is self-adjoint.
Hence, by Theorem 3 in Chapter VII, 3,
||T[|2= sap (Tx,Tx) = sup |(Г*Тл,х)|= ||Я|[ = ||7’*Г||= ЦТТ*
hdlsi	1И1£1
Since (T*)a = (T2)*, T2 is normal with T. Thus, as above, we obtain
|| T2||2 = [I T*2 T2|[, which is, by the commutativity TT* = T*T, equal
is self-adjoint, we obtain, again by
to || (Г* Г)21| = ||H21[. Since H2
Theorem 3 in Chapter VII, 3,
H||2 = sup (Hx, Hx]^= sup |(№%, x) | = ||7T21|.
IW[<i	Ildlsi
Therefore, ||7^[)2 = || H21| = 11H |2 = (ЦТЦ2)2, that is, 11 T211 = ||T\\2.
We have, by (7), ||T|| = lim [ W*1 ||lj/”, because we know already that
the right hand limit exists (see (3) in Chapter VIII, 2). Hence, by Theo-
rem 4 of the preceding section, the representation (3) is isomorphic and
(4) is true.
Proof of (5). Let a self-adjoint TfB satisfy, for a certain /0C {/},
T(/0) =a + ib with b-T 0. Then the self-adjoint operators = (T—aT}fb^B
satisfies (I + S2) (Jo) = 1 + i2 = 0, and so (I + S2) does not have an
inverse in B. But, by Theorem 2 in Chapter VII, 3, (I + S2) has an
inverse which surely belongs to B. Thus, if T E В is self-adjoint,
T(J) must be real-valued. Let T^B be not self-adjoint, and put
Then, since the first term on the right is self-adjoint, the self-adjoint
operator (T — T*)/2 i must be у 0. Thus, by the isomorphism of
representation (3), there must exist a J0£{J} such that——- (Jo) 0.
T T*	T___T*
Hence T (JQ) =-----------(/0) + i —— (/0) is not real. For, as proved
above, self-adjoint operators are represented by real-valued functions.
Proof of (6). We first show that
/'•(/) r(J) on {7|.	(«)
304
XI. Normed Rings and Spectral Representation
This is clear, since self-adjoint operators (T + T*)/2 and (T — T*)/2i
are represented by real-valued functions. Therefore, by (4) and the result
of the preceding section, the ring В is represented by the ring of all con-
tinuous complex-valued functions on {/} satisfying (5) and (8). Let
T(J) 77 0 over {J}. Then S(J) = T(J)1/2 is a continuous function on
{/}. Hence, by the isomorphism of the representation (3), S2 = T. By
(5), we have S = S*. Hence (Tx, x} = (S2x, x) = (Sx, Sx) 5: 0. To
prove, conversely, that the condition {Tx, x) 77 0 for all x G X implies
T (J) over {J}, we set Тг (/) = max (T (/), 0) and T2 (J) = Tx (J) — T (J).
Then, by what we have proved above, 1\ and 72 are both G В, self-ad-
joint and positive: (T^x, x) > 0 for all x € X (/ = 1, 2). Moreover, we
have TtT2 = 0 and T2 = 7\ - T. The former equality is implied by
Л (/) T2 (J) = 0.
Therefore, we have
0 (TTzx, T2x) = {—Tlx, T2x) = — (T2x, x) = — (T2T2x, T2x) <Z 0.
Thus (T^x, x) = 0 and so, by Theorem 3 in Chapter VII, 3, we must
have T2 = 0. Hence, by = lim j] T% H1^, we obtain T2 = 0. We
Я—НЭО
have thus proved T = Тг and hence T (J)	0 on {/}.
We shall thus write 7' 7>0 if 7is self-adjoint and positive. We also
write S 7> Г if (S - - 7') T 0.
Theorem 2.	Let {Т„} £ В be a sequence of self-adjoint operators such
that
0	7\ rg T2 g 7,.	S G B.	(9)
Then, for any x G X, s-lim Tnx = Tx exists, i.e., s-lim Tn = T exists and
n—>co	*oo
TeB,S^T^Tn (« = 1, 2, ...).
Proof. We first remark that, by (6),
E, F G В and E > 0, F > 0 imply E + F 7: 0 and EF > 0. (10)
Thus 0 X Tf < 7? X    < 7'^ X  • • S2. Hence, for any xG X, a
finite lim (T^x, x) exists. Since, by (6), T^+k S* Tn+kTn T'l, we also
have
lim (Г2+Лх, lim (Tn+kTnx, x) = lim (T^x, x).
>oo	п,Л—кю	л-+oa
Therefore, lim ({Tn — Tw)2 x, x) — lim [iT„x —7'wxI,'2 = () so that
s-lim T„% = Tx exists. That T E В and S 7> T 77 T„ is clear from the
n—>oc
process of the proof.
Theorem 3.	Let a sequence of real-valued functions {Tn(J)}t where
Tn£ В, satisfy the condition
0 Tx (J) X T2{J)	• • • X Tn{J)  -  77 a finite constant on {/}. (11)
3. The Spectral Resolution of Bounded Normal Operators 305
Then, by (6) and Theorem 2, s-lim Tn—T exists. In such a case, we can
П—i-CO
prove that
*> = {7е{7};Г(/)^ lim
*OQ
is a set of the first category, and so Dc ~ {/} — D is dense in {/}.
Proof. By Theorem 2, T Tn and so T(J) lim Tn(J) on {/}.
n^co
By Baire’s theorem in Chapter 0, 2, the set of points of discontinuity of
the function lim Tn(J) is of the first category. Hence, if the set D is not
П—KO
of the first category, then there exists at least one point JQ 6 D at which
lim Tn (J) is continuous. In other words, there exists a positive number <5
л—
and an open set V (Jo) Э JQ of {J} such that
T(J)	6 + lim Tn (/) whenever J € V (/0).
n—>OO
Since the compact space {J} is normal, and since T(J) lim Tn[J) on
{/}, we may construct, by Urysohn’s theorem an open set (Jo) 3 Jo
and a function IT (J) cC ({J}) such that 0 W (J) d on {/},
7,(7,)“ S	W{J) = <5/2 on 7,(Л) and W [J) = 0 on V{J^.
Hence T(J) — W(J) lim Tn(J) on {]}, and so, by (6), T — W Tn
(n = 1, 2, . . .). И7 (J) й 0 implies, by the isomorphism (3), that W =£ 0,
W 0. Thus, again by (6), T ~ W s-lim Tn, contrary to T = s-lim Tn.
№—>oo
Finally, since {/} is a compact space, the complement Dc = {J} —D
of the set D of the first category must be dense in {/}.
We are now able to prove (K. Yosida [12])
The spectral resolution or the spectral representation of operators € B.
Consider the set C ({/}) of all complex-valued bounded functions
T' (J) on {J} such that T' {J) is different from a continuous function T (J)
only on a set of the first category. We identify two functions from C' ({J})
if they differ only on a set of the first category. Then C' ({/}) is divided
into classes. Since the complement of a set of the first category is dense
in the compact space {J}, each class T' contains exactly one continuous
function T(J) which corresponds, by the isomorphism В ** C ({/}), to
an element T С B.
For any T^B and for any complex number z == Л -j- ip, we put
Ez = the element С В which corresponds to the class E' containing the
defining function E’t (J) of the set {J £ {/}; Re T (J) < Л, Im T (J) < p}.
It is clear that there exists a monotone increasing sequence of continuous
functions fn of complex argument such that EZ(J) — lim fn (T(J)) and
20 Ycmklft, Funrlloimt Anatynin
306
XI. Normed Kings and Spectral Representation
so £'{/) € £'({/}). We Have then
|Т(Л -	+ ,ft) (л;+,„(7) +	(7) -£^1+<и(Л
-£X-+wJJ))! e on {J} i*
- — a----E=- T_ a2 •-  •  - - zn - . a ~ sup
/2	W}
Д1 = - £ ~ yJ=-	= £ = sup
/2	jt{j}
/sup (Л,- — Я?_1)2 + sup (jUj —№-i)2)1/2	£•
Thus, by the definition of we have
I T<J) -£ M, + in1 (Е„1+1„(Л + Ел,.,+^,(7)
- £^+w(/) ~ Eij+wJJY) I e
on {J}, since the complement of a set of the first category is dense in the
compact space {/}. Therefore, by (4), we have
Ii E~~ (Ay + i/ij) (E^j+i/1. +	Ел^+чч E*i+44--) II — £>
3 = 2
which we shall write as
T = ffzdEtl	(12)
and it is called the spectral resolution of the normal operator T.
4. The Spectral Resolution of a Unitary Operator
И T is a unitary operator 6 В, then, by
T(J)T*(J) = T(J)T(J) = 1,	(i)
we see that the values taken by the function T(J) on {]} are complex
numbers of absolute value 1. From this fact, we can simplify the spectral
resolution f J z dEr of T.
The defining function E'9(J) of the set {J £ {/}; arg(T(/)) £ (0, 0)},
0 < 0 <_ 2 л:, belongs to C and we have, by setting Ef0(J) = 0,
IT U) - i (Е'в1 (J) - e„l t (/)) I « max |	|
(0 = в„<в1<-.-<9, = 2л).
Let Ев(Л be the continuous function on {/} which is different from
E'6 (/) only on a set of the first category, and let E9 be the operator € В
which corresponds to Ee(J)by the isomorphic representation B^T+*T(J).
Then, as in the preceding section,
11 T — Д ei6i (E9j — E9i J 11	max I — eie^ |.
7^=1	j
4. The Spectral Resolution of a Unitary Operator
307
This we write, in view of the fact e2** = 1,
T — f e*edF(6), where
о
F(0) = E6+q-E+0 for 0< 0 < 2 л, F(0) = 0, F(2n) =1.	(2)
Here Ee+o is defined by Ee+0%= s-lim Eex, the existence of this limit
will be proved below.
Theorem 1. The system of operators Е(д), 0^0^ 2 л, satisfies the
conditions:
each F (0) is a projection operator, commutative with
every bounded linear operator commutative	with T,	(3)
F(0)F(0')=F(min(0,0')),	(4)
F(0) = 0, Г(2л)=Д	(5)
F{6 4- 0) = F{$), 0 в < 2л, in the sense that
s-lim F (0') x = F (0) x for every x £ X.	(6)
Proof. It is sufficient to prove that E0, 0	0	2л, satisfies the
conditions:
each Eo is a projection	operator € В,	(3')
E6Eff =	(“П
Eq^^E^ = I,	(У)
Eex = s-lim Evx for every x € X and 0 в < 2л.	(6')
We have E'e(J) =E'e (J) and E’e(J)2 = E'g(J). Hence, by the result of
the preceding section, we obtain Ев = E* and E0 — E6. This proves (3').
(4') is proved similarly from E? {J} ==	(J), and (5') is
proved similarly. Next let 0„ j. 0. Then E0n(/) > E0„+1(/)	E'g(J}, and
so, by the result in the preceding section, s-lim Eg = E exists and
E (J) = E'e (J) = lim Egn (/) on {J} except possibly on a set of the first
category. Thus E = Eg.
Example 1. Let a linear operator T defined by
Ty(s) = ezsy(s}, where y(s)G£2(—oo, oo;.
T is unitary. We define, when 2пл 2(я 4- 1)я,
F(0) у(s) = у(s) for s < 0 2ял < 2(я 4- 1) л,
F(0) y(s) = 0 for 0 4- 2пл < s.
2л
It is easy to see that T = f et0 dF (0).
о
20*
308
XL Normed Kings ;ind Spectral Representation
Example 2. Let a linear operator Tj be defined by
T\x(£) = x + 1) in £2(—oo,oo).
I\ is unitary. By the Fourier transformation
y(s) = Ux(t) = l.i.m. (2 л) 1/2 f e~is/x(t) di,
w—>oo	“L
- T*
we obtain Ux (t + 1) = eIS Ux(i} = ег''у (s). Thus
T\x(t} = x(t + 1) = U'^yis} = U-'Tyts) = U^TUxtt),
that is, T\ = (d ] T U. Therefore, we have
2л
Тг= f fPdFM, where F^B) = U^F^B) U.
о
The uniqueness of the spectral resolution. Since 7_|	I'* and
71 (7) =	(7) = T (7)-1> we easily see that
2л
T J c'^dFif)).	(7)
0
Let max — г*®*-11 < c. Then, from
т = j;(<y-F(e,^) + а, цац<e,
we obtain, by (4),
Г = eziei(F(B3) ~ F^)) + <5', where
][(Т-<5)б|| + ||d(T — d) || + ||<52||
g (||T|| + £) £ +	+ e) + e2‘
2л
Hence we have T2= f e2^dF(0), and, more generally,
о
2л
T* = J ^вЛР(в) (я = 0,±1,±2,...).	(8)
О
2л
Therefore, if there exists another spectral resolution T = f e^dF1(B)
о
satisfying (3) to (6), then, for any polynomial p(B) in and e-t9,
2л
/ #(е)<г((2*(в)х,у)-(Л(е)«,у)) = о, (x,yex).
0
Thus, by continuity, the above equality holds for any continuous func-
tion p {6} with p (0) = p (2л). Let 0 < 60 <	< 2л, and take
pn(B) = 0 for 0 < В 0O and for	В 2л,
'К
= ifore0 + |g8ge1,
= linear for 9n < 0 <	+ — and for 0-.	0	0, -h —.
и - --- U n	1 —	— 1 n
5. The Resolution of the Identity
309
Then letting n -> oo, we obtain, by (6),
lim / pn (8) d [(F (8) x, y) - (F, (8) x, y)] = [(F (8) x, y) - (F, (8) x, y)$ = 0,
Я—НЭС q
which is valid for all x, у E X. Hence, letting 0O J, 0 and making use of
conditions (5) and (6), we see that F (0J = F^O). Therefore, the spectral
resolution for a unitary operator is uniquely determined.
5. The Resolution of the Identity
Definition 1. A family of projections E (A), — oo < A <; oo, in a Hil-
bert space X is called a (real) resolution oj the identity if it satisfies the
conditions:
E (Л) E fa) = E(min (Л,	,	(1)
E (—oo) = 0, E (4- oo) = I, where E (— oo) x ~ s-lim E (Z) x and
U“°°	(2)
E (+ oo) x -= s-lim E (Z) x,
Л-foo
E{7. + 0) = E{ty, where E(X + 0) x = s-lim Efa) x. (3)
/'4-A
Proposition 1. For any x, у С X, the function (E (A) x, y) is, as a func-
tion of A, of bounded variation.
Proof. LetA1<Aa<---<An. Then, by (l),E(a,0] =Е(Д)—E(a)
is a projection. Thus we have, by Schwarz’ inequality,
X | (E Я-,, Я,.] x, y)| = _£| (E (Л,-_„ %] X. E (Я,.,, Л,] y) |
3	J
^X HEft-i.y *|| ||Е(Й,-1,Й,-] y||
J
<IX llEft-,. Л,] X112)1'2  (X ||Е(Я,-1. %] y|]V'2
= (||Е(Я1,4]х||2)1'2-(ПЕ(А1,Я„]у ||2)1/2S 1И1-||y||.
For, by the orthogonality
E(A/_,,%]-E(Aj_1,Ai] = 0	(i^/)	(4)
implied by (1), we have, for m > n,
m—1
|И|2Ь ||Е(Л„,Л„]*||2 = ^ ||Е(Л,.Л(+1] <.	(5)
Corollary. For any Л, -oc < z < oo, the operators E(Z + 0) =
s-lim E (A') and E (A — 0) = s-lim E (Z) do exist.
г I а	г t *
Proof. From (5), we see that, if z„ j4 A, then
lim j|E(Zj, Aa] %||2 = 0,
jth-К»
and the same is true for the case Z„ | Я.
310
XI. Normed Rings and Spectral Representation
Proposition 2. Let /(Я) be a complex-valued continuous function
on (—00,00), and let x £ X. Then we can define, for—оо<дс<^<оо,
P
f j (Я) dE (Я) x
A
as the s-lim of the Riemann sums	'
-5/W)E (V Л+i]x- where « = ^1 < 4 <  • • < К = x-<= (Яр ^+1],
when the max }Яу+1 — Яу | tends to zero.
Proof. /(Я) is uniformly continuous on the compact interval [a,/}].
Let I/(Я) — /(Я') I S e whenever |Я — Я' |	<5. We consider two partitions
of [a, 
X =	•  • < zn- Д, max |Я,-+1 — Яу [ < <5,
J
<  • • <	= /3, max |/z/+1 — /Zy | < <5,
and let
x ~	<<. Vp = p < m -f- n,
be the superposition of these two partitions. Then, if (/zA, /zft+1], we
have
2 E (%> ^+11 x~£ f&k) E (Ek. Ek+i\ x
3	«
= 2? £s £(vs, vs+1] x, with |es | 2e,
and so the square of the norm of the left side is, as in (5),
OO
Corollary. We may define f f (Я) dE (Я) x as the s-lim
. 00	o4-00,0too
P
j тлЕЩх,
<x
when the right side limit exists.
Theorem 1. For a given x X, the following three conditions are
mutually equivalent:
.	co
f f (Я) dE (Я) x exists,	(6)
—00
co
f [fW ? d ||Е(Л) *||2 < 00,	(7)
—00
00
E(y) ~ f /(Я) d(E(A) y, x) defines a bounded linear functional. (8)
Proof. We shall prove the implications (6)	(8) -> (7) -> (6).
(6) -> (8). The scalar product of у with the approximate Riemann
5. The Resolution of the Identity
311
co
sum of f / (2) dE (2) x is a bounded linear functional of y. Hence, by
—oo
(у, E (2) %) = (E (2) y, x) and the resonance theorem, we obtain (8).
(8)	(7). We apply the operator E (a, /J] to the approximate Riemann
P___
sum of у = f f{tydE(X)x. We then see, by (1), that у = E (a, ^]y.
ft
Thus, again by (1),
_____ OO_______	P' ___
F(y) = f/(A)d(E(A)*,y) = lim f j(X)d(E (A)x,y)
-bo	a';-oo,/J'fooT'
= lira f f(A)d(E(A) x>E(a,p] y)
«'l-oo.p'too
F____
= lim J/(A)<i(E(a,fflE(A)x,y)
*4-00,0 too
0____
= jx.y) = ||y||».
Hence ||y||2^ lie'll* |[у|1> i-e-, ||у1! = lie'll’ On the other hand, by
Д________________________
approximating у = f /(2) dE{%) x by Riemann sums, we obtain, by (1),
ft
2	0
ft
||y||»= Jf(X)dEW*
*
p
so that f |/(2) |2 d ||E(2) #||2	||Е||2. Therefore, by letting л | —
ft
oo,
ДI wc see that (7) is true.
(7) -> (6). We have, for «' < a < Д < /Г,
0'	0
j /(2) dE(b) x-f /(2) dE(A)x
ft'	ft
2
л	p'
= f |/(A)|M ||E(A) x||2 + J |/(A)|M||E(A)x|p
ft'	p
as above. Hence (7) implies (6).
Theorem 2. Let /(2) be a real-valued continuous function. Then,
a self-adjoint operator H with D(H) = D is defined through
OO
(И X, y) = f / (2) d (E (2) x, y), where
(9)
— 00
|/(2) |2 d || E (2) x ||a < ool and any у £ X,
and we have H E (2) 2 E (2) H, that is, HE(1) is an extension of E (2) H.
312	XI. Normed Rings and Spectral Representation
Proof. For any у X and for any e > 0, there exist oc and /5 with
— oo < л < j5 < oo such that j | у — E (ос, j5] у || < e. Moreover, we have
OO	0
f |/(A) p Й||£(Л) £(«,/!] y||*= / |/(Л)|М||Е(Л)у||«.
— co	Л
Hence E(oc, fl]y£D and so, by (2), Da = X. H is symmetric by
/ W = ж (E (Л) X, y) = (E(X)y,x).
If у £ D (H*) and H*y == y*, then, by E{a, fl] z C D and (1),
= (НЕ(<х,Д] г,у) = //(Z)i(E(;.)z,y).
c3c
Thus, by the resonance theorem,
co
lim (z,£(«,fly)= / /(AH(E(A)Z,y)=F(Z)
а4.-оо,Д1оо
is a bounded linear functional. Hence, by the preceding theorem,
J |/(A) j2 d |(Л) у ||a < oo, that is, у € D.
—oo
Therefore, D = D(fi) □	Since H is a symmetric operator, we
have H £ H* and so H must be self-adjoint, i.e., H == H*.
Finally, let x £ D (H). Then, by applying E {ji.} to the approximate Rie-
OO
mann sums of Hx = J /(Л) dE (Л) x, we obtain, by (1),
—OO
E (^ j? f(A)d(E (f£) E (2) x)
—OC
= f /(A)d(E(A) E(ji)x) = НЕ[ц) x.
— OQ
Corollary 1. In the particular case / (Л) = Я, we have
(Hx,y)= f Ad(E(A)x,y), for x £ D(H), у G X.	(10)
—oo
We shall write/it symbolically
H= f XdE(A)t
— OQ
and call it the spectral resolution or the s-pectral representation of the self-
adjoint operator H.
Corollary 2. We have, for H = / f(X) dE (Л) given by (9),
—oc
/ |/(A) |2 d 11Я (л) x |[2 whenever x^D(H).	(11)
—00
6. The Spectral Resolution of a Self-adjoint Operator
313
In particular, if H is a bounded self-adjoint operator, then
co
(tf«x,y) = f f(Wd(E(X)x,y) for x,yEX (* = 0,1,2,...). (12)
“OO
Proof. Since E (Z) Hx = HE(X) x for % £ D (H), we have, by (1),
(Hx, Hx) = jf(tyd(E (Z) x, Hx) =ff(A)d (HE (Z) x, x)
= f i W d, {//Cu) d^E W E W z.x)}
=	/ /(д)^(ЕДОх,у1 = //(Л)«<г||£(Л)ж||».
—OO	I
The last part of the Corollary may be proved similarly.
Example. It is easy to see that the multiplication operator
Hx(t) tx (t) in E2(—oc, oo)
OO
admits the spectral resolution H = f s.dE (Z), where
—oo
E (Z) % (£) = % (£) for
= 0 for t > Z.
For,
pPd ||E(Z) %[|3 = f kWI2^= 7|%(*)M = 1РЧГ
—OO	—' oo	—oo	— oo
f Z^(E(Z)x,y) = f /.dx f x(t)y(t)dt~ f t- x(t)y(t)dt= (Hx,y).
—oo	—oo	—*oc	—co
6. The Spectral Resolution of a Self-adjoint Operator
Theorem 1. A self-adjoint operator H in a Hilbert space X admits
a uniquely determined spectral resolution.
Proof. The Cayley transform U = UH = (H — i I) (H + i I) 1 of the
2л
self-adjoint operator His unitary (see Chapter VII, 4). Let U = f e*edF(&)
о
be the spectral resolution of U. Then we have
F(2 л — 0) =s-lim F (2л — 0) = Е(2л) = I 
If otherwise, the projection F (2 л) E (2 л — 0) would not be equal to the
zero operator. Thus there exists an element у 7= 0 such that
(F(2л) — F (2л — 0)) у = у.
314	XI. Normed Kings and Spectral Representation
Hence, by F{&) F(0') = F(min(0, 0')),
2л
Uy= f ?^(F(0)(F(23t)-F(23r-O)))y
о
= (F(2?r) “ F{2n~ 0))y = y.
Thus (y, z) = {Uy, Uz) = (y, Uz) and so {y,z— Uz) = 0 for every
z(_ X. U being the Cayley transform of a self-adjoint operator H, we
know (see Chapter VII, 4) that the range R (I — U) is dense in X. Hence
we must have у = 0, which is a contradiction.
Thus, if we set
A = — cot 0, E(A) = F{6),
then 0 < 0 <. 2 л and — oo < A < oo are in a topological correspondence.
Hence E (A) is a resolution of the identity with F {&). We shall show
that the self-adjoint operator
#'= f ME{X)
—OO
is equal to H. Since H = i[I + U) {I — U)^1, we have only to show
that
(H' (y — Uy), V) = (i (y + Uy}, %) for all x, у С X.
But, since D{H')a ~ X, we may restrict x to be in the domain D{H‘).
Now, by F{6) • F{0') = F (min (8, &')),
2л
(у - Uy, F{ff) X) = J (1 - г‘у) (F(0') y, F(0) x)
0
2л
= j
0
fl
= f	>.*)
0
Hence
{y—Uy, H'x)= f Ad(y-.Uy,E(X)x)
— OO
I
*	2л	f fl
= f - cot 6^1/{l-e^dtFiO') y,x)'
o	[o
2л
= f i(lFeis)d(F{6)y,x) = (i(yFUy),x).
0
oo
uniqueness of ihe spectral representation. Suppose H. = f A dE (A)
— OO
oo
admits another spectral representation H == / XdE’ (A) such that
— oo
6. The Spectral Resolution, of a Self-adjoint Operator 315
E’ (Яр) yt E (Ло) for some Then, by setting
Л = —cotG, Е'(Я) = F'(0),
we have F' (Go) F (Go), where Ло - cot Go. By a similar calculation
OO
to the above} we can prove that the Cayley transform of f XdE* (A) is equal
“OO
2n
to f е*в dFf (0). Hence the unitary operator U admits two different
о
2я	2n
spectral representations U — f e*edF{6) and U = J dF'(&), con-
o	о
trary to what we have proved in Chapter XI, 4,
We have thus proved (see Chapter VII, 3 and 4) the fundamental
result due to J. von Neumann [1]:
Theorem 2. A symmetric operator // has a closed symmetric extension
H**. A closed symmetric operator H admits a uniquely determined spec-
tral representation iff H is self-adjoint. H is self-adjoint iff its Cayley
transform is unitary.
Remark. It sometimes happens, in applications, that H is not self-
adjoint but H* is self-adjoint. In such a case, H is said to be essentially
self-adjoint. In this connection, see T. Kato [7] concerning Schrodinger’s
operators in quantum mechanics.
The spectral representation of the momentum operator fTp.
Яхх(г) =|^%(£) in L2(—OQ,Oo).
The Fourier transform U defined by
x(t) = U  у (s) = l.i.m. (21/2 f e15iy(s) ds
я-к»	Jn
is unitary and t7-1x(£) = U*x(t) = Ux[— t). Hence, denoting by Е(Л)
the resolution of the identity given by (13) in Chapter XI, 5, we obtain a
resolution of the identity {E' (Л)}, E' (Л) = UE (Я) U~\ If both y(s) and
sy(s) belong to L2(—oo,oo) П L1 (—00,00), then
f eisly(s)ds}
\	—OO	/
= (2л)-1/2 f eKtsy(s)ds = U(sy(s)) = (7sL7-1x(i),
—00
or, symbolically,
‘4 t/sU-’.	(1)
316	XL Normed Кiti^s and Spectral Representation
oo
Hence, for the self-adjoint operator H — s • = f AdE(A), we have
— OO
U'XH1U  у (л-) - - 5 • y(s) = Hy(s) whenever y(s),
sy(s) both belong to L2(—00,00) p| Lx(—00, 00).
For any у (s) C D (H) = D (s •), set
yn (s) = у (s) or = 0 according as | s | < n or | s J > n.
Then surely yn(s), syn(s) both £ Lz(— 00,00) p L1^— 00,00) and, more-
over, s-lim yn = y, s-lim Hyn- Hy. Thus, since the self-adjoint opera-
«—►00	«—>00
tors U^HjU and H are closed, we have, by (U~1H1U) yn = Hyn>
{U~* H1U) у = Hy whenever у E D (Я),
that is, U^H^U is a self-adjoint extension of the self-adjoint operator
H. Hence, by taking the adjoint, we see that HX = Я is also an exten-
sion of (Я^1Я1Я)* = U~xHtlJ’, consequently U~1H1 U = Я and so
Ht = UHU-* = 7C7-1)= 7Л dE>№ 
—00	—00
7. Real Operators and Semi-bounded Operators.
Friedrichs’ Theorem
Real operators and semi-bounded operators, defined below, have self-
adjoint extensions. Thus we can apply von Neumann’s theorem to these
extensions to the effect that they admit spectral resolutions.
Definition 1. Let Лг - J,2(S, SB, m) and let Я be a symmetric operator
defined in X into X. H is said to be a real operator, if i) x (s) E D (H) then
x(s) € Я(Я), and ii) Я maps real-valued functions into real-valued func-
tions.
Example. Let /($) be a real-valued continuous function in (—00,00).
Then, for X = L2(—00,00), the operator of multiplication by /(s) is a
real operator.
Theorem 1 (J. von Neumann [1]). A real operator H admits a self-
adjoint extension.
Proof. Let U = UH be the Cayley transform of H. Then D{U) =
{(Я + ii) x?x£ D(H)} consists of the functions obtained by taking the
complex-conjugate of the functions of R(U) = {(Я — ii) x; D(H)}.
Thus, if we define an extension Ux of U through
Vx = U in D(U),
слул, where {7Ц is a complete
orthonormal system of the Hilbert space D (U)3-,
then иг is a unitary extension of U. Therefore the self-adjoint extension
Нг of H exists such that Ux = UHi (see Chapter VII, 4).
7. Real Operators and Sem.i-bou.nded Operators
317
Definition 2. A symmetric operator H is said to be upper semi-
bounded (or lower semi-bounded) if there exists a real constant oc such that
(Hx, x) oc ]|x||2 (or (Hx, x) S oc ||x ||2) for all D(H).
If (H x, x) 0 for all x 6 D (H), then H is said to be a positive operator.
Example. Let ^(s) be continuous and non-negative in (—oo, oo).
Consider the operator H defined for C2 functions x(s) with compact
support by
(Hx) (s) = — x" (s) + q (s) x (s).
Then H is a positive operator in the Hilbert space Z.2 ( oc, oo), as may be
verified by partial integration.
Theorem 2 (K. Friedrichs [3]). A semi-bounded operator H admits
a self-adjoint extension.
Proof (due to H. Freudenthal [1]). —H is lower semi-bounded if H
is upper semi-bounded. If H is lower semi-bounded as above, then
= H + (1 — ос) I satisfies the condition that (Hxx, x) 2: ||%||2 for all
D(Hy. Therefore, since ocl is self-adjoint, we may assume that the
symmetric operator H satisfies the condition
(Hx,x) 2: ||x||2 f°r x£D(H).	(1)
We introduce a new norm j | x | ]' and the associated new scalar product
(x, y)' in D(H) through
\\x\\'= (Hx,x), (х,у)' == (Hx,y).	(2)
Since H is symmetric and satisfies (1), it is easy to see that D (H) becomes
a pre-Hilbert space with respect to ||x ||' and (x, y)'. We denote by D (H)f
the completion of this pre-Hilbert space.
We shall show that D(H)' is, as an abstract set without topology, a
subset of the set X which is the original Hilbert space. Proof: A Cauchy
sequence {xn}' of the pre-Hilbert space D(H) satisfies ]|xw — xm||'
|!x„ —xm||and lim |[xn — xm |j' = 0; consequently {xn} is also a Cauchy
sequence of the original Hilbert space X. If we can show, for a Cauchy
sequence {y„} of D(H)', that
lim Hyjr 0 does not imply lim ||y„|| = 0,
then the correspondence
W' ->
{#»}
(4)
is one-one from the Cauchy sequences of D (H) into the Cauchy sequences
of X. Two Cauchy sequences {zn}' of D(H) (of X) being identified
if lim |lx„ —- zn|' = 0 (if Jim ||xn — zn\ =0). Since X is complete, we
may thus identify its Cauchy sequence {%„} with the element x( X such
318	XI. Normed Rings and Spectral Representation
that lim jlxn — xII — 0. Therefore, D(Hy may, as an abstract set with-
ft—>00
out topology, be identified with a subset of X by the correspondence (4).
The proof of (3) is obtained, remembering the continuity of the scalar
product in D(H}' and in X, as follows, lim \\xn—хж1Г = 0,
>0O
0 and lim [ | x„ ] | = 0 imply a contradiction
n—*oo
ft—>00
a2 = lim (xn> = lim (Hxn, xm) = lim (Hxn, 0) = 0.
изо	ню	«r->co
We next set
5 = /)(//*)
(5)
Since D(H) C D(H*), we must have D (H) g D Q />(#*). Hence we can
define an extension H of H by restricting H* to the domain D = D (H}.
We have to show that H is self-adjoint.
We first show that H is symmetric. Suppose x, у G D\ there exist two
sequences {хл}', {yw}' of D(H) such that ||x — xM -> 0, ||y — yn ||' -> 0
as w->oo. Hence, by the continuity of the scalar product in D(Hy,
we see that a finite lim (x„, ym)' = lim (Hxnym),= lim (x„,Hym)
>-cc	n.nwoo
exists. This limit is equal to
lim lim (Hxn, yw) = lim (Hxn, y) = lim (^,Яу) = (x.Hy)
m—кю, m—изо	»—нзо	ft—изо
and also to
lim lim (Hx„, ym) == lim (x, Hym) = lim (Hx, ym) = (Hx,y).
m—>oo ft—>00	jwoo	m—изо
Hence H is symmetric, that is, H £ (H)*.
Next let x £ D (H), yGX. Then, by
!(*,?)! ^ INI ||y||1ИГ• IMh
we see that /(x) = (x, y) is a bounded linear functional on the pre-Hilbert
space D(H). Hence /(x) can, by continuity, be extended to a bounded
linear functional on the Hilbert space D(H)'. Hence, by F. Riesz' repre-
sentation theorem as applied to the Hilbert space D(Hy, there exists a
uniquely determined y' £ D (H)' such that
/(x) = (x, у) = (x, у’У = (Hx, y') for all x E D (H).
This proves that у’ E D (H*} and H*y' = y. Hence у' E15 and Hy' = y.
We have thus proved that R (H) = X, and so, by the Corollary of Theo-
rem 1 in Chapter VII, 3, H must be self-adjoint.
8. The Spectrum of a Self-adjoint Operator	319
8. The Spectrum of a Self-adjoint Operator. Rayleigh’s
Principle and the Krylov-Weinstein Theorem. The Multiplicity of
the Spectrum
Theorem 1. Let H = f /. dE (Я) be a self-adjoint operator in a Hilbert
space X. Let <г(Я), Pa{H), Ca{H) and Ra{H) be the spectrum, the point
spectrum, the continuous spectrum and the residual spectrum of H,
respectively. Then (i) a (H) is a set on the real line; (ii) Яо f Pa (H) is
equivalent to the condition E (Яо) # E — 0) and the eigenspace of H
corresponding to the eigenvalue Яо is 7? (E (Яо) — E (Яо — 0)); (iii) Яо € Ca (H)
is equivalent to the condition E (Яд) = E (Яо — 0) in such a way that
E (Лх) E (Я2) whenever Ят < Яо < Я2; (iv) Ra {H) is void.
Proof. We know already that, for a self-adjoint operator H, the resol-
vent set q (//) of H comprises all the complex numbers Я with Im (Я)	9
(see Chapter VIII, 1). Hence (i) is clear. We have /^1 = Яо f dE (Я) by
—OO
the definition of the resolution of the identity {E (Я)} and so {H—Яо1) =
OO
J* (Я — Яо) dE (Я). Hence, as in Corollary 2 of Theorem 2 in Chapter XI, 5,
—oo
we obtain
||(H - ЯоЯ) %||2 - /°(Я-Яо)2^ ||Е(Я)ж||2, x£D(H).	(0
—OO
Thus, by E (— oo) = 0 and the right continuity of | j E (Л) x j j2 in Я, we see
that Hx = 2^x iff
E (Я) x = E (Яо + 0) x ~ E (Яо) x for Я > Яо and
E (Я) x — E (Яд — 0) x = 0 for Я < Яо,
that is, Hx = ^x iff (E(Яд) — E (Яо — 0)) x = x. This proves (ii). Next
we shall prove (iv). If Яо£ Ra(H), then, by (i), Яо is a real number. By
the condition R (H — ^1)* = D ((H -/^/) -T)<z 7r- X, we see that there
exists ay/ 0 which is orthogonal to R (H — ЯдЯ), i.e., ((H — ЯдЯ) x, y) =0
for all x C D (H). Hence. {Hx, у) = (^x, у) — (x, Ящу) and so у C D (Я*),
H*y = Яду. This proves that Hy — ЛуУ, i.e., Яо is an eigenvalue of H.
Hence we have obtained a contradiction Яо G Ra {H) П Pa (H), and so
Ra{H) must be a void set.
Let Яд be a real number not belonging to a {H). Then the resolvent
(Яо/-ЯГ exists. Hence H^ = (H — Яо/) has a continuous inverse
{H — Яо-I)-1. The last condition is, by (iv), equivalent to Яо€ q (Я) and
to the condition that there exists a positive number <x such that
| {H — Я,,!) x [| a ||x (I for all x € D (H).
320	XL Normed Rings and Spectral Representation
This condition is by (1) equivalent to
f (Л-Vd |[Е(Я) x||a a2 ||x||2 for all x^D(H).	(2)
—oa
Suppose that we have, for 2j < Д, < z2 with /Ч) - Я1 = Лг—Л0<сх,
E (2j .=. E (2g). Then, contrary to (2), we obtain
7 (Я-Я^Я ||Е(Я)л-||2 < a2 7 d\\Е(Л) x||2 = a21| %||2.
— 00	-oo
This proves (iii) by (i), (ii), (iv) and (2),
Remark. The Example in Chapter XI, 5 gives a self-adjoint operator H
for which all real numbers are in the continuous spectrum of H.
Theorem 2. Let Я be a bounded self-adjoint operator. Then
sup Л= sup [Hx, x), inf Я- inf (Hx, x).	(3)
Proof. Since (Hx, x) — (x, Hx) -	- real number, we can
consider
- - inf (Hx, x) and <x2 — sup (Hx,x).
Ildlsi	l!di£i
Let ^Еа(Н). Then, by Theorem 1, there exists, for every pair
(Ят, Я2) of real numbers with Ях < Я,, < Я2, а у =	=/= 0 such that
(E(Яй) — E(Ях)) у — у. We may assume that ||y || = 1. Hence
(Яу,у)=р<г(Е(Л)у,у) = р*||Е(Л)у||’
A,
= f л<г||(Е(Л)-Еад)у||«.
A
Thus, by letting Я2 j 4, we obtain lim (Hy^, уЛ1Л) = 4- This
proves that sup Я= sup a2.
Леа(Я)	_
Let us assume that a2 £ <r (H). Then, by Theorem 1, there exists a pair
(Яр Я2) of real numbers such that лх < л2 < Я2 and E (Я2) = E (2J. Hence
I = I - - ^(4) + E (Яг), (I — E (Я2)) E (\) = E (Ях) (I - E (Я2)) = 0 and
so either (I — E (2g)) or E (2J is not equal to the zero operator. If
(I — E (Я2)) Ф 0, then there exists a у with 11 у 11 = 1, (I — E (Яа)) у = у.
In this case, we have
(Ну,у) = /Л<г ||Е(Л) y||® = fXd ||EW (£- E(Wlla
= J Xd ||EW y||2 Л, > a,;
A>
and in the case E (Ях) 0, we obtain
{Hz, z) g < л2 for a z with ||г|| = 1, E(2j) z = z.
8. The Spectrum of a Self-adjoint Operator
321
Therefore the assumption a {H) is absurd and so we have proved that
sup Я = sup, H {x, %), Similarly, we can prove that inf Я = inf
ЛСа(Я) IHl^l	Л€в(Я) IMI<1
{Hx,%).
Theorem 3 (Krylov-Wein stein). Let H be self-adjoint, and define,
for any xQ D (77) with | | x |1 = 1,
ixx = {Hx, %),	= ||Я*||-	(4)
Then, for any e > 0, we can find а Я^С. о (Я) satisfying the inequality
«ж —	— «x)1' - 6 Ло S «X + (^ — <x2)1/2 + E. (5)
Proof. We have
$ = {Hx, Hx) = {H2x, x)= fAzd\\E{A) x ||2,
<xx = {Hx,x) = f Ad ||£(Я) x||2,
||%||2 =/ d ||Е(Я)%||2,
and so
/Я2 d ||£(Я) %||2 - 2« xf A d ||Я(Я) %||2 + a2 fd ||Е{Я) %||2
= J'{A — ax)2d \\E{A) <
Therefore, if ||Е(Я) x||2 does not vary in the interval given by (5), we
would obtain a contradiction
К - «1^	- ^)*'2 + <02 > r. - «г, 
Remark. The so-called Rayleigh principle consists in taking ocx as
an approximation to the spectrum of the operator H. If we calculate
then Theorem 3 gives an upper bound of the error when we take
as an approximation of the spectrum of H. For a concrete application of
such error estimate, we refer the reader to K. Yosida [1].
The Multiplicity of the Spectrum. We shall begin with the case of a
self-adjoint matrix H = J A dE (Я) in an «-dimensional Hilbert space X„.
Let Яь Я2, . . ., Ap {j> n) be the eigenvalues of H with the multiplicity
   ,PHp> respectively ( Д'	= n 1. Let x}i, xit, . . ., xjm be ortho-
normal eigenvectors of H belonging to the eigenvalue A^{Hx^ = Я,-*/) so
that{%;<; s = 1, 2,.. .,>»,•} spans the eigenspace EXj = R(E (Я,) — E (z; 0))
of H belonging to the eigenvalue Aj. Then the set {xjt; / = 1,2, ...
and s = 1, 2, . . ., is a complete orthonormal system of vectors of
the space Xn, and hence every vector у of Xn is represented uniquely as
the linear combination of x.:’s:
(6)
21 Yoaldii, Ful>h»пнI Aiuilynh
XL Normed Kings and Spectral Representation
322
Thus, denoting by PA. the projection (E (A,) — E (Ay — 0)) upon the
eigenspace EA , we have, for any <x < /3,
(EOT-E(«))y= X Х<х,Л.) = SPt,y. (7)
Wj	•
PAj. (E (/3) — E (ос)) у = £ ocj Xjt or = 0 according as <x < Ay /5 or not.
(8)
Hence, for fixed oc < /5 and a fixed linear subspace M of X, the set
{(E^j-E^yjyeM}
does not contain Ел. if the dimension of M, dim (M), is < w Moreover,
for a suitable M with dim(M) = Wy, the set {(E(/?) — E(a)) у; уб M}
with ос < А^ 3 contains EA>. In fact, the statement is true for M con-
taining Xyy, xjt, . . ., Xj . In particular, = m2 =  • • = mp = 1 with
p = n iff there exists a fixed vector у £ Xn such that the set of vectors
{(EOT-E(«)) у;«<Д
spans the whole space Xn.
These considerations lead to the following definitions:
Definition 1. The spectrum of a self-adjoint operator H == J A dE(E)
in a Hilbert space X is said to be simple, if there exists a fixed vector
yt X such that the set of vectors {(E(/l) — E (oc)) y; oc<zp} spans a linear
subspace strongly dense in X.
Definition 2. Let H = J A dE (A) be a self-adjoint operator in a Hilbert
space X. For fixed a < /3, consider linear subspaces M of (E(/3) —
E (oc))  X such that
(E (?) - E (oc))  M = (E (?) - E (oc))  X.	(9)
We may take, for example, M = (EQS) —- E(oc)) • X to meet condition
(9). The minimum of the set of values dim(M), where M satisfies
condition (9), will be called the total multiplicity oj the spectrum of H
contained in the interval (oc, /3].
Definition 3. The multiplicity of the spectrum of a self-adjoint operator
H = J A dE (A) at A = Ao is defined as the limit, as n oo, of the total
multiplicity of the spectrum of H contained in the interval (Ao — п~г,
Ao + J  >
Example. The coordinate operator H, i.e., the operator H defined by
H  x(t) = t • x(/) in £2(—oo, oo), is of the simple spectrum.
Proof. We know that the spectral resolution H = f AdE(A) is given by
E (A) x(i) ~ x (t) or = 0 according as t rg A or t > A.
Let у (Z) be defined by
у (/) = ck > 0 for k — 1 < t k (k ~ 0, 4~ 1, dr 2, . . .)
with £ cl < oo so that у (if) С E2 (— oo, oo),
9. The General Expansion Theorem
323
Then it is easy to see that the linear combinations of vectors of the form
(E (/?) — E («))	are strongly dense in the totality of step functions
with compact support and consequently strongly dense in L2{—cyoo).
The Problem of the Unitary Equivalence. Two self-adjoint operators
Hy and H2 in an и-dimensional Hilbert space Xn are said to be unitarily
equivalent to each other if there exists a unitary matrix U in Xn such
that Hy = UH2U~X. It is well-known that Hx and H2 are unitarily
equivalent to each other iff they have the same system of eigenvalues
with respectively the same multiplicities. Thus the eigenvalues together
with the respective multiplicities are the unitary invariants of a self-ad-
joint matrix.
The investigation of the unitary invariants for a self-adjoint opera-
tor in an infinite dimensional Hilbert space goes back to a paper by
E. Hellinger [1] published in 1909. See, e.g., M. H. Stone [1]. There
the Hilbert space is assumed to be separable. For the non-separable Hilbert
space case, see F. Wecken [1] and H. Nakano [1] and also P. R. Hal-
mos [2]. K. Yosida [13] proved the following theorem;
Let H be a self-adjoint operator in a Hilbert space X, and let us
denote by {HY the totality of bounded linear operators £ L {X, X) which
are commutative with H. Then two self-adjoint operators Hy and H2 in
X are unitarily equivalent to each other iff there exists a one-to-one
mapping T of (Hy)' onto (H2)'‘ such that T defines a ring-isomorphism
of the ring (Hy)' with the ring (H2Y in such a way that (T • В)* = T • B*
for every В G (HyY-
Thus the algebraic structure of the ring (Hy)' is the unitary invariant
of Hy.
9. The General Expansion Theorem. A Condition for the Absence
of the Continuous Spectrum
Let H == J' Л dE (Л) be a self-adjoint operator in a Hilbert space X.
Then, by E (+ oo) = I and E (— oo) == 0, we have the representation
x = s-lim f dE (Л) x = s-lim (Е(Я3) — (Я-J) x for every X.
(1)
We shall call (1) the general expansion theorem associated with the self-
adjoint operator H. In concrete cases, it sometimes happens that the
resolvent (Л/ — is obtainable more easily than the spectral resolu-
tion H ~ J AdE{X). In such a case, the general expansion theorem (1)
21*
324
XI. Normed Kings and Spectral Representation
may be replaced by

x = s-lim s-lim a--'
&
1 x du
a
Proof. If v 0, then
OQ
1 x - -
dE(X)x, for every x£X.
—oo
For, by approximating the integral by Riemann sums and remembering
the relation E (A) E (A') = F(min(A, A')), we obtain, for Im (fi) 0,
Therefore, we have
9. The General Expansion Theorem
325
which tends, when v | 0, strongly to
0-0
J 2 ztidE (2) x + 7ii (E fl) — E fl — 0)) x + 7ii (E (oc) — E (oc — 0)) x
a +0
= 7ii (E fl) + E fl — 0)) x — 7ii (E (oc) + E(oc — 0)) x.
This proves formula (I').
Remark. The eigenfunction expansion associated with the second
order differential operator
d2
-d^ + чм
with real continuous q(x) in an open interval (a, b) was inaugurated by
H. Weyl [2], further developed by M. H. Stone [1], and completed
by E. C. Titchmarsh [2] and K. Kodaira [1] who gave a formula which
determines the expansion explicitly. The expansion is exactly a
concrete application of (1). The crucial point in their theory is to give the
possible boundary conditions at x = a and x = b so that the operator
d2 i , x
becomes a self-adjoint operator H in the Hilbert space L2(a, b). Their
theory is very important in that it gives a unified treatment of the
classical expansions in terms of special functions, such as the Fourier
series expansion, the Fourier integral representation, the Hermite poly-
nomials expansion, the Laguerre polynomials expansion and the Bessel
functions expansion. We do not go into details, and refer the reader to
the above cited book by Titchmarsh and the paper by Kodaira. Cf.
also N. A. Naimark [2], N. Dunford-J. Schwartz [5] and K. Yosida [1].
The last cited book gives an elementary treatment of the theory.
If the continuous spectrum Ca(H) is absent, then the expansion (1)
will be replaced by a series rather than the integral. We have, for in-
stance, the following
Theorem 1. Let Z/ f 2 dE (z) be a self-adjoint compact operator in
a Hilbert space X. Then (i) Ca (H) contains no real number except possibly
0; (ii) the eigenvalues of H constitute at most a countable system of real
numbers accumulating only at 0; (iii) for any eigenvalue 0 of H,
the corresponding eigenspace E^ is of finite dimension.
Proof. Suppose a closed interval [2', 2"] on the real line does not
contain the number 0. Then the range R (E (2")  E (2')) is of finite
dimension. If otherwise, there exists, by E. Schmidt’s orthogonalization
in Chapter III, 5, a countable orthonormal system {x;} contained in
R(E (2") — E(2')). We have w-lim Xj = 0, since, by Bessel's inequality,
OO
Д I (/, xfl |a L. ||/||2 for any /е X.
326
XL Normed Rings and Spectral Representation
Hence, by the compactness of the operator H, there exists a subsequence
{xA such that s-lim Hx,- = да-limHx:> = 0. On the other hand, we have
j—ню	j—>oo
I j Hx, 1|2 = рм 11E (Л) Xj 113 = f A2 d I ] E (Л) (E (A") - E (Л')) Xjj |2
A"
= f	||x,||’-inin(p'|!. |Л"|«)
= min(H|2. P"P),
which is a contradiction.
If Са{Н) contains a number Ao 0, then, by Theorem 1 in the preced-
ing section, s-lim (E(Aq — s}— E (Aq — g)) x = 0 for any x E X. As proved
e 0
above, the range R(E(A$ + e) — E(Aq — e)') is of finite dimension and
this dimension number is monotone decreasing as e | 0. Hence we see
that (E (Лд + e) — E (Ao — g)) = 0 for sufficiently small e > 0, and so,
by Theorem 1 in the preceding section, z() cannot be contained in Ca(H).
This proves our Theorem.
Corollary 1. Let {2J be the system of all eigenvalues of H different
from 0. Then, for any x 6 X, we have
x = (E (0) — E (0 — 0)) x -f~ s-lim (E (L) — E (L — 0)) x.	(2)
я-мзо 7 = 1
Proof. Clear from (1).
Corollary 2 (The Hilbert-Schmidt Expansion Theorem). For any x£ X,
we have
Hx = s-lim J L (E (L) - E Q - 0)) x.	(3)
KX3 pl
Proof. Clear from the continuity of the operator H and the fact that
H (E (0) -E(0- 0)) = 0 and	— EQ — 0)) x = A, (E Q)
— E(Aj — 0)) x, the latter being implied by R(EQ) - EQ — 0)) = ЕЯу,
the eigenspace of H belonging to the eigenvalue A?.
Remark. The strong convergence in (3) is uniform in the unit circle
{%; ||x||	1}. For, we have
Hx— f AdE(A)x 2 =
£
2= f A2d ]|E(A) x||2
— 8
e
£2 f rfj|E(a) x jj2 g £2 f d \\E(A) % I)2 = s2 fi2-
— 8
10. The Peter-Weyl-Neumann Theory
Let G be a totally bounded topological group, metrized by a
distance satisfying the condition (see Chapter VIII, 5)
dis (%, y) = dis (axb, ayb) for every x, y, a and b^G. (1)
10. The Peter-Weyl-Neumann Theorem
327
Let /(g) be a complex-valued bounded uniformly continuous function
defined on G. For any e > 0, set
7 = {y € G; sup | / (x) — / (y^ x) | < e}.
(2)
Then, by the continuity of /, the set V is an open set containing the unit
e of the group G. Hence the set U = V А У-1, where V^1 = {y~l; у € I7},
is also open and Э e. If we put
k{x} = ^(^(x)	where
k-.(x} =	—----------------tt-	(dis(x,	Uc)	= mf dis (%, y)\,
1V ’	dis{x,	e} + dis(x, Vе}	(, x 7
then we have the results:
k(x) is bounded uniformly continuous on G, and
A(x) -- k(x~1')t 0 < k (x) < 1 on G, k(e) = 1 and (4)
k(x) ~ 0 whenever x £ Uc.
Hence, for all x, у £ Gt we obtain
Ш (/(*) — /(У1*)) | e k(y].
By taking the mean value (see Chapter VIII, 5) of both sides, we obtain
|М,(Л(y)) f(x)-M„(k(y)	I g e Mr(k!y)).
We have My (k (y)) > 0 by k (y)	0 and k (у) Ф 0. Hence
|/(%)- My (kQ (y) / (y-1 x)) [ <L e, where k0 (x) = k (х)/Мх (k (x)).	(5)
Thus, by virtue of the invariance My (g (y-1)) =	(g(y)) = (g (лу)) =
My(g(ya)) of the mean value, we obtain, from (5),
j/(x) — Му(Л0(ху-1)/(у))| e for all xeG. (6)
Proposition 1. We shall denote by C (G) the set of all complex-valued
bounded uniformly continuous functions h(g) defined on G. Then
C (G) is a В-space by the norm |[Л ||0 = sup |h{g) Then, for any b and
h^C(G'),
(bxh) W =М,(5(ху-1)Л(у))	(7)
also belongs to C (G).
Proof. By dis(x, z} = dis{iixc, azc) and the uniform continuity of the
function &(g), there exists, for any <5 > 0, an 77 = 77(d) > 0 such that
sup jf^xy-1) — &(%' y-1) j	d whenever dis (%, x')	77.
у
Hence, as in the case of Schwarz’ inequality, we obtain
\M,(b{xy^ h(yS) - My(fi{x'y-^ Л(у)) |2
£ M,((6(xy-‘) -	 V,(|A(y) |2) й й2М,(|Л(у) |2)
whenever dis(x, x')7/. This proves our Proposition 1.
328
XI. Normed Rings and Spectral Representation
Proposition 2. C (G) is a pre-Hilbert space by the operation of the
function sum and the scalar product
(6, h) = (6хЛ*) (e) = My(b(y~l) A(y-1)) = My(b(y) h(y)), where
—-	(9)
Л* (y) = h (у г). We shall denote this pre-Hilbert space by C (G).
Proof. Easy.
Proposition 3. We shall denote the completion of the pre-Hilbert space
C (G) by C (G), and the norm in the Hilbert space C (G) by [|Л || = (h, Л)1'12.
Then the linear mapping T on C (G) into C (G) defined by
(Th) (%) = ^h) (x), x^G,
(10)
can, by the continuity in C (G), be extended to a compact linear operator
T on C(G) into C(G).
Proof. By virtue of My(l) = 1, we obtain
И|| = (M)1/2 = My(h{y) /г(у))ш Asup |A(y)| = ||<.	(11)
У
The continuity of the operator T in C(G) is clear from the Schwarz
inequality for (7), and so, by the denseness of C(G) in C(G), we can
extend T to a bounded linear operator T in C (G). On the other hand, we
see, by (8), that T is a compact operator on C(G) into C(G). We prove
this by the Ascoli-Arzehi theorem. Thus, by (11), we easily see that T is a
compact operator on C (G) into C (G).
Therefore, by the denseness of C (G) in C (G), we see that the extended
operator T is also compact as an operator on C (G) into C (G).
We are now ready to prove the Peter-Weyl-Neumann theory on the
representation of almost periodic functions.
It is easy to see that, by Ao (%y4) = kQ (y x”1), the compact operator Tis
self-adjoint in the Hilbert space C(G). Hence, by the Hilbert-Schmidt
expansion theorem in the preceding section, we obtain
Th — s-lim)	A uniformly in h satisfying I h | A 1,	(12)
if we denote by {A»,} the system of all eigenvalues of T different from
0, and by P^ the projection upon the eigenspace of T belonging to the
eigenvalue /.m.
Since /(g), introduced at the beginning of this section, belongs to
C (G), we have T f = T/ 6 C (G). Since the eigenspace R (Рцт) — P^m  C (G)
is of finite dimension, there exists, for every eigenvalue Am, a finite
system	°* elements € C(G) such that each hQ R(PAm) =
10. The Peter-Weyl-Neumann Theorem	329
P^ • C (G) can be represented as a uniquely determined linear combina-
tion of hmj’s {j = 1, 2, . . nm). Let
]
P>.mh = £ Cjhwhere tfs are complex numbers. (13)
j—i
Then, since hm]^ R(l\y we have fhmj = and so, by applying
(8) to the operator T given by (10), we see that — ^(Th^) must
belong to C (G). Hence, by (13), we see that, for each eigenvalue of T,
the eigenspace R (P^) = Рцт • C (G) is spanned by the functions hmJ^ C (G).
Therefore, we have, by (12),
(T/) (x) = s-lim £ Лт/т(х) hi the strong topology of C (G),
n—ЮО w—1
where /m = P^ - f£C(G) for each m.
By applying (8) to the operator T given by (10), we see that
(T2/) (%) = lim Му(ло(ху-1)  £ Am/m(y)) uniformly in x. (14)
я—*oo	\	m=l	/
On the other hand, by (6) and My(k0(y)) = 1, we obtain
IM/Ao^-1) /(2)) —Mz(A0(x2_1) My(£0(zy-1)/(y))| E,
and so, combined with (6), we have
\f(x)-M,[k0(xz-1) M,(ka(zy-1)	2s.	(16)
The left hand side is precisely |/(x) — (P2/) (%) j 2a.
Since T • P (Рдт) £ R (T\«)» Wl’ have proved
Theorem 1. The function /(x) can be approximated uniformly on G by
linear combinations of the eigenfunctions of T belonging to the eigen-
values which are different from 0.
We shall take a fixed eigenvalue Л 7^ 0 of T, and shall denote the
base {hj} £ C (G) of the corresponding eigenspace PA • C (G) by (x),
e2(x), . . ., ek(x). Then, by the invariance of the mean value, we obtain,
for any a C G,
My (A W1) ei (У «)) = My (At (*« • e, (ya)) = Mx (k0 (xa  z~') (z))
= (T&j) (xa) = «Ц (xa).
Since the left hand side is equal to the result of applying the operator T
upon the function a3(ya) of y, we see that, for any given a £ G, the func-
tion ej(xa) of x must be uniquely represented as a linear combination of
the functions'(x), e2(x), . . ., eft(x). We have thus
A
e,(xa) V</,.;(«) c,(x) (/=1,2------------A),	(16)
< — I
330
XI. Normed Rings and Spectral Representation
or, in vectorial notation,
e(xa) = D(a) e(x).
(16')
By e(x  ab) = D(ab) e(x), e(xa  b) = D(b) e(xa) = D(b) D(a) e(x)t we
see, remembering the linear independence of ^(x), e2(x), . . ., en(x), that
D (ab) = D (b) D (a), D (e) = the unit matrix of degree k.	(17)
By applying E. Schmidt's orthogonalization, we may assume that {еДх)}
constitutes an orthonormal system in C (G). Then, by (16),
Mx (ej (x a) e* (x)) =	(a)	(18)
and so the elements ^(«) of the matrix D(a) belong to C(G). By the
invariance of the mean value, we see that
М/гДул) e/(ya)) = Му(е^(у) ei (у)) = <5y.
Hence the matrix D (a) gives a linear mapping transforming the ortho-
normal system {ej (%)} onto the orthonormal system {е-Дхд)}. Therefore
D(a) must be a unitary matrix. Hence the tranposed matrix D(a)' of
D (#) is also unitary and we have
D(ab)r = D(a)' D(b)', D(e)! = the unit matrix of degree k. (17')
Hence D (a)' gives a unitary matrix representation of the group G such that
its matrix elements are continuous functions of a. Letting x = e in (16),
we see that each (a) is a linear combination of the matrix elements of
the representation D(a)’.
We have thus proved
Theorem 2 (Peter-Weyl-Neumann). Let G be a totally bounded
topological group, metrized by a distance satisfying dis(x, y) =
dis (a x b, ayb). Let / (g) be any complex-valued bounded uniformly
continuous function defined on G. Then, /(g) can be approximated uni-
formly on G by linear combinations of the matrix elements of unitary
uniformly continuous matrix representations D(g)' of the group G.
Referring to A. Weil’s reduction given in Chapter VIII, 5, we obtain
the following
Corollary. Let G be an abstract group, and /(g) an almost periodic
function on G. Then//(g) can be approximated uniformly on G by linear
combinations of the matrix elements of unitary matrix representations
D(gY of the group G.
Remark 1. Let the degree of a unitary matrix representation D(g}'
of the group G be d, i.e., the degree of the matrix Z)(g)' be d. Then each
D(g)' gives a linear mapping of a fixed (/-dimensional complex Hilbert
space Xd onto itself. The representation D (g)' is said to be irreducible if
there is no proper linear subspace {6} of Xd which is invariant by
applying the mappings P(g)',_	Otherwise, the representation
10. The Peter-Weyl-Neumann Theorem	331
D(gY is called reducible, and there exists a proper linear subspace
7^ {0} invariant by every D[gY, g^G. Then the orthogonal com-
plement Х£г of Xai in Xd is, by the unitarity of the representation
D(g)', also invariant by every D(g), g G G. If we take for the base of the
linear space Xd the orthonormal system of vectors composed of one
orthonormal system of vectors of l and one of Х£г, then, by this
choice of an orthonormal base of Xd, the representation D{gY will be
transformed into the form
UD(gY	r, ? J, where U is a fixed unitary matrix.
X 0	^2 \Sf /
Hence a reducible unitary representation D(gY is completely reducible
into the sum of two unitary representations Dx(g)' and />2(g)' of the
group G acting respectively upon Xai and Xdtl. In this way, we finally
can choose a fixed unitary matrix Ud such that the representation
UaD(gY U,1 is the sum of irreducible unitary representations of’ the
group G. Therefore, in the statements of Theorem 2 and its Corollary,
we can impose the condition that the matrix representations D(gY are all
irreducible.
Remark 2. In the particular case when G is the additive group of real
numbers, a unitary irreducible representation D(gY is given by
D (gY = where a is a real number and i = j/— 1 .	(19)
For, by the commutativity of the unitary matrices D(gY> g£_G, the
representation Z>(g)' is completely reducible into the sum of one-dimen-
sional unitary representations /(g), that is, complex-valued solutions
Z (g) °*
%(gi + ёг) = z(gi) -%(g2). IzUl) I = 1 fen &eG),Z (0) = 1.	(20)
It is well known that any continuous solution of (20) is of the form
/(g) = etas. Hence, any continuous almost periodic function /(g) on the
additive group G of real numbers can be approximated uniformly on G
by linear combinations of with real a’s. This constitutes the fun-
damental theorem in H. Bohr’s theory of almost periodic functions.
According to the original definition by Bohr, a continuous function f (%)
(— oc < x < oo) is called almost periodic if for each e > 0 there exists a
positive number p = P(e) such that any interval of the form (t, t -|- p)
contains at least one r such that
/(% + t) — /(%)
< £ for ' OC <'. X <C OO.
See H. Bohr [1]. S. Bochner [4] has shown that a continuous function
f[x) (—oo <Z x < oc) is almost periodic in Bohr’s sense iff the following
condition is satisfied: For any sequence of real numbers the system
of functions {/„„(i); /„,,(') f{x | «„)) is totally bounded in the topology
332	XI. Normed Rings and Spectral Representation
of the uniform convergence on (— 00,00), It was extended by J. von
Neumann [4] to almost periodic functions in a group. Neumann’s result
contains as a special case, the Peter-Weyl theory of continuous representa-
tion of a compact Lie group (Peter-Weyl [1]). According to our treat-
ment, Bohr's result is easily proved by observing that lim |s —	= 0
implies lim supj/(es&)—f{atb}[1 = 0.
afi
11. Tannaka's Duality Theorem for Non-commutative Compact
Groups
Let G be a compact {topological} group. This means that G is a compact
topological space as well as a group in such a way that the mapping
(x, y) xy-1
of the product space G X G onto G is continuous.
Proposition 1. A complex-valued continuous function f{g} defined
on a compact group G is uniformly continuous in the following sense:
for any e > 0, there exists a neighbourhood U {e} of
the unit element e of G such that [/(%) -f{y) | < e 
whenever xy 1 £ U {e) and also whenever x^y E 17(e).
(1)
Proof. Since f(x) is continuous at every point a£G, there exists a
neighbourhood Va of a such that x£Vt implies |/(x)—/(a)[<e/2.
If we denote by Ua the neighbourhood of e defined by Ua = VacrT =
{va~l; v£ P], then Ua implies \f{x) — f{a) j < e/2. Let us denote
by Wa the neighbourhood of e such that W% C where W% =
{w1w2; ry(~ Wa {i = 1, 2)}. Obviously, the system of all open sets of
the form Wa • a, where a is an arbitrary element of G, covers the whole
space G. G being compact, there exists a finite set {a,; i = 1, 2, . . n}
such that the system of open sets -iit {i = 1, 2, . . ., n) covers G.
We denote by U {e} the intersection of all open sets of the system
Then U{e} is a neighbourhood of e. We shall show that if xy-1 E G(e),
then \f{x) — f{y} [ _£ e. Since the system • a^ covers G, there exists
a number k slich that yak1 6 Wak G Uak and therefore |/(y) — f{ak} | < e/2.
Furthermore we have xa^1 = ху'"'уакл E U {e) Wak £ W%k Q Uaks° that
f{x} — /(«*)[ < e/2. Combining these two inequalities we get
/(*) —/(y)| < e-
If we start with a neighbourhood Ua of e such that %-1 a 6 Ua implies
\f{x) - - f{a) j < e/2, we would obtain a neighbourhood U {e} of e such
that \f{x) — fly) j < £ whenever хл у E U{e). Thus taking the inter-
section of these two U (e)’s as the U (e) in the statement of the Proposi-
tion, we complete the proof.
11. Tannaka's Duality Theorem
333
Corollary. A complex-valued continuous function f(%) on a compact
group G is almost periodic on G.
Proof. Let U (e) be the neighbourhood of e given in Proposition 1.
For any a £ G, U (e) a is a neighbourhood of a. The compact space G is
covered by the system of open sets U {e} a, a £G, and therefore some
finite subsystem { U (e) i = 1, 2, . . n} covers G. That is, for any
a 6 Gr there exists some ak with k < n such that a a*1 € U{e}. Hence, by
{ax} {акх}~х — aa*1, we have sup \f{ax) — /(дАх)| < e. Similarly, we
can find a finite system {bjt j = 1, 2, . . ., m} such that, for any b£G,
there exists some bj with j m satisfying the inequality
sup \/(а^Ь} — {{a^fy} j
Therefore, for any pair a, b of elements E G, we can find and bj
{k n, j m} such that
sup\f{axb} —/{azxbj}
2e.
This proves that the system of functions {/а>ь{%}', fa,b(x} = /{axb}, a and
G} is totally bounded by the maximum norm 11 h j | — sup | h (%) |. Hence
f{x} is almost periodic on G.
We are now able to extend the Peter-Weyl-Neumann Theory of the
preceding section to complex-valued continuous functions f{x} on a
compact group G. For such a function f{x} and e > 0, we set
V = {y€ G;sup \f{x} — /(у-1*)
Then, by the continuity of /, the set V is an open set containing e. By
Urysohn’s theorem as applied to the normal space G, we see that there
exists a continuous function kt {x} defined on G such that
0 < k{x} < 1 on G, кг(е} — 1 and kr{x} = 0 whenever Vе.
Then the continuous function
k {x} = 2-1 (йх (%) + kt (x-1))
(2)
satisfies the condition k (x-1) = к {x} and
0 < k{x} 1 on G, k{e} = 1 and k(x} = 0
whenever x E Gc, where U = TV F”1.
Therefore, if we denote by C (G) the В-space of all complex-valued
continuous functions h{x} defined on G normed by the maximum norm,
we can define a linear operator T on C(G) into C(G) by
{Th} (x) - {kiyxh} (x), xE G.
(3)
334	XL Xormed Rings and Spectral Representation
Here, as in the preceding section,
k0 (x) = k (x)jMx (k (%))	(4)
and C (G) is the the space C (G) endowed witht the scalar product
(b, h} = My(b(y) h{y)) = (bxh*) (e), Л*(у) = Лёг1). (5)
Thus, as in the preceding section, we obtain
Theorem 1. Any complex-valued continuous function f(g) defined on a
compact group G is almost periodic, and /(g) can be approximated uni-
formly on G by linear combinations of the matrix elements of unitary,
continuous, irreducible matrix representations of G.
We shall say that two matrix representations /4j(g) and Az(g) of a
group G are equivalent if there exists a fixed non-singular matrix В
such that B^AjJg) В = for all g£ G.
Proposition 2 (I. Schur’s lemma). If the representations Ar(g), A2(g)
are irreducible and inequivalent, then there is no matrix В such that
A1{g) В BAz(g)	(6)
holds identically in g, except В ~ 0. In (6), the matrix В is assumed
to be of nY rows and n2 columns, where nlt w3 are the degrees of Ar(g),
H2(g), respectively.
Proof. Let and Xz be the linear spaces subject to the linear
transformations Л-Jg) and A2(g), respectively. В in (6) can be inter-
preted as a linear mapping ,r2 >	of X2 onto Хг. The linear
subspace of consisting of all vectors xv of the form Bx2 is invariant,
for H1(g)	~Bx'2 with	x2- By virtue of the irreducibility
of Ax (g), there are only two possibilities: either В x2 = 0 for all x2 in A'2,
i.e. В = 0, or Xx = BX2. On the other hand, the set of all vectors x2
in X2 such that Bx2 = 0 is an invariant subspace of X2, for В A2 (g) x2 =
A2 (g) Bx2 = 0. From the irreducibility of X2(g) we conclude: either
Bx2 = 0 for all x2 in Xz, i.e. В = 0, or x2 = 0 is the only vector in X2
such that Bx2 = 0, so that distinct vectors in X2 go into distinct vectors
in X± under the linear mapping B. Hence if В 0 we conclude that В
defines a one-on linear mapping of X2 onto But this means that В
is a non-singular matrix (nr = m2) and so ^(g) and A2(g) would be equi-
valent.
Proposition 3 {the orthogonality relations). Let A1(g) ~ («^(g)) and
.42(g) = (4(b)) be unitary, continuous, irreducible matrix representa-
tions of a compact group G. Then we have the orthogonality relations:
^g(4'(s) 4(g)) = 0, if Xx(g) is inequivalent to ^2(g),
Mg (alj (g) 4, (g)) = «j1	4 where nL is the degree of A x (G).
Proof. Let nlr n2 be the degrees of Hjfg), Xa(g) respectively. Take any
matrix В of пг rows and n2 columns, and set A (g) = A1(g) Bd2(g“1).
11. Tannaka's Duality Theorem
335
Then the matrix А =Мё(А(ёУ) satisfies Ay(g) A =AAz[g). For, we
have, by the invariance of the mean value,
/МУ) AAz(y~^ = Mg(AM a} (g) МГ) МЛ)
=	BA^yg)'1) = A .
"By Schur's lemma, A must be equal to a zero matrix. If we take В = (b#)
in such a way that only the element Ьц is not zero, then, by the unitarity
condition 32(f4) = /Mg)', we obtain
^(4(g) 4(g)) = o.
Next we have, as above, Ay (g) A = А Ay (g) for A ~ Мг(Лх (g) В Ay (g'1))-
Let tx be any one of the eigenvalues of the matrix A. Then the matrix
(A — a/nj), where Ini denotes the unit matrix of degree nlt satisfies
A(g)	=	/Mg).
Therefore, by Schur's lemma, the matrix (A — oclnp is either non-singular,
or (A — aZ„.) = 0- The first possibility is excluded since a is an eigen-
value of A. Thus A = <xlni. By taking the trace (the sum of the diagonal
elements) of both sides of A = Ms (Ay (g) BAyfg^1)), we obtain
nyoc = trace (Л) = Mg (trace (A у (g) 2?/Mg)-1) = Mg (trace (£>’))
= trace (Bj.
Therefore, if we take В — (fy;) in such a way that = 1 while
the other elements are all zero, then, from Me (Ay (g) BT1(g-1)) = ny1
trace (В) Ine we obtain
ms (4 (g) 4 (g)) = ni1	
Corollary. There exists a set U of mutually inequivalent, continuous,
unitary, irreducible matrix representations 17(g)	(u^g)) of G satisfy-
ing the following three conditions:
i)	for any pair of distinct points gt, g2 of G, there exists an U (g) C U
such that U (gy) U (g2),
ii)	if U(g)Q U, then the complex conjugate representation U (g) of
V (g) also belongs to 11,
iii)	if Uy(g), Uz(g) are in U, then the product representation Lj(g)x
2J2(g), explained below, is completely reducible to a sum of a
finite number of representations € U.
Proof. The definition of the product representation Uyi^xU^g)
is given as follows. Let (e7, e%,. . ., e^) and (ву, ef, . . ., 4) be orthonormal
bases respectively of the finite dimensional complex Hilbert spaces
subject to the linear mappings Uy(g) and U%{g], respectively. The product
space of these two Hilbert spaces is spanned by the product base con-
336
XL Normed Kings ;md Spectral Representation
sisting of nm vectors е- X ej {i 1, 2, , . ., n; / = 1, 2, . . ., m). Referring
to this base, the product representation Ui(g)xU2(g) of Uy(g) =
(4 U)) and U2(g) = («w(g)) given by
(?7i (g) X U2 (g)) (ej X ef} = £	(g) u$ (g) (ej X ej).
Let us take a maximal set U of mutually inequivalent, continuous,
unitary, irreducible matrix representations U (g) satisfying the condition
ii). Then, by Theorem 1, condition i) is satisfied. Condition iii) is also
satisfied since the product representation of two unitary representations
is also unitary and hence is completely reducible.
We are now ready to formulate T. Tannakas duality theorem. Let JR
be the set of all Fourier polynomials'.
that is, finite linear combinations of u^f (g), where («^(g)) € U and y$
denote complex numbers. Then JR is a ring with the multiplicative unit
и (и (g) = 1 on G) and with complex multipliers; the sum and the multi-
plication in the ring JR being understood as the function sum and the
function multiplication, respectively. Let be the set of all linear
homomorphisms T of the ring JR onto the field of complex numbers such
that
Tu — 1, Tx = Tx, the bar indicating the complex-conjugate. (8)
St is not void since each g C G induces such a homomorphism Tg:
Tsx = x(g).	(9)
By the condition i) for U, we see that
gi^g2 implies Tgi^ TSi.	(10)
Proposition 4. X may be considered as a group which contains G as
a subgroup.
Proof. We shall define a product Tr  T2 = T in % as follows. Let
у <"> (g) = (4> И) = 1,2..............»)
be members of 11. We put, for	>
k
Tu*y = z	.	(11)
k
By the orthogonal relations (7), we see that the functions u$ (g)'s, where
(«|*J(g)) € IL are linearly independent on G. Hence, we see that T may
be extended linearly on the whole JR. It is easy to see that the extension T
is also a member of % and that St is a group with the unit Te (e — the
unit of G) and Т~л = Tg-i. G is isomorphically embedded in this
group St by the correspondence g Tg, as will be seen from (11) and (10).
11. Tannaka’s Duality Theorem
337
In truth, we have
Theorem 2 (T. Tannaka). 2 = G, viz. every T £ 2 is equal to a
certain
Tx — %(g) for all xC 91.	(12)
Proof. We introduce a weak topology in the group 2 by taking the
sets of the form
{T € 2; j Тхг — TeXi \< Ei (i = 1, 2, . . ., и)}	(13)
as neighbourhoods of the unit Te of the group 2. Then 2 is a compact
space. For, by | T uff (g)|2 = (T  T) (1) = 1 implied by (8) and (11),
s
St is a closed subset of the topological product of compact spaces:
П {z; |z[ dd sup [Tx|},
and so we can apply Tychonov’s theorem. It is easily seen that the
isomorphic embedding g <+ Tg is also a topological one, because a one-
one continuous map of a compact space onto a compact space is a topo-
logical map. Therefore G may be considered as a closed subgroup of the
compact group 2-
By the above weak topology of 2, each %(g)E9i gives rise to a
continuous function x (T) on the compact group 2 such that x (Tg) = x (g).
We have only to set x (T) = T • x. The set of all these continuous func-
tions x(T) constitutes a ring Э1 (2) of complex-valued continuous func-
tions on X satisfying the conditions:
1)	1 = м(Г)у fR(2),
2)	for any pair of distinct points Tv T2 of 2, there exists an x (T) 6
91 (2) such that x (7^) x (Г2),
3)	for any x(T) £ 91 (2), there exists the complex-conjugate function
x(T\ = x(T) which belongs to 91 (2).
Now let us suppose that 2 — G is not void. Since a compact space 2
is normal, we may apply Urysohn’s theorem to the effect that there exist
a point To £ (X — G) and a continuous function у (T) on 2 such that
y(T) Sr 0 on 2, y(g) = 0 on G and y(T0) = 1.	(M)
By the Stone-Weierstrass theorem in Chapter 0, as applied to the ring
91(2) satisfying 1), 2) and 3), we see that there exists, for any e > 0, a
function x(g) = 2J (g) € 91 such that |y (T) —	(T) | < r
on 2, and, in particular, [y (g) — £	(g) | < £ on G. Let w(ue,(g)
= и (g) = 1. Then, by taking the mean value and remembering the ortho-
gonality relation (7), as applied to the compact groups 2 and G, we obtain
|MT(y(O)	(If>)
22 Yoalibi, F'incihuiiil Лгийуи1н
338	XI. Normed Rings and Spectral Representation
We have thus arrived at a contradiction, since Мт(у[Т)) > 0 and
^(y(g)) = 0 by (14)‘
Remark 1, The above proof of Tannaka’s theorem is adapted from
K. Yosida [14]. The original proof is given in T. Tannaka [1]. Since a
continuous, unitary and irreducible matrix representation of a compact
abelian group G is precisely a continuous function % (g) on G satisfying
X (gi) X (£2) = X (gi St) and \x te) I = 1.
Tannaka’s theorem contains, as a special case, the duality theorem of
L. Pontrjagin [1]. For further references, see M. A. Naimark [1].
Remark 2. To define mean values of continuous functions f^(g) [k =
1, 2,..w) on G by the method given in Chapter VIII, 5, we have only
to replace the dis (gp g2) there by dis (gp g2) = sup (g gx h) -
Л (& АЛ1
12- Functions of a Self-adjoint Operator
Let H = J AdE (Л) be the spectral resolution of a self-adjoint operator
H in a Hilbert space X. For a complex-valued Baire function /(A), we
consider the set
where the integral is taken with respect to the Baire measure m deter-
mined by w((Ax, Ag]) = ||Е(Л2) x |j2 — x |j2. As in the case of a
continuous function /(Л), treated in Chapter XI, 5, we see that the inte-
gral
OO
f	(2)
—CO
exists and is finite with respect to the Baire measure ж determined by
ж((Ях, ZJ) = (E (Я2) x, у) — (£(ЯХ) x, у). Further we see that (2) gives the
complex conjugate of a bounded linear functional of y. Hence we may
write (2) as (j(H) x, y) by virtue of F. Riesz’ representation theorem in
Chapter III, 6. In this way, we can define functions of H;
/	00
/(№= / /(Л)Ж(Л)	(3)
-oo
through (1) and (2).
ft
Example 1. If H is bounded self-adjoint and /(Л) ~	then,
as in Chapter XI, 5,
oo	«
= f f(A}dE(A) =
— OO
12. Functions of a Self-adjoint Operator
339
Example 2. If /(2) = (2 — i) (2 + I)"1, then f(H) is equal to the
Cayley transform UH of H. We have, in this case, /)(/(#)) = X since
[/(2) | = 1 for real 2. It is easy to see, by E (2!) £“(22) = £'(min(21, A^),
that if we apply the bounded self-adjoint operator (H — ii)-1 =
J (2 — i)-1 dE (2) to the operator / (H) = jf / (2) dE (2) then the result is
equal to (Я + tV)"1 = f (2 + f)"1 dE(X). Hence /(Я) = UH.
Example 3. As in Chapter XI, 5, we have
11/(Я) x |ja = f |/(A) |2 d |[£(2) x ||2 whenever x£_D(f(H)). (4)
“ OQ
Definition. Let A be a not necessarily bounded Enear operator, and
В a bounded linear operator in a Hilbert space. If
x£D{A) implies Bx£D(A) and ABx = BAx, (5)
that is, if AB 2BA, then we shall write В G (A)' and say that В is
commutative with A. We shall thus write the totality of bounded Enear
operators В commutative with A by (A)'.
Theorem 1. For a function f(H) of a self-adjoint operator /I =
J /.dE (2) in a Hilbert space X, we have
(/(Я))'2(Я)',	(6)
that is, /(H) is commutative with every bounded Enear operator which
commutes with H. (Since E (2) С (Я)' by Theorem 2 in Chapter XI, 5, we
see, in particular, that /(Я) is commutative with every E (2).)
Proof. Suppose S £ (Я)'. Then we can show that S is commutative
with every E(2). We first show that S is commutative with the Cayley
transform UH of H. For, if x C D(H), then we have, by S 6 (Я)',
S(H EiI)x=(H + ii) Sx, (H-il) Sx =	— x.
By putting (H + ii) x = y, we see, from the first of the above relations,
that
(Я + iFp1 Sy = S(H + И)-1 у for all у С X = R (H + ii).
Hence
S(H — ii) (H + iBr1 = (H~ ii) (H + H)-1 S, that is, SUH = UffS.
2л
Therefore, S is commutative with (UH)n = f ente dF (&) (» = 0, ih • • •)
о
and so
/ enie d(SF(0)x,y) = (s f e№iedF(0) x,y}= f d(F(6) Sx, y).
Hence, as in the proof of the uniqueness of the spectral resolution of a
unitary operator, given in Chapter XI, 4, we obtain SF(0)	F(0) S.
This proves S£"(2) =£"(2) Ssince Я(--cot 0)—F{0). Thus, for x£D(J(H’)'),
22*
340
XI. Normed Ringh ;md Spectral Representation
we obtain
X, y) = f/m<i(SE())x,y) = ff(l}d(EWSx,y),
that is, 5/(Я) Q f(H} S.
The above Theorem 1 admits a converse in the form of
Theorem 2 (Neumann-Riesz-Mimura). Let Я be a self-adjoint opera-
tor in a separable Hilbert space X. Let T be a closed linear operator in X
such that D (T)a = X. Then a necessary and sufficient condition in order
that T be a function / (H) of H with an everywhere finite Baire function
/(Я) is that
(TO(tf)'.	(7)
Proof. We have only to show that condition (7) is sufficient. We may
assume that H is a bounded self-adjoint operator. If H is not bounded,
then we consider Hy = tan1 H. Since [ tan1/. | < л/2, we easily see that
Hy is bounded self-adjoint. By Theorem 1, the operator H = tan Hy is
commutative with every operator g Thus, by hypothesis, we have
(ТУ 2	2 (HJ.
Therefore, if Theorem 2 is proved for bounded Hlr then T = fy(Hy)
4 (tan-1 H) = /2(H), where /2 (Л) = A (tan-1 Л).
We may thus assume that H is bounded and self-adjoint.
The first step. For any fixed xQ£ D(T), we can find a Baire function
F (Я) such that T x$ = F (H) xQ. This may be proved as follows. Let
М(я0) be the smallest closed linear subspace of X spanned by xQ, Нхй,
Н^хй,. . . , H^Xq, . . . We denote by L the projection upon M(xQ). Then
(Т)'э£. For, by HM(x0) Q M(xQ), we obtain HL — LHL and so
LH = (H£)* -- (LHL)* = LHL = HL, that is, L G (Я)'. Hence, by
hypothesis, (Г)' Э L.
Therefore TxQ=TLxQ = LTx0£M(x0), and so there exists a
sequence {А*(Я)} of polynomials such that
Txq = s-lim (Я) xQ.	(8)
л—изо
Hence, by (4), we obtain
CO
— 00
As in the proof of the completeness of L2(S, SB, m), we see that there
exists a Baire function _F(A) which is square-integrable with respect to
the measure m determined by w((Alt Л2]) = Ц-Е^) %0|[2— xoI)2
such that
OO
lim / р(Л) -Д,(Л)|М ||Я(Л) «о||2 =: 0.
12. Functions of a Self-adjoint Operator
341
Hence, for F (ZZ), we have
(Xj
lim | |CF W)xa ||s = lim J \F(X)-/>„(Я)	||£ (А) х„|Р = 0.
n-HX	n->OO _
This proves that Txa~ F (H) xQ. Since F (Я) is finite almost everywhere
with respect to the measure ж determined by m((Xv Яй]) =
||E (Л2) xQ j|2 — 11E (Zj) %01]2, we may assume that F (Л) is a Baire function
which takes a finite value at every Л; we may put F(A) = 0 for those A
for which | F (Л) | = oo.
The second step. Since X is separable and D{T)a — X, we may choose
a countable sequence {g„} g D(T) such that is strongly dense in X.
We set
1
/1 = gi> /2 = £2 — Li    > fn = gn — Lkg„, where Lk
(9)
is the projection on the closed linear subspace M(fk).
By the first step, we have (T)' Э Lk and so
Lkgn£ which implies fn£ D(T) (и = 1,2,...).	(10)
We may show that
ЦЕк = 0 (for i=£ k)	(11)
and
CO
/ =	(12)
Suppose that (11) is proved for i, k < n. Then, for i < n,
= »*%/. = 0.
Thus M (fn} is orthogonal to M(Д). This proves ЦТп ~ ТпЦ = 0.
00
Next we put £ Lk = P and show that Pgn = gn (n — 1, 2, . . .).
*=i
Then, since {g„} is dense in X, we obtain P = I. But, by (9), we have
We also have Pfn — fn by fnQM (/„) Hence, by PLk = Lh implied by
(11), we obtain Pg„ = gn (n = 1, 2, . . .).
The third step. Take a sequence {c,,} of positive numbers such that
Й	k
s-lim S cnfn, s-lim	cnT/„
i-XX) rt=l	k—>00 n=l
both exist. For instance, we may take cn = 2_” (||/„j| T ||T/W||)~'1.
Since T is a closed operator, we have
oo	co
x0 = cnfn Q D (T) and Txv = y0 = cnTfn. (13)
342
XI. Normed Rings and Spectral Representation
Hence, by the first step,
Tx^F(H)X(i.	(14)
Let В € (H)' be a bounded self-adjoint operator. Then, by hypothesis,
В € (Ту. By Theorem 1, F(H) is commutative with B. Hence
F(H) Bx9=BF{H)x0 = BTx0 = TBx0.	(15)
Let г„(А) be the defining function of the set {A; |F(A) ]	n}, and put
В = c”1 PwH!iLm, where Pn == en (H).
Then we can show that
TPn = F(B)P„.
In fact, we have LmxQ = Л? cnLmfn = cmfm, by (11) and
Hence, by (15),
F(H) PnH“f„ = F(Я) с^Р.НЧ^ = F(fl) Bx0 = TBx0
= TtfPJi'L.i, = TPnH*f„.
that is, for h spanned by Hkfm with fixed m, we have
F(H}Pnh = TPnh.
(16)
(169
But such A's are dense in M(/w) and so, by (12), we see that if we let m
take all positive integers then the h’s are dense in X. Hence we have pro-
ved (16') for those h's which are dense in X.
Now Pn is bounded by (4). By the operational calculus given below,
the operator F (77) Pn is equal to the function Fn (77) where
FM = F(A) eM = F(A), for |F(A) ( n,
= 0, for |F(A)| > n.
Thus Fn (H) = F (77) Pn is bounded.
Let h* £ X be arbitrary, and let A* = s-lim A, where h‘s are linear
j—x>c J	J
combinations of elements of the form Hkfm, Such a choice of hj is possible
as proved above. By the continuity of the operator F{H}Pn, we have
F (Я) Pn A* = s-lim F (77) Pn F.
j—J
Hence, by s-lim Pnh: = P„A* and (16'), we see that the closed operator T
satisfies (16).
The fourth step. Let y£D(F (77)) and set yn — Pny. Since F (A) is finite
everywhere, we must have s-lim l’n = I. Thus s-lim y„ = s-lim Pny — y.
Hence, by (16), we obtain TXyF (H). For, s-lim F(H) Pny — s-lim Fn (77)
у = F (77) у whenever у C D(F (77)).
12. Functions of a Self-adjoint Operator
343
Let у 6 D (T) and set yn = Pny. Then
Ту. = TP.y = F(H) P.y (by (16)),
TP.y = P.Ty (by P. = г,(Н)е (H)').
By the operational calculus given below, the function F(H) of Я is a
closed operator. Hence, letting n oo in the above relations, we see
that F(H) T. We have thus proved that T — F(H).
An Operational Calculus. We have
Theorem 3. (i) Let /(/) be the complex-conjugate function of /(Л).
Then Р(/(НУ)	and, for v, у £ D(/(H)) == D(/(H)), we have
(/(Я)х,у) = (ж,/(Н)у).	(17)
(ii)	If D(/(H)), ye Dfe(H)), then
(/(H)x,g(H)y) = f /(2)g(2)i(£(2)x,y).	(18)
—OO
(iii)	(tx /) (H) x = af{H} X if x£ D Ifx^D (/(H)) Л D (g(H)), then
(/ + £) (H)x=t/(H)v + g(H)x.
(19)
(iv)	If л t	then the condition /(H) x£D(g{H)) is equivalent
to the condition x£D(j  g(H)), where /  g(A) = /(Л) g{A), and we have
g(H)/(H)v = (g-/)(H)x.	(20)
(v)	If /(Я) is finite everywhere, then /(H) is a normal operator and
/(H)* =/(H).	(21)
In particular, /(H) is self-adjoint if /(Л) is real-valued and finite every-
where.
Proof, (i) D(/(H)) = D(/(H)) is clear and
00	oo
(/(H) x, y) = / /(2) </(£(2) x, y) = f /(2) a.(x, £(2) y)
—oo	— oo
= (/(H) y,x) = (%, /(И) y).
(ii)	We know, by Theorem 1, that Н(Я) is commutative with g(H). Thus
OO	00
(j(H}x,g(H)y) = f f(i) <1{E(X) x, g(H) y) = f
— 00	—00
oo____________	00	/ oo___
= f /ma(g(/l)EWy.x)= f /(2M f g^)d(E(f,)EWy,x)
— OC	— OO	\” 00
oo / Л______________ \ oo _____________ _
= /ДОHfWiMW'i) = / /(2)H2W(E(2)x,y).
—co	\—oo	/	—oo
344	XL Normed Rings and Spectral Representation
(iii)	is clear, (iv) Let л satisfy f |/(2) j2 d ||£(2) x ||2 < сю. Then, by
— OO
7? (2) H (/z) =7s (min (2,/i)), the condition J |g(2) |a7]|2i(2) f(H) #|J2<oo
-00
implies, by virtue of the commutativity of E (2) with /(H),
00	oo
°°> f	/ ^«^и/дажлх
—00	— oo
00	/ 00	\
= f MM f
—OO	\ —00	/
oo	/ A	\ oo
= I kfflM I |/М!‘Ф(М = f IgW/WMI-EWxll2-
—oo	\—00	/	— oo
Since the above calculation may be traced conversely, we see that, under
the hypothesis %£Д(/(Я)), the two conditions f[H) x £	and
% C D(J • g{H}) are equivalent and we have
oo
(g(H)/(tf)*,y) = J g«<«(E(2)/(ff)x,y)
—co
= Ггй4//иадх.у))
—OO	\ —00	/
00
= f g(A) /(Л) d(E(X) x. у) = ((g - /) (H) X, y).
—OO
(v) Let us put A(2)-|/(2)|+ «, Л(Я) = Л(Я)-\ g(2) =/(2) Zs(2)“V
where a is any positive integer. Then Л(2) and g(2) are both bounded
functions. Hence D(k(H}) =	= X. Thus, by (iv),
/(H)=A(H)g(H)=g(H)A(H).	(22)
We have, by (i) and Z)(£(H)) = X, k(H)* = k(H), i.e., k(H) is self-
adjoint. By (iv) we have x = h (H) k (И) x for all x £ X and x~k (77) h (H) x
for all x C D(h(HY). Hence h(H) — £(H)-1. Thus, by Theorem 1 in
Chapter VII, 3, h{H} is self-adjoint. Therefore D (/(77)) = D(Л(77)) is
dense in X and so we may define /(H)*. We shall show that /(H)* = 7(H).
Let a pair {y, y*} of elements E X be such that (/ (H) x, y) = (%, y*) for
all x£ 7)(/(77)). Then, by g(H)* = g(H) (implied by (i)) and (22),
(/ (Я) x, y) = fe (H) h (H) x, у) = (Л (Я) X, g (Я) y).
Hence, by x £ D (/(H)) = Dfyi (H)) and the self-adjointness of A(H),
we obtain
g(H)yCB(A(H)) and A(H)g(H)y = y*.
Thus, again by (22), we obtain / (H) у = у* so that /(H)* = f(H). There-
fore, by (iv), we see that /(77) is normal, that is,/(H)*/(H) = /(H)
13. Stone’s Theorem and Bochner’s Theorem
345
Corollary. If / (A) is finite everywhere, then f{H) is closed.
Proof. Clear from /(Я)»*	/ (Я)* =/=(Я) = /(Я).
A historical note. Theorem 2 was first proved by J. von Neumann
[7] for the case of a bounded self-adjoint operator T. Cf. F. Riesz [5].
The general case of a closed linear operator T was proved by Y. Mimura
[2]. The exposition given above is adapted from Y. Mimura [2] and
B. Sz. Nagy [1].
13. Stone’s Theorem and Bochner’s Theorem
As an example of functions of a self-adjoint operator, we give
Theorem 1 (M. H. Stone). Let {I7J, — oo < t < oo, be a one-para-
meter group of class (Co) of unitary operators in a Hilbert space X. Then
Ut = where /t(A) = exp(fiA) and iH = A, H* = H,
is the infinitesimal generator of the group Ut.	(1)
Conversely, for any self-adjoint operator H in X. Ut = ft (7/) defines a
one-parameter group of class (Co) of unitary operators.
Proof. We have, by the representation theorem of the semi-group
theory,
Utx = s-lim ехр(й77 (7 —	x.
H—ЮО
Since the function = exp (й 2(1 — M-1iA)-1) is smaller than
exp((— п1л2)/(п2 + A2)) in absolute value, we have, for H = J A<AE(A),
ехр^'Н^-и-Ч’Я)-1)
= / exP G Arm) W
—OO	K
and, moreover,
2
2
1 d ||F(A)	= 0.
—CO
—tA3\
n— t Л/
This proves that Ut = ft(H) = f exp (itA) dE(A).
—oo
For the converse part of Theorem 1, we observe that, by the opera-
tional calculus of the preceding section,
A (H)* = A-f (Я) and А(Я) A (H) == A+s (H), /о (H) = I.
We also have the strong continuity of /t (H) at t = 0 by
||4(AZ) x — x[|2 = J" |ехр(г/А) — 1|27 11 AT (A) xjj2-> 0 as Z->0.
—OO
Hence Ut = tt(H) is a one-parameter group of class (Co) of unitary
operators.
346	XI. Normed Rings and Spectral Representation
Remark. For the original proof, see M. H. Stone [2], Cf. also J. von
Neumann [8]. Another proof given by E. Hopf [1] is based on a
theorem due to S. Bochner:
Theorem 2 (Bochner). A complex-valued continuous function /(tf),
— oc < t < oc, is representable as
oo
fit) = f e*a dv(X) with a non-decreasing,
—oc
right-continuous bounded function »(Л),	(2)
iff f(t) is -positive definite in the following sense:
/ / / ““ s) (^) (s) dt ds 0
—OO —oo
for every continuous function у with compact support.	(3)
The proof of Theorem 1 given by E. Hopf starts with the fact that
f(t) = (Utx, x) satisfies condition (3) as may be seen from
f f (Ut_sx, x) (p(t) tpis) dt ds == f f (Utx, Usx) <p(t) <p(s) dt ds
—oo —oo	—oo —oo
= \ f <p(i)Utxdt, f <p(s) Usxds) 0,
We shall show that Bochner's theorem is a consequence of Stone’s theo-
rem.
Deduction of Theorem 2 from Theorem 1. Consider the totality
of complex-valued functions x(t), —oo < t < oo, such that x(t) = 0
except possibly for a finite set of values of t; the finite set may vary with
x. is a pre-Hilbert space by
(x + y) (t) = x[t) + y[t), (»x) (tf) = ax(i) and
(*>У) =	fit — s) x(t) y(s) for x,ye %,	(4)
—00 <(,$<00
excepting the axiom that (x, x) = 0 implies x = 0. That (x, x) 0 for
any x £ is a simple consequence of the positive definiteness of the func-
tion / it).
Let us set 9J = (rC (x, x) — 0}. Then the factor space is a
pre-Hilbert space with respect tA the scalar product (x, y) — (x, y) where
x is the residue class mod fft containing x£ Let X be the completion
of the pre-Hilbert space X = The operator C’T defined by
(l7Tx) it) = x{t — r), x£	(5)
surely satisfies the conditions
(U,x, U,y) = (x,y), UtU. = U,+, and Ua = I. (6)
14, A Canonical Form of a Self-adjoint Operator
347
Therefore, it is easy to see that naturally defines a unitary operator
UT in X in such a way that {C7T}, -oo<r<oo, constitutes a one-
parameter semi-group of class (Co) of unitary operators in X; the strong
continuity in t of Ut follows from the continuity of the function /(/).
Hence, by Stone's theorem, Ut= f e*a dE (Я). Let xG (t) € <5r be defined
— OC
by %oW = 1 and == 0 whenever 17^ t. Then, by (4) and (5),
/(t) = (UT xG, Xq). Therefore,
OO
/(r)= [ ^d ||E(2) 50||2.
— OO
which proves Bochner’s theorem.
Remark. The idea of using a positive definite function to define a pre-
Hilbert space as in (4) was systematically applied by B. Sz. Nagy [3] to
various interesting problems concerning the Hilbert space.
14. A Canonical Form of a Self-adjoint Operator
with Simple Spectrum
Let H = f Я dE (Я) be a self-adjoint operator in a Hilbert space X
with simple spectrum as defined in Chapter XI, 8. Thus there exists an
element у С X such that the set {(E (/J) — E (a)) у; x < /?} spans a dense
linear subspace of X. We put
а(Л)=(Е(Л)у,у).	(1)
Then ст(Я) is monotone non-decreasing, right-continuous and bounded.
We shall denote by cr(B) the Baire measure determined on Baire sets
on E1 from 6]) = a(b) — o(«). We denote by 00, 00) the tota-
lity of complex-valued Baire-measurable functions /(Я), — о© < Я < oo,
/ 00	\l/2
such that ||/||ff=l f j / (Я) |ao (с£Я) I	< 00. Then (-00,00) is a Hil-
\-oo	/
00	___
bert space by the scalar product (/, g)a = f /(Я) g (A) a (t/Я) with the
—OO
convention to consider f = g in iff /(Я) = g(A) <r-a.e.
Theorem. With any /(Я) €	—oq,<x>) we associate a vector f of
X defined by
OO
7= J tW^EWy.	(2)
—oo
Then the correspondence /(Я) -> /is a one-one linear isometric mapping
of L^(— cxj, 0©) onto X. Let this mapping be denoted by V, i.e., / = Vf.
Then Ute operator V 1H Г in сю,00) is precisely the
348	XI. Normed and Rings Spectral Representation
operator of multiplication by Я:
D^) ----- D(V'4/V] = {/(Л);/(A) and A/(A) both € L*(-oo, oo)}
and (H-J) (A) = A/(A) whenever /(A) £ ЩН1) •	(3)
Proof. We have, by E (Л) E (u) = E (min (A, ^)),
oo_________________	00 ___
(E (A) y, f) = f /(p)	(Л) у, E (p) y) = / /d^E (p) E (A) y, y)
— 00	—oo
A__	A_____
= f /(e) d(E(p) y,y) = f/(p)(r(dp)t
— 00	—oo
and so
OG	OO	____
O = l/WW)y,?)= f f[X)gW am = (f,g),.	(4>
—oo	—oo
Therefore, V maps D (V) = L2 (— oo, oc) onto {/',/ = f /(A)dE(A)yr
—oo
/С L2 (—oc ,oo)} one-to-one, linearly and isometrically. Hence, in particular,
7?(R) is a closed linear subspace of X. But R(V) surely contains the
elements of the form f dE (А) у = (E (8) — E (oc)) y, — <x> <C oc <Z fi < oo,
a
and so, by the hypothesis that the spectrum of H is simple, we see that
7?(F) = R (F)a = X. Thus the first half of the Theorem is proved.
Next we have
OO	CO
Е(Л)/" = £(А) /	j
—oo	— oo
= J / (e) dE (e) у ’
—oo
and so, by (4),
л________________________________________
(£(l)/,S> f	(5>
—oo
Hence the condition J A2d(E(A) /, /) <oo, equivalent to f£D(H),
—co
00
is equivalent to the condition f A2 \ f (A) |2 a(dty < oc. Moreover, in the
-oo/
last case, we have, by (20) in Chapter XI, 12,
— f ArfE(A)7,
—co
and hence, by (4) and (5),
№/. Й, = Vi, Й. = (H V  /, V  g) = (H  t, g)
CO	00	___
= f H<l(EWi,g) = f H/Wi^am-
—OO	“OO
15. The Defect Indices of a Symmetric Operator
349
On the other hand, we have
—oo
Therefore we must have
(Нг/) (Я) = Л/(Л) ст-almost everywhere.
Remark. There is a close connection between self-adjoint operators
with simple spectrum and the Jacobi matrices. See M. H. Stone [1],
p. 275. For a canonical form of a self-adjoint operator with not necessarily
simple spectrum, see J. von Neumann’s reduction theory. Neumann [9].
15. The Defect Indices of a Symmetric Operator.
The Generalized Resolution of the Identity
Definition 1. Let U = UH = (H — ii) (H + ii)-1 be the Cayley
transform of a closed symmetric operator H in a Hilbert space X. Let
Xj = D (ин)г, Хд -	and let m = dim(X^), n = dim(X^)
be the dimension numbers of X#, Xj respectively. Then H is said to be
of the defect indices (m, n). H is self-adjoint iff it is of the defect indices
(0, 0) (see Chapter VII, 4).
Proposition 1. The defect indices (m, n) of a closed symmetric operator
H may be defined as follows: m is the dimension number of the linear
subspace {%E X;H*x = ix}; n is the dimension number of the linear
subspace {Al; H*x = — ix}.
Proof. Clear from Theorem 3 in Chapter VII, 4.
Example 1. Let X = L2(0, 1). Let D be the totality of absolutely
continuous functions x(t) E L2(0, 1) such that x(0) = %(1) = 0 and
x'(t) E L2(0, 1). Then the operator Тг defined by T1x(t) — i^x'ty) on
D — D(Tj} is of the defect indices (1, 1).
Proof. As was shown in Example 4 of Chapter VII, 3, T* = T2 is
defined by
T2x(t) = i^x'(t) on D(T2) = {x(0 E L2(0, 1); x(t) is
absolutely continuous such that x’ (t) E L2(0, 1)}.
Thus the solution у E L2(0, 1) of T*y = T2y = iy is a distribution solu-
tion of the differential equation
y(Z) (у,/СЛг(0, 1)).	(1)
Then z(t) = у (t) exp (£) is a distribution solution of the differential equa-
tion
/(0-0 (2,/EL2 (0,1)).	(2)
Wc shall show that, there exists a. constant C such that z(t) =C for a.e.
IC (0, 1). To this purpose, take any function x()(i) E Cq°(0, 1) such that
350
XI. Normed Rings and Spectral Representation
i
f x0 (t) dt = 1 and put
о
1
% (t) — хй (t) f x (t) dt
0
t
= и [t], w[t} = f U (s) ds ,
0
where x(t) is an arbitrary function from C£° (0, 1). We have w £ Cjf (0,1)
i
by f u(s) ds = 0.’Therefore, by (2), we have
о
i	r
— f z(t) w! (t'j dt = — J" z (if) u (t) dt = 0,
о	0
that is,
it	i
J* z{t} x(t} dt = С У x(t) dt„ where C = f z(t) %0(i) dt.
oo	о
This proves, by the arbitrariness of x{t} £ C£J°(0, 1), that z{t] ~ C for a.e,
(0, 1).
Hence any solution of T*y = iy is of the form y(£) ~ C exp(—/).
In the same manner we see that any solution of T*y = — i у is of the
form y{t] ~ C exp(t). Thus T is of the defect indices (1, 1).
Definition 2. A symmetric operator H in a Hilbert space X is called
maximal symmetric if there is no proper symmetric extension of H.
Proposition 2. A m aximal symmetric operator H is closed and H =
A self-adjoint operator H is maximal symmetric.
Proof. By Proposition 1 in Chapter VII, 3, //** is a closed symmetric
extension of H. Thus the first half of Proposition 2 is clear. Let HQ be a
symmetric extension of a self-adjoint operator H. Then, from H QHg,
//0 £ Ho*, we obtain Ho £ H* £ H* and so, by H = Я*, H £ Ho £ H.
This proves that a self-adjoint operator H is maximal symmetric.
Corollary 1. Every maximal symmetric extension of a given sym-
metric operator H is also an extension of H**.
Proof. The relation H £ Hoimplies the relation	QH* and//** £ //**.
Since, by the preceding proposition, Ho = H**, Corollary 1 is true.
Corollary 2. If H is a symmetricjoperator such that H* = H**, then
the self-adjoint operator //* is the only maximal symmetric extension
of H.
Proof. Being self-adjoint, //** is maximal symmetric. Thus any maxi-
mal symmetric extension of H, which is also a symmetric extension of
//** by Corollary 1, is identical to H** = H*.
We are thus in a position to state
Definition 3. A symmetric operator H such that H* = //** is said to
be essentially self-adjoint. A self-adjoint operator H is said to be hyper-
maximal. The latter terminology is due to J. von Neumann.
15. The Defect Indices of a Symmetric Operator
351
Example 2. Let X — L2(—oo,oo), and define an operator H by
Hx(t) = t • x(t) for x(t) £ Cg(- oo, oo), H is surely a symmetric operator
in X. It is easy to see that H* is the coordinate operator defined in
Example 2 in Chapter VII, 3 so that H is essentially self-adjoint. The
operator H defined by Hx(t) = i-1 x' (t) for x(t) C Cq(— oo, oo) is also
essentially self-adjoint in X = La (—00,00). For, in this case, 7У* is the
momentum operator defined in Example 3 in Chapter VII, 3.
Theorem 1. Let the defect indices (m, n) of a closed symmetric opera-
tor H satisfy
m = m' + p, -и = n + p {p > 0).
Then there exists a closed symmetric extension H' of H with the defect
indices (m', n).
Proof. Let {«pj, <p2,  • •, <pP, fPp+i,   , <Рр+т-}, {Vh> V2>   > Vp> Vp+i- • • •
• • > Ур+п'} be complete orthonormal systems of X£ = D(UH)\ XPj =
^(C/^)-1-, respectively. Define an isometric extension V of UH by
Vx=UHx for x£D(UH),
p	p
V  X oci<pi = X
*=1
We have R{I — !7я)а = X by Theorem 1 in Chapter VII, 4. Thus, by
R{I —V) 2R(I- UH) and Theorem 2 in Chapter VII, 4, there exists a
uniquely determined closed symmetric extension H' of H such that
V — (Hr — il) (H' -j- il)-1. The defect indices of H' аге (ж', n') as may
be seen from dim^fF)1-) = m , dim^fF)1) = n'.
Corollary. A closed symmetric operator H with the defect indices
(m, n) is maximal symmetric iff either m = 0 or n = 0.
Proof. The “only if” part is clear from Theorem 1. If, for instance,
-m — 0, then, by D{UH) = X, we must have UHo = UH for a closed
symmetric extension of H. This proves Ho = i (I + UHa) (I —
= i (I + UH) (I — UH)-1 = H. Also, in the case n = 0, we see that there
is no proper closed symmetric extension of H.
Example 3. Suppose that {(рх,	is a complete ortho-
normal system of a (separable) Hilbert space X. Then, by
OQ	OO	OO
U- Xat(?i = Xa-i<pi+i> X |«i
1=1	t-1	1=1
2 < oc
we define a closed isometric operator U such that D(U) = X and
dim (R (D)1) = 1. If R(I — U)a =£ X, then there exists an x 0 such
that x C R (I — U)^. Consequently ((/ — U) xt x) — Q and so (Ux, x) =
||x||2 = j[ 17%j|2. This implies that
||(Z- U) %]|2 = ]|x||2- (Ux,x) - (x, Ux) 4- ||D%|j2

352
XI. Normed Rings and Spectral Representation
that is, U x = x and so, by the above definition of U, x must be zero. This
contradiction shows that R(I ~ U}a = X. Hence, by Theorem 2 in
Chapter VII, 4, U is the Cayley transform of a closed symmetric operator
H. H is of the defect indices (0, 1) since D (U) = X and R (U)1 is spanned
by Thus H is maximal symmetric without being self-adjoint.
Theorem 2 (M. A. Naimark [3]). Let a closed symmetric operator
in a Hilbert space be of the defect indices (m, n). Then we can con-
struct a Hilbert space X, containing X2 as a closed linear subspace, and
a closed symmetric operator H in X with the defect indices (m + w,
m + n) such that
Нг = P(X-j) fiP(Xy), where P(X1) is the projection of X onto Xx.
Proof. Consider a Hilbert space X2 of the same dimensionality as
We construct a closed symmetric operator in X2 with the defect
indices In, m). For instance, we may take H2 = — supposing that X2
coincides with Xr Then we would have {x 6 X2;	= ix} = {x 6
H* x — — ix}, {a 6 X2; H*x = — ix} = {x £ A\; H*x = ix} and so the
defect indices of H2 must be (и, m).
We then consider an operator H defined by
Я{%, у] = {Hrx, H2y} for {x,y}e Х]ХХ2 with xe Р{Нг}, ye D(H2}.
It is easy to see that H is a closed symmetric operator in the product
Hilbert space X = Хг>с.Х2. The condition
Я*{%, y} = i{x, y} (or the condition F*{,r, y} = —i{x, y})
means that Я*x — ix, H*у = iy (or H*x = — ix, H*у ~ — iy\ Hence
we see that the defect indices of H are (m + n, m + и).
Corollary. Let, by Theorem 1, H be a self-adjoint extension of H,
and let H = f A dE [A} be the spectral resolution of H. Since H and a
fortiori H are extensions of Ят when is considered as an operator in
X, we have the result:
if xe QXl^P(X1)X, then x = P(X1) xe D(H} and
= f AdF(A) x, where -
—co
The system {F(2); —oo < A surely satisfies the conditions'.
F{A) is a self-adjoint operator in
(3)
< ^2 iniplies that (F (Лг) x, x} (F (Л2) x, x) for every x^Xt,
F(A + Q)^F(A),
F (—oo) x = s-lim F(A} x - 0,
F (oo) x = s-lim F (Л) x — x for all x E X.
15. The Defect Indices of a Symmetric Operator
353
Remark 1. For the closed symmetric operator Hy, we have
HyX = f AdF (Л) x for all x £ D ,
—OO
where {F(A); —oc < A < oo} satisfies (4). In this sense, H1 admits a
generalized s-pectral resolution.
Example 4. Let Хг == £2(—oo, 0), and let Dy be the totality of
absolutely continuous functions	—°0, 0) such that %(0) = 0
and x'(t)^L2(—oc, 0). Then the operator Hy defined by Hyx(i) =
i~lx (/) on Dy = D (Hy) is a maximal symmetric operator with the defect
indices (0, 1). This we see as in the above Example 1. Let X2 = £2(0, oc),
and let £>2bethe totality of absolutely continuous functions x (t) g£2(0,oo)
such that x (0) = 0 and x' (t) C £2(0, oc). Then the operator H2 defined by
Я2 x (t) = i~2 x' (t) on D2 = D (H2) is a maximal symmetric operator with
the defect indices (1, 0). In this case, X = XxxX2 = £3(—oo,oo), and
the operator H in Theorem 2 is given concretely by Hx(t) —
for those x(t) € £2(—oo, co) for which x(0) = 0 and x (t) 6 L2(— oo, oo).
Remark 2. Since H f Я dE (A) is an extension of Hy, we have, for
xED(Hy),
||Я1Л:||2 = ||Я*||2= f A24||E(A) x||2 = f №d(E(X)x,x)
—co	—oo
= 7;2^(Е(Я) P(Xy) x, P(Xy)x) = f ^d(F(A)x,x).
“OO	—oo
oo
However, the condition f A2 d(F(E) x, x) О oc does not necessarily
—OO
imply that x t D (Hy). Concerning this point we have
Theorem 3. In the case of a maximal symmetric operator Hy, we
have, for the corresponding generalized spectral resolution J A dF (A)
in Xp
OO
x 6 D (Hy) is equivalent to the condition J A2 d(F(A) x, x) <_ oo. (5)
— OO	oo
Proof. The reasoning in Remark 2 shows that J №d(F (E)x,x)
—OQ
< oo implies that f A2 d ||E(A) x|]2< oo, i.e., x£D(H). The operator
—OQ
H’ =P(Xy)HP(Xy),
when considered as an operator in Xy, is a symmetric extension of Hy
and D (H') — D (H) C\ Xr Thus, by the maximality of Hy, we must have
Hr = H’. Thus D(Hy) = D(H') = D(H) DXy and so, by Hf =
P (Xj) HP (Xj) , the condition f A2 d | ] E (Л) x 112 < oo with x С Хг implies
— nu
23 Yfwida» Functltmnl Лпп|ун(н
354
XL Normed Kings and Spectral Representation
oo
f №d(F (Л) xt x) < oo; and conversely, the condition x £ D implies
—oo
OO	00
J A2 d(F(A) x,x} = f A2d ||E(A) x ||2 < oo.
—oo	—co
Remark3. Since, by Theorem 1, any closed symmetric operator can be
extended to a maximal symmetric operator //р we can apply our Theo-
rem 3 to the effect that (5) is valid. For a detailed exposition of the gene-
ralized spectral resolution, see N. I. Achieser-L M. Glasman [1] or
B. Sz. Nagy [3]. The spectral representation of a self-adjoint operator
in a Hilbert space can be extended to a certain class of linear operators
in a Banach space with an appropriate modification. This result is due to
N. Dunford and may be considered as an "elementary divisor theory” in
infinite dimensional spaces. N. Dunford-J. Schwartz [6].
16. The Group-ring L1 and Wiener’s Tauberian Theorem
The Gelfand representation admits another important application in
functional analysis, namely, an operator-theoretical treatment of the
Tauberian theorem of N. Wiener.
The linear space L1 (— oo, oo) is a ring with respect to the function sum
and the product X defined by
OO
M(^) = (/*g)(/)= f — s) g($) ds.	(1)
—OO
For, by the Fubini-Tonelli theorem,
OO
/(* —f |g(s)|&
**00
00	oo
= f	f
—oo	—oo
We have thus proved
j|/Xg||	||/||  ||g||, where || || is the norm in Lx{~oo, oo). (2)
Therefore we have proved
Proposition 1. We can introduce formally a multiplicative unit e, in
the ring L1 consisting of all elements z given formally by
z = Ae -Ь x, x£ oo, oo).	(3)
In fact L1 is a normed ring by the following rule:
(V + *1) + (V + Xi) = (Ar + Аз) e + (Xj + %2),
л(Ае + x) = хЛе 4- xx,
(Ai & -j- %<) X (Ал c -j- ж»)	A, Ao & 4~ Ai #o	Ao x, 4" x-i x x«,
]|Ae 4~ x|| = |A| 4- ||я||.
16. The Group-ring L1 and Wiener’s Tauberian Theorem	35&
This normed ring Ll is called the group-ring of the group 7?1 of real
numbers written additively. We shall find all maximal ideals of this ring
L1. One is Zo = L1 (— oo, oo) from which L1 is obtained by adjunction
of the unit e. We shall find all maximal ideals I 7^ Zo.
For any maximal ideal I of the normed ring L1, we shall denote by
(z, I) the complex number which corresponds to the element z by the ring
homomorphic mapping L1 —> L1/!. Thus (z, Zo) = Л for z = Ae + x,
x 6 Zo.
Let I be a maximal ideal of L1 different from Zo. Then there exists a
function LT{—00,0c) = Zo such that (x, Г) 7^ 0. We set
Z(«) = (xa, Г), where xjt) =-- x{t + a).	(5)
Then /(0) = 1, and, by | (%a, I) | |jxa]| = ||x we see that (a) |
I x I |/j (%, Z) j. Thus the function %(#) is bounded in a. Moreover, by
|x(a + <3)-х(л)| IK+5 —	||/| (%, Z) |,
the function % (a) is continuous in a. For, by (2) in Chapter X, 1, we have
lim J" | x (tx + d -j- i) — x (a + t') | di = 0.
<r~*o
On the other hand, we have, by (^я+^Х%) (if) = (%aXXp) (t),
(xa+p> I) (x, Z) = (%a, Z) (xfi, I)
so that
Z(<* + /5) =/(«)/(/?).	(6)
From this we can prove that there exists a uniquely determined real
number £ = £ (Z) such that
% (a) = exp (г - f(Z) • a).
(?)
In fact, by /(ил.) = %{a)w and the boundedness of the function x> we
obtain |x(<x) j 1. Consequently, by / («)/(—a) = /(0) = 1, we must
have |x(a) | = 1- Thus, as a continuous solution of (6) whose absolute
value is one, the function ^(<x) must be given in form (7). That the
value £ (Z) is determined by Zindependently of the choice of L1 (—00,00)
may be seen from % (л) (у, Z) = (уя, Z) which is implied by %/Xy =
Every continuous solution of (6) with the absolute value identically
equal to one is called a continuous unitary character of the group I?1 of real
numbers written additively.
We have thus constructed the (continuous unitary) character %(<x)
of the group 7Z1 with respect to the given maximal ideal I 7^ Zo.
We next show how to reconstruct the ideal I with respect to this
character or, what amounts to the same thing, to reconstruct the value
(z, Z) from /.
23*
356
XI. Normed Kings and Spectral Representation
For any y(Z) £ /?(—oo, oo), we have
oo	oo
(xXy) (/) = f x(t — s) y(s) ds = f x_s(t)y(s)ds.
—OO	—00
Hence, by (5) and the continuity of the ring homomorphism Z.1	L1/!,
we obtain
OO
(*xy, I) {x, Z) (y, Z) = (x, I) f %(—s) y(s) ds.
—00
Thus, by (x, I) 0, we obtain
OO
(y,Z)= / y(s) exp(— i • £(Z)  s) ds.	(8)
— OO
Therefore, for any z = Ле + x with x 6 L1 (— oo, oo),
OO
(z, I) = (Ле, I) + (%, Z) = Л + J x(s) exp(— i • £ (I) • s) ds. (9)
“OO
Conversely, any (continuous, unitary) character ^(«) = exp (г • £ • л)
of the group R1 defines a ring homomorphism
OO
z -> Л + f x (s) exp (— i • £ • s) ds (z ~ Ле + x, хУ L1 (—00,00))) (10)
—OO
of L1 onto the ring of complex numbers. For, we have, by the Fubini-
Tonelli theorem.
/ (*1 X x2) (i) exp (—i  £ • t) dt
—00
00	00
— / x1(i) exp (—г £ • t) dt f x2(t) exp (—г - £ • t) dt.
—co	—co
We have thus proved
Theorem 1 (Gelfand [4] and Raikov [1]). There exists a one-to-one
correspondence between the set of all maximal ideals I Zd(—oo, 00)
of the group ring L1 of the group R1 and the set of all continuous, unitary
characters / (a) of this group R1. This correspondence is defined by for-
mula (9).
We shall show that the normed rijag Z,1 is semi-simple or, what amounts
to the same thing (see Chapter XI, 2), that the following theorem is true.
Theorem 2.	The normed ring Z.1 has no generalized nilpotent ele-
ments other than 0.
Proof. Let xE Z.x(— 00, 00) and у 6 L2(— 00, 00). Then, by Schwarz'
inequality,
16. The Group-ring L1 and Wiener’s Tauberian Theorem 357
and so, by the Fubini-Tonelli theorem, we see that the left hand side
belongs to £2(— oo, oo) and
||xXy ||2	||x|| ||у ||2, where || ||2 is the norm in L?'{— oo, oo).	(11)
Hence we can define a bounded linear operator Tx on Z,2(—oo,oo) into
L2 (— oo, oo) by
(Txy) (/) = f x(t— s)y(s)ds whenever x 6 L1 (— oo, oo); (12)
—oo
and, moreover, we have
llniks ||x||,	(13)
T* = Tx* where x*(£) = x(—t).	(14)
Thus, by applying the Fubini-Tonelli theorem again, we obtain
TXT* ~ Txyx, =	— T* Tx, i.e., Tx is a normal operator. (15)
Hence, by Chapter XI, 3,we have | TJ|2 = lim (|j I 2)1/w. Therefore,
п—ЮО
by
11 x X x X x X •   X x | [	117^ 1|2,
n factor
we see that, if x is a generalized nilpotent element, then | T\||2
= 0.
From this fact we easily prove that x must be a zero vector of £r(—oo,oo).
Let now z — Ле + x with x E L1 (— oo, oo) be a generalized nilpotent
element of the normed ring L1. Then, for any maximal ideal I of L1, we
must have (z, I) = Я — (x, Z) = 0. Hence the Fourier transform
(2 л) 1/2' f x(t) exp(— i • £- £) dt
—OO
must be identically equal to — (2л)-1^2 Л. From this we prove that Л
must be 0. For, we have (the Riemann-Lebesgue theorem)
00
J" x(Z) exp (—i • $  t) dt
— OO
x (t) — x (t 4- -r
\ €
dt 0 as f > oc.
Thus a generalized nilpotent element z £ L1 must be of the form z = x
with x £ L1 (— oo, oo) and so, by what we have proved above, z = x = 0.
We are now in a position to state and prove the Tauberian theorem of
N. Wiener [2]:
Theorem 3.	Let x(t) E ZT(— oo, oo) be such that its Fourier transform
OO
(2л)-1/2 J x(f) exp(- -i  $ - t) dt
— CO
358
XI. Normed Rings and Spectral Representation
does not vanish for any real %. Then, for any у(/)бТт(—oo, oo) and
s > 0, we can find the real numbers /Ts, the complex numbers <%'s and a
positive integer in such a way that
oo
oo
N
y(t) - Л? «,%(/-£)
г.
(16)
Proof. It suffices to find и £ I.1 in such a way that
We first show that
||y — xxz ||	e/2.
(17)
OO
I*111 f I?(*) —	(^) 1^=0» where
1 Oo	-
(л)	1 f	\ 1 COS (XS у
v)=^ J y(<-s) ——&
OO
(18)
oo
For, by J (1 — cos as) (as2)4 ds = я (a > 0), the left side is
—oo
W-lim
<»~M5O-5o	5
Next we have
(Зя)-1'2 f РУ'2	e-*t i№
J \гс/ a«2
—oo л '
1 —|^|/a (|f| <«),
.0 (Ifl^a)-
(19)
For, we have
(Зя)-1'2 /	= (4)1/2/(i—l)cos«f
— ot	'	0 '
-2X1/2 1  - cos uoc
Л /	M2(X
HOC
and so we have only to apply Plancherel’s theorem in Chapter VI, 2.
Hence, by the Parseval relation of the Fourier transform,
y(a) (t) satisfies f	• £  /) dt = 0 if |£| a. (20)
—oo	*
Therefore, we may assume that у C L1 (— oo, oo) in (17) satisfies the
condition
OO
f y(i) exp(—i - £ • t) dt = 0 whenever |£ | oc.
—00
(21)
Next we choose a positive number and a sufficiently large positive
number у such that [—/3 — y, —4- y] and \p -- y, p : y] both contain
16. The Group-ring L1 and Wiener's Tauberian Theorem
359
Let Gj (£) and Cs(£) be the defining functions of the intervals
[—y, y] and АЬ respectively. Then
« (£) = (2/3)-1 f C^— y) Cz(t]) drj = 1 for £ € [—a, tx],
—OO
= 0 for sufficiently large |£ [,
0	«(£)	1 for all real £.	(22)
1И
By the Parseval relation of the Fourier transform, we have
£(i)=^(2*)l'2Cx(i)C2(0.
Thus, by Plancherel’s theorem, we see that u (t) belongs to L1 (— oo.oc) A
L2(— oo, oo). Hence we may apply Theorem 1 so that
/(/) = u(t) = (2л)-1/2 («, It) with a certain maximal ideal It=£ To. (23)
Moreover, by Plancherel’s theorem, the inverse Fourier transform of /(/)
is equal to «(£), i.e., we have
«(£) = (2 л)-1'2 (/./_,).	(24)
We next set g = e — (2лг)-1/2 /. Then, by what was proved above,
0	(g, /f)	1; (g, if) = 0 whenever £ € [—a, a];
(g. Ц) — 1 for all sufficiently large [£ .
(25)
On the other hand, we have, for x*(£) = x(—t), the relation (x, I() =
(x*, lf). Thus, by hypothesis, we have (%* xx, Tf) = | (x, Tf) |2 > 0 for all
real £. Hence the element
g + x* X% t T?
satisfies the condition that (g + x* х x, T) > 0 for all maximal ideals I
of L1. Consequently, the inverse (g + x*xx)-1 in L1 does exist.
We define
z = (g + xXx*)-1xx*Xy.	(26)
Then, by (4), the element x\z belongs to L1 (— oo, oo). Moreover, we have
for every real number £,
(xx z, T() = (x, Tf) (z,Tf) = (x, I() &	.
Hence, by (25) and the hypothesis that (y, Tf) = 0 whenever |£ |	oc, we
obtain
(xxx, Is) = (y, Tf) for all real £.
The normed ring L1 is semi-simple by Theorem 2. Hence we must have
xXx = y. We have thus proved Theorem 3.
Corollary. Let k^t) belong to L1{—00,00), and let its Fourier trans-
form vanish for no real argument. Let &a(£) belong to L1^00,00). Let
360
XI. Normed Rings and Spectral Representation
f(t) be Baire measurable and bounded over (—00,00). Let, for a con-
stant C,
lim f kr(t— s)f(s)ds — C f k-,(t)dt.	(27)
Z-«O ..go	go
Then
OO	OO
lim f fe2 (f — s) / (s) ds = C f
_QQ	— OO
k2 (2) dt.
(28)
Proof. We may plainly suppose that C = 0. And (28) holds for k2(i)
of the form k2(t) — (ххкг) (I) with x(i) £ L1 (—00,00). It is easy to
prove that (28) holds for k2(t) if k2(t) = 5-lim (i) in L1 (—oo, 00) with
k^(t) 6 L1 (—00,00) for which (28) holds (n = 1, 2, . . .). Hence, by
Theorem 3, we see that (28) holds for every k2(t) £ L1(—00,00).
Remark. N. Wiener ([1], [2] and [3]) has applied the above Corollary
to a unified treatment of classical results concerning the limit' relations
in series and integrals which include a new proof of the prime number
theorem. Cf. also H. R. Pitt [1]. The above proof of Theorem 3 is adapted
from M. Fukamiya [1] and I. E. Segal [1]. Cf. also M. A. Naimark [1]
and С. E. Rickart [1] and the bibliographies cited in these books. For
the sake of comprehension of the scope of the above Corollary, we shall
reproduce Wiener’s deduction of the special Tauberian theorem. The
special Tauberian theorem, as formulated by J. E. Littlewood reads:
OO
Theorem 4.	Let anxn converge to s(x) for
n—0
lim $(%) = (?.
.r->l-0
Let, moreover,
sup n I an j - К < oc .
Then
OO
c *
л = 0
w
Proof. Put f(%) = an. Then we see, by
w=0
x I < 1, and let
(29)
(30)
(31)
DO
S 2 К + К f e~uu~1du = constant,
1
that f(x) is bounded.
16. The Group-ring L1 and Wiener's Tauberian Theorem 361
Hence, by partial integration,
OO	00	oo
s(<T*) = 2? ane~nx ~ f e~“xd/(u) = f x<Tuxf(u) du.
n-°	-o	0
Thus we have
C = lim f xe~U£/(u) du = lim f	/ (e*1) e}idrj.	(31')
X-^0 Q	>O0
This may be written as
lim f кг(1 — s) f(es) ds = C f ^(Z) dt, where kx (i) = e^le~e \ (32)
oo	—00
since
f dt ~ j e~te.~‘'tdi= f eTxdx—l.
—oo	— oo	0
Furthermore, we have
j k^t) e~iMdt = J xiue~xdx = Г(1 + iu) 0.
—oo	0
Thus we can apply the Corollary and obtain, for
M) = ° (t<0);	(/>0),
the limit relation
C~Cf e~ldt—C f k2(t) dt ~ lim f k2(t — s) f(es) ds
0	—oo	(“**°° -oo
X
= lim x~r f f(y) dy.
X-^OO Q
It follows that, if Л > 0,
(l-U)*
x
On the other hand, we have, by (30),
(1 + Л)л
1 Г
X
af
362
XII. Other Representation Theorems in Linear Spaces
for sufficiently large value of x. Hence, by (33),
lim Шх)~С 2ЯК,
x—too
and since A is any positive number, we get
lim /(%) = C.
Thus we have proved (31).
XII. Other Representation Theorems in Linear Spaces
In this chapter, we shall prove three representation theorems in
linear spaces. The first one, the Krein-Milman theorem says that a non-void
convex compact subset К of a locally convex linear topological space is
equal to the closure of the convex hull of the extremal points of K. The
other two theorems concern the representations of a vector lattice as point
functions and as set functions.
1. Extremal Points. The Krein-Milman Theorem
Definition. Let Я be a subset of a real- or complex-linear space X.
A non-void subset M £ К is said to be an extremal subset of K, if a
proper convex combination ockr + (1 — <x) k2, 0 < a < 1, of two points
kY and k2 af X lies in M only if both k-^ and k2 are in M. An extremal set
of К consisting of just one point is called an extremal point of K.
Example. In a three dimensional Euclidean space, the surface of a
solid sphere is an extremal subset of the sphere, and every point of the
surface is an extremal point of the surface.
Theorem (Krein-Milman). A non-void compact convex subset К of a
locally convex linear topological space X has at least one extremal point.
Proof. The set К is itself an extremal set of X. Let 9Й be the totality
of compact extremal subsets M of K. Order 9T by inclusion relation. It
is easy to see that, if is a linearly ordered subfamily of ЭД1, the non-
void set П M is a compact extremal Subset of К which is a lower bound
for the subfamily
Thus, by Zorn’s lemma, 5W contains a minimal element MQ. Suppose
that MQ contains two distinct points x0 and y0. Then there exists a con-
tinuous linear functional f on X such that /(x0) =£ /(y0). We таУ
assume that Re f(xQ) =/= Re /(y0). Mo being compact, the set Мг~
{x С Re f(x) = inf Re f(y)} is a proper subset of Mo. On the other
hand, if kr and h2 are points of К such that a 4~ (1 — a) k2 € for
1. Extremal Points. The Krein-Milman Theorem
363
some a with 0 < a < 1, then, by the extremal property of MQ, and k2
both G Mo. It follows from the definition of Мг that k2 and k2 both €
Hence MY is a closed extremal subset properly contained in MQ. Since 21f0
is a minimal element of iUt, we have arrived at a contradiction. Therefore
Mo must consist of only one point which is thus an extremal point of K.
Corollary. Let К be a non-void compact convex subset of a locally
convex real linear topological space X. Let E be the totality of the
extremal points of K. Then К coincides with the smallest closed set
containing every convex combination X	0,	of
points E, i.e., К is equal to the closure of the convex hull Conv (£) of E.
Proof. The inclusion E Q К and the convexity of К imply that
Conv(E)e is contained in the compact set K. Suppose that there exists
a point Ao contained in (K — Conv (E)a). We can then take a point
c 6 Conv (E)a so that (Ao —с) G (Conv (E)a — c). The set (Conv (E)a — c)
being compact convex and Э 0, there exists, by Theorem 3' in Chapter IV, 6,
a continuous real linear functional / on X such that
/ (Ao — c) > 1 and f{k — c) 1 for (k — c) G (Conv (E)tt — c).
Let K-l = K\ f(x) = sup/(y)|. Then, since A0G K, the set КгГ\ E
I	yCK J
must be void. Moreover, since К is compact, Kjis a closed extremal subset
of К. On the other hand, any extremal subset of Кг is also an extremal
subset of К and hence any extremal point of Klt which surely exists
by the preceding Theorem, is also an extremal point of K. Since КХГ\Е
is void, we have arrived at a contradiction.
Remark. The above Theorem and the Corollary were first proved
by M. Krein~D. Milman [1]. The proof given above is adapted from
J. L. Kelley [2]. It is to be noted that for the unit sphere S = {x G X;
||x||	1} in a Hilbert space X the extremal points of S are precisely
those on the surface of S, i.e., those of norm 1. This we easily see from
(1) in Chapter I, 5. For applications of the notion of the extremal points
to concrete function spaces, see, e.g.,K. Hoffman [1].
A simple example. Let C [0,1] be the space of real-valued continuous
functions x{t) defined on [0,1] normed by x = max |x(if) |. The dual
space X = C [0, 1]' is the space of real Baire measures on [0, 1] of boun-
ded total variations. The unit sphere К of X is compact in the weak*
topology of X (see Theorem 1 in the Appendix to Chapter IV). It is
easy to see that extremal points of К are in one-to-one correspondence
with the linear functionals G X of the form <%, Д)> = G [0, 1].
The above Corollary says that any linear functional / 6 V is given as
the weak* limit of the functionals of the form
П	n
where ’> 0,	-- 1 and tj & [0, 1].
; — 1 ) 1
364
XII. Other Representation Theorems in Linear Spaces
Recently, G. Choquet [1] proved a more precise result; If X is a
metric space, then E is a G$ set and, for every x С K, there exists
a non-negative Baire measure p,x (B) defined for Baire sets В of X
such that fix(X~E) = 0, p,x{E) = 1 and x — J y[ix(dy). As for the
E
uniqueness of and further literature, see G. Choquet and
P. A. Meyer [2].*
2. Vector Lattices
The notion of “positivity” in concrete function spaces is very im-
portant, in theory as well as in application. A systematic abstract treat-
ment of the “positivity” in linear spaces was introduced by F. Riesz [6],
and further developed by H. Freudenthal [2], G. Birkhoff [1]
and many other authors. These results are called the theory of vector
lattice. We shall begin with the definition of the vector lattice.
Definition 1. A real linear space X is said to be a vector lattice if X
is a lattice by a partial order relation x < у satisfying the conditions:
x А у implies	-\-z,	(1)
x А у implies <xx QQocy (or ocx	ay) for every a A 0 (or a 0). (2)
Proposition 1. If, in a vector lattice X, we define
x+ = x V 0 and x~ = x A 0,	(3)
then we have
x V у = (% — y)+ + y, x Д у = — ((— x) V (—у)) •	(4)
Proof. The one-one mappings x x + z and x ocx (a > 0) of X
onto X both preserve the partial order in X.
Example. The totality A (S, 93) of real-valued, o-additive set functions
x (B) defined and finite on a сг-additive family (S, S3) of sets В Q S is
a vector lattice by
(* + у) (B) = x(B) + y(B), (cxx^B) = ocx(B)
and the partial order x А у defined by x(B) А у (В) on 93. In fact, we
have, in this case,
(B) = sup x (A) = the positive variation V {x; B} of x on В.	(5)
Proof. We have to show that V(x; В) = (x V 6) (B). It is clear that
V(x; B) A 0 and x(B) A V(x; B) on 93. If 0 A y(B) and %(B) A y(B)
on S3, then, for any N QB, y(B) = y(N) + у (В — N) A x(N) and so
у (B) A V(x; B) on 93.
* Sec also Supplementary Notes, p. 4G7.
2. Vector Lattices
365
Proposition 2. In a vector lattice X, we have
x\y +	(* + *) л (У + *h	(6)
« (^b) = M л (ЛУ) for	л> °,	(7)
Л (# Д У) = («*) V (ЛУ) for	(8)
X л y = — (— x) v (—y), x~ = ~( — x)+, x+ = — (—%)-.	(9)
Proof. Clear from (1) and (2).
Corollary.
х_|_у = х\/у_|_хДуДп particular, x = %+ + x~.	(10)
Proof. % V У — x — у = 0 V (у — х) — у — (— у) V (— х) — — (у Л х),
Proposition 3. We have
х д у = у д х (commutativity),
(11)
% V (у V 2) = (% V у)
х Л (у Л z) = (% Л у)
V г = sup (%, у, z)
Л Z = inf (х, у, z)
 (associativity),
(12)
(13)
(% Л у) V z = {X v z) Л (у V Z) 1	(14)
(х V у) Л z = (х Л z) V (у Л z) J		(15)
Proof. We have only to prove the distributivity. It is clear that
(x Д у) V z < x V z, у V z. Let w S x V z, у V z. Then w A x \J z =
x + z — x A z and so x + z A w + x A z. Similarly we have у + z A
w + у A z. Hence
w + (x A z) A (y A z) = (w 4- x A z) A (w + у A z)
(x + z) A (y + z) = x Л у + z,
and so
® A (x A y) v z - (t A y) A z = (x Л у) V z.
We have thus proved (14). (15) is proved by substituting — x, —y, —z for
x, y, z, respectively in (14).
Remark 1. In a lattice, the distributive, identity
x Л (у V z) = (x А у) V (x A z)	(16)
does not hold in general. The modular identity:
x A z implies x V (y A z) = (x V у) Л г	(17)
is weaker than the distributive identity (15). Let G be a group. Then the
totality of invariant subgroups V of G constitutes a modular lattice, that
is, a lattice satisfying the modular identity (17), if we define V
and A N2 as the invariant subgroup generated by and N2 and
the invariant subgroup A N2, respectively.
Remark 2. A typical example of a distributive lattice is given by the
Boolean algebra', a distributive lattice В is called a Boolean algebra if
it satisfies the conditions: (i) there exist elements I and 0 such that
0 x / for every x е В, (ii) for any x С В, there exists a uniquely
366 XII. Other Representation Theorems in Linear Spaces
determined complement x' g В which satisfies x V x' = I, x Л x' = 0.
The totality В of subsets of a fixed set is a Boolean algebra by defining
the partial order in В by the inclusion relation.
Proposition 4. We define, in a vector lattice X, the absolute value
|x| = x v (— %).	(18)
Then
|x| A 0, and |zj = 0 iff x = 0,	(19)
|% + у | A |x| + |у|, |a%| = |<x| |ж|.	(20)
Proof. We have
x+ л (-%-) = xA A (-%)+ = o.	(21)
For, by 0 = x —	(—x) + x Д (—x) > 2 (x Д (—x)) and the
distributive identity (14), 0 = (x Д (— %)) V 0 = x+ A (—x)+. Thus
х+~Х~ = х+ + (— x)+ = x+ V (—%)+ •
On the other hand, from x V (—x) A x A (—%) A — ((— %) V я), we
have x \j (—x)	0 and so by (21)
X v (~x) = (% V (— %)) V 0 = a V (— Л')+ =	+ M+.
We have thus proved
%+ — x~~ ~ %+ 4- (—%)+ = x+ V (~x)+ = x V (—x).	(22)
We now prove (19) and (20). If x =	+ x~ Ф 0, then	or (—%-)
is	> 0 so that | x | = %+ V (—x~) '> 0.	\ocx | = (ocx)	V	(~ocx) =
ос	(x V (—x)) = [a | |x|. From |x | 4- |y | /:> x	+ y, —x — y,	we obtain
x	4- | У | A (x + у) V (—x — y) = ] x T у .
Remark. The decomposition x = x+ + xr, %+ A (—xr) = 0, is called
the Jordan decomposition of %; x+, xr and correspond to the positive
variation, the negative variation and the total variation, of a function x (t)
of bounded variation, respectively.
Proposition 5. For any у 4 X, we have
\x — xi| = [x V y — xt V y[ + А у — хг A y|.	(23)
Proof. We have
| a — b | = (a — b)+ — (a — b}~ = a V b~b — (а Д b — b)
= a V b — а Д Ь.
Hence the right side of (23) is, by (10), (14) and (15),
= (x V у) V хг — (x V у) A (%i V у) + (x Л у) V (xi A y) — (x A y) Ax,
= (x V xr) V у — (x A Xj) v у + (x V x^ Л у — (x Л хг) Л у
= %V Xj + у — (x A Xj + у) = X V Xj—X A Xj = \x — Xj .
Definition 2. A sequence of a vector lattice X is said to O-convergc
to an element x 4 X, in symbol, O-lim xn = x if there exists a sequence
{да„} such that [% — x„\ wn and wn ф 0. Here wn J, 0 means that
2- Vector Lattices
367
w, w2  and Л a>n - 0. The O-lim if it exists, is uniquely
Я-НЮ
determined. For, let O-lim xn = x and O-lim xn = y. Then lx ~ xn I wn,
WK>0	ft-KX)
w„ J 0 and I у — xn I	unt un j 0. Thus [x — у | < | x — xn | + {—- у | <
wn + un and (w„ + wn) j 0 as may be seen from + Д un
Л (w„ + «я). This proves that x = y.
Proposition 6. With respect to the notion of O-lim, the operations
% -j- y, x v у and % Л у are continuous in x and y.
Proof. Let O-lim xn = x, O-lim yw — у. Then | x + у — xn — yn I <
rtWOO	WOO
% —
I* Л У — xn Л y«l I* л У ~xn л у! + \%n \ y — Xn \ yn|
x — xn
У — Уп
and so O-lim (x„ Y yn) = /О-lim zn) д /О-lim ул).
\ П-КЮ /	\ Я—ЮО /
Remark. We have
O-lim я •	= <x - O-lim xn.
WOO	WOO
But, in general,
O-lim tx„x	/ lim ccw) x.
n—>00	\ n—>oo /
The former relation is clear from \ocx — ocxn\ —
a I I x — xn j. The latter
inequality is proved by the following counter example: We introduce in
the two-dimensional vector space the lexicographic partial order in which
(fi, T7i) A (f2, %) means that either or = £2, г}г ?/2. It is
easy to see that we obtain a vector lattice. We have, in this lattice,
n-i(l, 0)	(0, 1) > 0 = (0, 0) (n = 1, 2, . . .).
Hence O-lim «^(l, 0) 7^ 0. A necessary and sufficient condition for the
WOO
validity of the equation
O-lim ocn x = tx x	(24)
is the so-called Archimedean axiom
O-lim = 0 for every x 0.	(25)
WOO
This we see by the Jordan decomposition y y+ * У 
Definition 3. A subset {%a} of a vector lattice X is said to be bounded
if there exist у and z such that у < х& z for all xa. X is said to be
complete if, for any bounded set [t'J of X, supxa and infxa exist in
a	a
X. Here sup хл is the least upper bound in the sense of the partial order
in X, and inf x№ is the largest lower bound in the sense of the partial order
oc
in X. A vector lattice X is. said to be o-complcte, if, for any bounded
368 XI1. Other Representation Theorems in Linear Spaces
sequence {x„} of X, sup x„ and inf x„ exist in X. We define, in a <t-
complete vector lattice X,
O-lim xn =milsup xA , (9-lim xn = sup (inf x„),	(26)
w	/ л-юа	w	/
Proposition 7. (9-lim xn = x iff (9-lim xn = (9-lim xn = x.
П-ЮО	Я—ЮО	я—wo
Proof. Suppose j % — xn j w„, wn | 0. Then x — wn X xn x + wn
and so we obtain (9 1im (x — w„) = x < (9-Um xn < (9-lim (x + wn) = x,
«—X5O	Л—*<X>	Я—ХЭО
that is, (9-lim xn — x. Similarly, we obtain (9-lim xn = x.
n—>oo
We next prove the sufficiency. Put 7/rt = supxw, vn = inf xm,
m^n	”‘?'п
un — vn = wn. Then, by hypothesis, wn | 0. Also, by xn Al w„ = x +
(«„ — x) A x + (un — vn) = x + and xn > x — obtained similarly,
we prove |x — xn I '> wn. Thus (9-lim xn = x.
Г»—ЮО
Proposition 8. In a o-complete vector lattice X, xx is continuous in
л, x with respect to (9-lim.
Proof. We have \xx — xnxn | \xx — xxn [ + \xxn - xnxn ] =
|a ] \x - - xn | + I® — a» | | xn |. Hence, if (9-lim x„ = x; lim xn - - a, then
П-+ОС
the (9-lim of the first term on the right is 0. Therefore, by putting
sup\x„ | = y, sup |a — I = fin, we have to prove (9-lim/?rty = 0.
m;>n	n-K»
But, by у AO and /?n|0, we see that (9-lim/J„y = z exists and
Я-+СО
(9-lim 2~1fi„y = 2-1г. Since there exists, for any и, an nQ such that
we must have z = 2~rz, that is, z — 0.
Proposition 9. In a o-complete vector lattice X, (9-lim xn exists iff
Я—>OQ
(9-lim | xn - xm |	0.	(27)
>oo
Proof. The necessity is clear from | x„ — xw | ±9 [ xrt — x | 4- | x — xm |.
If we set |x„ - xm [ = ynm, then (9-lim xn xm + (9-lim ynm, (9-lhn xw
«—►CO	tt—«Г-КЮ
xm — (9-hm ynM. Hence
Я—ЮО
0 S (9-lim x„ - (9-hm xw < (9-lim ((9-lim ynnt — (9-lim ynm 1=0.
n—*"00	-ft—ЮО	#И^>0О у >O0	ц—>.0Q	/
Thus we have proved the sufficiency. '
Proposition 10. A vector lattice X is cr-complete iff every monotone
increasing, bounded sequence {xw} С X has sup xw in X.
Proof. We have only to prove the sufficiency. Let {zw} be any bounded
sequence in X, and set xn = sup zm. Then, by hypothesis, sup xn~ z
m^n
exists in X, and z = sup zn. Similarly, we see that inf zn = inf / inf zM\
also exists in X.
3. В-lattices and F-lattices
369
3.	В-lattices and F-lattices
Definition. A real 5-space (F-space) X is said to be a В-lattice (F-
l attic e) if it is a vector lattice such that
|x|	|у | implies j|x
(1)
Examples. C(S), LP(S) are В-lattices by the natural partial order
x < у which means x(s) y($) on S (%(s) y(s) a.e. on S for the case
Z/(S)). M{S, 93, m) with m(S) <oo is an F-lattice by the natural
partial order as in the case of LP(S). A (5, 99) is a В-lattice by
[ j x 11 = ] x | (S) — the total variation of x over 5.
We have, by (1) and |x| = | (j% |) |,
=	(2)
In A (S, 99), we have, moreover,
x 0, у 0 imply | [% 4- у |[ = ||x|| + ||y ||.	(3)
S.	Kakutani called a В-lattice satisfying (3) an abstract LA-space. From
(3) we have
| x | < |y | implies ||x|[< [|у||-	(4)
The norm in A (S, 99) is continuous in (9-lim, that is
(9-lim xn - x
fW-OQ
implies
lim
я-юо
Mi-
(5)
For, in A (S, 93), (9-lim xn = x is equivalent to the existence of ул£ Л (S, 99)
п—КЮ
such that [ x — xn | (S) yn (S) with y„ (S) | 0. It is easy to see that
M(S, 93, m) satisfies (4) and (6).
Proposition 1. A o-complete F-lattice X satisfying (4) and (5) is a
complete lattice. In particular, A (S, 99) and Lp (S) are complete lattices.
Proof. Let {ла} С X be bounded. We may assume 0 < xa x for all
a, and we shall show that supx, exists. Consider the totality {^} of
tx
П
Zg obtained as the sup of a finite number of xe’s:	= V хЛ}- Set
j=i
у = sup 11 Zp 11. Then there exists a sequence {zpJ such that lim 11 z^} | j = y.
If we put zn
definition of y, we have
Suppose that xa V
’'	J—400
= sup Zp., then O-lim zn — w exists, and, by (5) and the
w | j = y. We shall prove that w = sup xa.
&
w for a certain xa. Then, by (4), ||xw V z^|[ >
||w|| = y. But, by xavW = (9-lim (xa V zn}> xa \/ zn<= {^} and (5), we
n—KJO
have ||ял V = lim ||xa V z„\\A^y, which is a contradiction. Hence
n—K3O
we must have шАхл for all x„. Let x„ < и for all xa. Suppose да Л и < да.
24 YoHldu, Functional Апд1ун1н
n—ИЗО
370
XII. Other Representation Theorems in Linear Spaces
Then, by (4), ||w Л «||<y, contrary to the fact that w Д и Zp for all
2$. Hence we must have w = sup xa.
a
Remark. In C (S), O-lim xn = x does not necessarily imply
n—too
s-lim%„ = %. In M(S, ЭЗ, m), s-limx„ = % does not necessarily imply
Л—X3O	rt—ХЭ0
O-limxn = x. To see this, let xx(s), x2(s), . . . be the defining functions of
the intervals of [0, 1]:
Then the sequence g M([0, 1]) converges to 0 asymptotically but
does not converge to 0 a.e., that is, we have s-lim xn == 0 but O-lim x„ = 0
H—H3O	n—too
does not hold.
Proposition 2. Let X be an F-lartice. Let a sequences {x„} С X satisfy
s-lim xn = x. Then {.r„} *-converges to x relative uniformly. This means
that, from any subsequence {y„} of {#„}, we can choose a subsequence
and a z E X such that
— *[	* 4 (A = 1, 2, . . .).
(6)
Conversely, if {x„] *-converges relative uniformly to x, then s-lim xn = x.
Л—ЮО
Proof. We may restrict ourselves to the case x = 0. From lim 11 yJ I = 0,
w^>oo
we see that there exists a sequence {n (A)} of positive integers such that
OO
7 A-3. Then (6) holds by taking z — j?
versely, let condition (6) with x = 0 be satisfied. Then we have
!|К(А) || = Н^~1-г11 from I = Hence s-limуи(А) = 0. There-
Л—>00
fore, there cannot exist a subsequence {y„} of {x,J such that lim I yn I > 0.
ИЗО
Remark. The above proposition is an abstraction of the fact, that
an asymptotically convergent sequence of ЛГ (S, m) with m (S) < oo
contains a subsequence which converges ж-а.е.
4.	A Convergence Theorem of Banach
This theorem is concerned with the almost everywhere convergence
of a sequence of linear operators whose ranges are measurable functions.
See S. Banach [2]. A lattice-theoretic formulation of the theorem reads
as follows (K. Yosida [15]):
Theorem. Let X be a real 5-space with the norm || || and Y a tr-
complete F-lattice with the quasi-norm || J|x such that
0-limyn = y implies lim l|y„l|1 = IMh
ft—H3Q	ft—H5O
(1)
4. A Convergence Theorem of Banach
371
Let {/„} be a sequence of bounded linear operators € L(X, Y). Suppose
that
O-lim T„x\ exists for those x’ X which form a set
G of the second category.
Then, for any x~ X, O-limTnx and O-lim Tnx both exist and the (not
^“*00
necessarily linear) operator T defined by
Tx = [ O-lim Tnx)— (O-lim Tnx
\ Я-—H5O	/ \ Я—>OQ
(3)
is continuous as an operator defined on X into Y.
Remark. The space M (S, 58, m) with m{S) < oo satisfies (1) if we
write уг у2 when and only when yx(s) Уг(5) w-a.e., the quasi-norm
|(y Ik being defined by ||y |k = J |y(s) | (1 + |y (s) I)'1 m(ds). Likewise
s
Z?(S, 58, wi) with m(5) oo also satisfies (1) by the same semi-order.
Proof of the Theorem. Set Tnx = y„, yn = sup |yw |, / = sup |y„|,
and consider the operators Vnx = у» and Vx — y’ defined at least on G
into Y. By (23) of the preceding section 2, each Vn is strongly conti-
nuous with Tk. Since lim |l V„x —	— 0 by (1), we have
rt—>oo
lim | k~xVnx | = Ikr1 Vx I and further lim	= 0- These are
Я-Н» 1 «II	I	fe—изо	111
implied by the continuity in ос, у of ocy in the F-space Y. Hence for any
e > 0, oo	,	.	,
GC U Gk where Gk = {xCX; sup |JA-1	)|x Si e}.	(4)
*=i	l «Й1	J
By the strong continuity of V„, each GA is a strongly closed set of X.
Thus some G^ contains a sphere of X, in virtue of the hypothesis
that G is of the second category. That is, there exist an x0 £ X and a
<5 > 0 such that ||x0— x|| S d implies sup V»x||i ~ £- Hence, by
putting z = x() — %, we see that
sup ll^o1 sup Ц&01 V„xo111 + SUP IHo 1 vnx111
that is, since lzM(/vo M = kQT Vnz, we have
sup |] Vnz ||i g 2e whenever ||г|| <i <5/&0.
»ai
This proves that s-lim V„ * z = 0 uniformly in «.
G being dense in X, V  x is defined for all x € X and V  x is strongly
continuous at x = 0 with V • 0 = 0. Hence, by
|f -x|	27  x
and
11TTx21|i	ЦТ"(Xi	xa) 11,
we see that T  x is strongly continuous at every x € A'.
372
XII. Other Kepresentation Theorems in Linear Spaces
Corollary. Under condition (1), the set G = {x ё X; O-lim Tnx exists}
n—>OQ
either coincides with X or is a set of the first category.
Proof. Suppose the set G be of the second category. Then, by our Theo-
rem, the operator Tis a strongly continuous operator defined on X into Y.
Thus G — {x € X; Ту = 0} is strongly closed in X. Moreover, G is a
linear subspace of X. Hence G must coincide with X. Otherwise, G would
be non dense in X.
5.	The Representation of a Vector Lattice as Point Functions
Let a vector lattice X contain a "unit” I with the properties:
1 > 0, and, for any /G X, there exists an a > 0 such
that -л/ U / -G oil.
(I)
For such a vector lattice X, we can give an analogous representation
similar to the representation of a normed ring as point functions.
An element /G X is called “nil-potent” if n |/| < I (n = 1, 2, . . .).
The set R of all nilpotent elements f£X is called the “radical” of X.
By (20) of Chapter XII, 2, R constitutes a linear subspace of X. More-
over, R is an “ideal” of X in the sense that
/G R and |g|	|/| imply gG R.	(2)
Lemma. Let XT and X2 be vector lattices. A linear operator T defined
on A\ onto X2 is called a lattice homomorphism if
=	(Ту}.	(3)
Then (3) implies that N ~ {xХг; Tx = 0} is an ideal of Xx. Conversely,
if a linear subspace N of Xj is an ideal of X1} then the mapping x-> Tx
— x = (the residue class mod A7 containing %) satisfies (3) by defining
x у through x у.
Proof. Let T be a lattice homomorphism. Let лТ N and |y| gG |x
Then, by Г(|%|) = T(x V —%) = (Tx) V $T(—x)) = 0, we obtain,
0 Ty+ = T(y+ Л | % |) = Ty+ А T | x | = 0 and so y+ G N. Similarly
we obtain yXN and thus у = y+ + y~ G N.
Next let a Unear subspace N ~ {x G Xj; Tx = 0} be an ideal of Xx.
If у = z, i.e., if у - z G X, then, by (23) in Chapter XII,2, we have
g |y - z\ G N
so that the residue class (x д у) is determined independently of the choice
of the representative elements x, у from the residue classes x, y, respect-
ively. Hence we can put хчлу — x^y by defining z > 0 through z v 0 - z.
5. The Representation of a Vector Lattice as Point Functions 373
Remark. The above Lemma may be phrased as follows. Let N be a
linear subspace of a vector lattice. Then the linear-congruence a = Ъ
(mod N) is also a lattice-congruence:
a = b, a' = bf (mod N) implies a J( b = а' д b' (mod N),
iff N is an ideal of X.
Now an ideal A is called “non-trivial’ ’ if N =/= {0}, X. A non-trivial
ideal N is called “maximal" if it is contained in no other ideal =4 X.
Denote by Ж the set of all maximal ideals N of X. The residual
class X/N of X mod any ideal N e 9T is “simple", that is, X/N does not
contain non-trivial ideals. It will be proved below that a simple vector
lattice with a unit is linear-lattice isomorphic to the vector lattice of real
numbers, the non-negative elements and the unit I being represented by
non-negative numbers and the number 1. We denote by f(N) the real
number which corresponds to f 6 X by the linear-lattice-homomorphism
X^XfN, Newt.
After these preliminaries we may state
Theorem 1. The radical R coincides with the intersection ideal
П N.
Proof. The first step. Let X be a simple vector lattice with a unit I.
Then we must have X = {ocl; — oo <z oc < oo}.
Proof. X does not contain a nilpotent element f =/= 0, for otherwise X
would contain a non-trivial ideal N = {g; )g | rj |/| with some 77 < 00}.
Hence, by (1), X satisfies the Archimedean axiom'.
order-limn 1 Ix =0 for all x£X.	(4)
Suppose there exists an fQ G X such that у I for any real number y.
Let
oc = inf oc', 0 = sup 0'.
Then, by (4),< /0 <ocland^< oc. Hence (/0 - c5A)f_ 0, (/0 — <5Z)_5*^ 0
for 0 < <5 < oc. Thus, by л (—x~) = 0, the set Ao = {g; |g | <: 77 (/0 — dF)+
with some 77 < 00} is a non-trivial ideal, contrary to the hypothesis.
The second step. For any non-trivial ideal Ao, there exists a maximal
ideal TVj containing Ao.
Proof. Let {;V0} be the totality of non-trivial ideals containing Ao.
We order the ideals of {Ao} by inclusion relation, that is, we denote
Nai < if is a subset of N^. Suppose that {-V.J is a linearly or-
dered subset of {A3 and set Ns — U N„. We shall then show that
Nfi is an upper bound of {Лтл}. For, if x, у £ Nfi, there exist ideals Nat and
Ал> such that x £ A^ and у £ N^. Since is linearly ordered, Na £ A^
(or Naj T NaJ and so x and у both belong to АЯ1. This proves that
374
XII. Other Representation Theorems in Linear Spaces
(yx + i5y) € AT^ g Np and that |z[ jx| implies 26 A^ g Np. Since
the unit I is contained in no N*, I is not contained in Np. Therefore Np
is a non-trivial ideal containing every Na, i.e., Np is an upper bound of
{N«}. Thus, by Zorn’s Lemma, there exists at least one maximal ideal
containing Af0.
The third step. R Q Г\ N. Let / > 0 and nf < I (n = 1, 2, . . .).
Then; for any N £ 9К, we have nf(N) 5g I (N) = 1 (n ~ 1, 2, . . .), and
hence f(N) = 0, that is, f£N.
The fourth step. R 2	2V. Let f > 0 be not nilpotent. Then we have
to show that there exists an ideal N G such that N. This may be
proved as foUows.
Since / > 0 is not nilpotent, there exists an integer n such that n/ I.
We may assume that nf I, since otherwise f £ W for any N € 9K and so
we have nothing to prove. Thus suppose p = I — (n • /) A I > 0. Then,
for any positive integer m, we do not have m  p^. I. If otherwise, we
would have wr11 I — (n f) T\I and hence
[n.f] A	А
Thus, by (6) in Chapter XII, 2,
(n • / — (1 — ntr1) T) /\ т~г1 = (n • / — (1 — ш"1) T) A 0^0,
and so, by the distributivity of the vector lattice,
0 = {(n • f — (1 — ж-1) 7) A m-1/} V 0 = (n • f — (1 — m-1) A rrr1!,
that is, (n  / - (1 — ш”1) Г)+ Л I = 0. Put b = (n - f — (1 — nr1') 7)+
and assume that b > 0. By hypothesis (1), we have b < <xl with some
oc > 1. Then 0 < b = b A ocl and so 0 < (л-1 6) Л I ^b A I, contrary
to & A I ~ 0. Therefore 6 = 0, i.e., n  /	(1 — m-1) I. This contradicts
the fact that n  / I. Hence the set Ar0 = {g; jg|	?? \p [ for some
Tj < oc} is a non-trivial ideal. Af0 is contained in at least one maximal
ideal 2V, by the second step. Then 0 = p (2V) = 1 — (n  /(AQ) A 1 which
shows that f(N) > 0, that is, /Е N.
We have thus proved our Theorem 1.
The vector lattice X = X/R is again a\vector lattice with a unit I.
By Theorem 1, the intersection ideal О N of all the maximal ideals N
__	_ jv
of X is the zero ideal and X contains no nilpotent element =£ 0.
Hence X satisfies the Archimedean axiom
order-limit и”1 l/| = 0 for all /t X.
n too
(5)
Let N be any maximal ideal of X. Then the factor space X/N is a
simple vector lattice, and so, by the first step in the proof of Theo-
rem 1, X/N is linear-lattice-isomorphic to the vector lattice of real
6. The Representation of a Vector Lattice as Set Functions 375
numbers; the non-negative elements and the unit being represented by
non-negative numbers and 1. We denote by /(TV) the real number which
corresponds to J by this homomorphism X -> X/Nl We also denote by
50? the set of all maximal ideals of X. We have thus
Theorem 2. By the correspondence /->/(X), X is linear-lattice-iso-
morphically mapped on the vector lattice F(50?) of real-valued bounded
functions on 50? such that (i) |?| —> [/(TV) (ii) 7 (TV) = 1 on 50? and (iii)
F (50?) separates the points of 50? in the sense that
for two different points TV,, ?V2 of 50?, there exists at
- -	- -	- -	(6)
least one / C X such that / (TVa)	/ (2TV).
Remark. We introduce a topology in 50? by calling the sets of the form
{TV e 50?; I /" (TV) - 7t (TV0) ] < £i (г = 1, 2, ...,«),
where —7 f I for all i}
neighbourhoods of TV0. Then is compact since it may be identified
with a closed subset of a topological product (of the same potency as
the cardinal number of the set of elements f€X which satisfy -7 /g 7)
of the closed intervals [—1, 1]. The proof is entirely similar to the case
of the set of all maximal ideals of a normed ring in Chapter XI, 2.
Moreover, each function f(N) £ F(50?) is continuous on the compact space
50? topologized in this way. Thus, by the Kakutani-Krein theorem in Chap-
ter 0, 2, we see that F (50?) is dense in the В-space C (50?). The above two
theorems are adapted from K. Yosida-M. Fukamiya [16]. Cf. also
S. Kakutani [4] and M. Krein-S. Krein [2].
6.	The Representation of a Vector Lattice as Set Functions
Let X be a o-complete vector lattice. Choose any positive element
x of X and call it a “unit" of X and write 1 for %; when not ambiguous
we-also write tx for a • 1. A non-negative element e£ X is called a “quasi-
unit" if e A (1 — e) = 0. A finite linear combination £ of quasi-
units is called a “step-element", and we call the element у С X "ab-
solutely continuous" (with respect to the unit 1) if у can be expressed as
the O-lim of a sequence of step-elements. An element z С X is called
“singular" (with respect to the unit 1) if |z| A 1 = 0.
We shall give an abstract formulation of the Radon-Nikodym theorem
in integration theory.
Theorem. Any element of X is uniquely expressed as the sum of an
absolutely continuous element and a singular element.
Proof. The first step. If / > 0 and / A 1 ф 0, then there exist a
positive number л and a quasi-unit / 0 such that /	We can,
376
XII. Other Representation Theorems in Linear Spaces
in fact, take
= V	Л 1)A 1}.	(1)
»Z>1
Proof. Put ул = -x1/ - а1/ Л 1- Then we obtain
2еа Л 1 = J V (2nya Л 2)1 Л 1 = e^,
J
and so ea is a quasi-unit. We obtain / A txea from
wy* A 1 = пог1} Л [1 + «(а-1/ Л 1)] — «(tx-1/ Л 1)
< (n + 1) a-1/ A (w + 1) — «(tx-1/ Л 1)	a-1/ A л-1/.
If we can show that y№ A 1 > 0 for some a > 0, then ея > 0 for such a.
Suppose that such positive a does not exist. Then, for every tx with
0 < tx < 1, we have
tx 1(tx 1/ —<x V A 1) A tx”1 = 0.
Hence (/ — / A a) A 1 = 0 and, letting a ф 0, we obtain f A 1 = 0,
contrary to the hypothesis f A 1^0.
The second step. Let /	0, and f A <xe where tx > 0 and e is a
quasi-unit. Then, for 0 < a' < a, ea> A e and f A o' ex> where ел. is
defined by (1).
Proof. For the sake of simplicity, we assume that <x = l.For0<<5< 1
1 — <5	1 — <5	1	(1 — <5+l —	+	<5
1 - <5 A 1	1 — <5 ’
Since e is a quasi-unit, we have 2e A 1 = e, e Л 1 = e. Hence me A 1
— e when w >1. Thus, by 1 < (1 — d)-1, we have (1 — Й)-1 e A 1 - &•
Therefore, we obtain, from the above inequality,
and so, by (1)., e <5-
The third step. The set of all quasi-units constitutes a Boolean
algebra, that is, if ev and e2 are quasi-units, then eY V e* and A A are
also quasi-units and 0 A % A 1- The quasi-unit (1 — e) is the comple-
ment of e, and 0,1 are the least and the greatest elements, respectively
in the totality of the quasi-units.
6. The Representation of a Vector Lattice as Set Functions 377
Proof. The condition e Л (1 — e) = 0 is equivalent to 2e A 1 = e.
Hence, if e1, e2 are quasi-units, then
2(^1 A tfj) A 1 = (2«! A 1) A (2£;2 A 1) = A e2,
V %) A 1 = (2et A 1) V (2e2 Al) = q V
so that A e2 and V are also quasi-units.
The fourth step. Let / > 0, and set / — sup ре$ where sup is taken
for all positive rational numbers ft. Because of the third step, the sup of
a finite number of elements of the form fap is a step-element. Hence/
is absolutely continuous with respect to the unit element 1. We have to
show that g = / — / is singular with respect to the unit element 1. Suppose
that g is not singular. Then, by the first step, there exist a positive number
a and a quasi-unit e such that g > ле. Hence / and so, by the second
step, there exists, for 0 < <хг < ос, a quasi-unit eai > e such that /	oc^^.
We may assume that is a rational number. Thus / A' and so
/ = / + g A 2»!^. Again, by the second step, there exists, for 0 <
oj < a1( a quasi-unit е2м' A eai such that / > 2л,1е2а;- We may assume
that 2oq is a rational number so that / > 2oc'1e2a/i. Hence /=/ +	3a{ e.
Repeating the process, we can prove that, for any rational number л„
with 0 < ocn <_ л,
/ A (n + 1) ocne (n = 1, 2, . . .).
If we take ocn A <x/2, we have (и + 1) ocne > 2~1noce. Hence / A note
(и = 1, 2, . . .) with oc > 0, e > 0. This is a contradiction. For, by the <j-
completeness of X, the Archimedean axiom holds in X.
The fifth step. Let / = /+ + f~ be the Jordan decomposition of a
general element / С X. Applying the fourth step to /+ and /" separately,
we see that / is decomposed as the sum of an absolutely continuous ele-
ment and a singular element. The uniqueness of the decomposition is
proved if we can show that an element h £ X is = 0 if h is absolutely
continuous as well as singular. But, since h is absolutely continuous, we
have h = Q-iim hn where hn are step-elements. hn being a step-element,
there exists a positive number ocn such that |йя| < <xn  1. Since h is
singular, we have |A| Л |ЛЯ| = 0. Therefore [A] = |Al A |A| =
O-lim () A | A IM = 0.
ft—M5O
Application to the Radon-Nikodym theorem. Consider the case X =
A (S, S3). We know already (Proposition 1 in Chapter XII, 3) that
it is a complete lattice, and that in A (S, SB),
xA(B) = sup x(B') = the positive variation V(x; B) of x on B. (2)
B'QB
We shall prepare a
378	XI1. Other Representation Theorems in Linear Spaces
Proposition. In A (S, S3)t let x > 0, z 0 in such a way that x Д z
= 0. Then there exists a setB0E 53 such thatx(B0) = 0and.z(S--Bo) = 0.
Proof. Since x Д z = (x — z)~ z, we know, from (2), that
(x/\z)(B) = inf (x — z)(B') + z(B) = inf [x(B') + z{B — B')]. (3)
B*QB	B' QB
Hence, by the hypothesis x Д z = 0, we have
inf (z(B) + s(S - B)J = (x Л 2) (S) = 0.
xft io
Hence, for any «>0, there exists a BecS3 such that x(BB) Д e,
z (S — B^j e. Put Bo = Д / V B2-n\ . Then, by the cr-additivity of
*;>i	.1
x(B) and z(B),
0 x(BQ) = lim x i V B2-»\ 'L lim £	= 0,
A—hxj	>oa u = k
0 z(S — Bo) - lim z(S - V B2-n) = lim ~ B2-k) = 0-
Corollary. Let & be a quasi-unit with respect to x > 0 in A (S, 53).
Then there exists a set Bt £ 53 such that
e(B) = x(B П BJ for all В 6 SB.	(4)
Proof. Since (% — e) Д £ = 0, there exists a set B0C 53 such that
e [B^ = 0, (x — e) (S — Bo) = 0. Hence e (S — Bo) = x (S — Bo) = £ (S),
and so e (В) = x (В — Bo) = x (В П BJ where BL = S — Bo.
We are now able to prove the Radon-Nikodym theorem in integration
theory. In virtue of the above Corollary, a quasi-unit £ with respect to
x > 0 is, in A (S, 53), a contracted measure e(B) = x(B Q Be). Hence a
step-element in И (S, 53) is the integral of the form;
Xk J x(ds),
* Br-.Bi
that is, indefinite integral of a step function (= finitely-valued function).
Hence an absolutely continuous element in A (S, 53) is an indefinite
integral with respect to the measure x(B). The Proposition above says
that a singular element g (with respect to the unit x) is associated with a
set Bo C 53 such that x(B0) = 0 and g(B) =Ag(B П Bo) for all В C 53. Such
a measure g(B) is the so-called singular measure (with respect to x(B)).
Thus any element /С A (S, 53) is expressed as the sum of an indefinite
integral (with respect to x(B)) and a measure g(B) which is singular (with
respect to x(B)). This decomposition is unique. The obtained result is
exactly the Radon-Nikodym theorem.
Remark. The above Theorem is adapted from K. Yosida [2]. Cf.
F. Riesz [6], H. Fkeudenthal [2] and S. Kakutani [5]. For further
references, see G. Birkhoff [1].
1. The Markov Process with an. Invariant Measure
379
XIII. Ergodic Theory and Diffusion Theory
These theories constitute fascinating fields of application of the
analytical theory of semi-groups. Mathematically speaking, the ergodic
t
theory is concerned with the “time average" lim Z-1 [ Ts ds of a Semi-
co0 o'
group Tt, and the diffusion theory is concerned with the investigation of a
stochastic process in terms of the infinitesimal generator of the semi-group
intrinsically associated with the stochastic process.
1.	The Markov Process with an Invariant Measure*
In 1862, an English botanist R. Brown observed under a microscope
that small particles, pollen of some flower, suspended in a liquid move
chaotically, changing position and direction incessantly. To describe
such a phenomenon, we shall consider the transition probability x; s, E)
that a particle starting from the position x at time t belongs to the set E
at a later time s. The introduction of the transition probability P(t,x;s,E)
is based upon the fundamental hypothesis that the chaotic motion of the
particle after the time moment t is entirely independent of its past
history before the time moment l. That is, the future history of the particle
after the time moment t is entirely determined chaotically if we know
the position x of the particle at time t. The hypothesis that the particle
has no memory of the past implies that the transition probability P satis-
fies the equation
P(i\ s, E) =	dy) P[ut v; S, £) for	(1)
s
where the integration is performed over the entire space S of the chaotic
movement of the particle.
The process of evolution in time governed by a transition probability
satisfying (1) is called a Markov process, and the equation (1) is called
the Chapman-Kolmogorov equation. The Markov process is a natural gene-
ralization of the deterministic process for which P(t,x', s, E) = lor = 0
according as у С. К от yC E; that is, the process in which the particle at
the position x at the time moment t moves to a definite position у —
y(x,t,s) with probability 1 at every fixed later time moment s. The
Markov process P is said to be temporally homogeneous if P[t, x\ s, E)
is a function of (s — Z) independently of t. In such a case, we shall be
concerned with the transition probability P (t, x, E) that a particle at
the position x is transferred into the set E after the lapse of t units of
time. The equation (1) then becomes
Pit -I- 5, ,v, /') f P(t, V, dy) P(s.y, E) for t, s>0.	(2)
* Helt also Supplementary Noles, p. 4(17 .nut p 4(iH.
380	XIII. Ergodic Theory and Diffusion Theory
In a suitable function space Xt P(^ xf E) gives rise to a linear transfer-
mation T,:	(Г,/) (») = f P(i, x, dy) f(y), /<= X,	(3)
s
such that, by (2), the semi-group property holds:
Tt+s = TtTs	(4)
A fundamental mathematical question in statistical mechanics is
concerned with the existence of the time average
lim;-1 /TV*.	(5)
In fact, let S be the phase space of a mechanical system governed by the
classical Hamiltonian equations whose Hamiltonian does not contain the
time variable explicitly. Then a point x of S is moved to the point yt (x)
of S after the lapse of t units of time in such a way that, by a classical
theorem due to Liouville, the mapping x —> yt(x) of S onto S, for each
fixed t, is an equi-measure transformation, that is, the mapping x -> yt(x)
leaves the "phase volume” of S invariant. In such a deterministic case,
wehave	(L I) (x) = / (y,(x)) ,	(6)
and hence the ergodic hypothesis of Boltzmann that
the time average of any physical quantity = the space
average of this physical quantity
is expressed, assuming f dx < oo, by
s
t
lim E1 f f (ys (x)) ds = f f(x) dx j f dx for all f £ X,
'I00 о	s / s
dx denoting the phase volume element of S.	(7)
A natural generalization of the equi-measure transformation
x yt(x) to the case of a Markov process P(t, x, E) is the condition of
the existence of an invariant measure m (dx):
J m(dx) P(t, x, E) ~ m(E) for all i > 0 and all E.	(8)
s
We are thus lead to the
Definition. Let SB be a u-additive family of subsets В of a set S such
that S itself G 93. For every t > 0, x 6 S ancl E £ S3, let there be associated
a function P (t, x, E) such that
P(t,x,E)^0, P(t,x,S) — l,	(9)
for fixed t and x, P (t, x, E) is cr-additive in E £ 93,	(10)
for fixed t and E, P(t, x, E) is SB -measurable in x,	(11)
P(t + s, X, E) = /P(t, x, dy) P(s, у, E)
s
(the Chapman-Kolmogorov equation).	(12)
1. The Markov Process with an Invariant Measure
381
Such a system P{t, x, E) is said to define a Markov process on the phase
space (S, S3). И we further assume that (S, S3, яг) is a measure space in
such a way that
fm(dx) P(i,x,E) = m(E) for all £^93,	(13)
s
then P[t, x, E) is said to be a Markov process with an invariant measure
m[E).
Theorem 1. Let P(t, x, E) be a Markov process with an invariant
measure m such that m(S) < oo. Let the norm in Xp = Lf (S, S3, яг) be
denoted by p 1- Then, by (3), a bounded linear operator
Tt £ L(Xp, Xp) is defined such that Tt+s = TtTs [t, s > 0) and
Tt is positive, i.e.,	(x) ^OonS яг-а.е. if /(x) 2? 0 on S яг-а.е., (14)
Trl-1,
(15)
ЦЛ/Ц, ||< for /€ xp = LP[S, S3, яг) with p = 1, 2 and oo. (16)
Proof. (14) and (15) are clear. Let /6 L°°[S, S3, яг). Then, by (9), (10)
and (11), we see that ft[x) = (Tt f) (x) € £°°(5, S3, m) is defined and
|A|foo |[/l!oo- Hence, by (9) and (13), we obtain, for	SB, яг)
with p = 1 or p — 2,
pu/p
m [dx)
f P(t,x, dy) f(y)
s

dy} fP(t. I'')}1"’
5	I
= {/«>W IW}W =
S	J
We put, for a non-negative /С //(S', S3, яг) with p = 1 or p ~ 2,
„/(s) = min(/(s), n), where « is a positive integer.
Then, by the above, we obtain 0 <, (nf(s))t („+i/(s))< and |[U)J|/,
|И!/> Thus> if we Put A(s) = frjj, then, by the Lebes-
gue-Fatou lemma, 11 jt |11/1that is, Д € Lp (S, 93, яг). Again by the
Lebesgue-Fatou lemma, we obtain
ft[x) = lim fP[t, x, dy) („/(?)) f P(f, x, dy) ( Um («/(y)))
Я—*СО	CJ	/
= fP[i, x,dy)f[y).
s
Thus f[y) is integrable with respect to the measure P{t, x0, dy) for those
x0's for which /f(x0) oo, that is, for яг-а.е. x0. Hence, by the Lebes-
gue-Fatou lemma, we finally obtain
J P(t, x0, dy) (lim (n/(y))) = lim f P[t, x0> dy) (J(y)).
X	\П'-*©О	/
382
XIIL Ergodic Theory and Diffusion Theory
Therefore, ft(x0) = f P(t, x0, dy)/ (y) m-a.e. and ||/X= Н/П/’- For a
s
general / < LP(S, SB, m), we obtain the same result by applying the posi-
tive operator Tt to /+ and separately.
Theorem 2 (K. Yosida). Let P(t, x, E) be a Markoff process with an
invariant measure m such that m(S) < oo. Then, for any / £ Z/(S, SB, m)
with p — 1 or p = 2, the mean ergodic theorem holds:
I
s-lim n-1 £ Thf = /* exists in LP(S, SB, m) and l\f* = /
n—K5C
whenever /Е LP(S, SB, w),
and we have
f / (s) m (ds) = f /* (s) m (ds).
s	s
(17)
(18)
Proof. In the Hilbert space L2(S, SB, да), the mean ergodic theorem
(17) holds by (16) and the general mean ergodic theorem in Chapter VIII,3.
Sincem (S) <oo, we see,by Schwarz’ inequality, that any/CZ?(S, SB, да)
belongs to L1 (S, SB, m) and Ц/^	||/||2 - m(S)1/2. Hence the mean
ergodic
theorem lim f f* (s) — п-1 £ (Tk f) (s) m(ds) = 0, together
«—►CO §	k = l
with Tx/* = /*, hold for any /£ L2(S> $8, m). Again, by m(S) < oo and
0 = lim ||x = lim f \f(s)~tJ(s)\m(ds) with J(s) = min(/(s),w),
«—►CO	rt-ЮО s
we see that L2(S, SB, m) is £1-dense in £1(S, SB, да). That is, for any
/Е £1(S, SB, m) and e > 0, there exists an /e£ £2(S, SB, m) А £T(S, SB, да)
such that ||/ — Д ||i < £- Hence, by (16), we obtain
1	n	n
i	T„t., _ и/—aII, a*
I	Л=1	fc=l	1
The mean ergodic theorem (17) in £T(S, SB, m) holds for fe and so, by
the above inequality, we see that (17) in £X(S, SB, m) must hold also
for /.
Since the strong convergence implies the weak convergence, (18) is a
consequence of (17).
Remark 1. The above Theorem 2 isMue to K. Yosida [17]. Cf. S. Kaku-
n
tani [6], where, moreover, the ж-а.е. convergence lim n-1 £ (Tkf)(x)
«—►CO	A = 1
is proved for every/С £°° (S, SB, m). We can prove, when the semi-group T{
n
is strongly continuous in t, that we may replace s-lim n! Tkf in (17)
A = 1
t
by s-lim P1 f T; fds. We shall not go into the details, since they are
/foo 0
given in the books on ergodic theory due to E. Hopf [1] and
K. Jacobs [1]. We must mention a fine report by S. Kakutani [8] on
2. An Individual Ergodic Theorem and les Applications 383
the development of ergodic theory between Hopf’s report in 1937 and
the 1950 International Congress of Mathematicians held at Cambridge.
In order to deal with the ergodic hypothesis (7), we must prove the
individual ergodic theorem to the effect that
lim n-1
Л—>oo
n
(Л/) (x) =/*(%) ж-а.е.
Л = 1
In the next section, we shall be concerned with the ж-а.е. convergence
tt
of the sequence w-1 £ (7\/) W- Our aim is to derive the ж-а.е. con ver-
A = 1
gence from the mean convergence using Banach’s convergence theorem
in Chapter XII, 4.
2.	An Individual Ergodic Theorem and Its Applications
We first prove
Theorem 1	(K. Yosida). LetAx be a real, o-complete F-lattice provi-
ded with a quasi-norm ||%||x such that
O-lim xn = x implies lim j xn
и—>oo	И—+OO
\x 111 •
(1)
1 —
Let a linear subspace X of Xx be a real В-space provided with a norm
| x j| such that
s-lim = x in X implies s-lim xn = x in Xx.	(2)
n—>CQ	H—>OO
Let {Tn} be a sequence of bounded linear operators defined on X into X
such that
O-lim | Tnx | exists for those x’s which form a set S of
n—КЮ
the second category in X.	(3)
Suppose that, to a certain z € Xi, there corresponds a z E X such that
lim || Tnz — ~z || = 0,	(4)
T n1 ~~z (n = l, 2, ...),	(5)
O-lim — TnT^z) — 0 for k = 1, 2, . . .	(6)
Я--УОО
Then
O-lim Tnz — z.	(7)
Proof. Put г = г + (z-г). We define an operator T on 5 into Yx by
Tx = O-lim T„x —O-lim Tnx.	(8)
м->ос	П-ХЮ
Then, by (5),
0 <. Tz < T(z — z).
384
XIII, Ergodic Theory and Diffusion Theory
We have T(z — Tkz) = 0 (k = 1, 2,...) by (6), and lim [I (z~z) — (z—Tkz) I
k—KX
= 0 by (4). Hence, by Banach's theorem in Chapter XII, 4, we have
T(z~z) = 0. Hence 0 5^ Tz 0, that is, 0-lim Tnz = w exists.
И—ЮО
We have to show that w =2, We have,by (4) and (2), lim 11 T'^z—z^--- 0.
n->OO
Also we have, by (1) and O-lim Tnz ~ w, lim jTnz — wlk ~ 0. Hence
П-ЮО	fl+OC

Specializing X} as the real space M(S, ЯЗ, w) and X as the real space
£X(S, 53, m), respectively, we can prove the following individual ergodic
theorem.
Theorem 2	(K. Yosida). Let T be a bounded linear operator defined
on 58, w) into £1(S, 58, m) where m(S) < 00. Suppose that
|T"||^C<oo (w=l, 2, ...),	(9)
-1 v
Gm n
«—>00
Suppose that, for a certain z C £1(S, 58, m),
lim n ' (Tnz) (s) — 0
K>o
00 m-a.e.
л	I
and the sequence U-1	contains a subsequence
m*=l	J
which converges weakly to an element z f £*(£, SB, m).
Then
s-lim n'1 £
«—►oo	m-1
(10)
(11)
(12)
(13)
and
lim и"1 (Tmz)(s) ~z(s)
ffl = l
(14)
Proof. Consider the space M(S, 58, m), and take the ^'-lattice =
M(S,58,w) with j|%||1 = f |%(s) ] (1 + |%(s) |)-1 m(ds), the B-lattice
s
X = L1 (S, 58, m) with ||%|| = f |%(s) | m(rfs) and Tn = и"1 ^7^*.
s	m=1
Then the conditions of Theorem 1 are satisfied. We shall, for instance,
verify that (6) is satisfied. We have
Tnz — TnTkz = n~1(T -TT2 H--------+ Г") z
— п~г(7*+1 + Tk+2 + •  - + Tk+n) z
and so, by (11), lim (Tnz — T„Tkz) (s) — 0 m-a.e. for k — 1, 2, . . .
n—>OO
Hence, by taking the arithmetic mean with respect to k, we obtain
Gm {Tnz — TnTkz) (s) = 0 m-a.c. for k = 1, 2, . . . Conditions (4) and
(5), that is condition (13) is a consequence of the mean ergodic theorem
in Chapter VIII, 3.
2. An Individual Ergodic Theorem and Its Applications
385
Remark. The above two theorems are adapted from K. Yosida [15]
and [18]. In these papers there are given other ergodic theorems im-
plied by Theorem 1.
As for condition (11) above, we have the following result due to
E. Hopf [3]:
Theorem 3.	Let T be a positive linear operator on the real L1(S, SB, m)
into itself with the Id-norm |[ T || 1. If	93, m) and ;!>Eld(St 93, w)
be such that p (s) > Om-a.e., then
/Я-1
lim (T**f) (s) £ (T^p) (s) = 0 w-a.e. on the set where p(s) > 0.
n—юо	/ ;=0
If m(S) < oo and T  1 = 1, then, by taking p(s) = 1, we obtain (11).
Proof. It suffices to prove the Theorem for the case /	0. Choose
e> 0 arbitrarily and consider the functions
n—1
g„=T»/-£.^ Tip, g0 = f.
j=0
Let x„(s) be the defining function of the set {s £ S;gn(s) S: 0}. From
xngn  gn = max(g, 0) and + Ер = T gn, we obtain, by the posi-
tivity of the operator T and
f gn+1 -m(ds) + sf Xn+1p 'm(ds) = f x„+1(gn+1 + ep) m(ds)
s	s	s
= /xn+1Tgn-m(ds) Jxn+1Tg+ -m(ds) J Tg+-m(ds) g f g+-m(ds).
s	s	s	s
Summing up these inequalities from n = 0 on, we obtain
/ gn • w(tfs) + e j p- .E xk -m(ds)	f gj • m(ds),
s	s ft=1	s
and hence
0°
I p •	xk-m (ds)	с”1 f go  m (ds).
s ft=1	s
00
Hence xk(s) converges ж-а.е. on the set where p (s) > 0. Thus, on the
Л = 1
set where p(s) > 0, we must have gw(s) < 0 m-a.e. for all large n.
Therefore, we have proved that (T*f) (s) < e (Tkp) (s) m-a.e. for
/j-0
all large n on the set where p(s) > 0. As e > 0 was arbitrary, we have
proved our Theorem.
As for condition (10), we have the following Theorem due to
R. V. Chacon-D. S. Ornstein [1]:
Theorem 4 (R. V. Chacon-D. S. Ornstein). Let T be a positive
linear operator on the real L1 (S, S3, m) into itself with the Id-norm
25 Yoalda, Functional Analysts
386
XIII. Ergodic Theory and Diffusion Theory
jlTJh 1. If f and p are functions in 99. m) and p (s)	0, then
( n	I n
(3) L^p) Й
, _ (*=o	l A=o
is finite m-a.e.
OQ
on the set where j? (Г”/>) (s) > 0.
n=0
If m(S) < oo and T • 1 = 1, then, by taking p(s) = 1, we obtain (10).
For the proof, we need a
Lemma(CHACON-ORNSTEiN[l]).If/=/++/" and if lim £ (T*/)($)> 0
*OQ fc=0
on a set B, then there exist sequences {dk} and {fk} of non-negative func-
tions such that, for every 2V,
N
f dk-m(ds) + ffN-m(ds)^ f f+ • m(ds),	(15)
s *=°	s	s
£dk(s) = — /"(s) on B,	(16)
6 = 0
TN/+ = J TN~kdk + /N.	(17)
6=0
Proof. Define inductively
dv = /о /-1 = о,
4+1 = fi “b / + dQ + ' • • + d^ +, a!i+1 == Tfi — x.	(18)
Note that
+ t~ + d§ -f- •   + di 0,	(19)
and that the equality holds on the set where /Д$) > 0, for
fi = (Г A-i	/“ -J- d0 +  •  + ^-i)+
= (ГA-i — fi + f~ + dQ + •   4- (/,_! + /J+
= (di + f~ + dQ +    + di-X + /{)+.
It follows from (18) that
r/+= Х+-Ч + /,-.	(20)
By the definition, 4 is non-negative, and so is d{ by the last two equa-
tions of (18) and (19). From (20), we hav£
We then prove
(21)
(22)
To this purpose, we note that
л J . .	»	. / n-j \
2: ^T~kdk= STisdA
/=0 6=0	J=o \^0	/
2. An Individual Ergodic Theorem and Its Applications
387
and that, by (19)	and the positivity of T,
 I n~i \
— Vj	£ dk\	for
~~ 0	/
Rewriting (22), we have
ft	fl-
£ T>(f+ + /-) g 4; ц + 4) + /- •
7=0	J=o
We will now prove that
OO
£ dj (s) -J- /“ (s)	0 m-a.e. on В.
j=0
(23)
(24)
(25)
It is clear from the remark after (19) that (25) holds with equality m-a.e.
on the set C = {$ 6 S; fj (s) > 0 for some j > 0}. It remains to show that
(25) holds on the set В —C. This is proved by noting that (24) implies
that on B, and on В — C in particular, we have the inequality
OO
(*) +Л(*)^о.
Now we note that (17) is exactly (20) and that (16) is implied from
(19) and (25). To see that (15) holds, note that we have, by the assump-
tions on T and (18),
Hence we obtain (15) by induction on /, since d0 /0 = f+.
Proof of Theorem 4. It is sufficient to prove the Theorem under the
hypothesis that f (s) Sr 0 on S, and only to prove the finiteness of the
indicated supremum when p (s) > 0. The first remark is obvious, and
the second is proved as follows. By the hypothesis that the indicated
supremum is finite at point s where p (s) > 0, we have
___ n	/ я .	j
lim 2	(T+kf)	(s) / Д	(Ti+kp)	($)}	is finite m-a.e.
n—K5O U=0	/ 7=0	J
on the set where (TA/>) (s) > 0,
(26)
___ n	n ,
and this implies that lim Д (T*/) (s)l (T*P) (s)r is finite m-a.e. on
и—юо (y=o	/j=o
the set where (Tkp) (s) > 0.
Now assume the contrary to the hypothesis. Then
__ H	In
lim .2 О (s) Д(Р>)(з)
I j —0	/ J=0
is infinite m-a.e. on a set E of positive m-measure, and p(s) > ft > 0 on
E for some positive constant ft. We have therefore, that for any positive
constant a,
sup Д (Г ((/ — ocp)+ +’(/ — <%/>) )) (s) -
« u=o
0
25*
388	XIII. Ergodic Theory and Diffusion Theory
ж-а.е. on E. Applying the Lemma with / replaced by (/ — a.*), we obtain
from (15) and (16) that
J (/—«/>)+ж (t/s) f dkm(ds)^ — f (f — ocp)~ m(ds).
s	s A=o	£
However, the extreme right term tends to oo as « f oo, and the extreme
left term is bounded as a f oo. This is a contradiction, and so we have
proved the Theorem.
Я
We have thus proved
Theorem 5. Let T be a positive linear operator on L1(S, S3, m) into
itself with the TA-norm ||	1. If m(S) < oo and T • 1 = 1, then, for
any LX(S, S3, ж), the mean convergence of the sequence in
implies its ж-а.е. convergence.
As a corollary we obtain
J=1
Theorem 6. Let P(i, x, E) be a Markov process with an invariant
measure ж on a measure space (S, S3, ж) such that ж($) < oo. Then,
for the linear operator Tt defined by (Tt f) (x) = f P(t, x, dy) f(y), we
s
have i) the mean ergodic theorem:
for any /6 LP(S, S3, ж), s-lim п~г £ Tkf = /*
A=1
exists in Lp (S, S3, ж) and T\ /♦ == /♦ (/> ~ 1, 2),	(27)
and ii) the individual ergodic theorem:
for any f£Lp (S, S3, ж) with /> = 1 or = 2,
n
finite lim и-1 Л (ГА /) (s) exists ж-а.е. and is -	ж-а.е.,
я—юо	Л=1
and moreover, f f(s)m (ds) = J /* (s) ж (ds).	(28)
s	s
Proof. By Theorem 2 in the preceding section, the mean ergodic theo-
rem (27) holds and so we may apply Theorem 5.
Remark. If Tt is given by an equi-measure transformation x->y( (x)
of S onto S, the result (27) is precisely the mean ergodic theorem of
J. von Neumann [3] and the result (28) is precisely the individual
ergodic theorem of G. D. Birkhoff [1] and A. Khintchine [1].
A Historical Sketch. The first operator-theoretical generalization of
the individual ergodic theorem of the Birkhoff-Khintchine type was given
n
by J. L. Doob [1]. He proved that и-1 (Tft/)(x) converges ж-а.е.
when Tt is defined by a Markov process P (t, x, E) with an invariant
measure ж on a measure space (S, S3, ж) such that m(S) — 1 and / is the
defining function of a set £ S3. It was remarked by S. Kakutani [6] that
3. The Ergodic Hypothesis and the H-theorem	389
Doob’s method is applicable and gives the same result for f merely a
bounded SB-measurable function. E. Hopf [2] then proved the theorem
assuming merely that / is /« integrable. N, Dunford-J. Schwartz [4]
extended Hopf’s result by proving (28) for a linear operator Tt which
increases neither theZ1- nor Z°°-norm, without assuming that Tt is posi-
tive but assuming that Tk = and Tx  1 = 1. It is noted that Hopf's and
the Dunford-Schwartz arguments use the idea of our Theorem 1. R. V. Cha-
con-D. S. Ornstein [1] proved (28) for a positive Unear operator Tt of
Zd-norm < 1, without assuming that Tt does not increase the £°°-попп,
and, moreover, without appealing to our Theorem 1. Here, of course, it is
assumed that Tk — and • 1 — 1. We will not go into details, since,
for the exposition of this Chapter which deals with Markov processes,
it sufficies to base the ergodic theory upon Theorem 6 which is a con-
sequence of our Theorem 1.
3. The Ergodic Hypothesis and the H-theorem
Let P[t, x, E) be a Markov process with an invariant measure ж on a
measure space (5, SB, ж) such that ж($) = 1. We shall define the ergodi-
city of the process P(t,x, E) by the condition:
the time average /*(%) = lim w-1 £ (Tkf)(x)
tWOO £=1
= lim tr1 £ f P(k, x, dy) f (y)
я^оо	Л = 1 £
= the space average J" f(x) m(dx) w-a.e.	(1)
s
for every / C LP(S, S3, ж) (/> = 1, 2).
Since J /*(a) m(dx) = f f(x) m(dx), (1) may be rewritten as
s	s
/* (%) = a constant ж-а.е. for every / £ Lp (S, S3, m) (/> = 1,2). (1')
We shall give three different interpretations of the ergodic hypothesis
(1) and (1').
1.	Let Xb(x) be the defining function of the set В € 33- Then, for
any two sets Br,	the time average of the probability that the points
of Br will be transferred into the set B2 after k units of time is equal to the
product miBp m{B^. That is, we have
w(B2).	(2)
Proof. If /, g G L2 (S, SB, ж), then the strong convergence и-1 JS Tft/—>/*
in L2(S, S3, ж) implies, by (1'),
= (f*>g) =f*(
П—*OO	A = 1	5
= ff(x)m(dx)- fg(x)m(d%) for ж-а.е. %.
s ‘ s
By taking f(x) = Хяг(х)> g(x) = Xb^x), we obtain (2).
x) J g(x) m(dx)
lim «-1 £{Thf, g)
390
XIII. Ergodic Theory and Diffusion Theory
Remark. Since linear combinations of the defining functions %b(x)
are dense in the space L?(S, S3, m) for p — 1 and ф = 2, we easily see
that (2) is equivalent to the ergodic hypothesis (1'). (2) says that every
part of S will be transferred uniformly into every part of S in the sense
of the time average.
2.	The mean ergodic theorem in Chapter VIII, 3 says that the mapp-
ing / —> T*/ = /* gives the eigenspace of belonging to the eigenvalue
1 of T1; this eigenspace is given by the range R(T*). Hence the ergodic
hypothesis (1)' means precisely the hypothesis that R(T*) is of one-
dimension. Thus the ergodic hypothesis of E) is interpreted in
terms of the spectrum of the operator Тг.
3.	The Markov process P{t,x, E) is said to be metrically transitive
or indecomposable if the following conditions are satisfied:
S cannot be decomposed into the sum of disjoint sets
Bp B26 93 such that m(Bp) > 0, w«(S2) > 0
and P(l, x, E) = 0 for every x E Bir E C Bj with ip j.	(3)
Proof. Suppose P (t, x, E) is ergodic and that 5 is decomposed as in
(3). The defining function (%) satisfies	by the condition
fP{l,x, dy) %Bi(y)=P(l, x, BJ = 1 = %B1(X)> for
5
= 0 = ZbiW> for xtBi-
But, by ж(Вг) m(B2) > 0, the function %Bl(x) = /вДя) cannot be equal
to a constant wi-a.e.
Next suppose that the process P (t, x, E) is indecomposable. Let T\f = f
and we shall show that /*(%) = f{x) equals a constant m-a.e. Since
Тг maps real-valued functions into real-vaiued functions, we may assume
that the function f(x) is real-valued. If f(x) is not equal to a constant
?n-a.e., then there exists a constant a such that both the sets
Br = {$ C 5; f (s) > a} and B2 = {s € 5; / (s)	a}
are of m-measure > 0. Since TJf—tx) = f—a, the argument under The Angle
Variable of p. 391 implies that (/—«)+ = (f — oc)+, Г, (/ a)-- = (/—a)~.
Hence we would obtain P (1, x, E) 0 if x 6 Bit E C Bj (i y).
Remark. The notion of the metric transitivity was introduced by
G. D. Birkhoff-P. A. Smith [2] for the case of an equi-measure trans-
formation x -> yt (x) of S onto S.
An Example of the Ergodic Equi-measure Transformation. Let S be a
torus. That is, S is the set of all pairs s = {x, y} of real numbers x,y such
that s = {%, y} and s' — {x'} y'} are identified iff x = x’ (mod 1) and
у = y' (mod 1); S is topologized by the real number topology of its
coordinates x, y. We consider a mapping
s = {%, y} -> Tts = st = {% + toe, у + ip}
3. The Ergodic Hypothesis and the H-theorem
391
of S onto S which leaves invariant the measure dx dy on S. Here we
assume that the real numbers oc, are integrally linearly independent
mod 1 in the following sense: if integers n, k satisfy noc 4- kft = 0
(mod 1), then n = k == 0. Then the mapping s -> Tts is ergodic.
Proof. Let /(s)££?(S) be invariant by Tlt that is, let /($) = /(sr)
dx dy-a..e. on S. We have to show that / (s) = /* (s) = a constant dx dy-
a.e. on S. Let us consider the Fourier coefficients of /(s) and /(sj =
/(7\$), respectively.
i i
// /(s) exp (—2лл(А1 x + Aay)) dxdy,
о о
1 i
f J f (Тгs) exp (— 27ii x + k2y)) dx dy.
о о
By TjS = {x 4- ос, у + /?} and the invariance of the measure dx dy by the
mapping s —> /p', the latter integral is equal to
i i
// /(«) exp(— %7ii(kYx 4- &2y)) ехр(2лг(й1а + k2$))dxdy.
о о
Hence, by the uniqueness of the corresponding Fourier coefficients of
/(s) and f(T\s), we must have
exp (2лг(А1а 4- Л2^)) = 1 whenever
i i
J J /(s) exp(— 27ii(kxx 4- A2y)) dx dy 0.
о о
Hence, by the hypothesis concerning oc, /5, we must have
i i
f f f(s) exp(—2ni(k1x + Л2у)) dx dy = 0 unless k± = k2 = 0.
о 0
Therefore / (s) = /* (s) must reduce to a constant dx dy-a.e.
The Angle Variable. Let Tt be defined by the Markov process
P (t, x, E) with an invariant measure m on (S, S3, m) such that m (S) = 1.
Let /(s) £ £2(S, 39, m) be the eigenvector of Тг pertaining to the eigen-
value z of absolute value 1 of 7\:
T,f = i.f, |Л| = 1.
Then we have Тг [ / ] = |/1. For, we have, by the positivity of the operator
л, (Л|/|)ИЬ1(Л/)И1 = |/И|. that is, fP(l,x,dy'l\f(y)\^
s
|/(%) [. In this inequality, we must have the equality for m-a.e. x, as may
be seen by integrating both sides with respect to m (dx) and remembering
the invariance of the measure m. Hence, if P(i, x, E) is ergodic, then
/(x) must reduce to a constant wz-a.e. Therefore, if we put
f(x) — \f(x) | exp (z0(x)), 0 < &(x) < 2л,
392
XIII. Ergodic Theory and Diffusion Theory
we must have
Tj. exp («& (x)) = A exp (i & (%)).
In case A 1, such 0(x) is called an angle variable of the ergodic Markov
process P(t, x, E).
The Mixing Hypothesis. Let Tt be defined by a Markov process
P(t, x, E) with an invariant measure m on (S, m) such that m(S) = 1.
A stronger condition than the ergodic hypothesis of P(t, x, E) is given by
lim (7\ f,g) = (f*, g) = f f(x) m(dx) • f g(x) m(dx)
for every pair {/, g} of vectors £ L2(S, 93, m).	(4)
This condition is called the mixing hypothesis of the Markov process
P(t, x, E). As in the case of the ergodic hypothesis, it may be interpreted
as follows: every part of S will be transferred into every part of S uniformly
in the long run. As for the examples of the mixing equi-measure transfor-
mation x—> yt (%), we refer the reader to the above cited book by E. Hopf.
H-theorem. Let P (t, x, E) be a Markov process with an invariant
measure m on (S, S3, m) such that m(S) = 1. Let us consider the function
H(z) = —z log z, z > 0.	(5)
We can prove
Theorem (K. Yosida [17]). Let a non-negative function f(x) G
L1(5, 93, m) belong to the Zygmund class, that is, let us assume that
f /(%) log4- f(x) m(dx) < oo, where log+ ] = log |z | or =0 according
s
as |z|	1 or |z] < 1. Then we have
/Я(/(х)) ».(&)£ / H ((T, f) (*)) m (dx).	(6)
s	s
Proof. Since H (z) = — z log z satisfies H" (2) = — 1/z < 0 for z > 0,
H (z) is a concave function. Hence we obtain
the weighted mean of H (/(ж)) H (the weighted mean of /(%)),
and so
/P(t, x, dy) H(J{y)’)	P(t,x, dy) f{y)y
Integrating with respect to m (dx) and remembering the invariance of the
measure m(E), we obtain (6).
Remark. By virtue of the semi-group property Ti+s = TtTs and
(6), we easily obtain
У H((TtJ) (x)) m (dx)^ f H((TtJ) (x)) m(dx) whenever < t2.	(6')
s	s
This may be considered as an analogue of the classical H-theorem in
statistical mechanics.
4. The Ergodic Decomposition of a Markov Process	393
4.	The Ergodic Decomposition of a Markov Process
with a Locally Compact Phase Space
Let S be a separable metric space whose bounded closed sets are
compact. Let S3 be the set of all Baire subsets of S, and consider a Markov
process P(t, x, E) on (S, 93). We assume that
/	(x) G Cg (S) implies ft (x) = f P (t, x, dy) f (y) £ Cg (S).	(1)
s
The purpose of this section is to give the decomposition of S into ergodic
Parts and dissipative part as an extension of the Krylov-Bogolioubov case
of the deterministic, reversible transition process in a compact metric
spaceS (see N. Krylov-N. Bogolioubov [1]). The possibility of such
an extension, to the case where S is a compact metric space and with
condition (1), was observed by K. Yosida [17] and carried out by
N. Beboutov [1], independently of Yosida. The extension to the case
of a locally compact space S was given in K. Yosida [19]. We shall
follow the last cited paper.
Lemma 1. Let a linear functional L [f) on the normed linear space
Cg(S), normed by the maximal norm [|/)| = sup \f(x) |, be non-negative,
viz., L (/) > 0 when f(x) > 0 on S. Then L (f) is represented by a uniquely
determined regular measure <p(E), which is a-additive and > 0 for Baire
subsets E of S, as follows:
£(/) = f/W р(Л) for all /€ Cg(S).	(2)
5
Here the regularity of the measure tp means that
tp (E) = inf (p (G) when G ranges over all open sets 2 E.	(3)
Proof. We refer the reader to P. R. Halmos [1].
We shall call a non -negative, o-additive regular measure tp (E) defined
on 93 such that <p (S)	1 an invariant measure for the Markov process if
<p{E) — J<p(dx) P(t, x, E) for all t > 0 and EG 93.	(4)
s
We have then
Lemma 2. For any /G Gg (S) and for any invariant measure <p{E),
и-1	Д (x) = /♦ (x)	(x) = / P (k, x, dy) f (y) j exists (5)
(p-а.е. and
/ /* (x) ?	= J f(x)<P (dx) •	(6)
394
XIII. Ergodic Theory and Diffusion Theory
This is exactly a corollary of Theorem 6 in the preceding section.
Now, let -J, 2f, 3/, ... be dense in the normed linear space (S). The
existence of such a sequence is guaranteed by the hypothesis concer-
ning the space S. Applying Lemma 2 to J, d, . and summing up the
exceptional sets of ^-measure zero, we see that there exists a set IV of 9-
measure zero with the property:
ft
for any xfN and for any Cq(S), lim w-1 £ fk(x) = /*(%) exists.(7)
ft—ЮО	6 = 1
A Baire set S' C S is said to be of maximal 'probability if <p (S — S') = 0
for every invariant measure tp. Thus the set S' of all x for which /♦ (x) =
ft
lim к-1 £ Д (%) exists for every t e c° (S) is of maximal probability.
Я^-ЮО 6=1
Hence if there exists an invariant measure with <p (S) > 0, then there
exist a g £ C° (S) and a point x0 such that
___	ft
g** (*o) = lim gk Ы > 0 	(8)
Я--Н5О fc=l
For, if otherwise, we would have /* (%) = 0 on S for all / C (S) and
hence, by (6), f /(x) <p(dx} = 0.
s
Let, conversely, (8) be satisfied for a certain g£ Cq(S) and for a cer-
tain x0. Let a subsequence {«'} of natural numbers be chosen in such a
ft'
way that lim (w')-1	= g**(xo)- By a diagonal method, we
>OO
ft"
may choose a subsequence {и"} of {«'} such that lim
л—>oo	6=1 J
exists for / = 1, 2, . . . By the denseness of {3/} in Cq(S), we easily see
that
ft"
lim («"A1 £ /A(#o) = /*** (%o) exists for every / 6	(S).
n—ЮО	6=1
If we put /*** (%0) = L%a(/), then
V) = LXB (A).	(9)
For, by condition (1), we have Д 6 Cg(s/and so
ft"
£,.(/) = lim (яТПшй =/Г*(х0) =Z„(/i).
ft —>O0	6=1
By Lemma 1, there exists a regular measure (px<i{E) such that
£J/)=/’**W=	(10>
s
Surely we have
0£n(E)£l,	(11)
4. The Ergodic Decomposition of a Markov Process
395
and, by (9),
f t (*) 94 {dx) = Ц f P(l,x,dy) f {y)\ <px*(dx).
By letting / (%) tend to the defining function of the Baire set E, we see
that <рХв{Е) is an invariant measure. We have <pXa(S) > 0, since, by (8)
and (10), we have LXB(g) = g***(x0) = f g(x) tpx(dx) > 0. We have
5
thus proved
Theorem 1.	A necessary and sufficient condition for the non-exist-
ence of non-trivial invariant measures is
lim w-1 £ fk(x) = /* (%) = 0 on S for any /G Cq(S) .	(12)
Я^ОО Л=1
Definition. We will call the process P(t, x,E) on (S, 93) dissipative
if condition (12) is satisfied.
Example. Let S be the half line (0, oo) and 93 the set of all Baire
sets of (0, oo). Then the Markov process P{t, x, E) defined by
P (t, x, E) = 1 if (x + £) G E and = 0 if (x -j- t) G E
is dissipative.
We shall assume that P (t, x, E) is not dissipative. Let D denote the set
n
< x 6 5; /*(%) = lim n-1 £ Д (%) =0 for all / G Cq (-5) ’ 
W“>0O	6 = 1
Since it is equal to
{* G S; (,/)* (%) = 0 for ? = 1, 2, . .
D is a Baire set. We shall call D the dissipative part of S. We can prove
below that <p(D) = 0 for any invariant measure <p, and so, since the
process P {t, x, E) is assumed to be not dissipative, So = S — D is not
void. We already know that there exists a Baire set 5г C So with the
property:
to any x G Slf there corresponds a non-trivial invariant measure
cpx{E) such that /*(%) = lim я-1 £ fh(x) = f f(z) <px(dz)
n">cc й = ]	s
and <p(S0— Sx) = 0 for any invariant measure <p.	(13)
For any invariant measure (p, we have (6) and hence, by (13),
/ f{y)<p{dy) = f ( f /(2) <Py (dz)\tp(dy).
S	Sr \s	J
Thus we obtain
<?(£)= /«M#) <p{dy),	(14)
5,
and so we have proved
396	XIII. Ergodic Theory and Diffusion Theory
Theorem 2.	Any invariant measure <p may be obtained as a convex
combination of the invariant measure <py (E) with у as a parameter.
We see, from (11) and (14), that the set
S2 =	5;	<ря(51) = 1}
is of maximal probability. Hence So is of maximal probability.
For any / 6 Cq (S) and for any invariant measure <p, we have
= 0,	(15)
s. \s,	/
since the left hand side is equal to
/ W ( f f* (у)2 9^ W) — 2 f /* (*) 4> (<**) (ft* (у) <P* №)]
\$i	/	Sj	\Sf	/
+ f /* W2 9? GM  f 9^ GM
S,	s,
= f /* (y)2 <P GM - 2 f /* (%)2 <P (dx) + f /* (%)2 (dx),
S>	Si	Si
by (13), (14), (6) and the definition of S3. By applying (15) to rf,
we see that the set
S, = J*eS,; f (/*(y)-/*«)a?,(<fy)=O for all /€Cg(S)l
I s»
is of maximal probability.
We shall give the ergodic decomposition of S. For any x £ S3t put
E. = {yeSe-, /‘(y) = /*(») for all /eCg(S)}.	(16) 
We can then show that each Ex contains a set Ex with the property:
<pAEx) =<px(Ex) and ^(^У, Дг) = 1 for any y^Ex. (17)
Proof. By the definition of S3, we see that /* (y) = /* (x) if the measure
<px (E) has variation at the point y. Hence cpx(Ex) = <px(Ss) = 1- Thus, by
the invariance of the measure we obtain
l=^(£z)= /P(l, z, Ex) <px(dz) = J P(l,z,Ex)<px(dz).
Sa	j	&X
Since 0 < P(l, z, Ex) 1, there exists a Baire set E1 C Ex such that
(p^E1) = (px(Ex) and z£ E1 implies P(l, z, Ex) ~ 1.
Put
E2 ^{zeE\P(l,z, E1) = 1}.
Since
f P(l,z, E1) <px (dz) = <px (E1) = (px (Ex) = J P (1, z, Ex} <px (dz),
(18)
4. The Ergodic Decomposition of a Markov Process	397
we must have
J (P (1, z, E„) - p (1, z, E1)) <px (<Zz) = 0.
As z£E2 implies P(l, z, E1) = 1 by the definition of E\ we obtain
Next put
E3 = {z€E2;P(M, £2) = 1}.
Then, as above, д?ж(Е3) = «рДЕД In this way, we obtain a sequence {En}
such that
E, 2 El 2 E2 2  • , рДЕ”) = p,(E,) and
P(l, z, E") = 1 if zeE”+* (ИЙ О, 1,E° = E„).
л oo
We have thus obtained the set Ex = П En with the demanded property.
»=1
Therefore, we have
Theorem 3. P{t, у, E) defines a Markov process in each Ex which is
ergodic, in the sense that
Ex is not decomposable in two parts A and В such
that <p (A)	> 0 for a certain invariant
measure cp and P(l,a,B) = 0 when a^A and
P(l,b, A) = 0 when b^B,
Proof. Let <p be any invariant measure in Ex with <p(E^ = 1. Then,
by (14), cp(E} = J(p[dz) y, (E) whenever E £ Ex. Since <p*(E) = tpx(E)
Ex
for every z £ Ex by the definition of Ex we have
<P (£) = <Px (£) / 4>	(£) •
Ex
Thus there is essentially only one invariant measure tpx(E) in Ex.
This proves the ergodicity. For, let Ex be decomposed as in (18). Then,
by the invariance of <p,
(p(C) = f P(l, z, C) (p(dz) whenever C Q Ex.
Ex
Thus the measure ip defined by
¥>(C) =<p(C)l<p(A) if CQA
= 0 ii C QB
is invariant in Ex and differs from the unique invariant measure <p.
398
XIII. Ergodic Theory and Diffusion Theory
5. The Brownian Motion on a Homogeneous Riemannian Space
There is an interesting interplay between Markov processes and
differential equations. Already in the early thirties, A. Kolmogorov
proved, under a certain regularity hypothesis concerning the transition
probability P(i, x, E), that
u(t,x) = f P (t, x, dy) f (y)
s
satisfies the equation of the diffusion type\
= bij (x)  82u- + a^x) —- = Au, t> 0,	(1>
where the differential operator A is elliptic in the local coordinates
(%1; x2, . . ., x„) of the point x of the phase space S. Following Einstein’s
convention of the tensor notation, it is understood that, e.g., ab(x) d/Sxi
ft
means £ al(x) д/дхг. For the derivation of equation (1), see A. Kolmo-
1 = 1
gorov [1]. The leit-motif in his research was the investigation of local
characteristics of Markov processes.
Following work done by E. Hille at Scandinavian Congress in 1949
(Cf. E. Hille [9]) and K. Yosida [29] in 1948, W. Feller [2] began in
1952 a systematic research of this new field in probability theory using
the analytical theory of semi-groups. His investigations were further deve-
loped by E. B. Dynkin [1], [2], K. Ito-H. McKean [1], D. Ray, [1]
G. A. Hunt [1], A. A. Yushkevitch [1], G. Maruyama [1] and many
younger scholars, especially in Japan, USSR and USA. These investiga-
tions are called the theory of diffusion. A comprehensive treatise on the
diffusion theory by K. Ito-H. McKean [1] is in the course of printing1.
We shall sketch some of the salient analytical features of the theory.
Let 5 be a locally compact space such that S3 is the totality of the
Baire sets in S. To define the spatial homogeneity of a Markov process
P(t, x, E) on the space S, we suppose that S is an тг-dimensional, orien-
table connected C°° Riemannian sjjace such that the full group (SJ of the
isometries of S, which is a Lie group, is transitive on S. That is, for each
pair {.X, y} of points £ S, there exists an isometry M £ ® such that
M • x = y. Then the process P (t, x, E) is spatially homogeneous if
P (t, x, E) = P(t, M - x, M • E) for each x £ S, E € 53 and M 6 ®. (2)-
A temporally and spatially homogeneous Markov process on S is
called a Brownian motion on S, if the following condition, known as the
continuity condition of Lindeberg s type, is satisfied:
lim P1 J" P (t, x, dy) = 0 for every £ > 0 and x £ 5.	(3)
dis(x,y)>e
1 Berlin/Gottingen/Heidelberg: Springer.
5. The Brownian. Motion on a Homogeneous Riemannian Space 399
Proposition. Let C(S) be the В-space of all bounded uniformly
continuous real-valued functions / (x) on S normed by 11 /11 = sup | / (x) .
Define
(Г,/) (ж) =	if Z>0,
s
= /(x), if t = 0.
(4)
Then {Tj constitutes a contraction semi-group of class (Co) in C (S).
Proof. By P(t, x, E) 0 and P(t, x, S') — 1, we obtain
\(Tt /) (x) | sup |/(y)
у
and the positivity of the operator Tt. The semi-group property Tt+s = Tt Ts
is a consequence from the Chapman-Kolmogorov equation. If we define a
linear operator Mr by (M' f) (x) = f(M  x), M £ ®, we obtain
TtM' = M'Tt, 0.	(5)
For,
(ЛГ T, f) («) = (Z, /) (M  %) = f P (t, M • x, dy) t ly)
5
= fp(t, M  x,d(M  y)) f(M • y)
= {P(t,x, dy) f(M  y) = (TtM'f) (%).
5
If M E ® be such that M  x = x', we have
(T, D (x) -(T,f) M = (T, f) (x) - (M' T, f) (x) = T,(f - M'f) (x).
Hence by the uniform continuity of /(x), we see that (Ttf) (x) is uniformly
continuous and bounded.
To prove the strong continuity in t of Ttf it suffices by the
Theorem in Chapter IX, 1, to verify weak right continuity of Tt at t= 0.
Therefore, it is surely enough to show that lim (P/) (x) = /(x) uniformly
t 0
in x. Now
|(Г,/)(х)-/(х)|=|/P(t.x,dy) [/M-/W]
I s
/ P(t,x,dy) lf(y)-/{x))
d(xty) ge
f P[t,x, dy) [/(y) — /(x)]
/ P(t,x,dy)	+2
II/Ц / P(t,x,dy}-
The first term on the right tends to zero uniformly in x as e 0 and, for fixed
e > 0, the second term on the right tends to zero uniformly in x as t j. 0.
The latter fact is implied by (3) and the spatial homogeneity. Thus
lim (Ttf)[x) = /(x) uniformly in x.
qo
400	XIII. Ergodic Theory and Diffusion Theory
Theorem. Let x0 be a fixed point of S. Let us assume that the group of
isotropy ®0 = {M 6 G\ M - x0 = л'о} is compact. Being a closed subgroup
of a Lie group ®, ®0 is a Lie group by a theorem due to E. Cartan. Let A
be the infinitesimal generator of the semi-group Tt. Then we have the
following results.
[1]. If f C D (A) Cl C2 (S), then, for a coordinate system (x$,	. . .,
at xQ,
(6)
where
(x0) = lim Г1 J (xi — %*) P (t, x0, dx},	(7)
'I0
(%0) = lim Г1 / (x* — 4) (V — x30) P (L xQ, dx),	(8)
the limits existing independently of sufficiently small e > 0.
[2]. The setD(A) П CZ(S) is “big" in the sense that, for any C°°(S)
function h(x) with compact support there exists an f(x) € D(A) П C2(S)
such that Sf/dx^, д^Цдх^дх^ are arbitrarily near to h{x^, dhfdxQ,
d2h/8xQ 8xi, respectively.
Proof. The first step. Let h (%) be a C00 function with compact support.
If /С T)(A), then the “convolution”
(/®Л)(х)= f l(My-x)h(M,-x„)dy	(9)
(My denotes a generic element of ® and dy a fixed right invariant Haar
measure on ® such that dy = d (y • M) for every M £ &) is C°° and
belongs to D(A). The above integral exists since the isotropy group @0 is
compact and h has a compact support. By the uniform continuity of f and
the compactness of the support of h we can approximate the integral by
k
Riemann sums £ f (Myt • x) C, uiyforrnly in x:
(/ ® A) (x) = s-lim 2 f (My - x)
Since Tt M' = M'l\, M' commutes with A, i.e., if fCD(A), then
M'fcD(A) and AM'f = M'Af. Putting g (x) = (A f) (x), we obtain
g € C (S) and
(A	\ Л	k
2:f(MVi-x) CA= 2(AM'yi.f)(x) Ci= £(М'у<А!)(х)С<
»=1	/	1—1	1=1
k
= lg{Mn.X) cit
5. The Brownian Motion on a Homogeneous Riemannian Space 401
and the right hand side tends to (g ® h) (%) = (Л / ® h) (%). Since the
infinitesimal generator A is closed, it follows that f®h£D(A) and
A (f ® h) - Af ® h. Since S is a homogeneous space of the Lie group (JJ,
i.e., S = ®/®0. we can fiHd a coordinate neighbourhood U of x0 such
that for each x C U there exists an element M = M (x) 6 ® satisfying the
conditions:
i) M (x)  x = x0,
ii) M (x) x0 depends analytically on the coordinates (x^x2, . . ., xn)
of the point x C S.
This is so, since the set {My 6 My  x — xQ} forms an analytic sub-
manifold of ($; it is one of the cosets of ® with respect to the Lie subgroup
(SJ0. Hence, by the right invariance of dy, we have
(/ ® h) (x) = f / (My M (x) x) h (My M (x) Xq) dy
G
=	x0) h (M, M (x)  xa) dy.
G
The right side is infinitely differentiable in the vicinity of and
(/® A) (x) = j f(M,- x0}D’,h(MyM(x}  x^dy. (10)
G
The second step. Remembering that D (Л) is dense in C (.S') and choosing
/ and h appropriately, we obtain:
there exist C°° functions F1 (x), F2 (x), . . ., Fn (x) € D (A) such
,	, T ,. д(РЦх)............F"(x)) л f
that the Jacobian -y !-------- .> 0 at x0,
and, moreover
there exists a C°° function Fo (x) £ D (Л) such that
« <12>
We can use L'1 (x), F2 (x), . . ., F* (x) as coordinate functions in a neigh-
bourhood {x; dis(x0, x) < f} of the point xQ. We denote these new local
coordinates by (x1, x2, . . .,%”). Since F^(x) £ D<A}, we have
lim r> f P(t, x0, dx) (P(x) - P(x0)) = (A F) (x0)
f Ю s
= lim Г1 J P(t,xa, dx) (F (x) — F> (x0))
HO dis(x,x0)
independently of sufficiently small 8 > 0, by virtue of Lindeberg's
condition (3). Hence there exists, for the coordinate function x\ %2,... ,xn
= F?), a finite
limt-1 f (xJ ~ x^) P(t, x0, dx) — a? (x0)	(13)
HO dis(^^0)^e
26 Yosida, Functional Analysis
402
XIII. Ergodic Theory and Diffusion Theory
which is independent of sufficiently small e > 0. Since Fo £ D(A}, we
have, again using Lindeberg's condition (3),
(Л Fo) (x) = lim t-1 f P (t, x0, dx) (Fo (x) — Fo (x0))
= lim/-i f P(t,xls,dx)(F0(x)^Fls(xl,))
= lim
Go

dis(*Oj*J

ж=Хф4-б(ж—*0)

8F
where 0< 6 < 1. The first term on the right has a limit a*(x0)-z-f
Hence, by the positivity of P(t, x, E} and (12), we see that
lim Z‘1 f £ (xl — Xq)z P (t, xa, dx} < oo.	(14)
*4° disCw) < s *=1
The third step. Let /G D(A} П C2. Then, by expanding /(%) — /(x0),
we obtain
—,/>(*7~ /Ы = z_, J	p(t ix}
s
= t 1 f (/ W — / (*o)) P xo> dx}
dis(x,x0) >e
+ r1 J (%* - ^) P (Z, хй, dx}
dis(jr,xq)^e	0
+ 1-1	/	(?-x‘)(?-*’)
d2f
+ Z 1 J (x1 —4)(xJ —4) С^(е) P(Z,x0,<Zx)
distal)
= C^Z, e) + C2{t, e) C3(Z, s) + CJp, s), say,
where C^(e}	0 as e 0. We know, by (3), that lim e) = 0 for
fixed £ > 0, and that lim C2{t, e) = af(x0)	, independently of suffi-
t I 0
ciently small e > 0. By (14) and Schwarz' inequality, we see that
lim C4 (t, e) = 0, boundedly in t > 0. The left side also has a finite Emit
(A f) (x0} as 11 0. So the difference
lim C3 (Z, e) — lim C3 (Z, s)
'4°	f|0
can be made arbitrarily small by taking e > 0 sufficiently small. But,
by (14), Schwarz' inequality and (3), the difference is independent of
6. The Generalized Laplacian of W. Feller
403
sufficiently small £ > 0. Thus a finite limit lim C3 (t, s} exists independent-
ly of sufficiently small £ > 0. Since we may choose F £D(A} (*1 C00 (S) in
such a way that 82Е/8х*0 8х& (i, 7 = 1, 2, . . ., n) is arbitrarily near
where «у are arbitrarily given constants, it follows, by an argument
similar to the above, that
a finite lim tr1 f (x1 — Xq) (x3 — x$ P(i, x0, dx} =	(x0) exists (15)
GO dis(«,x,)£e
and
limC3(Z,£) = 6«W^.
This completes the proof of our Theorem.
Remark. The above Theorem and proof are adapted from K. Yosida
[20]. It is to be noted that 6*J (x) = 5Jt(x) and
5°’(*o) tify 0 for everY real vector (£x, £2, . . ., £„),	(16)
because (ж* — Xq) (x3 — x$ = I (%* — Хц) &) .
\i=l	/
The Brownian Motion on the Surface of the 3-sphere. In the special
case when S is the surface of the 3-sphere S3 and is the group of rota-
tions of S3, the infinitesimal generator A of the semi-group induced by a
Brownian motion on S is proved to be of the form
A = С Л , where C is a positive constant and Л is the Laplacian
1 8	8	1 82	(I?)
= -—„ --- sin fl , д -z-z on the surface of S3.
sin 0 8q 88 ' sin2 в 8<p2
Thus there exists essentially one Brownian motion on the surface of S3.
For the detail, see K. Yosida [27].
6. The Generalized Laplacian of W. Feller
Let S be an open interval fo, r2}, finite or infinite, on the real line,
and SB be the set of all Baire sets in (r1; r2). Consider a Markov process
P(t, x, E) on (S, SB) satisfying Lindeberg’s condition
lim/1 f P[t,x,dy}~0 for every x£ fo, r2) and £>0. (1)
GO |x_ >B
Let C [rlt r2] be the В-space of real-valued, bounded uniformly continuous
functions f(x) defined in fo, r2) and normed by ||/|| = sup J/(x) [. Then
z
(T, f) W = f Ptf.x. dy) /fy) (I >0) ;=f (x) (Z — 0)	(2)
defines a positive contraction semi-group on C fo, r2]. We prove by (1) the
weak right continuity of Tt at t = 0 so that Tt is of class (Co) by the
Theorem in Chapter IX, 1.
26*
404
XIII. Ergodic Theory and Diffusion Theory
Concerning the infinitesimal generator A of the semi-group T,, we
have the following two theorems due to W. Feller.
Theorem 1.
A  1 = 0;	(3)
A is of local character, in the sense that, if f^D(A)
vanishes in a neighbourhood of a point x0 then (4)
(Af) (x0) = 0;
If f £ D (A) has a local maximum at xn, then
(5)
(X/)(xo)g0.	1 Z
Proof. (3) is clear from Tt  1 = 1. We have, by (1),
(Л f) (хй) = lim 1-г f P (t, x0, dy) (J (y) — f (x0))
independently of sufficiently small e > 0. Hence, by P (t, x, E) 0, we
easily obtain (4) and (5).
Theorem 2. Let a linear operator A, defined on a linear subspace
Z)(A) of C [rlt r2) into C[rJtr2], satisfy conditions (3), (4) and (5), Let
us assume that A does not degenerate in the sense that the following
two conditions are satisfied:
there exists at least one /0EZ)(A) such that (A /0) (x)
(6)
> 0 for all % C (гг, r2),
there exists a solution v of A  v = 0 which is linearly (7)
independent of the function 1 in every subinterval
(%r, x2) of the interval (rx, r2).
Then there exists a strictly increasing continuous solution s = s(x) of
A  s = 0 in (r-p r2) such that, if we define a strictly increasing function
m = m(x), not necessarily continuous nor bounded in (rlf r2), by
X
(8)
we obtain the representation:
(Л /) (x) = Z)+ £>+ f(x) in (rx, r2) for any f E D (A).	(9)
Here the right derivative Dp with respect to a strictly increasing function
p = p(x) is defined by

at the point of continuity of p,
+ 0) —	0)
p(x + 0) — p(x — 0)
(10)
at the point of discontinuity of p.
6. The Generalized Laplacian of W. Feller
405
Proof. The first step. Let и £ D (A) satisfy A  и = 0 and и (xj = и (x2),
where r± < xx < x2 < r2. Then и (x) must reduce to a constant in {xr x2).
If otherwise, u(x) would, for example, assume a maximum at an
interior point x0 of (xlt x2) in such a way that и (x0) — s и (xx) = w (x2)
with some £ > 0. Let /0 be the function E T)(A) given by condition (6).
Then, for a sufficiently large 5 > 0, the function F (x) = /0(х) + 5и(х)
satisfies F(x0) > F(x*) (i = 1, 2). Hence the maximum of F(x) in
[xv x2] is attained at an interior point x'Q although (A F) (x'o) = (A f0) (x(j) >0,
This is a contradiction to condition (5).
The second step. By (7) and the first step, we see that there exists a
strictly increasing continuous solution s(x) of A  s = 0. We reparametrize
the interval by s so that we may assume that
the functions 1 and x are both solutions of A  / — 0.	(11)
We can then prove that
if (A h) (x) > 0 for all x in a subinterval (xv x2) of the
interval (rlt r2), then h(x) is convex downwards in (12)
(Xj, x2).
For, the function w(x) = h(x)—ax— ft, which satisfies (A«) (x) =
(A h) (x) > 0 for all x in (xx, x2), can have no local maximum at an interior
point of {xp x2).
The third step. Let /€ В (A). Then, by (12) and (6), we see that, for
sufficiently large 3 > 0, the two functions f-^x) = /(x) + <5/0 (x) and
/2(x) = <5/0(x) are both convex downwards. Being the difference of two
convex functions, / (x) = /x (x) — /2 (x) has a right derivative at every
point x of (/j, r2). Let us put
A • / = q>, A  /о = ?0, D+ /0(x) = /z(x).	(13)
For gf = f — tfQ, we have A  gt = A f — tA  /0 = tp — t(p0. Hence, by
(12),
t < min 9>(«)/(p0(x) implies that gt{t] is convex downwards 1
in (xlt x2) and so Dfpgt(x) is increasing in (xx, x%) 
Applying the same argument for t > max <p (x)/<p0 (x), we obtain:
for any sub-interval (xp x2) of(r1,Xa)>we have
min Жх(0,+/оЫ-В,+/«Ы)£й,+/Ы-0.+/Ы .
XjCtfCXa Wo W
— max x (^+/оЫ~^7о^1))-
406	XIII. Ergodic Theory and Diffusion Theory
The continuity of the function 9? (x)/^ (x) implies, by the above inequality,
that
Dtf(x2)~D}f(x,) = f
which is precisely the integrated version of A  f = D* Df~ /.
Remark 1. We may consider the operator D* Z)s+ as a generalized
Laplacian in the sense that Ds+ and correspond to the generalized
gradiant and the generalized divergence in one dimension, respectively.
Feller called s = s(x) the canonical scale and m = m(x) the canonical
measure of the Markov process in question. We easily see, from the first
step above, that the function 1 and s constitute a fundamental system of
linearly independent solutions of A • v = 0 to the effect that any solution
of A  v = 0 can be expressed uniquely as a linear combination of 1 and
s(x). Thus the canonical scale of P(t, x, E) is determined up to a linear
transformation, i.e., another canonical scale .q must be of the form
= as + /? with л ?> 0; hence the canonical measure mY which corre-
sponds to must be of the form mt = oc^m.
Remark 2. Theorem 2 gives the representation of the infinitesimal
generator A at the interior point x of r2). To determine the operator
A as the infinitesimal generator of a positive contraction semi-group Tt
of class (Co) on C r2] into C [rlt r2], we must consider the lateral
condition, that is, the boundary condition of the operator A at both
boundary points r, and r2 in order to describe the domain D(A) of A
concretely and completely. According to Feller [2] and [6] (cf. E. Hille
[6]), the boundary points (or r2) are classified into the regular-boundary,
the exit-boundary, the entrance-boundary and the natural-boundary. To
this purpose, we introduce four quantities:
Oj =	Jf dm(x)ds(y), p-i=	fj	ds (%) dm (y),
4	r1<y<x<r;
CT2 =	//	dm(x}ds(y}, /z2=	JJ	ds (x) dm(y).
r,>y>z>r'	r,>y>x>rJt
The boundary point rt (i = 1, 2) is called
regular in case < oo, < oo
exit in case crt < co, = oo
entrance in case cr, сю, м, < oo
natural in case p =oo, .oc
(the conditions are
 independent of the
choice of and rg).
We shall illustrate by simple examples.
6. The Generalized Laplacian of W. Feller	407
Example 1.	= d2/dx2, S = (—oc, oc). We can take s = x,
m = x. Hence
co
ff2 = ff dxdy = f (y — r2) dy = oo,
0O>y>^>ri	ri
fa = ff dxdy co,
оо>у>ж>г£
and so oo is a natural boundary. Similarly -oo is also a natural bound-
ary.
Example2.	— d^dx2, 5 = (—oo, 0). We can take s = x,
tn = x. In this case, - oo is a natural boundary and 0 is a regular boun-
dary.
d^ d
Example 3.	D~s x2 ,	, S = (0, oo). A strictly increasing
cJ/v (лгЛ
continuous solution s — s (x) of D+ Ds' s = 0 is given by s (x) = f e~1(t dt
so that
d%
2rl/x d evx
dx dx'
Therefore ds = e 1;T dx, dm = x 2 el/x dx. Hence
ct1 = ff x 2 ellx dx e llydy = f [— e1/x]y & livdy<oo,
о
i
J f e~lfx dx y~2 e1{y dy — f e~1/x [—^1/y]o dx — co.
о<у<*<1	о
Thus 0 is an exit boundary. Similarly we see
cr2 — ff e1/x x~2 dx e~1/y dy = f [— e1/x^ • e-1/y dy = oo,
0O>y>X>l	1
=	J J	e-1/* dx elly y~2 dy - f e~llx [—dx «= oo.
oo>y>x>l	1
That oo is a natural boundary.
d2 d
Example 4.	Z\+ = x2 + -j-, S = (0, 2). As above, we see that
ZZJf“ WJV
ds = e1!x dx, dm = x~2 e~1/x dx, and so we can verify that 0 is an entrance
boundary and 2 is a regular boundary.
Feller’s probabilistic interpretation of the above classification is as
follows:
The probability that a particle, located at first in the interior of the
open interval (г1л гг), will, after a finite lapse of time, reach a regular
boundary or an exit boundary, is positive; while, the particle can, after a
finite lapse of time, neither reach an entrance boundary nor a natural
boundary.
408
XIII. Ergodic Theory and Diffusion Theory
References
Feller's original paper was published in 1952: W. Feller [2]. The
proof of Theorem 2 given above is adapted from W. Feller [3] and [4],
who also delivered an inspiring address [5]. We also refer the readers
to the forthcoming book by K. Ito-H. McKean [1] and E. B.Dynkin [3],
and the references given in these books.
7.	An Extension of the Diffusion Operator
Let the possible state of a (not necessarily temporally homogeneous)
Markov process be represented by the point x of an w-dimensional C00
Riemannian space R. We denote by P(s, x, t, E), s the transition
probability that a state % at a time moment s is transferred into a Baire
set E £ R at a later time moment t.
We shall be concerned with the possible form of the operator As
defined by
(A J) (x) = lim Г1 f P(s,x,s 4 t, dy) (Ду)-- /(*)), /GCg(R), (1)
GO #
without assuming the Lindeberg type condition:
lim P1 f P(s, x, s 4 dy) ~ 0 for all positive constants e
(d(x, y) = the geodesic distance between two points x and y).	(2)
We can prove
Theorem. Let there exist an increasing sequence {k} of positive
integers such that, for a fixed pair {s, x},
lim k  J" P(s, x, s 4 fo-1, dy) — 0 uniformly in k,
k j j- -- P (s, x, s 4 k~~\ dy) is uniformly bounded in
R	Х,У	,
Suppose that, for a function /(x) € C‘t(R),
a finite lim A J P($, x, s 4 A-1, dy) Ду) — f (x) exists
a	«
k.
(3)
(4)
(«)
We have then, in any fixed local coordinates (xlf x2, . . ., x„) of the point
x£R,
(^4 J) (x) = (s> x) + by <4 x)
+ djy}^ lZ M ~ W “ 1 +	“ Xj}	(6)
^MG(S,x,.y).
7. An Extension of the Diffusion Operator
409
where
G (s, x, E) is non-negative and ст-additive for Baire sets
(7)
E Q R and G(s, x, R) < oo,
q(x, y) is continuous in x, у such that q(x, y) = 1 or
0 according as d (x, y) (5/2 or > <5 (5 is a fixed (8)
constant > 0),
the quadratic form b^s, x) is 0.	(9)
Remark. Formula (6) is given in K. Yosida [26]. It is an extension
of the diffusion operator of the form of the second order elliptic differen-
tial operator discussed in the preceding sections. The third term on the
right of (6) is the sum of infinitely many difference operators. The
appearence of such a term is due to the fact that we do not assume the
condition (2) of Lindeberg's type. Formula (6) is an operator-theo-
retical counterpart of theP. Levy-A. Khintchine-K. Ito infinitely divisible
law in probability theory. About this point, see E. Hille-Phillips [1],
p. 652.
Proof of Theorem. Consider a sequence of non-negative, cr-additive
measures given by
G*(s, «,£) = *	P(S,x,s + k~\dy).	(10)
E
We have, by (3) and (4),
GA(s, x, E) is uniformly bounded in E and k,	(11)
lim f Gk(s, x, dy) = 0 uniformly in k.	(12)
“too dlx,y}^a
Hence, for fixed {s, ж) ,the linear functional Lk given by
П(Й = jG^.x.dy'jgty), g<iC4,{R'l,
R
is non-negative and continuous on the normed linear space Cq(R) with
the norm 11 g 11 = sup | g (%) |; and the norm Lk is uniformly bounded in k.
X
Therefore, by the separability of the normed linear space Cq (7?), we
can choose a subsequence {A7} of {/?} such that lim Lk- (g) = L (g) exists
А—изо
as a non-negative linear functional on C'/j(A). By virtue of a lemma in
measure theory (P. R. Halmos [1]) there exists a non-negative, <r-additive
measure G{s, x, E) such that G(s, x, R) < oo and
lim J Gk (s, x, dy) g(y) = f G (s, x, dy) g (y) for all g C-Cg (R). (13)
ft = ft*—>0O	Д
410
XIII. Ergodic Theory and Diffusion Theory
We have г p
k J P(s,x,s + k-1, dy} f{y) — f(x)
= /{[/W	4(^5-}(* x- Ы
R
+	(14)
R
The term { } in the first integral on the right is, for sufficiently small
d(y, x),
(x?	*;) qXj + b>i xi) <Уу Xj) \3Xi8Xj d{y, x)2	’
where Xj = Xj + 0 (у, — Xj), 0 < 0 < 1.
Thus { } is bounded and continuous in y. Hence, by (12) and (13), the first
term on the right of (14) tends, as k = k' ->oc, to f { } G(s, x, dy).
Therefore, by (5),	R
a Hnite	G‘ <s’ X' = “i <S' X
R
8f
dx}
exists. Hence, by (3), we obtain (6) by taking
b{j (s, x) = lim lim A f (y^ — xj (yy — xy) P (s, x, s -f- A-1, dy).
e j. 0 h — k ~>°o diy,z} fi®
8.	Markov Processes and Potentials
Let{Tf} be an equi-continuous semi-group of class (Co) on a function
space X defined by (Ttf) (x) = f P(t, x, dy) f(y) where P(t, x, E) is a
s
Markov process on a measure space (S, S3). Let A be the infinitesimal
generator of Tt. Suggested by the special case where A is the Laplacian,
an element / 6 X is called harmonic if A • / = 0. Then / is harmonic
iff Л(Л/— Л)-1/= / for every^ л > 0. An element f^X is called a
'potential if / = lim {Л1 — A) 1 g with some g. Because, for such an element
л Ф 0
f, there holds by the closedness of A the Poisson equation
A • / = lim Я (Я/ — A)^g = lim{— g + Я(Я7 — ^)-1g} = —g.
A 0	A 0
Suppose that X is a vector lattice which is also a locally convex linear
topological space such that every monotone increasing bounded se-
quence of elements € X converges weakly to an element of X which is
greater than the elements of the sequence. We also assume that the
resolvent	--(/.I—A )-1 is positive in the sense that / 0 implies
The situation is suggested by the special case where A is the Laplacian
considered in a suitable function space.
9. Abstract Potential Operators and Semi-groups
411
We may thus call subharmonic those elements /£ X for which the
inequality A - / <7 0 holds. By virtue of the positivity of a sub-
harmonic element / satisfies Аf for all A > 0. We shall prove an
analogue of a well-known theorem of F. Riesz concerning ordinary sub-
harmonic functions (see T. Rado [1]) :
Theorem. Any subharmonic element % is decomposed as the sum of a
harmonic element xh and a potential xp, where the harmonic part xh oi
x is given by xh = lim AJ^x and xh is the least harmonic major ant of x in
the sense that any harmonic element xH x satisfies xH xh.
VTOQi (K. Yosida [4]). By the resolvent equation
7д-7, = 1«-А)Л7,.	(i)
we obtain
(/-Я7д) = (/ + 0‘-^7.)(/-/'7я)-	(2)
Since x is subharmonic, we see, by the positivity of JXt that
z>u implies А]лх^> fij ; Л > x.
con-
Hence the weak-limA/лх = xh exists by virtue of the hypothesis
A|0
cerning bounded monotone sequences in X. Therefore, by the ergodic
theorem in Chapter VIII, 4, we see that xh = lim Л J^x exists and xh is
harmonic, i.e., AJxxh = xh for all A > 0. We also have xp = (x - xA) =
lim (L — л/А) x = lim (— А) (А/ — A)-1 x = lim (Al — A)-1 (— A x)
А Ф 0	Л 0	Л Ф 0
which shows that xp is a potential.
Let a harmonic element xH satisfy xH x. Then, by the positivity of
AJX and the harmonic property of xH, we have
and hence xH limA/;.% = xh.
9.	Abstract Potential Operators and Semi-groups
Let {Tt; t 0} be an equi-continuous semi-group of class (Co) of
bounded linear operators on a Banach space X into X, and let A be its
infinitesimal generator. Thus D(A)a = X and the resolvent (Al — A)^1
of A exists for A > 0 as a bounded linear operator on X into X satisfying
the condition
sup [|Л(А I -	< oo	(1)
л> о
so that, by applying abelian ergodic thoerems in Chapter VIII, 4, we
obtain
R(A)a = R(A (^ I - A)-1)tt = {x С X; s - lim Л(Л I - A)-1 x = 0}
A A 0
for all fi > 0	(2)
412
XIII. Ergodic Theory and Diffusion Theory
and
D(A}a =	7 — A)-1)“ X;s - lim Л(Л J ~ A)~x x = x) = X
A -f OG
for all /z > 0 .	(3)
We shall be concerned with the situation in which
the inverse A^1 exists with its domain 7)(A~'1) = R (A) strongly
dense in X,	(4)
and in case (4) we shall call V = — A-1 as the “abstract potential
operator” associated with the semi-group Tt so that the “abstract
potential V x" satisfies the “abstract Poisson equation”
A V x = - x	(5)
with respect to the “abstract Laplacian operator” A.
Proposition 1 (K. Yosida [34]). (4) is equivalent to the condition
s — lim Л (Л I — A)-1 x = 0 for every x £ X .	(6)
Proof. The condition A x = 0 is equivalent to Л(Л/ — A)-1 x~x
and so (6) implies the existence of A-1. Moreover, by (2), the denseness
of R(A) is equivalent to (6) so that (4) must be equivalent to (6).
Corollary. In case (6), the abstract potential operator V = - Л is
defined also by
V x = s — lim (Л I — A)-1 x .	(7)
qo '
Proof. For x £ Z)(A-1), we have
-A'1*- (Al - A)~lX= —A~1x — A (A I - A)-1 A-1 x
= A(AI - A)-1 A^x
so that, by (6), we obtain s - lim (A I — A\~l x = - A-1 x. On the other
я|о	'
hand, let s — lim (A I — A) x = у exists. Then A(AI-A]~1x
= — x у A(A I — A)^1 x implies, t)y the closure property of the infinitesi-
mal generator A, the equality A у = - x, that is, у = - /1 1 x and so we
have proved (7).
From now on we shall be concerned with the case
D(AY = X and sup||A(A/-А)-г|| g 1	(8)
л > о
which characterises the infinitesimal generator A of a contraction semi-
group {Tt 0} of class (Co). We can prove
Theorem 1. The abstract potential operator V and its dual operator
V* must satisfy the following inequalities, respectively:
9. Abstract Potential Operators and Semi-groups	413
||APx + %||	||A7%|| forall x£D(V) and A>0,	(9)
||A ]7* /* + /*|| £ ||A V* /*|| for all /* £D{V*) and A > 0 . (10)
Proof. We put /д = (A Z — A)-1. Then Ji satisfies the resolvent
equation
7л-Уд=(д-Я)Л7,-	(ii)
Letting fj, | 0 in (11), we obtain
/л x — V x = — А /д V x for all XD (У) and all A > 0. (12)
Hence /а(А V x + x) = V x and so, by ||A JA\\ S 1, we obtain (9).
Since D(A) and 7?(A) are both strongly dense in X, we have
(A*)-1 = (A'1)* or equivalently V* = ( — A-1)* = (—A*)-1 (13)
by R. S. Phillips’ theorem in Chapter VIII, 6. Therefore, we have, when
V* /* - (AZ* - A*)-1 /* = ( — A*)-1 /* -(AZ*- A*)-1 /*
= (—A*)-1 /* - A* (AZ* - A*)-1 (A*)-1 /* = A(AZ* - A*)-1 (A*)"1 /* .
Hence we obtain
(14)
since 7£'* — lim A J* g* = 0 for all g* £ X* by (6). Thus, as in the case
Л ’I' о
of V, we prove (10) by
and sup ||A/*|| ^ i 
A > 0
(11)'
Corollary. The inverse (A V + Z)^1 and the inverse (A V* + Z*)-1
both exist as bounded linear operators.
Proof. The existence of (A V + Z)-1 and of (A P* + Z*)^1 are clear
from (9) and (10), respectively. We shall prove
R(A V + Z) = X for all A > 0 .	(15)
The existence of the inverse (A V* + Z*)-1 implies, by the Hahn-Banach
theorem, that
7?(A V + I)	(15)’
is strongly dense in X.
Hence, for any у £ X, there exists a sequence {%„} £ ZlfP) such that
s — lim (A V xn + xn) — y. Thus, by (9), we have [] A V (xn — xm)
H —> O0
+ (x„ — xm) ||	|| A V (xn — xm) || and so s — lim V xn = z exists, proving that
M—> oc
s — lim xn = x exists. Since an abstract potential operator is closed by
n—> co
definition, the operator V must be closed. Hence we must have V x = z,
414
XIII. Ergodic Theory and Diffusion Theory
that is, у = 2 Vx + x. This proves (15), and thus (2 V + I)is a bounded
linear operator by the closed graph theorem. Hence, by the closed range
theorem in Chapter VIII, 5, we have
R(AV* + I*'} = X* for all 2>0,	(16)
and so, again by the closed range theorem and (15), (2 V* + Z*)-1 is a
bounded linear operator on the dual space X* into X*.
We can now give a characterization of abstract potential operators'.
Theorem 2 (K. Yosida [35]). Let a closed linear operator V with
domain D(V) and range 7?(V) both strongly dense in X be such that V
and its dual operator l-z* satisfy (9) and (10), respectively. Then V is an
abstract potential operator, i.e., — V is the inverse of an operator A which
is the infinitesimal generator of a contraction semi-group of class (Co) in X.
Proof. We first remark that the Corollary above holds good for V
and for V*, and hence (2 V + Z)^1 is a bounded linear operator on X
into X. Moreover, the linear operator defined through
J^Vx + x)=Vx [for all % £D(V) and all 2 > 0]	(17)
is a bounded linear operator on X into X such that
Л = JZ(2 V + Z)-1 and sup ||2 g 1 .	(18)
Я > 0
We can prove, by (17), that JA is a pseudo-resolvent. In fact, we have
JA(2 Vx + x) -	(2 Vx + %) = Vx -	(— Qw V x + %) 4- (1 ——) xj
= V X - — Vx —
fl	\ flj V \ fl / x	V
and
0* - Л) 4Л(Л Их + X) = (/X - Я) Vx
=	- A) (их + yx- ]-x)
= 0» - A) Vx - (Л Я) 13, x = (1 - 4) (Fx - Л x) 
А*	А*	\ Iй /
Because of (18), we can apply abelian ergodic theorems:
Z? (J„)a = {x £ X: s — lim 2 x = x} for all ц > 0 ,	(19)
A oo
R(I — /a JH}a = {% £ X; s — lim 2 x = 0} for all /z > 0 . (20)
Since R(V)a = X, we see R(JfPa = X by (17). On the other hand, the
null space of «Д is independent of 2 (see Proposition in Chapter VIII, 4).
Hence, by (19) and = X, the null space of consists of zero
vector only. Therefore is the resolvent of a linear operator A :
Л = (2 Z - Л)-1 , where A - 2 I - J~' .	(21)
9. Abstract Potential Operators and Semi-groups
415
Hence we obtain
D(A)a =	= {% С X', s - lim A x = x} for all /г > 0	(19)'
Л j ОС
R(A)a = R (I - д J4)° = {x £ X; s - limА Л x = 0} for all fx > 0 .
(20)'
By (19)' and R{Jfi)a = X, we see that Г) (Л)а = X. We can also prove that
R(A)a = X. In fact, we have, by (17) and (21),
(A A - А) Л(А V x + x} = XV x + x = (A. I — A] V x kV x — A V x ,
that is,
— A В x = x whenever x £ D(V) .	(22)
Hence R(A) ?L>(F) is strongly dense in X, that is, R(A)a = X and so,
by (20)', s — lim A J; x = 0 for all x Q X. Therefore, by Proposition 1,
Л •J' 0
—A~* is an abstract potential operator.
That V = — A-1 may be proved as follows. Firstly, the inverse У-1
exists since, by (22), V x = 0 implies x = 0. Thus, by (18) and (21), we
obtain
A I - A = Jf1 = (A V + 1] V-1 = A I + У-1
which proves that — A"1 = V.
Remark. The above proof shows that (10) is used only to prove (15)'
and hence (15).
As in Chapter IX, 8, we introduce a semi-scalar product [%, y] in X
satisfying conditions:
[x+y, z] = [x,z] A [y, z], [Аж, y] = A [%, y] ,	(23)
[%, x] = ||< and [[x,y]| A ||%[|  Ы 
Then a linear operator V, with domain D(V} and range A? (У) both in X,
is called accretive (with respect to [%, y]) if
Re[x, V x] Й 0 whenever x £Z7(F) .	(24)
We shall prepare three propositions.
Proposition 2. An abstract potential operator V must be accretive.
Proof. Let {7'( A > 0} be the contraction semi-group of class (Co)
whose infinitesimal generator A is given by A = - V~\ We have, by
inn s i.
Re [7~, x - x, x] = Re[T x, x] - |x|* g ЦТ, *11' 1*1 - M2 S 0 .
Hence, for x £ D (A), we obtain
Re [A x, x] = lim [7-1 (Tt x — x), x] £ 0
416
XIII. Ergodic Theory and Diffusion Theory
so that, for x0 £ D{V) = D (A-1),
Re [A V x0, V x0] = Re [- x0, V x0] S 0 , that is , Re [%0, V %0]	0 .
Proposition 3. V satisfies (9) if V is accretive.
Proof. We have, by (23) and (24),
ЦЯ V\%||2 = [Л V x, Л Vx] g Re{[A V x, A
= Re[A Vx + x, Л Vx] ||Л V x + x|l -Ц V x\\ ,
that is, we have proved (9).
Proposition 4, Let V be an abstract potential operator associated
with a contraction semi-group	0} of class (Co) with the infinitesi-
mal generator A so that V = — A~l. Then the dual operator V* must
satisfy, for any semi-scalar product [f*,g*] in the dual space X* of X,
the inequality
[/*, L+/*]S0	(25)
where
V+ is the largest restriction of V* with domain and range both in
R(P*)«
(26)
Proof. Let {Tt+; t 0} be the dual semi-group (see Chapter IX, 13) of
{Tt; t 0), so that, {T(+; t 0} be a contraction semi-group of class
(Co) in X+ = £>(/*)“ = 7?(L*)a. Thus the infinitesimal generator A+ of
the semi-group Tt+ is the largest restriction of A * with domain and range
both in X+ = D(A*)a. Then V+ = ( —Л+)-1, and so (25) is proved as in
the case of Proposition 2.
We are now able to prove
Theorem 3. Let V be a closed linear operator with domain and range
both strongly dense in X. Then V is an abstract potential operator iff V is
accretive and V+ is accretive.
Proof. The "only if part’’ is proved by Proposition 2 and Proposi-
tion 4. We shall prove the “if part". Firstly, V satisfies (9) by Pro-
position 3. Next let A V* /♦ + /* = 0. Then /* £7?(P*) and so V* /♦
= -Л-1/* СЛ(У*). Thus f*£D(V+) and A V+	Therefore,
by the accretive property of V+, we obtain
Д/*, -/*]=	о, that is, /* = 0.
Hence the inverse (A V* + J*)-1 exists and so, by the Hahn-Banach
theorem, (15)' and consequently (15) both hold good. This proves that V
is an abstract potential operator.
Comparison with G. A. Hunt’s theory of potentials. Consider a special
case in which X is the completion C00(S), with respect to the maximum
9, Abstract Potential Operators and Semi-groups
417
norm, of the space C0(S) of real- or complex-valued continuous functions
x(s) with compact support in a locally compact, non-compact separable
Hausdorff space S. In this case, the semi-scalar product may be defined
through
[%,y] = %(s0)  у (s0), where s0 is any fixed point of S such that
(27)
1Ж)1 = sup|y(s)|.
$ £ s
Therefore the accretive property of the linear operator V is given by
i?e(%(s0) • (И %) (s0))	0 whenever |(Fx) (s0)| - sup ((I7 x} (s)| . (28)
s £ S
This may be compared with the principle of positive maximum for V in
Hunt's theory of potentials (G. A. Hunt [1], cf. P. A. Meyer [1] and
K. Yosida [32] and the bibliography referred to in these). Assuming
that the space C00(S) consists of real-valued functions only, Hunt
introduced the notion of the potential operator U as a positive linear
operator on the domain D{U)CCOO{S) into Сж(5) satisfying three
conditions: i) D{U) 2C0(S), ii) U  C0(S) is strongly dense in C^fS) and
iii) the principle of positive maximum given by
for each %(s) £C0(S), the value x(s0)	0 whenever (U x) (s)
. .	(29)
attains its positive supremum at s = s0.
(In the above, the positivity fo U means that U maps non-negative
functions into non-negative functions.) Hunt then proved that there
exists a uniquely determined semi-group {Tt;t 0} of class (Co) of
positive contraction operators Tt on CTC(S) into C0O(S) satisfying the
following two conditions:
A U x = — x for every x(s) £ C0(S), where A is the infinitesimal
generator of Tt,
(30)
{U x) (s) = f (Tt x) (s) dt for every x(s) £ C0(S), the integral being
°	(31)
taken in the strong topology of the В-space COO(S).
It is to be noted that our abstract potential operator 7 is defined
without assuming the positivity of the operator V. Moreover, in our
formulation, (30) can be replaced by the true Poisson equation
-A-P
(30)'
In this connection, it is to be noted that the smallest closed extension V
of the operator U restricted to C0(S) satisfies (30)'. See K. Yosida,
T. Watanabe, H. Tanaka [36] and K. Yosida [37]. Furthermore, in
our formulation of the abstract potential operator V, (31) is replaced by a
27 Yosida, Functional Analysis
418
XIV. The Integration of the Equation of Evolution
more general (Ия) (s) = s- lim ((2.1 — Л)-1х) (s) which is based upon (7).
n-> CO
This formulation has the advantage that it can be applied to essentially
more general class of semi-groups than (31). See, e.g., K. Yosida [37],
[38], A. Yamada [1], and K. Sato [1], [2]. It is to be noted here that
F. Hirsch [1], [2] has also developped an abstract potential theory
directed by essentially the same idea as the present author.
XIV. The Integration of the Equation of Evolution *
The ordinary exponential function solves the initial value problem
dyjdx — (xy, y(0) = 1 .
We consider the diffusion equation
du/St = Zlw, where 1^.' d^jdxj is the Laplacian in Rm.
j=i
We wish to find a solution и = u(x, /), t > 0, of this equation satisfying
the initial condition u(x, 0) = f(x}, where f(x) = f(x1, . . xm) is a
given function of x. We shall also study the wave equation
02w/5Z2 = Au, — oo < / <; oo,
with the initial data
u(x, 0) =/(x) and (8u/dt}t=0 = g(x),
/ and g being given functions. This may be written in vector form as
follows:
with the initial condition
/«(%, 0)\ _ //(x)\
0)/ ~ \g(x)/
So in a suitable function s^ace, the wave equation is of the same form
as the diffusion (or heat) equation—differentiation with respect to the
time parameter on the left and another operator on the right—or again
similar to the equation dy(dt = ay. Since the solution in the last case is
the exponential function, it is suggested that the heat equation and the
wave equation may be solved by properly defining the exponential func-
tions of the operators
/0 1
A and (d 0
in suitable function spaces. This is the motivation for the application
of the semi-group theory to Cauchy's problem. It is to be noted that the
* See also Supplementary Notes, p. 468.
1. Integration of Diffusion Equations in £2 (7?m)
419
Schrodinger equation
i~l 8uj&t = Hu = (d + U(%)) u, where U(x) is a given function,
gives another example of the equation of evolution of the form
dujdt = Au, t > 0,	(1)
where A is a not necessarily continuous, linear operator in a function
space.
The equation of the form (1) may be called a temporally homogeneous
equation of evolution. We may integrate such an equation by* the semi-
group theory. In the following three sections, we shall give typical
examples of such an integration. We shall then expound the integration
theory of the temporally inhomogeneous equation of evolution
8uf8t = d (/) и, a < t < b.	(2)
1. Integration of Diffusion Equations in £2 (
Consider a diffusion equation
du/dt = Au, t > 0,	(1)
where the differential operator
A = « S17 + ‘' W S' + 1W M W (2)
is strictly elliptic in an w-dimensional euclidean space 7?’”. We assume that
the real-valued coefficients a, b and c are	functions and that
max /sup | av (x) |, sup | d1 (%) |, sup | c (x) |, sup ] a^k (%) |,
\ X	X	X	X
sup Ib\(x) I, sup Ia*kXi (x) |\ = < oo.	(3)
X	X	/
The strictly elliptic hypothesis concerning A means that positive con-
stants Ло and jaQ exist such that
m
(4)
m
Ло	.
J=1	1 = 1
vector £ = (£n £»,
Let Hq be the space of all real-valued Cq°(2?”') functions f(x) normed
/ „	rn	\ 1/2
b	it	/	/* l-b	_ Л E“\	1	.
(5)

and let Hq be the completion of Hq with respect to the norm 11 f 1^. Let
similarly Hq be the completion of H^ with respect to the norm
1!/!!о = П/2&у'2.	(6)
27*
420
XIV. The Integration of the Equation of Evolution
We have thus introduced two real Hilbert spaces H^ and Hq, and H^
and Hq are || ||0-dense in Hq. We know, from the Proposition in Chap-
ter I, 10, that Hq coincides with the real Sobolev space IV1 (Rm); we know
also that Hq coincides with the real Hilbert space L2 (7?”). We shall denote
the scalar product in the Hilbert space Hq (or in the Hilbert space Hq)
ЪУ (Л g)i (or by (/, g)0).
To integrate equation (1) in the complex Hilbert space L~(Rm) under
condition (3) and (4), we shall prepare some lemmas. These lemmas will
also play important roles in the following sections.
Lemma 1 (concerning partial integration). Let f, g£ Hq. We have then
M /, g)0 = - f a^fXigXjdx — fa”. fx.gdx
Rm Rm
+ fb*jx.gdx + fc/gdx,
Rm	Rm
that is, we may partially integrate, in (Л. /, g)0 ,the terms containing the
second order derivatives as if the integrated terms are zero.
Proof. By (3) and the fact that / and g both belong to H^> we see
that a” fXiX}g is integrable over Rm. Hence, by the fact that / and g are
both of compact support,
f fXiXjgdx = — f a” fXigXjdx— f a”fx.gdx.
Rm	Rm	Rm
Remark. The formal adjoint A * of A is defined by
Я2	Й
(Л‘Л W = aVarO’W /W)(»‘W /W) +‘W/W- (8)
C/Л f l/Л j	l/Л f
Then, as above, we have the result: If f,g£ Hq then we may partially
integrate, in {A*f, g)Q, the terms containing the first and the second
order derivatives as if the integrated terms are zero. That is, we have
M*f'g)o = — } дЧ dx— f «X/gx.dx
Rm
J bligXidx+ f cfgdx.
(7')
Corollary. There exist positive constants x, у and 5 such that, for all
sufficiently small positive constants oc,
«5	(/ — ocAj, f)Q<, (1 + ay) 5|/||i when /^Hq,
«5 ||/[|?^ (/ — aA*f, /)0	(1 + ay) |[/||? when fEH^,
](/ — aAf, g)0|	(1 + ay) ]|/||i |[g[|x when
|(/ —аЯ*/, g)0|	(1 + txy) ll/lli Ugjli when f.gEH^,
|W,g)o— (/, -4g)o|	« 11/111 ||g)|o when /,g(E#o-
(9)
(10)
(H>
1. Integration of Diffusion Equations in Lz(Rm)
421
Proof. (9) and (10) may be proved by (3), (4), (7) and (7') remem-
bering the inequality
2 a | a b | a (r |«|2 + r-1 | b |2)
(12}
which is valid whenever oc and v are > 0. In fact, we can use an estimate
such as
/ Sdx
Rm
< J?
1=1
We also obtain (11) from
И/, g)e ~ (Л ^g)0 = — f (2</*,£ +	— Zb'/^g - byg) dx.
Rm
Lemma 2 (concerning the existence of solutions of м— ocAu = /).
Let a positive number txobe so chosen that the above Corollary is valid for
0 < a < л0. Then, for any function /(%) G Bq, the equation
и — ocAu - / (0 < oc < Oq)	(13)
admits a uniquely determined solution и G Hq C\ C°° (Rm) 
Proof. Let us define a bilinear functional B(u, v) = (« — ocA*u, u)0
defined for functions w, v G Hq. From the above Corollary, we have
|B(«, v) I (1 + ocy) ||«111 IIwill* a<5,||«111 ^ £(«,«) 	(14)
Hence we may extend B(u, v), by continuity, to a bilinear functional
В («, v) defined for «, v G Hq and satisfying
|#(«, w)|	(1 + осу) ЦмЦх llwllp a<5||w|||	B(u,u).	(14')
The linear functional F (w) = (w, /)0 defined on Hq, is a bounded linear
functional by |(«, /)0| ||wllo ||/|lo = IIм 111 II/Ho- Hence, by F- Riesz'
representation theorem in the Hilbert space Hq, there exists a uniquely
determined v = v (J) G H^ such that («, /)0 = (u, v (f))v Thus, by the
Milgram- Lax theorem applied to the Hilbert space Hq, we have
(«. /)o = (M> v (/))1 = B (u> Sv (/)) for all w G H%, where
1	(lb)
S is a bounded linear operator on Щ onto Hq .
Let w run over	(jRw) , and let vn G Hq be such that lim 11 vn — 5 v (/) |
twoc
= 0. Then
В (u, Sv(f)) = lim B(u, vn) ~ lim B(u, vn)
W-K)0	n—НЭС
= lim (« — ocA*u, vn)0 = (« — ocA*u, Sv(f))0,
because the norm || ||г is larger than the norm || ||0. Hence
(«-/)o= (« — ocA*u, Sv(/))0,	(15')
that is, Sw(/) G Hq is a distribution solution of the equation (13). Hence,
by the strict ellipticity of {I — ocA) and by the fact /G C^0(/?*”), we see,
422 XIV. The Integration of the Equation of Evolution
from the Corollary of Friedrichs' theorem in Chapter VI, 9, that we may
consider и = Sv (f) £ Hl to be a CaD(R’”) solution of (13). Hence и =
Sv{f)eH1ar\C(Xi(Rm).
The uniqueness of such a solution « of (13) may be proved as follows.
Let a function и £ Hq A C°° (7?w) satisfy и— ocAu — 0. Thus Au{
Hl A C°°(Rm) C Hq and so the expression (w - -ocA w, «)0 is defined and
= 0. Let C Hl be such that lim ||« —	|[j = 0. We obtain, by partial
integration as in (9),
0 - (« — ocAu, w)0 = lim (w — ос А и, «и)0 oc& I w I 2, that is, и -- 0.
n—юо
Corollary 1. Positive constants ocq and /;() exist such that, for any
/ 6 Hl, the equation
оси — Au = f (0 < a0 + Ao + rj0 A oc)	(16)
admits a uniquely determined solution = uf £ Hl A C°° (Rm), and we
have the estimate
l^llo (a —Ao—?70)-1 |[/||0.	(17)
Proof. By Schwarz’ inequality we have
||(ai — A) «||o 
|«||o	~A)u,u)0
for « G Hl.	(18)
By partial integration, we have
((<*/ — A) u,u)0 =a ||«||o + f avuXiuXfdx + J a%uXjudx
nm	Km
— f RuXiudx — f cuudx.
Rm	Rm
Hence, we have, by (3), (4) and (12),
((«/ — A) u, u)Q ;> oc |[«||o + Л) (||«||i— IMIo)
— m2 Tj |>(l|M||i — ||w||o) + v-1 |l«||o + "A2 ||w||o]
= (a — Ao — m2 y] (v-1 — v + nA2)) ||«||o + (Ao — w2 rj v) ||« |j2.
Thus by (18) we have, for^0 = m2 гДг”1 —v + w2),
|| («У — A) «||o (a — Ao— r]0) ||«||0 whenever u£Hl, (17')
by taking v > 0 so small that (Ao — m2 rjv) and tjq are > 0. We then take
«0 so large that for ос а0 + Ло + r/0 we can apply Lemma 2
in solving (16).
Since the solution и =	£ Hq A C°° (Rm) of (16) is approximated by
|| Uj-norm by a sequence of functions £ Hl, we obtain the estimate (17)
from (18) and (17').
Corollary 2. Consider A as an operator on 71(A) = (л I — A)~l Hl QHq
into //(j. Then the smallest closed extension A in Hq of A admits, for
1. Integration of Diffusion. Equations in L2 {Rm}	423
a > a0 4- Яф + the resolvent (a/ — .4) 1 defined on into Hq
such that
||(al — Л)-1||о^ (a —Яо —	(19)
Proof. Clear from Corollary 1 remembering the fact that D(A) and
Hq both are || [|0-dense in Hq.
Thus, by Corollary 2 in Chapter IX, 7, A is the infinitesimal generator
of a semi-group Tt of class (Co) in the В-space Hq such that [| Tt 1|0
for t 0.
Actually we can prove
Theorem 1. Let Complex-/^ be the space of all complex-valued
functions /CC£°(Rm) with
Let Hq and Hq be the completions of Complex-/^ with respect to the
norm ||/1Ij and ||/|]0, respectively. We know that Hq = (complex) Sobo-
lev space W1 (jR”1) (see Chapter I, 10). It is also clear that Hq = (complex)
Hilbert space L2 (R*). We consider A as an operator defined on D (Л) =
(al — A)~YHl C L2(Rm) into L2(Rm). Then the smallest closed
extension A in L2 (Rm) of A is the infinitesimal generator of a holomorphic
semi-group Tt of class (Co) in L2(Rm) such that ||	|[0 e(Ao+’!o)i for
t > 0.
Proof. By the preceding Corollary 2 and the reality of the coefficients
in the differential operator A, we see that the range R(al — A) —
(al — A)  D(A) is || ||0-dense in L2(Rm) for <x > <x0 +	+ ^0. More-
over, we have, for (u + iv) 6 L2(Rm) such that (u + iv) £
||(al —A) (w + j'w) ||o = || (а/— Л) « ||q + ||(a/- A) v ||§
= (a — Л) “ Ло)2 (11u I |o + 11v I |o) 
Hence the inverse (al— X)-1 is a bounded'linear operator on L2 (Rm)
into L2(Rm) such that || (al— X)"11[0 < (a — — ^o)”1 for a > a0+
Therefore, by the Theorem in Chapter IX, 10, we have only to show that
lim |т I || ((a + ir) I — A) 1||0<oo.
M too
(20)
We have, for w £
|((a + it) I — A) w||0 ||^||0	|(((a + it) I — A) w, w)0|, (21)
By partial integration, we obtain, as in the proof of (17),
(Real part (((a + ix) I — Л) W}
Ы IMIo + Real part / J arjw%iwXjdx +
\Rm
424
XIV. The Integration of the Equation of Evolution
— J b*wXiwd%— J cwwd%\
J
(л —Л) — *7o) IMIo + (^0 — II®111-
Similarly we have
| Imaginary part (((a + it) I — A) w, w)01
— M IIw llo — m2 ^(|)w ||i + m~2 Iloilo) | 4- (|т| — t?) ||w||q —m2?? ||w ||2.
Suppose there exists аго06Й(Я), ||M’ollo °, such that
| Imaginary part (((a/ + гт) I— A) w0, w^Q | < 2"1 (|т | — 77) 11 w01 |o
for sufficiently large т (or for sufficiently large —t). Then, for such large
т (or —t), we must have
*7 Il№o||i> 2 1 (|т| — ??) ||w0|]§
and so
I Real part (((a + ix} I— A) w0, и»0)0|	(^ -~m2r}v)
Hence we have proved Theorem 1 by (21).
r I — »?) I |2
2тгт] I w° I0’
Theorem 2. For any /С L2(Rm), u(t, %) = (7^/) (%) is infinitely diffe-
rentiable in t > 0 and in xC Rm, and и (t, x) satisfies the diffusion equation
(1) as well as the initial condition lim 11 и (t, x) — f (x) 110 = 0.
Proof. We denote by the А-th strong derivative in L2 (Rm) of
Tt with respect to i. Tt being a holomorphic semi-group of class (Co) in
L2(Rm), we have Zj*>/ = AhTtf £ L2{Rm) if t > 0 (k = 0, 1, . . .). Since
A is the smallest closed extension in£2(i?m) of A, we see that AkTtf^L2 (Rm)
for fixed t > 0 (k = 0, 1, 2, . . .) if we apply the differential operator Ak
in the distributional sense. Thus, by the Corollary of Friedrichs’ theorem
in Chapter VI, 9, и (I, x) is, for fixed / > 0, equal to a C'c’°(A>m) function
after a correction on a set of measure zero.
Because of the estimate || Tt |j0	e(z°+’?o);, we easily see that, if we
apply, in the distributional sense, the elliptic differential operator
any number of times to и (t, x), then the result is locally square integrable
in the product space {t\ 0 < t < oo}x7?m. Thus again, by the Corollary
of Friedrichs' theorem in Chapter VI, 9, we see that u(t, x) is equal to a
function which is C°° in (t,x), t > 0 and 7?m, after a correction on a
set of measure zero of the product space. Thus we may consider that
u(t, x} is a genuine solution of (1) satisfying the initial condition
lim ||«(Z, %)-/(%) ||0 = 0.
t 4- и
Remark. The above obtained solution u(t, x) satisfies the "forward
and backward unique continuation property" ;
2. Integration of Diffusion Equations
425
if, for a fixed tQ > 0, и (tQ, x) ~ 0 on an open set G £ Rm',
then «(t, x) = 0 for every t > 0 and every x € G.
(22)
Proof. Since Tf is a holomorphic semi-group of class (Co), we have
WOO
lim	£ (kl)-’-htAkTJ = 0
A = 0	0
for a sufficiently small h. Hence, as in the proof of the completeness of
the space Lz(Rm), there exists a sequence {n'} of natural numbers such
that
u(tQ-\-h, x) = lim (Л!) 1 hkAku(tQ, x) for a.e. x£Rm.
n —hx> fe=0
By hypothesis given in (22), we have Aku(l0, x) = 0 in G and so we must
have и (Zo + h, x) = 0 in G for sufficiently small h. Repeating the process
we see that conclusion (22) is true.
References
The result of this section is adapted from K. Yosida [21]. As for
the “forward and backward unique continuation property" (22), we have
the more precise result that и (/, x) = 0 for every t > 0 and every x G Rm.
For, S. Mizohata [3] has proved a “space-like unique continuation pro-
perty" of solutions u(t, x) of a diffusion equation to the effect that
и (/, x) = 0 for all t > 0 and all x £ Rm if и (t, x) = 0 for all t > 0 and all
x C G. Concerning the holomorphic character of the semi-group Tt ob-
tained above, see also R. S. Phillips [6]. It is to be noted here that the
unique continuation property of the solutions of a heat equation ди/dt =Au
was first proposed and solved by H. Yamabe-S. Ito [1].
There is a fairly complete discussion of parabolic equations from the
view point of the theory of dissipative operators. See R. S. Phillips [7].
2. Integration of Diffusion Equations
in a Compact Riemannian Space
Let R be a connected, orientable m-dimensional C°° Riemannian
space with the metric
dsz = gij(x) dx'dx*.
(1)
Let A be a second order linear partial differential operator in R with
real-valued C°° coefficients:
A = a? (x)	(x) Д.	(2)
v 8x* dx* 2 * 4 ’ dx
We assume that a? is a symmetric contravariant tensor and that 6* (x)
satisfies the transformation rule
426
XIV. The Integration of the Equation of Evolution
(3)
7i 7 i	S2xi
b' ~bk —r 4- ak3 —-—,
cjx*	dx* Ёх
by the coordinate change (x1, x2, ..., xm) -> (x1, x2, . . ., xw) so that the
value (Л /) (x) is determined independently of the choice of the local coor-
dinates. We further assume that A is strictly elliptic in the sense that there
exist positive constants z0 and such that
£ £2 S: «,J (x)	4: Ao £ for every real vector
M	J=1
(£i, • • •, £m) and every x 6 Rm.	(4)
We consider the Cauchy problem in the large on R for the diffusion
equation: to find solution w(C x) such that
&u/8t = Au, t > 0, w(0, x) = /(x) where /(x) is a given
function on R.	(5)
We shall prove
Theorem. If R is compact so that R is without boundary, then the
equation (5) admits, for any initial function /6 C°°(7?), a uniquely deter-
mined solution u(t, x) which is C°° in (t, x), t > 0 and x £ R. This solution
can be represented in the form
(6)
R
where P(t, x, E) is the transition probability of a Markov process on R.
Proof. Let us denote by C (7?) the 7^-space of real-valued continuous
functions /(x) on R normed by f]/If = sup |/(x) We first prove
X
for any C°°(R) and any n > 0, we have
max A (%)	/(x) > min A (%) where A (%) = /(%) — n~1(A /) (x). (7)
X	X
Suppose that / (x) attains its maximum at x = x0. We choose a local
coordinate system at x0 such that ai} (x0) =	(= 1 or 0 according as
i = j or i ф f). Such a choice is possible owing to condition (4). Then
o> <
j/ \	—Im’/ \	— 1 v*	//
since we have, at the maximum point x0,
—. = 0 and Q/	0.
ex ’0	e(x&2~
Thus max A(x) /(x). Similarly we have f(x} > min A(x).
X	x
We shall consider A as an operator on D(A) = C°°(R) C C (7?) into
C (R). Then, by (7), we see that the inverse (7 —	exists for n > 0
3. Integration of Wave Equations in a Euclidean Space RM
427
(7 — n^A) - D(A). This range is, for sufficiently large n, strongly dense
in C(R). For, as in the preceding section, we have the result: For any
g £ C°° (R) and for sufficiently large n > 0, the equation w — nr1 A и = g
admits a uniquely determined solution Mf	Because, by the com-
pactness of the Riemann space R, Lemma 1 concerning the. partial inte-
gration in the preceding section may be adapted to our Riemannian
space R without boundary. Moreover, C00 (R) is strongly dense in C (R)
as may be seen by the regularization of functions in C (R) (see Pro-
position 8 in Chapter I, 1).
Hence the smallest closed extension A in C (R) of the operator A
satisfies the conditions:
for sufficiently large n > 0, the resolvent (7 —
exists as a bounded linear operator on C (R) into C (R) 
such that 11 (7 — и-1 A )-111	1,
((/ n~Y Л)-1 K) (x) > 0 on R whenever h (%) f;> 0 on R,
(f-n-1^)-1 -1=1.
(8)
(9)
(10)
The positivity of the operator (7 — я-1 Л) 1 given in (9) is clear from (7).
The equation (10) follows from A - 1 = 0.
Therefore A is the infinitesimal generator of a contraction semi-
group Tt in C(R) of class (Co). As in the preceding section, we see, by the
strict ellipticity of A, that, for any /CC°°(R), the function u(t, x) =
(Ttf) (x) is a C°° function in (/, x) for t > 0 and x C R so that и (t, x) is a
genuine solution of (5),
Since the dual space of the space C (R) is the space of Baire measures
in R, we easily prove the latter part of the Theorem remembering (9)
and (10).
3. Integration of Wave Equations in a Euclidean Space Rm
Consider a wave equation
02«/0t2 - Au, — oo < t < oo,
(1)
where the differential operator
A = a“ w + b'^h + c « w =aii w <2>
is strictly elliptic in an m-dimensional Euclidean space R”1. We assume
that the real-valued C°° coefficients a, b, and c satisfy conditions (3)
and (4) in Chapter XIV, 1. As done there, we shall denote by the
space of all real-valued C£° (Rw) functions / (x) normed by
(„	m <•	\	1/2
//2 dx	I-	J	dx\
)	1 1{m	J
428
XIV. The Integration of the Equation of Evolution
and let (and Hq) be the completion of H'G with respect to the norm
Ц/U! /and with respect to the norm ||/||0 = / f dx\ll2\.
\	\7?«*	/	/
Lemma. For any pair {/, g} of elements € #o the equation
/ ZZ
\\°
0\	/0 Z\\ и \ f\
I \A 0 \v \g
(3)
admits, if the integer n be such that | n 1 | is sufficiently small, a uniquely
determined solution {u,v}, и and v E Hl A C°°(Rm), satisfying the esti-
mate
((и — >.„ А и, и)„ +«,(», в)0)”2
S (1 - Д I n ГТ1 ((/ - M f. f)0 + «0(g, g)o)1'2.	(4)
with positive constants a0 and fl which are independent of n and {/, g}.
Proof. Let iq C Z/q A C°° (/?”) and v1£H^C\ C°° (Rm) be respectively the
solutions of
u1 — n~2A ил - / and Vj - n~2A vt — g.	(5)
The existence of such solutions for sufficiently small | n-1l was proved in
Lemma 2 of Chapter XIV, 1. Then
U =	V = П~*А M-j + tq	(6)
satisfies (3), i.e., we have и — n~ ' v  f,v - nr1 Au = g.
We next prove (4). We remark that
A и = n (y - g) C Hl A C°° (Z?m) C and hence, by /, g €	,
A v = n(Au~ Af) EHl A C°°(Rm) £ Hq.
We have thus, by (3),
(/ — а0Л/, /)0 = (w — n — хХ)Л (и — n^v), и -
= (и — (х.0Аи, u)9 — 2 л'1 {«, y)0 + aon-1 (A u, y)0 + aow-1 (A v, u)9
and	+ n~2(y — KqAv, v)q
I
®0 (?> g)o = «о (у — n-1 A u,v — п~гА u)0
^^(v,	— Ли)о —*0«-1 Иu> w)o + ixon~z(Auf Au)0.
By a limiting process, we can prove that (9), (10) and (11) in Chapter XIV, 1
are valid for / ~ и and g = v. Thus, by (12) in Chapter XIV, 1, there
exists a positive constant /? satisfying
((/ - «оA /, /)0 + a,, (g, g)0)1/2	((и — а0А w, u)q + «(, (y, »)0
~ ло Iм"11 IИ v)0 — {A v, «)01 - 2 l^11 I («, v)01)1/2
(1 — I w''11) ((w — o^A u, u)q + л0 (w, v)0)1/2
3. Integration of Wave Equations in a Euclidean Space Rm 429
for sufficiently large |я|.
The above estimate for the solutions {«, v) belonging to Hq A C°° (Rm)
shows that such solutions are uniquely determined by {/, g}.
Corollary. The product space Hq А of vectors
(w \
I = {u, v}', where uQ Hq and Hq,	(7)
is a B-space by the norm
= ||{w, v}' || = (B(u, w) + a0(v, v)0)1/2,
(8)
where В (/, g) is the extension by continuity with respect to the norm
| | |i of the bilinear functional
В (J, g) = (f—aoAf, g)0 defined for /, g€ Hq.
We know that	is equivalent to the norm (see Chap-
ter XIV, 1):
a0(5 || w [j2 < B{u, w) < (1 + aoy)	(9)
Let the domain D(9l) of the operator
(10)
be the vectors {w, v}' ^HqXHq such that «, v £ Hq are given by (6). Then
the Lemma shows that the range of the operator
I
3 — и-1 where 3 = L I,
contains all the vectors {/, g}' such that f, g('_ Hq. Thus the smallest
closed extension 91 in Hq X Hq of the operator 91 is such that the operator
(3 — w-19() with integral parameter n admits, for sufficiently large |и |,
an everywhere in Hq X Hq defined inverse (3 — n~l 91)-1 satisfying
| ((3 - я-1 й)-111 (i - £ | «-1! r1.	(ii)
We are now prepared to prove
Theorem. For any pair{/(x),g(x)} of C£° (Rw) functions, the equation
(1) admits a C°° solution и (/, x) satisfying the initial condition
«(0, x) = f(x), «Д0, x) = g(x)	(12)
and the estimate
(B(u, «) + a0(wo ut)0)1/2	exp(/3 |*|) (£(/, /) + a0(g, g%)1/2. (13)
Remark. Formula (9) shows that B(u, «) is comparable to the poten-
tial energy of the wave (= the solution of (1)) u{t, x), and (uh ut)Q is
comparable to the kinetic energy of the wave u(t, x). Thus (13) means
430
XIV. The Integration of the Equation of Evolution
that the total energy of the wave и (t, %) does not increase more rapidly
than exp (ft 11 j) when the time t tends to °0- This is a kind of energy
inequality which governs wave equations in general.
Proof of Theorem. Estimate (11) shows that 21 is the infinitesimal gene-
rator of a group Tt of class (Co) in X H® such that
(14)
By hypothesis we have, for k = 0, 1, 2, . . .,
H	Z ) g Cf (Rm) x(Rm) Q xHf
\g/ \g/
Hence, if we put
/м«%)\ _	//(%)\
\v(t,x}/	z\g(%)/’
then, by the commutativity of 21 with Tt, we have
^/M(i!,%)\ _^T(//(x)\_	(u(t, x)\
\g(x)J	°	°
for k = 0, 1, 2, . . . Here we denote by dkTt[dtk the k th strong derivative
in HfxHf Therefore, by Hq Q H% = L2(Rm) and the strict ellipticity
of the operator A, we see, as in the proof of Theorem 2 in Chapter XIV, 1,
that и (t, x) is C°° in (t, x) for — oo < t < oo, x £ Rm, and satisfies equa-
tion (1) with (12) and estimate (13).
Remark. The result of the present section is adapted from K. Yosida
[22]. Cf. J. L. Lions [1]. P. D. Lax has kindly communicated to the
present author that the method of integration given in this section is
very similar to that announced by him in Abstract 180, Bull. Amer. Math.
Soc. 58, 192 (1952). It is to be noted here that our method can be modified
to the integration of wave equations in an open domain of a Riemannian
space. There is another approach to the integration of wave equations
based upon the theory of dissipative semi-groups. See R. S. Phillips [8]
and [9]. Thq method is closely connected with the theory of symmetric
positive systems by K. Friedrichs [2]. Cf. also P. Lax-R. S. Phillips
[3].
4. Integration of Temporally Inhomogeneous Equations of
Evolution in a B-space
We shall be concerned with the integration of the equation
dx (Z) jdt = A (t) x (Z), a '.A t < b.	(1)
Here the unknown x(t) is an element of a В-space X, depending on a
real parameter t, while A (Z) is a given, in general unbounded, Unear
4. Integration of Temporally Inhomogeneous Equations 431
operator with domain D (Л (/)) and range R (A (t)), both in X, depending
also on t.
T. Kato [3], [4] was the first to make a successful attack on the
problem of integration of (1). He assumed the following four conditions:
(i)	The domain D (A (£)) is independent of t and is strongly dense in
X such that, for oc > 0, the resolvent (7— a A (Z))-1 exists as a bounded
linear operator £ L(X, X} with the norm 1.
(ii)	The operator В (t, s) = (7 — A (Z)) (7 — A (s))”1 is uniformly boun-
ded in norm for t sg s.
(iii)	В [t, s) is, at least for some s, of bounded variation in t in norm,
that is, for every partition s = tQ <_	< •• • < tn — t of [a, &],
rt—1
£ ||7?(^+i, s) — B(t}, s) |l <; N(a, b) < oo.
j“0
(iv)	B(t, s) is, at least for some s, weakly differentiable in t and
8B(t, s)/8i is strongly continuous in t.
Under these conditions, Kato proved that the limit
о
U (Z, s) x0 = s-lim П exp ((/y+1 -Zy) A (Z,-)) xQ
щах] fj |	J — J.
exists for every x0C X and gives the unique solution of (1) with the initial
condition %(s) = x0 at least when .r0€ D(A (s)).
Kato’s method is thus an abstraction of the classical polygon method
of Cauchy for ordinary differential equation dx(t)/dt = a(t) x(t}. Although
his method is very simple and natural in its idea, the proof is somewhat
lengthy because it is concerned with a general partition of the interval
[s, /]. Kato [3] has shown that the proof can be simplified when the space
X is reflexive. For the reflexive TTspace case, see also K. Yosida [28].
In this section, we shall be concerned with an equi-partition of a fixed
interval, say, [0, 1], independently of s and t, to the effect that Kato's
original method be modified so as to yield a fairly simplified presentation.
We shall assume the following four conditions which are essentially
the same as Kato’s conditions (i) through (iv) above:
D(A (£)) is independent of t and it is dense in X.
For every Л 0 and t, 0 A t g 1, the resolvent (A7 — A (Z))-1
exists as an operator £ L(X, X) such that ||(Л7 — A (Z))-1|| A
£ A-1 for Л > 0 .
A (t) A (s)-1 e L (X, X) for 0 g s, t 1 .
(3)
(4)
For every x £ X,
432
XIV. The Integration of the Equation of Evolution
(i — s)^1 C (t, s) x = {t — s)-1 (A (tf)A (s)-1 — /) x is bounded '
and uniformly strongly continuous in t and 5, t 4= s, and
s-lim k • C U,t — —) x = C (Z) x exists uniformly in t so that
k—>oo	\	A /
C (t)	(X, X) is strongly continuous in t.
Remark, Condition (4) is stated for the sake of convenience. It is
implied by (2) and the closed graph theorem, since A (s) is a closed linear
operator by (3). Moreover, conditions (2) and (3) imply that A ($) is the
infinitesimal generator of a contraction semi-group {exp(M(s)); t 0}
of class (Co). Hence we have (see Chapter IX):
ехр(Ы (s))x = s-lim exp (7 4 (s) (I — w1 A (s))-1) x uniformly in CO	1;
Я-^-00	4	'
(6)
exp(CA (s)) • exp(t2A (s)) - exp((C + Z2) A (s));	(7)
£ехр(мр)Ьс = A exp(M ^y = ехр(/Л (s)) . A y,
у QD(A ($)), where d/dt means differentiation in the strong
topology of X;

 (8)
J
s-lim exp(C4 ($)) x = exp(i0A ($)) x .	(9)
i—s^o
We are now able to state our result.
Theorem 1. For any positive integer k and 0 s 4 t 1, define
the operator Uk(t, s) £ L (X, X) through
/	Ц — jn
l7ft(C s) = exp l(Z — s) A (—7— I) for
\	\ /* / /
i — 1	i
k~ <' s 7
(1 S f A),
' (io)
uk(t, r) = Uk(t, s) Uk(s, r)
for 0 <;" r s C t 1 .
Then, for every x £ X and 0	$ SA g 1,
s-lim Uk(t, s) x = U(t, s) x exists uniformly in t and s .	(11)
Moreover, if у £ D(A (0)) then the Cauchy problem
= A (() x(l), x(0) = у and x(t) € D(A (0), 0 S I S 1	(1)'
is solved by x(t) = U(t, 0) у which satisfies the estimate ||x(f)|| £ ||y|(.
Proof. By (3), (6) and (10), we obtain
ШИМ H (k = 1,2, . . .,; 0^ sg t 1;%CX) .	(12)
We also need, for the operator
Wk(i, s) =A(i) Uft(i,s) A(s)-\
(13)
4. Integration of Temporally Inhomogeneous Equations
433
the following estimate which is the key for the whole proof:
II5) x||	(1 + k^N)2 • exp(7V(/ - s))  И ,
N = sup || (Z — s)-1C(Z, s)|| .
0£s,l£l,s
•	(К)
To prove this, we shall rewrite Wk(t, s), remembering (10) and the com-
mutativity A (s)-1 exp((/ — s) 4(s)) = exp((/ — s) A (s)) A (s)'1:
Wk{t, s)
= A W А (Щ)'1 U, (t. И) А A	Uk (<£!,	') ' ' •
v ' \k ] я \ k /	\ k J \ A / л \ A k /
. . . A /[As]_/ 2\ A /[As] + 1 \ -1	/ [As ] + 2 [As]+_1 \/[As] + 1 \ A /[ks]\ -1
\ A J \ k / ft\ A ’ A / \ A / \ k /
Expanding the right side and again remembering (10), we obtain
that is,
И7,(t, s) = (/ + C («,^)) {Ut (t, s) + иу (t, s) + <> («, s) +   •}x
х(/ + с(И s)),
[ft/]	/	1\
иум= £ U^l.^ciu.u-jju^u.s),
ku— [ks]+l	'
(18)
[W]	/
Wt*+1)(M=	£ Uk{t,u)C(u,u
Aw = [As] 41
- i) Wy > (M, s),
(m = 1,2,..., M-l). <
C Iu, и —
We have
x ^IHI by (14), Hence from (12) we
/?
conclude that
11^1(1,3)^11 g (l-s)NW,
tie)
ii тми s
Combined with the definition of N, we have proved (14) by (12),
28 Yoalda, Functional Anulynh
434
XIV. The Integration of the Equation of Evolution
Incidentally, we have proved that
Uk (t, s) у £ D (Л (Z)) when у £ D (A (s)) .	(17)
Hence
) A
UK(t, s)y = exp ((z
\ \	rf
is differentiable at t т (* = 0, 1, . . ., k) and
Л
it/. ('.') у = л	(t| s) у, у e o (л (0)) .	(18)
Similarly, Uk(t, s) у is differentiable at s Ф у (i = 0, 1, . . k) and
fv
^1’5) y - - ys('.s)^	eO(A(0)).	(19)
These derivatives are bounded and are strongly continuous in t and s
i	i
except at Z , and s = —. For the proof, rewrite the right side of (18)
n	n
as = C , tj W*(t, s) A (s) А (О)"1 x,у = A (0)1 x, and make use of
(5), (9), (12), (14) and (15); similarly for (19).
Since Un(t, s) Uk(s, 0) A (0)-1 x is strongly continuous in s by (9),
we have, remembering the fact Ufc(s, 0) A (0)-1 x £ D(A (s)) together
with (18) and (19),
0) - Un(t, 0)) А (0)-1 X = [Z7„(Z, s) Uk(s, 0) A (0)-* x]ssz£
0
I
17» (s, OJAfO)-1*^
0
t
[MS]
n
-1 Wft(s, Q)xds.
о
Hence, (12) and (14) yield
||CA(1, 0) А (О)-1 X - U„(t, 0) A (0)-i *1
t
0
(/+ c(^,s)) TF,(s, 0) x
К J \	\ n	/ J
ds
Wzft(s, 0) x ds
4. Integration of Temporally Inhomogeneous Equations
435
t
о
[As] _ [ws]
k n
[As]
S-~T
) (1 + 4 n) exp (Ns) • || %|| ds.
/ \ ** /
This proves that
s-lim Uk(t, 0) A (Op1 x exists uniformly in t, 0 P t < 1,
A—MJQ
and so, by (2) and (12), we prove that, for every % (- A7 ,
s-lim Uh (t, 0) x — U (t, 0) x exists uniformly in t, 0 t A I
fe—ИОО
Similar argument shows that, for every x\ X,
s-lim Uk (t, s) x — U (t, s) x exists uniformly in t and s, 0 £ s sU 1.
(20)
Therefore, by (9), we see that U (i, s) x is uniformly strongly continuous
in t and s.
We also prove easily, remembering (15), (16) and (20), that as k ->oo
Wk(t, 0) x strongly converges boundedly in t such that
s-lim Wk (i, 0} x - W (i, 0) x - U (t, 0) x + (if, 0) x + IV<2> (/, 0) x
A—>oo
i
where 0) x = f U(t, s) C(s) U(s, 0) x ds,
0
(21)
i
11 (/, 0) X = f U(t,s) C (s) W'm) (s, 0) x ds (m-1,2,...).
о
Therefore, if у £ D(A (0)) then the limits
s-lim Uk [t, 0) у = U (I, 0) у and s-lim A (t) Uk (t, 0) у = W (Z, 0) A (0) у
k—*-00	jA—>00
both exist boundedly and uniformly in t and W (t, 0) A (0) у is uniformly
strongly continuous in t. A (t) being a closed linear operator by (3), wr
have proved that, for every у £ D(A (0)) ,
U(t, 0) у СП(Л(/)) and
A (Z) U(i, 0) у - s-lim A (t} A	Uk{t, 0) у
A—>-OO	\ rt /	\ Л /
= W(t, 0)A(0)y,
(22)
where the s-lim exists boundedly and uniformly in t, 0	1 .
Hence, by letting k ->oo in
t	t
=J{iu^y)is =fA
0	0
We obtain	t
U(t, 0) у - у f A (s) U(s, 0) у ds .
0
L/ft(s, 0) у ds
uh(t> 0 У — у
28*
436
XIV. The Integration of the Equation of Evolution
As proved in (22), the integrand is strongly continuous in s so that we
have proved that x(t) = U(t, 0) у solves Cauchy problem (1)'.
Remark. Theorem 1 and its proof are adapted from K. Yosida [30].
It is to be noted here that an inequality similar to our (14) was obtained
in T. Kato [3] by solving a Volterra type integral equation. The appro-
priateness of the equi-partition of the interval [0,1] as revealed in (10)
is suggested by a paper by J. Kisynski [1].
The approximation ya(/) = Uk(t, 0)y to the true solution x(t)
= U (t, 0) у of (1)' is a kind of difference approximation to the differen-
tial equation given in (1)'. However, the true difference approximation
would be to use the scheme
choosing the backward difference. This scheme gives
— (J — (t, C-i) (0) 1	•
Therefore, suggested by (10), we are lead to the approximation
From the point of view of the numerical analysis, this approximation
Fft(C s) would be much more practical than the approximation Uk(t, s),
since in the construction of the approximation Uk(t, s) we have to appeal
to the construction of exp(M (s)).
We shall prove (Cf. the last section in T. Kato [10]).
Theorem 2. Under the same conditions (2) through (5), the Cauchy
problem given in (1)' is, for every initial data у £/)(А(0)), solved by
x (/) = s-lim Vк (t, 0) у .
А—>сю
Proof. jFrom (3) it follows that
|| Vk(t, s) %|| < ||%|| (A - 1, 2, . . .; 0 g s g 1; x Q X} .	(12)'
We next see, from
Vk(t, $} = (c
[^] + lj ( [fed +J.
that V k(t, s) у is differentiable at s 4= у (z = 0, 1, . . ., A) and
s)y- = - V.(t, s) (l -	.	(19)'
ds	* v 1 \	/	\ я //
We see, by у £ D(A (0)) = D(A ($)), (3) and (12)', that this derivative
is bounded and strongly continuous in s except at s — у. Since
4. Integration of Temporally Inhomogeneous Equations
437
FA(t, s) C\(s, 0) A (0)-1 x is strongly continuous in s, wr <ihtain, remem-
bering the fact Uk(s, 0) A (0)"1 x £ D(A (0)),
- ИА(/,0))Л(ОГх = [П(М)	.1(0) < cX,
t
= f	uk(s, 0)Л(0И %) ds
0
- f vk{t, s) (.4 (^)- A	-h X
0
X 0) (0)“T x ds
= f vks) (/-(/_ (1^ -s) Л (/ + С s))x
0
x PKfc (s, 0) xds .
Thus we have
- v„(i, 0)) A (0)-' M	(24)
0
From (5) and (21), it follows that
s-lim 11 + C X—,	W\(s, 0) x = W (s, 0) x uniformly in s, 0	5	1 .
У k ”	h	(25)
On the other hand, we have, for z £ D(A (s)) ,
By (3) and (5), we see that this tends strongly to 0 as & ->oo. The integrand
in (24) is bounded in k and s. Thus, since D(A (s)) = D(A (0)) is dense
in X, we see from (25) that
s-lim (Uk(t, 0) -	0)) A (0)"1 x = 0 .
Л—
Therefore, by virtue of Theorem 1, we have proved Theorem 2.
Remark. The above proof is communicated from H. Fujita who is
suggested by the last section of T. Kato [10].
Other approaches. K. Yosida [23] and [28] devised the idea of
approximating Cauchy problem (1)' by
=5^00(0)),OSUfi 1.	(1)"
Applying the Cauchy polygon method to equation (1)", J. Kvsinski 11 ’]
proved the strong convergence of (	(Z)} to the solution of (1)', Another
438 XIV The Integration of the Equation of Evolution
method of integration of (1) has been devised by J. L. Lions [2]. He
assumes the operator A (t) to be an elliptic differential operator with
smooth coefficients depending on t, and seaks distributional solutions
by transforming equation (1) into an integrated form in concrete func-
tion spaces such as the Sobolev space	and its variants. We also
refer the reader to a paper by 0. A. Ladyzhenskaya! . M. Visik [1]
which is motivated by a similar idea as Lions’.*
5.	The Method of Tanabe and Sobolevski
Let X be a complex В-space, and consider the equation of evolution
in X with a given inhomogeneous term / (t):
dx{t)!dt = Ax(i) у tlf), a^t^b.	(1)
Then the solution x {t} £ X with the initial condition x (a) = x0 Q X is
given formally by the so-called Duhamel Principle from the solution
exp((£ — a) A) x of the homogeneous equation dxldt =- Ax:
x(i)—exp((i!— a) A) x0 + f exp((t— s) A) • /($) ds. (2)
a
This suggests that a temporally inhomogeneous equation in X:
dx^tydt = A(t)x[p,	(3)
may be solved formally as follows. We rewrite equation (3) in the form
dx{t)!dt = A {a) x(i) + (Л (if) — A (a)) x(t).	(4)
By virtue of the formalism (2), the solution x(t) of (4) with the initial
condition x (a) = x0 will be given as the solution of an abstract integral
equation
x [t] = exp ((£ — a) A (a)) x0
t	(5)
)	4 f exp ((/ — s) A (s)) (A (s) — A («)) x(s) ds.
a
Solving (5) formally by successive approximation, we obtain approximate
solutions:
*iW = exp((^~a) Л(а)) x0,
xn+i(t) = exp((z — a) A («)) x0
j
+ J exp ((£ — 5) A ($)) (Л (5) — A (a)) xn (s) ds.
a
Hence the solution of (5) would be given formally by
* See also Supplementary Notes, p. 468,
5. The Method of Tanabe and Sobolevski
439
t
x (t) = exp ((i — a) A (a)) x0 + / exp ((/ - s) A (5)) R (s, a) x0 ds,	(6)
A
where
R(t, s) = ^Rm(i,s)t
Ri{i> 5) = G4 (*) — A (*)) exP((* — s) A G)), s < t
= 0, s Z,
Rm (Л 5) = f Rx (t, a) 2?w_! (a, s) do (m = 2,3,.. .).
(7)
Justification for the above formal procedure of integration has been
given by H. Tanabe [2] using the theory of holomorphic semi-groups
as given in Chapter IX, 10. We shall follow Tanabe's approach, and
assume the following conditions:
For each U. [л, 6], A it) is a closed linear operator
with domain dense in X and range X such that
the resolvent set q (A (/)) of A it) contains a
fixed angular domain 0 of the complex Л-plane
consisting of the origin 0 plus the set {A;—
arg Л < в with в > я/2}. The resolvent (Л/ — A (/))-1
is strongly continuous in t uniformly in Л on any com-
pact set C &.
There exists positive constants M and N such that,
for А 6 0 and t £ [a, d], we have ||(Л/ — A (t))^1 [| TV
(|Л | — M)1 whenever |A | > M with N = 1 for real Л.
(8)
(9)
The domain D(A (/)) of A (t) is independent of t so
that, by the closed graph theorem in Chapter II, 6, the
operator A (t) A (s)^1 is in£(X, X). It is assumed
that there exists a positive constant К such that ' '
11A (t) A (s)-1 — A (r) A (s)-1 j I К 11 — r I for s, t and
ML
Under these conditions, we can prove
Theorem. For any %0 С X and s with a <? s < b, the equation
dx(t)jdt = A (t) x(t), x(s) = x0, s < t b	(3')
admits a uniquely determined solution x (t) С X. This solution is given by
x (t) = U(t, s) x($) = Uit, s) x0, where
U {i, s) = exp ((i — s) A ($)) + W (t, s),	‘
W(t, s) = f exp ((/ — a) A (a)) R(a, s) do with R(i, s) given 
by (7).
(И)
(12)
440 XIV. The Integration of the Equation of Evolution
For the proof, we prepare three lemmas.
Lemma 1. R(t, s) satisfies, with a constant C,
||K (t, s') || КС  exp (КС (£-$)),	(13)
and R{t, s) is strongly continuous in a s < t b.
Proof. By (8) and (9), we see that each A (s) generates a holomorphic
semi-group which is given by (see Chapter IX, 10)
ехр(£Л($)) = (2tu)-1 У (A7 — A (s))-1 d%, where C! is
c'	.	(14)
a smooth contour running from oo e ,e to oo elG in 0.
Hence, by A (s) (Я7—A (s)) =Я (Я7 — A (s))-1 — I, we have, for (6 — «) >
t> 0,
||ехр(£Л (s)) || S C and ||Л (s) exp (tA (5)) ||	C/-1,
where the positive constant C is independent of t > 0	(15)
and s E [«,	•
We have, by (7),
7?i (/, 5) = (A (t) — A (5)) A (s)”1 A (s) exp ((£ — s) A (s)), t > s,
and so, by (10) and (15),
||2?1(Cs)|| ^KC.	(16)
It is also clear, from (8) and (14), that R^t, s) is strongly continuous in
a s < t < b. Next, by induction, we obtain
i
||K„(L s) || ^ J II^JC о) ||  ||/?„,_! (cr, s) || do
5
i
J (КС)" (<7 - s)M-2(|m - 2)~1da
s
= (KC)m (г-s)”*-1 (|m- l)-1,
and hence (\з) is obtained. In the same way, we see that R (t, s') is strongly
continuous in a < s < £ < b.
Lemma 2. For s < т < t, we have
where Q is a positive constant which is independent of s, т and t.
Proof. We have, by (7),
s) — (T- 5) = (4 (0 — A W) exP ((^ — s) Л (s))
+ (A (^) — Л (s)) [exp ((£ — s) A (s)) - exp ((t — s) A (s)) ].
5. The Method of Tanabe and Sobolevski
441
By (10) and (15), the norm of the first term on the right is dominated by
KC[t — r) (t — s)'1. The second term on the right is
j
= (Л (t) - A (s)) J exp (aA (s)) = (Л (r) — A ($)) A (s)”1
t— s
X f A (s)2 exp (o A ($)) da,
T^S
and we have, by (15),
I	i—S
f A (s)2 exp (a A (s)) do\f l| (Л (s) exp (2 1 a A (s)))2 || do
Т-s	ii r-S
/ — 5
/ (2ОД2 da = 4C2
T— s
r___1“| 1 5
-^J	= 4C2 (/ - - t) (I - s)-1 (r - s)"1.
T — S
Therefore
R^t, s) - ^(r, s) || XC(1 + 4C)~
(18)
On the other hand, by (7),
CxC	co
Rjh (t, $
m = 2	m = 2
t
= f a) 7?(tr, s) do
s
T	t
— f R1 (t, a) 7? (a, s) da = f (/, a) J? (a; s) do
5	r
1
+ У (Rx(t, a) — Rx(t, ct)) R(a, s) da.
s
The norm of the first term on the right is dominated by
f 112?x(^, <т) 11 117? (<r, s) || da g A?2C2 exp (KG (b —a)) (t  -t) .
T
The norm of the second term on the right is, by (13) and (18), dominated
bv
f 117?г (t, о) — 7?x (t, or) 11 11R (o, s) | [ da
$
< /С2С2(1 + 4C) exp (KC(6 — a)) f (t — r) (^ — o)"1 da
s
= K, (( - t) log ^4 .
Therefore we obtain (17).
Lemma 3. For s < t, we have
||XW{exp((<-S)AW)-exp((<_S)XW)}i|SC„ where
C2 is a positive constant independent of s and t.
442 XIV, The Integration of the Equation of Evolution
Proof. We obtain, from (14),
Л (t) {exp ((Z — s) A (Z)) — exp ({Z — s) A (s))}
= f A (Z) (Л/ - A (Z))"1 (A (t) - A ($)) (Л/ - А .
c
On the other hand, we have A (i) (Л/ — A (Z))-1 = Л(Л/ — A (£))~T — I,
and so, by (9),
||A (t) (Л1 - A (Z))^||	+ 1 for ЛС (9 and/€ [a, 6].	(20)
Hence we obtain (19) by (10) and
11 (A (Z) - A (s)) (II - A (s))-> 11
.< II (A (Z) - A (s)) A (s)~* || ||И (5) (XI - A (s))-> ||.
Proof of the Theorem. We rewrite W(i, s) given in (12) as follows:
W (t, s) = f exp ((Z — t) A (Z)) R (t, s) dr
s
+ J {exp((Z — t) A (t)) — exp ((Z — r) A (Z))} R (t, s) dr
s
+ f exp ((Z — t) A (t\) (R (r, s) — R (t, $)) dr.
By approximating the integrals by Riemann sums and making use of the
closure property of the operator A (t), we see that we can apply A (Z) to
each term of the right side of the above equality. For, by (19), we can
apply A (t) to the second term of the right side; and also to the third term
of the right side by (15) and (17); we also have, by A (t) exp ((Z — r) A (Z))
= — d exp ((Z — t) A (t^ldr,
A (Z) f exp ((Z — t) A (Z)) R (t, s) dr = {exp ((Z — s) A (Z)) — 1} R (t, s).
Hence we obtain
A (Z) U (t, s) = A (Z) exp ((Z — s) A ($)) + {exp ((Z — s) A (Z)) — 1} R (t, s)
+ f A (Z) {exp ((Z — т) A (t)) — exp ((Z — t) A (Z))} R (t, s) dr (21)
+ f A (Z) exp ((Z — t) A (Z)) (R (T, s) — R (Z, s)) dr.
s
The above proof shows that A (Z) U (t, s) is strongly continuous in
a s <_ t b and
IM « W(t, S) II C, and || A (I) U(t, s) || g C3(Z - s)-‘.
where C3 is a positive constant independent of s and Z.
Comments and References
443
We next define, for s < A) < t,
t—h
Uh(t,s) = exp((i —s) Л(х)) + f exp((/ — т) A (т)) R(t, s) dr. (23)
3
Since a holomorphic semi-group exp (tA («)) is differentiable in t > 0,
we have
4 J7A (f, s) = A (s) exp ((/ — s) A ($)) + exp (hA (t — A)) R(t — h, s)
t-h
-r f A (t) exp ((t — т) A (t)) R (t, s) dr.
Hence we have, by (7),
£ U„(f. s)~A W Uk (f, s) = exp (h A (I - Л)) J? (t - h, s) - /?, (t, s)
Gv
(-Л	(24)
— f R-i {t, r) R (t, s) dr.
By (8), (13) and (14), exp (А Л (г1— A)) R (t — A, 5) tends ’strongly to
R (t, s) as A j 0. Thus we have
t	\
R (t, s) — Rx (t, s) — f R-l (t, a) R (a, s) da\xQ, x0 G X.
S	/
(25)
The right side must be 0 as may easily be proved by (7). Since we have
s-lim A (t) Uh{t, s) x0
= A (t) U(t, s) x0
by the reasoning used in proving (21), we obtain from (25)
-lim
4.0
t/A (t, s) xQ = A (^ U(t, s) r0 for t >
s and x0 £ X.
(26)
The right side of (26) being strongly continuous in t > s, we see, by
integrating (26) and remembering s-lim Vh[t, s) xQ~ U{t, s) x0, that
л 4,0
-^7 U (t,s) xQ = A (t) U {t, s) xQ for t > s and x0 G X.	(27)
С/И
Therefore, x(t) = U (t, s) x0 is the desired solution of (3'). The uni-
queness of the solution may be proved as in the preceding section.
Comments and References
The above Theorem and proof are adapted from H. Tanabe [2]. To
illustrate his idea we have made Tanabe's conditions somewhat
444 XIV. The Integration of the Equation of Evolution
stronger. Thus, e.g., the condition (9) may be replaced by a weaker
one:
11A (£) A (s)-1 — A (r) A (s)^311 Kr\t — with 0 < @ < 1.
For the details, see the above cited paper by H. Tanabe which is a refine-
ment of H. Tanabe [3] and [4]. It is to be noted that the Russian school
independently has developed a similar method. See, e.g., P. E. Sobolevski
[1] and the reference cited in this paper. Of. E. T. Poulsen [1].
Komatsu’s work. H. Komatsu [1] made an important remark re-
garding Tanabe’s result given above. Let A be a convex complex neigh-
bourhood of the real segment [a, considered as embedded in the com-
plex plane. Suppose that A (/) is defined for t £ A and satisfies (8) and (9)
in which the phrase [a, b]” is replaced by “t^A". Assume, more-
over, that there exists a bounded linear operator Ao which maps X onto
D, the domain of A (/) which is assumed to be independent of i C A, in a
one-to-one manner and such that B(t) = А (Z)A0 is strongly holomorphic
in t^A. Under these assumptions, Komatsu proved that the operator
U (t, s) constructed as above is strongly holomorphic in t 6 A if
[arg (2 — s) | < 60 with a certain i90 satisfying 0 < 0O < я/2.
The result may be applied to the "forward and backward unique con-
tinuation" of solutions of temporally inhomogeneous diffusion equations
as in Chapter XIV, 1. In this connection, we refer to H. Komatsu [2],
[3] and T. Котакй-М. Narasimhan [1].
Kato’s work. To get rid of the assumption that the domain D(A (£))
is independent of t, T. Kato [6] proved that, in the above Theorem, we
may replace condition (10) by the following:
Foracertain positive integerk, the domain D ((—A(i))1/A)
is independent of t. (Here (—A (())^A is the fractional
power as defined in Chapter IX, 11.) Further, there .	,
exist constants K2 > 0 and у with 1 — Л"1 < у 1	' '
such that, for s, t C [a, b], || (— A (t))1/A (—A (s))1/A — 11|
k x, |;-sy.
Comments and References
I 15
Tanabe’s and the Kato-Tanabe recent work. With the same uu>l iv.d imi
as Kato,H. Tanabe [1] devised a method to replace rondilion (l<>) by
A (t)~l is once strongly differentiable in u : I ' b
and such that, for positive constants K3 and л,
dA (t)~l dA (s)—1 i
dt	ds
<	!/ —s|\
-- I	I
Further, there exist positive constants N and q with
0 < 2 = 1 such that
(10)"
A (/))-!
/Р-1, o<l.
For details, see Kato-Tanabe [8]. Their point is to start with the first
approximation exp ((/—a) A (t)) x0 instead of exp ((t— a) A (a))x0.
Nelson’s work on Feynman integrals. The semi-group method of
integration of Schrodinger equations gives an interpretation of
Feynman integrals. See E. Nelson [2].
The Agmon-Nirenberg work. Agmon-Nirenberg [1] discussed the
behaviour as t f oo of solutions of the equation — Au = 0 in
some B-spaces.
6. Non-linear Evolution Equations 1 (The Komura-Kato Approach)
Let A be a real or complex Banach space, endowed with a semi-scalar
^product [x, y] such that (see Chapter IX, 8)
[«1+ oc2x2, y] = oq IX, y] + a2 (x2, y] , | [%, у] | й | И Ы ЫI and
[x, %] - ||%||2,	(1)
and consider a family {7\; t^ 0} of non-linear mappings of X into X
satisfying conditions:
(TtTs = Tt + „ To — the identity mapping I (semi-group property),
|| Ttx — Tty|[< ||% — у11 (contraction property) ,
and strong continuity in / of Tt. As in the case of linear mappings, we
define the infinitesimal generator A of {Tt; t> 0} by
A • x = s-lim A-1 (Ttx — %) .
A j. 0
Then A must be dissipative in the following sense:
Re [A x — Ay, x — y] g 0 .	(2)
The proof is easy, since we have
Re [h~r{Thx - x) - h-1 (Tky - y), x - y] - h~x Re [Thx - Thy, x - y]
446 XIV. The Integration of the Equation of Evolution
- Л-i [x -y,x-y]^ ^[T^x - 7\y||*||* -Jll - Ml* "Ml8
£ A_1||x — y||2 — ^-1||ж — y|)2 = 0.
A celebrated example due to Y. Komura [1] is this: Let X = R1 with
[%, y] = x • у and ||M| = |xf, and
(max (x — t, 0) for x > 0 ,	(- 1 for x > 0 ,
for x^O,	0 for x 0.
By virtue of (2) we can prove that, for all Я > 0, the mapping (I — Я A) of
D(A) into X has an inverse Jx = {I — Я A)-1, because
xx — ЯАхх = z = x2 - Я A x2
implies
0 = [(*i - AAxJ - (хг - ЛАхг), xY - хг]
= [%x — х2, хг — x2] — Я [А хг — A x2, x1 — x2] so that
0 = ||%1 — x2||2 — Я Re [А хг — Ax2, хг
*2] 11 *i ~ *2| l8Xe.,	= х2.
Therefore, the theory of linear contraction semi-groups (Chapter IX, 8)
suggests us to approximate the non-linear evolution equation
= Au(i) for t 0 with и (0) = x0 (A) by equations (Я > 0)
U* к
j? uft)
—j-—= AA«d)(f) for 0 with м<А)(0) = x0, where Ад= Я'1 (JA— 1),
Lv ***
under the assumption that D(Jn) = R{I — ЯА) = X, expecting that we
shall have w(£) = s-lim u^ (t).
A J.0
In the above example of Komura, we have D(Ji) = (— 00, 0] U (Я, 00)
which does not coincide with X = R1. To obtain D (JA) = X — R1, it will
be necessary to extend A to a multi-valued mapping A by
for
for
for
x > 0,
x ~~ 0,
x< 0,
for which the dissipative property is preserved in the following sense:
Re <yx -y2, хг - хг)^ 0 for any {x(-,yj with yf C Axt (i = 1, 2). (2)'
In this way, we are led to the following setup for our further discus-
sions. Let X be a Banach space whose dual space is denoted by X'. An
element of X x X will be written in the form {x, y} where x and у are
both (>Y. For a subset A of X x X, the following notations will con-
veniently be used:
1)	£>(A) = {x; {%,y} £A for some y},
2)	R(A) = {y; {%,y) £A for some %),
6. Non-linear Evolution Equations 1
447
3)	A-1 = {{y, x};{x,y) {A},
4)	A A = {{%, dy} ; {x,y} and for real A),
5)	A + В = {{я, у + z] ; (x, у] £A and {x, z} ,
6)	Xx = {y; {%,y} £ Л for a fixed x £ D (Л)},
7)	]|Лх|| = inf {||y||;y £/!%},
8)	JA = [I — A A)-1 = {{x — Ay, x} ; {x, y} ^A} for real A and
I = {{%, %}; % С X},
9)	Ал - {{x - Ay, у} ; {x, у} £Л} ;
we have
ААл = /л-1.	(3)
If A x consists of a single element for all x (F (Л), then A is the graph
of a uniquely determined function (= single-valued mapping) from D(A)
with values in X; in this case, we may identify the set A and the above
function.
To state the notion of dissipative sets in X x X, we give
Definition 1. By the duality map of X into X‘ we mean the multi-
valued mapping F from X into X' defined by
F(x) = {/ С X'; <x,/> = ||<=ll/il2}-	W
That F(x) is not void is clear from the Hahn-Banach theorem. If X is a
Hilbert space, then F. Riesz’ representation theorem asserts that F{x)
consists of x alone and <y, F(x)) = (y, x), the scalar product of у and x.
Definition 2. A set A in X x X is called a dissipative set if, for arbitrary
two points {%p yj and {x2, y2} of A, there exists an / £ F (aq — x2) such
that Re <У!-у2, />< 0.
%
Remark. A is a dissipative set iff — A is a monotonic set in the sense of
G. Minty [1].
We also give
Definition 3. Let D be a subset of X, and T be a function from D
into X. T is called a Lipschitzian mapping (function) with Lipschitz
constant k > 0 if ЦГй?! — Tx2|| A||xi ~ xz\\ f°r a^ xv x2 £ we can
take k = 1, then T is called a contraction mapping (function). We have
Lemma 1 (T. Kato [11]). Let x,y^X. Then ||x — dy||^ ||x|| for
all A > 0 iff there exists an f £F(x) such that Re (y,	0.
Proof. The assertion is trivial if x = 0. So we shall assume x 0 in
the following. If Re (y,	0 for some f£F(x), then ||x|]2 = (%, /)
= Re (x, />£ Re (x - Ay, f)^ || x - Ay ||  ||/||. Since ||x|| = ||/||, we
obtain ||%||	|| x - Яу||.
448
XIV. The Integration of the Equation of Evolution
Suppose, conversely, that |]x||g |[% — Яу|| for all Я > 0. For each
Л>0, let fxQF (x- Лу) and put gx = so that 1- Then
ll*IK ||%-Яу|| = <%- Лу, £я> = Re <x,gx> - Я Re <y, gx)^ ||x|| - Я Re
<y,g;) by 1Ы1 = ! Hence
lim inf Re (%, gx"} = |\x
Я 1 0
and - Я Re <y,g;>^ 0.
Since the closed unit sphere of the dual space X! is weakly* compact (see
Theorem 1 in the Appendix to Chapter V, 1) the sequence {&/*} has a
weak* accumulation point g £ X' with ||g,|] = 1- Thus we see that g must
satisfy-Re (x,	||x|| and Re (y, g)^ 0, that is, we must have ||g|| = 1
and (x, g) = ||%||. Therefore / = ||%||- g satisfies / and Re (y,	0.
Corollary. A is a dissipative set iff
IK^i ~ Яуг) — (*2 “	11= llxi ~ *2|| whenever Я > 0
and {х^ у J C A (i = 1, 2) .
We have the following
(5)
Proposition 1. If A is a dissipative set and Я > 0, then Jx and Ax are
both single-valued mappings, and
I I7i*i - 7л*2|Ш К “ *zl I for	*2 € D (7;) ,	(6)
1Ил*1 - A;X2\\ < IK - *all f°r *1> *2 £ D{AX) - D(JJ .	(7)
moreover, Ax is dissipative and
AJxx = A (Jxx) 3 Axx for %££>(JA),	(8)
Ря*1К|И%|| forall x £ D(A) r\ D(JX) .	(9)
Proof. That Jx and Ax are both single-valued is clear from (5). We
also have (6) by (5), and thus (7) is proved by (3) and (6). Next let
f (x1~ xz). Then we obtain, by (3), (4), and (6),
£e (Ллхх - Axx2, f) = Я1 Re - xt) - (Jxx2 - xs), f)
= Я-1 Re - Jxx2, /> - Я"1 <%x - x2, /> £ Я-1 ИЛ*! - 7я*2||' ll/ll
-	<*1 - *2, />^ Я”1 11%! - %2||2 - Я"1!^! - %2||2= 0.
This proves that Ax is dissipative. (8) is clear. (9) is proved as follows.
By (3) and (6), we obtain, for any у £ Ax,
лц-MHLM - ^\\=\\j^ - jx* - ад и s и* - (% - адп=wn.
Lemma 2 (Y. Komura [1]). Let A be a dissipative set СЛ" x X, and
assume that D (7л) = X for а Я > 0. Then D (7^) = X for every fi > 0
with 0^ K/л — Я)//*[< 1.
6. Non-linear Evolution Equations 1
449
Proof. Take any point x £ X, and consider a single-valued mapping T
defined by
Л и — л
— X -j---------2
Я Л
ХЭ2^Т? = /Л
Since Jx is a contraction mapping by (6), we obtain
||T2 - Тк/Цё

that is, T is a Lipschitzian mapping with Lipschitz constant
a = |(^ - W|< 1.
Thus, for n > m and for any point z £ X, we have
||Т"г — Tmz\\<L amj|Tn~mz — .z||= «m(||Tz ~ 2|[ + ll^2 — Tz\\ + ’ ' ')
am(l + a + a2 + •  •)  ||Tr — am(l — a)*1)] Tz — z||.
Hence, by the completeness of the space X, s- lim Tnz = у esists in X.
я—>0
T being continuous as a Lipschitzian mapping, we obtain T  у = у by
у = s-lim Tn+1z = s-lim T{Tnz). Hence
я^О	n—>o
/Я w — Л \	,/	, / 1	1 \\
у = a Lx+~~y)=Ji у -я Ly ~ уx)) 
,	. /I 1 \ _ .
that is, \ —	£Ay>
and so у — fiz = x with z £ Ay. This proves that Jpx = y. Since x was
arbitrary, we must have D (Ju) = X.
Corollary, Repeating the above argument, we can prove that
D (J^) = X for all p. > 0.
We are now able to give
Definition 4. A dissipative set A g X x X is called a hyperdissipative
set if D (J^) = X for some Л > 0 and hence for all Л > 0.
Proposition 2. A hyperdissipative set A g X x X must be maximally
dissipative in the sense that there does not exist a dissipative set В g X x X
which contains A as a proper subset.
Proof. Let us assume that a dissipative set В g X x X contains A as
a subset. Let у £ Bx. Then, since A is a hyperdissipative set, there exists
a point C A such that x — у = хг — Since A g B, we must
have {xj, yj £ В and hence we obtain x = x1 and у = уг by (5) applied to
the dissipative set B.
We have thus prepared tools for the exposition of Komura's ap-
proach to the integration in Hilbert space of non-linear evolution
equations. Let H be a real or complex Hilbert space with the scalar
29 Yosida, Functional Analysis
450
XIV. The Integration of the Equation of Evolution
product (x, z), and A Q H x H be a hyperdissipative set. We shall be
concerned with strong solutions of the initial value problem:
£Au(t) for almost every t on the interval [0, oo),
«(0) = x0^D(A) ,
where и (t) defined on [0, oo) with values in H is called a strong solution
of (10) if u(t) is strongly absolutely continuous in t, u{t} is strongly dif-
ferentiable almost everywhere on [0, oo) such that the strong differential
quotient du (t)fdt is Bochner integrable on any compact interval of t and и
satisfies (10). We shall approximate this problem (10) by
—(£) = 0 for all t on [0, oo) (Л > 0),
и(л) (0) = xA = x0 — 2ya with a fixed y0 £ A %0.
Since Ал with Л > 0 is a Lipschitzian mapping, we see that equation (10)'
admits a uniquely determined solution wO) (f) which can be obtained by
E. Picard’s successive approximation:
«»+	=^+ f AA^n}(s) ds (и = 0, 1 ... ; u^(t} = %A) .	(10)"
0
Lemma 3 (Y. Komura [1]). For the solution w<A) (/) of (10)', we have
the following estimates:

d
ds
for 0 s < t ,	(11)
u® (s)
and
- «W(f)|!as ((A-^ + W+^iiwr (12)
Proof. Putting wO) (£ + Л) =	(^ with h > 0, we obtain
d
dt
|hW(i) - u0)(i)||2 = 2Re (-£-гНлф) -
\ a v
(t), y0) (/) — «0) {£) j
= 2Re (ЛАуО)(/) _ ЛАм0)(/), w0)(/) - u^{t))	0
by the dissipative property of Ал. Thus ||u0) (t -p h) — u0) (z)|| is monoton
decreasing in t and so we obtain (11). Therefore we have
dt M<A) $
= | \A^ (Z) 11 | \A^ (0) 11 = 11Ал (x0 - Лу0) 11 = | Ы |.
(13)
By a similar argument as above, we obtain, by ЛАЛ = I,
||М(Я)	(;)||2_|[W(A)(0)-«W (0)||M|w(h (fy-uW (/)||2_||(Л-^)y0||2
||w<A)(s) — wW(s)||2ds
о
6. Non-linear Evolution Equations 1
451
= 2 f Re	— А^гА*1) (s'), «<A)(s) — wW(s)) ds
о
= 2 f Re (ЛАи<л) (s) - A^uW (5),	(s) -	($)) ds
0
— 2 f Re (Лд«(А) (s) — А^цЮ (s), ЛЛАм(л> (s) —	(s)) ds .
0
The first term on the extreme right is non-negative, because A is a
dissipative set in jT x H and ЛдМ<л) (s) £A (s) by (8). The second term
on the extreme right is, by (13) smaller than 4/(Л + /z)||y0||2 s0 that
1И)(0 - MW(OI12 - (/* - Я)21Ы12^ 4Ф + ^)|WI2-
This proves (12).
Corollary.
s-lim w<A) (2) = w (t) exists uniformly on every compact set of t. (14)
л 4.0
s-lim= u(t) uniformly on every compact set of t. (15)
л ;o
Proof. (14) is clear from (12). We also have (15) from (14), since we have
- ^)(/)|H A||TAMW(i)||g А1Ы1 by (13).
Lemma 4 (Y. Komura [1]). «(/) £D(A) for all 0.
Proof. By (8) we have, for fixed t > 0,
{7aw(a) (t), Ахи<М (£)} С Л for every Л > 0 .	(16)
Thus, setting Лд«<л> (/) = w(A) (/), we obtain ||^<A) (i)||^ ||y0|| by (13). Hence,
by the local weak compactness of the Hilbert space H, there exists, for
fixed t > 0, a sequence {Лм} of positive numbers satisfying Лп |0 and such
that
KnlimK'<A’,)(0 = w(t) H with ||K'(0ll^ IWl•	(^)
n—>00
Since Л is a dissipative set, we have, for every point {%, y} £ A, the
inequality Re(y —	(/), x —	(f)) £ 0. Hence, by letting nf 00,
we obtain Re(y — w(t), x — u(t))	0 by (15). This proves that
w(t}}£A ,	(18)
because Л is a maximally dissipative set С H x H as proved in Pro-
position 2.
Lemma 5 (Y. Komura [1]). w(/) is strongly absolutely continuous in
t, and it is strongly differentiable at almost every t 0.
2f>*
452
XIV. The Integration of the Equation of Evolution
Proof. By (13), we obtain, for 0 q < t2<  •  with £ (fi+1 — ff) < oo,
*i+i
IK&+i) -	/
ds^ (ti+1 ~ Q • ||y0||.
~	(s)
as	' '
Therefore, by letting Я |0. we obtain
2? ||«(<i+1) -«(Olis S - it)  Ы|
i	i
(19)
This proves the strong absolute continuity in t of и (f).
Next let 0< i0< oo. Then, by (19), the set {«(£); Og Zo} is com-
pact in H, hence it is separable. Therefore, without losing the generality,
we may assume that H is separable. Thus let {vA} be a strongly dense
countable sequence of H. Each numerical function vft(f) = xk) is
absolutely continuous by (19), and so there exists a set Nk of measure
zero such that vk(i) is differentiable on [0, f0] — Nk. In view of (19), we see
CO
that w(f) is weakly differentiable on [0,	— U Nk, and the weak
£ = 1
derivative u'(t) satisfies j|w'(f)j|^ H being assumed separable, the
weakly measurable function ur (/) is strongly measurable by Pettis’
theorem (Chapter V,4) and so, by the boundedness condition
||m'(f)||^ ||y0)|, «'(/) is Bochner integrable. Therefore, by Bochner’s
theorem (Chapter V,5),
«(t) — и (0) — f и' (s) ds	(20)
о
is strongly differentiable for almost every t £ [0, t0] with «' (Z) as the
strong derivative.
We are now able to prove Y. Komura’s
Theorem. «(0 is a strong solution of the initial value problem (10).
Proof. For a fixed positive number Zo, we define the space Z2 ([0, i0],H)
of Zf-valued strongly measurable function % = x(t) such that
f IH5)II2 d$< oo, and normed by ||%||2= f ||%(s)||2 ds. It is easy to
о	о
show that L2([0, Zo], H) is a Hilbert space with the scalar product
^0
(x,y} = f (x(s),y(s)) ds,	(21)
0
where (%(s), у (s)) is the scalar product in H of %(s) and у (s).
Thus our « = «(/) can be considered as an element of L2([0, Zo], H),
because u(t) is a strongly consinuous Zf-valued function. Moreover, the
hyperdissipative set A can naturally be extended to a hyperdissipative
set A £ £2([0, Zo], H) x Z,2(0, Zo], H) by defining
6. Non-linear Evolution Equations 1
453
Ax = {y £ Z,2 ([0, £0], H);y(i] £ A x(t) for almost every t £ [0, if0]}. (22)
We have, by (16) and (13),
’ Али<*>} 6 A for every Я>0.	(16)'
Again by (13), {Лди<А>; Л > 0} is a norm bounded subset of the Hilbert
space L2([0, t0], H). Hence there exists a sequence {A„} of positive
numbers satisfying |0 and such that
№-limi>> = w €£2([0, i0], H) .	(17)'
Moreover, we have, by (15),
s- lim = Й e L« ([0, У, H) .	(15)'
n~* OQ
A is a hyperdissipative set by the fact that A is a hyperdissipative set.
Thus, by Proposition 2, A is a maximally dissipative set. Therefore, we
have, as in the proof of Lemma 4,
w(t) £ A for almost every t £ [0, f0] .	(18)'
On the other hand, we have
t
и(л»)	__	(Q) — у AL- (s) ds
о
(for 0 f £ tQ),
and so, for every x QH,
t
(w(JW (/), x) - (хЛп> x) = f	(s), x) ds (for 0g К i0)
0
by Collollary 2 in Chapter V,5. Hence, we obtain, by (14) and (17)',
i	t
%) — (x0, x) = f (w(s), x) ds = ( f w(s) ds, x) .	(23)
о	о
Since x £ H and > 0 were arbitrary, we obtain, by (23), (18)' and (20),
that _	= u' (i) = w(t) (Au (i) for almost every t 0.
Remark. T. Kato [11] extended the above Theorem to the case
where X is a Banach space whose dual space is uniformly convex. An
almost complete version of the Hille-Yosida theorem for non-linear con-
traction semi-groups was devised and proved in Hilbert space by Y.
Komura [2]. Cf. M. G. Crandall-A. Pazy [1], T. Kato [12] and J. R.
Dorroh [1] cited in § 5 of Y. Komura [2]. In this paper of Komura, the
most crucial point is the proof that the infinitesimal generator is densely
defined, and this proof was later fairly simplified by T. Kato [12].
Recently, H. Brezis has published a comprehensive treatise on non-
454
XIV. The Integration of the Equation of Evolution
linear semi-groups in Hilbert space. This book (H. Brezis [1]) contains
an extensive bibliography which is not necessarily restricted to the
Hilbert space.
7. Non-linear Evolution Equations 2 (The Approach through the Crandall-
Liggett Convergence Theorem)
Let X be a real or complex Banach space, and A the infinitesimal
generator of an equi-continuous semi-group {7\; t 0} of linear operators
(L(X, X) of class (Co). Then the Lemma in Chapter IX, 12 gives the
proof of E. Hille’s approximation of T t:
For every x0 £ X, TtxQ = s- lim (I — n~1iA)~n • x0, and the
(1)
convergence is uniform on every compact interval of t.
The titled convergence theorem is suggested by (1). It reads as follows.
Theorem 1 (M. Crandall-Т. Liggett [2]). Let A be a hyperdis-
sipative set С X x X. Then, for all x0 £ D(A),
s- lim (J(/n)n  x0 exists uniformly on every compact interval of t,
n —> OO
where J^ = (I — ЯА)-1 as in the preceding Section 6.
The following proof is adapted from S. Rasmussen [1] which seems
to give a good modification of the original proof by Crandall-Liggett:
Proof. The first step. We shall prepare the following (2) —(4):
||Аля||£ ||A x|| forall x£D(A), and for all Я > 0 ,
tt
П Jh x - x
i = 1
(tt	\
»= i	/
for all % C D (A) and for all Я, > 0 (i = 1, 2, . . . ,n").
Jp (y x +	7л*) = Jzx f°r all x £ X and for all Я, ц > 0 .
(2)
(3)
(4)
(2) tyas proved in the preceding Section 6. The proof of (3) is obtained by
the contraction property of (2) and ААЛ —	— I as follows.
tt
ПJMx-x
t = 1
tt / tt
X П
i = I \/ — i
n	«
П J 1}х- П J^X
j = i	f =»4-1
tt
2\\h<x - *14 2 НМл(*14
»=1	i = l
tt \
2^ 1И<
i-i /
We next prove (4). Since A is a hyperdissipative set, there exists, for any
x^X and Я > 0, a point	such that x = хг — Яуг Thus
Jxx -- and so
7, Non-linear Evolution Equations 2
455
£4 A — U T U .	.	. , A — U
x H д = (xi — АУх) “I A ’	= %i —
proving that fi (xx -	= хг = fix.
The second step. Let x £ D (A) and Л > 0. Let furthermore {jun; 1}
be a sequence such that 0 < //„ Я for all n. We then define
Л
ТП
n
Hfix -fix
d=i
for n, m = 0, 1,2,...
(°	\
We set П J-- I = fi for convenience. I
i = i	/
Moreover, let
К
tn = Z>i(« = 0,l,2,...;/O = 0)
t = 1
and
a„ = //„/Л and fi = 1 —a„ = (Я — ^П)/Я for n = 1,2,..
Then we have
(5)
(6)
(7)
An,m^	[(тЛ - Z„)2 + Я t„] 1fi 11A xj [.
For the proof, we first show
л
for
n, m = 1, 2, , . . .
(8)
(9)
In fact we have, by (4) and the contraction property of Jtln,
n — 1
Xfix
t"l
#71 ^?i—i1	т” Pn^n— 1* m 
We shall prove (8) by (9) and induction. We start with the proof that
Л0)ТП satisfies (8) for all m = 0, 1, . . . In fact, we have
A0,m= ||x-J7%j|g жЯ||А%|| (by (3) and (5))
{[шЛ - i0)2+ тЛ^/2Т LM-^HA^jjjHxfKsince/o-O)
Let now n and m be arbitrary and assume that An,m and A™,™-, both
satisfy (8). We shall then show that An+Vm satisfies (8). In fact, we have
456	XIV- The Integration of the Equation of Evolution
— ^n+1 'Д-пгт—1 "I"	(^У
<*n+i{[((m - IM - Q2 + (m - 1)ЛТ2+ [((™ - IM - У2+
+ &+1{[(жЛ - Q2+ жят+ [(жя - tn)2+ ад/2}Р*!1
Ы((« - lM-^)2+ (rn - 1М2]1/2+&+1 [(тЛ-^2+жЯ2?/2}||Л%||
+ {a„+i [((m - IM - i.)2+ ^y/2 + $n+1 [(жЯ - Q2 + Я^} ||A x||
(an+l+ ^n+l)1/2{^n+lU(m - IM -	+ (m - IM2] +
+	- in)2 + жЯ^р/грхЦ
+ (a„+1+ &+1)1/2 {«n+i[((w - IM - U2 + Щ
+ ftn+i [(wl — tn)2+	x|| (by Schwarz' inequality)
= {т2Л2-ап+1 2m/.2 + a„+M2~ 2mUn + <хя+12Un + t2
+ тЯ2-ая+М2}1/21И^11
+ {ж2Я2-ап+12жЯ2 + а„+М2- 2жЯ/я + а„+12Я^П + Н- Я^}1/2 j|Л х||
(by = 1 — ая_х)
= {т2Я2 - ^п+12тЯ +/z„+M - 2 жЯг„ + 2/г„+1/п +/£ + жЯ2 -/г„+1Я}1/2| И xl I
+ {т2Я2 - //п+12тЯ + /хя+х Я - 2тМп + 2^я+1/я + % + ЯЦ1/21 |Л %| |
= {{^ ~ (^«+i + 2J)2~ f£+\+ тк2У!2\\А х\\
+ {(™Я - (/гя+х + /я))2 - ^2+ j + Я(2„ + j«„+1)} V211А х [ |
{ЦжЯ-/я+1)2+жЛ2?/2+ [(жЯ - *я+1)2 + Я/я+1]1/2}1И<
and we have thus proved that Ля+1)т satisfies (8). We can also prove that
ЛП10 satisfies (8) for all n = 1, 2, . . . In fact, we obtain by (3)
^n,o —
< Ёpc P*ll =
i= 1
П J«,.% - X
i 1
S {[(0  i. - tny + 0 •	+ [(0 • Л - („)> + Яf„]^} I \A x\I.
Therefore, we have completed the induction and hence (8) is proved.
The third step. Let t be a positive number, and consider a partition Д
of .the closed interval [0, £].
1
For this partition, we define
Ai = (tt - ti_-P (i = 1, 2, . . . , n) and |Zl | =
Take another partition
max (^ - tt-У .
4': 0 - to< t(<
and define
t;_r<	z;+1

d-^tj-t'j-!) (j = 1,2, . . . , k) and I4'\ = max (t' -t'_j)
1 ь
7. Non-linear Evolution Equations 2
457
Then, for x0 D (A), we obtain, by taking
Я = max(|d|, |d'|) and m = the largest integer^ t/2.,
the following inequality implied from (8):
Я	fe
ПJ
i=l	i=\
n
П J- J?x,
i = 1
k
ГЛХ.- nj^x9
7 = I
2 {[(жЯ - ty + rfy/2 + [(тЛ - /)2 + Я/]1/2}||Лх0||
(10)
g 2 {[Я2 + U]V2 + [Я2 + B]l/2}j|Xx|| .
Therefore, by takingd,- = //w (i = 1, 2, . . . , «) and Aj = //A (j = 1,2,
. . . , k) in (10), we obtain (1)'.
Remark. We have incidentally proved that
'Ttxf)= s- lim (I — n'} tA'j^x^ = s-lim (I — n~1A)-^-nt]x0 uniformly
w—> oo
 on every compact interval of/, where [nt] = the largest integer nt.
(1)"
(И)
The scope of this Theorem will be seen from Theorem 2, Theorem 3
and Theorem 4 given below.
Theorem 2. Let A be a hyperdissipative closed set of X x X. Then,
for every x0 £D(A), the following two conditions are mutually equivalent,
i): м (/) is a strong solution of the initial value problem
— £ Au(t) for almost every t^ 0 and ы(0) = x0,
as formulated in the preceding Section 6.
ii):
'«(/) = s- lim (J — w_1X)_[”t] x0 and w(/) is strongly
n-+ 00
differentiable at almost every t 0.
(12)
We first give the Brezis-Pazy [2] proof of the assertion i) -> ii), and the
proof of the assertion ii) -> i) due to Crandall-Liggett [2] will be given
later; after the proof of Theorem 3. We shall begin with a key Lemma due
to T. Kato [11].
Lemma. Let .S' be the set of those s > 0 at which the strong solution
«($) of (11) is strongly differentiable. Then, we have, at almost every
s ^5,
2->-A.||„($)|]*= ||«(s)|| £115гШ-= Re(^, /)whenever/fF(«(s)). (13)
Proof, «(s) being a strong solution of (11), the strong derivative
<fw(s)/<fs is Bochner integrable on every compact interval of s so that
/rfu(s) T
—d[~ds
0
458
XIV. The Integration of the Equation of Evolution
by Bochner’s theorem in Chapter V,5. Hence ||и($)|| isofbounded varia-
tion on every compact interval of s. Therefore, ||w (s)|| is differentiable at
almost every s Й 0.
Since Re	j|«(f)|| • ||w(s)|| an(i Re (M(s)> f) = ||m(s)||z by
/ £ F (w ($)), we obtain
Re <u(t) - u{sj, |H$)|| ( II^WH ~ IMS)II) 
Dividing the both side by (/ — s) and letting t -> s from above and from
below, we obtain at almost every s £ S,
Re , /) S 11« W11 FF and Re , /) a | |«(s) 11 FF '
We have thus proved (13).
Corollary (H. Brezis-A. Pazy [1]). For the strong solution w(i) of
(11), we have
du (/)
dt
||Ax0]| = inf | at almost every 0.
(14)
Proof. We put dw [t^dt =y (/) so that, by (11), у (t)	at almost
every 0. For such t and for every z C Ax0, there exists, by the dis-
sipative property of A, an /0 £ F (w (t) — x0) such that Re (y (t) — z, /0)	0.
Hence, by the same argument which enabled us to prove (13), we obtain
2-1411“ W - x»l I2 -1 Iя w - *»l I	11“ (') - *.lI “ Re (-ytr - °' a)
= Re (y (/) — z, /0) + Re (z, /0) sS Re (z, at almost every t 0.
Thus, at almost every t 0, we have
ll«W -*oll Wu& -%oll I <^-/оЖ IHI • |I«W — ^oll
L4 ir
and so, since z £ A x0 was arbitrary,
||w(Z) — ж0||	/||A%0|| at every 0 .	(15)
Next, let h > 0 and put u(t + Л) = v(t). Then
,	C A. ?F) for almost every 0 and v(0) = u(h) .	(11)'
Thus, similarly as above, we obtain, for a certain / £F(v(i) — w(i)),
2-1 -yr |]w (/) — «(^)||2= Re /^7- ~ X о at almost every t 0.
Therefore, we have
| |w (i + k) —w(0|| 11« (s + Л) — w (s) | [ whenever s 0 ,
and so, combined with (15), we finally obtain
|]w(f + h) —m(0|| |IMF) — u (°) 11 S h\И %0|| .	(15)'
This proves (14).
7. Non-linear Evolution Equations 2
459
Proof of the assertion i) -> ii) in Theorem 2. The existence of the right
hand limit of (12) is guaranteed by Theorem 1. Let T be an arbitrary
positive constant. Besides step functions un{t) = {I — n~1A)^nt^x0, we
define on [0, T] another sequence of functions vn(t) by
MO =	4- (f -jin) • и • [«„((; + l)/n) - «„(;/n)]
(for j/n^.	(У+ l)/w ;; = 0, 1, . . . ,[nT] ~ 1)	(16)
= un(/) (for nr1 [nT]^ti T) .
Clearly, vn{t) is strongly differentiable on [0, T~\ except for finite number
of points of the form t = j/n, and for j/n< t< (/+ l)/n we have
w K(0'+ i)/№) MH = «[(/- п~гА)-1ип(Лп) -un{j/n)]
= Allnun{yn) (by Ал = ^(7л~/))	(17)
,	Al/n	Al/n 0/^0 •
By (7), (8) and (9) of the preceding Section and by
M(jM) = AJi/n un((j~ IM Э A1/n и„ ((j - I)/«), we obtain
l)/w)||^ IM ММ l)/n)j|. (18)
J
Thus, by (17), we have
= IMi/nM7'M)IM IM M«(0)ll =11A *ofl for a'e-	(19)
di
We have incidentally proved from the proof of (17)—(19) that, whenever
jM (j+ Vtfn,
IMO - МОП = (HM)  n -||M(7 + i)M) - un(j[n)\\
and hence
IMO - МОП MIMII for all	(20)
We put du(t)jdt = y(t) and n[un(t) — un{t — l/и)] =yn(i). Then, by
(11), we have y(t) ^Au(t) for almost every 0. We have also
yn(t) £ A un(t) when t j/n (j = 0, 1, . . .), because
yn(t) =	~ MO CM - n-^A)-1^ - 1/n) = Aun(t)
by (8) of the preceding Section 6. A being a dissipative set, we obtain,
by (5) of the preceding Section 6, that for almost every i 0 the in-
equality
IIM  un(i) - yn (0] - M  M)—IIм ’ un(t) - • «(Oil 
Therefore we have, for almost every t 0,
460
XIV. The Integration of the Equation of Evolution
dUdt' ~пЫ1) -u(t- 1/n}] = ||n [«„(/) - «n(f- l/«)]
- n[u(t) — u(t - 1/n)] + y(t) — y„(OII
II [n-un(t)	-y(f)]||-||wMn(f- l/n) (21)
— n  и (t — l/и) > n  || (un (t) — м (f) || — n  |j«n (t — 1/n) — u(i
We here extend un (t) and и (/) for negative value of t by
{u„(f) = xQ ~n~*zQ, where zQ is any fixed point A xQ,
«(0 = x0.
Then, integrating the resulting inequality (21) on [0,where n-1g 6 g T,
we obtain
(22)
0
a
f ~wWII di
0 - 1/n
6
du (£)
dt
— n \u{t)	— 1/n)] dt,
that is,
n f
e—i/n
— i/n
du(t}
dt
w Ew(0 ~ «(* - !/«)] ^ + «_1lkll- (by (22)).
Adding these inequalities for 6 = 1/n, 2/n, , . . t N/nwithN = [nT] yields
Njn	r
n У* ||«„(f) - «(/)||ifz g N J
0	0
4
and therefore
dt +
Njn	T
f	T f
0	0
du(t)
dt
— n \u[t) — н {t — \fn)]
n [u(t) - u(t- 1/n)]
dt+ n-'T • ||z0||.
(23)
Since, by the assumption, s- lim n [u(t) — w(f — 1/n)] = du(i)jdt for
rt—> <x
T
almost every t on [0, T], we obtain lim f ||un(Z) —	= 0 by (23)
о
0
0
о
0
0
7. Non-linear Evolution Equations 2
461
and by
— n [u(t) — и (t — 1/w)]
^2-11^1,
t
which is implied by (14) and n [u(t) — и (t — 1 /и)] =
t—l/n
Therefore, combined with (20), we obtain
т
lim f ||M0 - u(t)\\dt = 0 .
о
ds.
(24)
dvn {^)	(0
dt dt
On the other hand, we have, by (14), (19) and by the same argument
which enabled us to prove (13),
2~4w) - IKW - «WII 
Й ||vn(Z) - w(0 II  2 • 1Ы1 for almost every t £ [0, Т].
Therefore, by y„(0) = «(0) = x0, we obtain
IKW -	4 f \\vn(s) - «(s)j^s 'IJ^oll for O^Z^T.
0
Hence, by (24), we obtains- lim = w(/) uniformly on [0, Т]. Thus,
n—* oo
by (20), we have proved that s- lim un(t) = u(t) uniformly on [0, Т].
Ы-+ OO
Theorem 3. Let A be a hyperdissipative set С X x X, and let x0 £ D (A)
and Я > 0. As in the preceding Section 6, the initial value problem
d"U^ (ii
— ,	= Ад«6) (Z) for 0 and (0) = x0 (25)
has a uniquely determined solution «(A)(Z), because Ax is a Lipschitzian
mapping. We have then
(26)
s-	lim nW (t) exists uniformly on every compact interval of t and
A 0
s-	lim WW(Z) = s- lim (I — M"1ZA)-nx0.
. Л 4 0	n—> co
Remark. If the strong solution w{Z) of (11) exists, then, as in the linear
case, и (Z) can be given by Hille’s type approximation s- lim (I —n-1t A)-" • x0
n—> OO
as well as by Yosida’s type approximation s- lim (Z).
Л 4 oo
Proof of Theorem 3. Since Ад = (Jx — I) is dissipative, we obtain, as
in the case of (15)',
||«<»(Z) - «<0 (0)|| S Z • ||л1«о>(0)11 - Z • ||ЛЛ|| = t • 11Л»,-x0||. (27)
On the other hand, it is easy to verify that the solution u^(t) of (25)
with Л 1 is given by
462	XIV. The Integration of the Equation of Evolution
и(1) (t) = e~iX(i + f JlUW (s) ds .	(28)
Hence
«Ю (0 - J\x„ =	(x„ - ГЛ) + j<:-< [/,«« (s) - fix,] ds
0
and so, by the fact that J1 is a contraction mapping, we obtain
t
fs-4\ud>(s}-J^x„\ds. (29)
0
From this we shall derive
||«<4(«) - J?x,|| s |/S‘ || (Л - /)x„||.	(30)
We may assume that (J1 — Г) x0 = Aixo 0. For, if otherwise, the
solution и® (t) of (25) with Л = 1 is given by ut1) (£) = x0 and hence (30)
would be trivial. The assumption (J1 — 1} xQ 0, combined with (29),
gives
t
(pn (t) пе~1 + rpn-i (s) ds, where	(31)
о
We shall prove, by induction with respect to n, that
<pn(t^{(n-ty + t}^.	(32)
This gives (30) by putting t — -и. Inequality (32) is clear for n = 0. If (32)
t
is true for (и — 1), then tpn(i) + f es~*{(n— 1 — $}2 + $}F2cZs _ Denoting
о
the right hand side by yn(t) e~t, we have to show that
Vn(t) e* {(« - t)2 + 01/2-
Since ipn (0) = n, it is sufficient to prove that
y>n(t) = e* {(« — 1 — /)2 + t}1/2^	е*{(п — I)2 + t}1/2.
Th^ last inequality is easily checked and so we have proved (30).
We are now prepared to prove (26). Put — w<A) (Л£). Then, by
Ai = Л"1 (/я - I), we have
^7 — = (}л - I) for all 0 with v<*> (0) = x0.	(25)'
CL f
Taking for and v6) for in (30) and making use of (9) of the preceding
Section 6, we obtain
||«ffl(n) - Дх0|| = ||«W(»A) -	|/«||(J, - I) x„||
S/иЛ||Л,х0|| g ]/йЛ||Лх0||.	(33)
7. Non-linear Evolution Equations 2
463
Let Aft J.0 and let nk = [i/Aft] so that t. Hence, by (1)' and (10), we
obtain 5- lim J^kxQ = s- lim (J — ?гЬЛ)'пл'о. Therefore, combined with
(33), we have
11|Az„|I.
Since was arbitrary, we have thus proved (26).
Remark. The above proof is adapted from H. Brezis-A. Pazy [2]. It
is to be noted here that inequality (30) is due to I. Miyadera-S.Oharu[1].
Proof of the assertion ii) -> i) in Theorem 2. Let u(t) = s-lim
n—>00
(I — n~1tA)~nx0 be strongly differentiable at t0 > 0 with strong
derivative у = tf«(f)/«!t|( = (0 so that
w (70 -f- h) = w(/0) + hy + o(h), where s-lim o(h)/h = 0 .	(34)
h | 0
Since (19) and (20) implies that t -> u(t} is a Lipschitzian mapping with
Lipschitz constant [JA^0|j, we obtain i) if we can prove
WoLj'JM-	(35)
The proof of (35) reads as follows. Let 0 < A < Zo. Then, by the strong
differentiability of и [t) at t = and by the hyperdissipative property of
A, there exists a uniquely determined point {xA, yA} £ A such that
4 - AyA = w (i0 - A) = м(^о) — Ay + o(A) .	(36)
We obtain (35) if we can show
s-lim%A = и (f0) and s- limyA = у ,
Л 10	A 0
(37)
because A is a closed set in X x X by the assumption. For the proof of
(37), we put
xA = x — Ay, where {x,y} is a point of A to be determined later. (38)
Then, by (13) and (25), we obtain a certain / £F(w(z)(£) — xA) such that
2-'-^- ]|«w W -	= Re (0, f) = Re <A^m (/) - A A, f) +
+ Re(AA, Re<y, /> .
because АлХд у and, moreover, Aa is dissipative by Proposition 1 in
the preceding Section 6. Thus
h
||«o> ((. + Л) - zj- I[«^> ((<,) z,||2 < 2 f <y, um (Z„ + t) -z,>, -dr, (39)
0
where
(w, z)s = sup Re (w, k} .	(40)
A£F(»)
464	XIV. The Integration of the Equation of Evolution
It will be proved later that
{w, is upper semi-continuous, i.e., s- lim wn = and (41)
Л —> 00
s-lim zn — z^ implies that lim sup (wn, zn \ S 2«>\.
co	n—> co
Hence, by s- lim it) = w (i) = s- lim (/ — n~1tA)~n x0 proved in (26) and
А ф co	л—> oo
by s- lim xx — x, we obtain
n—► co
IIм (to + Л) - *||2 — IIw (Q - *||2^ 2 f(y, и (t0 + t) - x)3  dr . (42)
0
For every j £F (m(/0) — x), we have
Re <w (tf0 + h) - w(/0), /> - Re <x - «(f0), /> - Re <w (t0 + h) - x, />
£ HM(*0 + ^ - *||  ||* - «(fo)ll^ 2-1 {IIw (fo+*) ~*l|2 + ||*- U(Z0)||a) .
Therefore, by Re (x — w(£0), /} = — ||* — w(t0)||a and (42), we obtain
2 . Re /«(^V «(<<) , /) s 2 / <;, « (<„ + hr) - x),  dz.
4	0
Hence, by the strong differentiability of u(Z) at t = Zo and (41), we have
Re<?./>S <?, «('»)-«>. 	(«)'
We may replace the supremum on the right hand side of (42)' by the
maximum, because the set F (w(t0) — *) contained in the strongly closed
sphere of X' is w*-compact in X' (see Thorem 1 in Appendix to Chap-
ter V, 1). Thus there exists an / £F («(t0) — x) such that
Re <y, A> Re <y, /> for all /, £F («(t0) - x) .
By taking x = xx> у = yx and fx = /, we obtain
Re <y, /> < Re <yA, />, that is, Re <ул -у />£ 0
On the other hand, we have, from (36),
-----A----+ — -У~У1.
andtfience, by (43), we obtain
ll«(*o) ~ *л!12 = Re <«(t0) - xx, /> Й Re <о(Л), /)й о (Л) •)|«(/0) - *А||.
Непсе ||w (t0) — *д|| < о (Я) and so we obtain (37) by (44).
Finally, the proof of (41) is given as follows. Since the set
{/ С X' ; ||/||£ ||*m||} is w*-compact (Theorem 1 in Appendix to Chap-
ter V, 1), we can find fm £ X' such that
= (wm, /m) and ||/w||^
We may assume, without loss of generality, that lim (wm, exists.
(43)
(44)
7. Non-linear Evolution Equations 2
465
Let £ X’ be any one of the ^-accumulation points of {/w}. Then we
have
lim (wm, ~ hm Re (wm, fmy = Re (да^, f&У > |[/0o||^|| z^ ] | ,
m > oo	m—> co
because s- lim wm = and да*- lim fm> = for a suitable subsequence
m—► oo	m'—> co
[m'} of the sequence {m}. Therefore we must have (41).
Remark. It is an unfortunate and a fortiori an interesting situation
that, although s- lim (Z — «-1/A)“n.r0 and s-lim u^(t) both exist and
n—> со	Я j 0
coincide each other under a general condition, this limit need not be
strongly differentiable in t anywhere. Such an example is given in
M. Crandall-T. Liggett [2]. Recently, P. Benilan [1] devised the
following remedy for the above situation. Let u(t) be a strong solution of
initial value problem (11). Let, further, {z, да} be any fixed point of the
hyperdissipative subset A of X x X. Then, as in the case of (42), we obtain
t
| (0 — z\|2 g |\u (s) — |s + 2 J" (да, и (т) — z)s  dx for all 0 s t
and with a fixed initial condition w(0) — x0 £ Z)(A).
P. Benilan [1] called a strongly continuous X-valued function u{t) as'
an integral solution о/ initial value problem (11), if u[t) satisfies the above
condition for all {z, да} £ A simultaneously, and he proved
Theorem 4. Let A be a hyperdissipative set С X x X. Then
u(t) — s-lim «W (/) = s- lim (Z — n^tA)-” • xQ is the uniquely determined
A 0	n—► oo
integral solution of (11).
For the proof of this theorem and further interesting extensions as
well as comments, we refer the reader to P. Benilan [2].
Remark. The scope of applications of Theorems in the present and
the preceding Sections may well be seen, e.g., by reading S. Aizawa [1],
M. G. Crandall [3], Y. Konishi [1] and [2], and В. K. Quinn [1].
Supplementary Notes
Chapter I and Chapter VI
1. For the distributions or the generalized functions due to S. L. Sobo-
lev, L. Schwartz and I. M. Gelfand, see the new and enlarged edition
L. Schwartz [6] and the English edition I. M. Gelfand [6] — [10].
2. For the recent development of the hyperfunctions due to M. Sato,
see M. Sato [2]—[3] and P. Schapira [1].
Chapter VI
1. (Section 7) Sobolev Spaces. The theorem {Sololev's Lemma) given
on p. 174 is a very special case of the so-called “Sobolev Imbedding
Theorems" in S. L. Sobolev [1]— [2]. A comprehensive description of the
development of this imbedding theorems is given in R. A. Adams [1]. See
also E. M. Stein [2].
Chapter X
1. (Section 1) The trace operator or the generalized boundary values.
We can prove
Theorem. Let £? be a bounded open domain in Rn such that its
boundary hypersurface dQ is C2. Then, for any /(%) £ И71,2^) and for
almost all ££ dQ, the limiting value <p(£) = lim/(%) exists if we let x
tend to £ along the normal at the point £ £ Moreover, this boundary
value of /(%) belongs to L2(5&) and satisfies the inequality:
n
IIOILw)^ c {ll/WILw + 2? ||awiL‘W} 
м
where the positive constant C is independent of the individual choice of
/ and the notation means differentiation in the sense of the
“distribution".
The linear map / -> is called the trace operator. For the proof as well
as generalizations of the above theorem, see, e.g., S. Mizohata [6],
R. A. Adams [1] and F. Treves [2].
2. (Appendix to Chapter X/ For extensions with a unified treatment
of the result of R. A. Minlos [1], see L. Schwartz [7] and the biblio-
graphy cited there.
Supplementary Notes
467
Chapter XI
1. For recent development of the theory of contraction operators in
Hilbert spaces, see B, Sz. Nagy-C. Foias [4].
Chapter XII
1. (Section 1) ForChoquet’s refinement of the Krein-Milman Thorem,
see R. R. Phelps [1].
Chapter XIII
1. (Section 1) We can prove (K. Yosida [17])
Theorem 1'. Let P(£, x, E) be a Markov process on the phase space
(S, 93) with an invariant measure m such that m(S) = 1. Then, for any
/ £ L1 (S, 93, m) and t > 0, there corresponds an £ L^S, 93, m) with
ll/lliin such a waythat
f m(clx) f(x) P(t, x E) = f f_t(x) m(dx) for all E £ 93 .	(19)
s	E
Proof. Since P(t, x, E) is a bounded function, f m{dx) f(x} P{t, x, E)
s
does exist for any / £ L1 (S, 93, m) and we have
sup\$m(dx}f(x) P(t, x, E)|£ sup f m(dx)[f(x}\P(t,x,E')^ ll/Hx- (20)
E S	E S
If f £ L1 (S, 93, m) is £ L00 (S, 93, ж), then, by (8), the u-additive function
f m(dx)f{x} P (t, x, E) of E £ 93 is m-absolutely continuous such that
s
\fm(dx)f(x)P(t, x, E) |	||/[]L°° • m{E)> where ||/||L°° = essential sup
S	x
|/(%)|. Next, let / ([ L1(S, 93, m) be real-valued non-negative and put
/(«)(%) = min (/(a), n). Since
fm(dx)f(x)P(t, %, E) = lim f m{dx)f^n')(x) P(t, x, E) for every E £ 93
the left hand side above is ж-absolutely continuous by the Vitali-Hahn-
Saks theorem on p. 70. Thus, in this special case for /, we can define
f_t £ L^S, 93, m) by (19), and we have Н/^Н^ ||/||г by (20). This proves
our Thorem Г.
Remark .Consider the deterministic case given by P (t, x, E) =CE (yt (%)),
defined through a one-parameter family {yt(x); — oo < t < oo} of one-
one transformation x ->yt(x) of S onto S such that m(E) = m(yt • E).
Then we see that the mapping / -> /г in Theorem 1 (p. 381) and the map-
ping / f_t in Theorem Г are respevtively given by /((%) = /(>*(«))
468
Supplementary Notes
and f_t(x) =	In this sense, a Markov process with an invariant
measure m is "time reversible".
2. (Section 1) As a corollary of Theorem Г above, we obtain, as in
the case of Thorem 2 (p. 382), the following
Theorem 2' (K. Yosida [17]). Under the assumption as in Theorem Г,
the following mean ergodic theorem holds good:
s-lim п~г f-к ~ /~* exists in L^S, 23, m) ,	(21)
ff(s)m(ds) = ff~*(s)m(ds) .
S	8
(22)
Chapter XIV
1. (Section 4) We can prove
Theorem 1'. Under the assumption x (0) = у £ D (A (0)) and (3) (p. 431),
the solution of (1)' (p. 432), if it exists, is uniquely determined.
Proof. We follow an argument given in T. Kato [3]. Thus we have,
for 5 > 0,
x (t + <5) = x (Z) + 5 A (Z) x(t) + o(d)
= (I + 5 A (Z)) (7 - dA (Z)) {I - dA (Z))-1* (Z) + о (d)
= (I — dA (t))~1x(Z) - d*A (Z) (I - dA {t))^A (Z) x(Z) + o(d)
= (I — d A (t))-1 x (t) — d ((I — dAty))-1 — !) A(t) x(t) + o(d).
Because of (3), we have s-lim (7 — <5 A (Z))-1 z = z for any z £ X [see (2)
<5^0
in Chapter IX,7]. Hence, again by (3), we have
II* (* + <3)||S||*(Z)|| + o(5)
and so d+ ||x(Z)||/7Z^ 0 which implies that ||x(Z)||	||x(0)|| .
Therefore, the difference between two solutions of (1)' each with the
same initial condition £ D (A (0)) must be 0.
Chapter IX and Chapter XIV
1.	For Linear evolution equations, see S. Krein [2], P. Lax-R. S.
Phillips [4], J. L. Lions [5] and F. Treves [2].
2.	For Linear as well as nonlinear evolution equations, see: V. Barbu
[1], F. Browder [2], J. L. Lions [5], R. H. Martin [1], К. Masuda [1],
I. Miyadera [4] and H. Tanabe [6]. For the last cited book of Tanabe,
an English translation is in preparation,
3.	Currently, convex analysis is playing an important role in non-
linear evolution equations. For this analysis, see, e.g., R. S. Rockafeller
[1] and I. Ekeland-R. Гемам [1],
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[1] A semigroup treatment of the Hamilton-Jacobi equation in one space
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[1] See Kantorovitch-Akilov [1].
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[1] (with H. Hopf) Topologie, Vol. I, Springer 1935.
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Index
absolutely continuous 17, 93, 135,
375
absorb 44
absorbing 24
abstract Lx-space 369
accretive operator 208
accumulation point 3
Achieser, N. I, 354, 439, 469
adjoint operator 196
a.e. 15
Agmon, S. 445, 469
Aizawa, S. 465, 469
Akilov, G, 175, 469
Alexandrov, Pt 469
algebra 294
Banach- — 294
Boolean — 365
almost everywhere 15
almost periodic function 218, 330
Bohr's theory 331
mean value of 222
angle variable 391
approximate identity 157
approximation
Stone^Weierstrass theorem 9
Weierstrass' polynomial — theo-
rem 8, 249
Weierstrass' trigonometric —
theorem II
of Со-functions by (^-functions
29
Aronszajn, N. 96, 469
-Bergman reproducing kernel 96
Ascoli-Arzel^ theorem 85
asymptotic convergence 38, 122
Baire, R.
category argument 11
function 19
-Hausdorff theorem 11
measure 18
set 18
theorem 12
Balakrishnan, V. 260, 266, 469
balanced 24
B-algebra 294
Banach, S. 23, 68, 73, 75, 79, 102,
104, 106, 112, 141, 205, 370, 469
-algebra(or B-) 294
’lattice (or B-) 369
-space (or B-) 23, 52
locally compact 85
reflexive 126, 141
closed graph theorem 79
generalized limit 104
open mapping theorem 75
theorem on a.e. convergence of
linear operators 370
barrel space 138
base, open 60
basis 21
Beboutov, M. 393, 469
Benilan, P. 465, 469
Bergman, S* 96, 469
kernel 96
Bernstein, S. 8
Bers, L. 178, 469
Bessel's inequality 86
bidual space 112
Birkhoff, G-, 364, 378, 470
Birkhoff, G. D. 388, 390, 470
B-lattice 369
Bochner, S. 132, 260, 266, 331, 346,
470
. integral 132
theorem 346
Bogolioubov, N. 393, 470
Bohnenblust, Ht F, 105
Bohr, H. 331, 470
Boolean algebra 365
Borel, E. 18
measure 18
set 18
bornologic space 44
bound
greatest lower 2, 367
least upper 2, 367
488
Index
bound
lower 2
upper 2
boundary
condition 286, 406
function 163
bounded
convergence topology 110
essentially — function 34
linear operator 43
set 6, 44, 367
totally — set 13
boundedness
uniform — theorem 68
Bourbaki, N. 23, 52, 145, 470
Brezis, H, 453, 457, 458, 463, 470
Browder, F. E. 470
Brown, R. 379
Brownian motion 398, 403
B-space 23, 52, 85, 126, 141
canonical
scale 406
measure 406
carrier (= support) 26
Cartan, E. 400
Cartesian product 3
category
first 11
second 11
Cauchy
convergence condition 11, 52
integral representation 128
integral theorem 128
sequence 11
Cayley transform 202
Chacon, К. V. 385, 389, 470
-Ornstein theorem 385
Chapman-Kolmogorov equation 379
character, continuous unitary 355
Choquet, G. 364, 470
Clarkson, J. A, 127,470
closable operator 77
closed
graph theorem 79
linear operator 77
range theorem 205
set 3
closure
convex 28
strong 125
weak 125
commutative 339
compact 4
group 332
in B-spaces 274
locally — 5
compact
locally sequentially weakly —
126, 141
operator 274, 277, 325
relatively — 5
compactification, one point — 5
complement 3, 366
orthogonal 82
complete 11, 52, 105, 367
orthonormal system 86
a- - 367
sequentially — 105
sequentially weakly — 120, 124
completely continuous operator 274
completion 56, 105, 113
complex measure 35
condensation of singularities 73
conjugate
exponent 33
operator 194
continuous
absolutely 17
mapping 4
spectrum 209, 323
unitary character 355
contracted measure 378
contraction semi-group 233
convergence 104
asymptotic — 38, 122
in measure 38, 122
mean — 34
O- - 366
relatively uniform *- — 370
strong — 30, 32
weak — 120
weak* — 125
convex 24
closure 28
hull 363
uniformly — space 126
convolution 148, 156, 354
coordinate operator 198, 351
countably additive (= a-additive) 15
covering 4, 60
scattered open — 60
Crandall, M. G. 453, 454, 457, 465, 470
Crumm, M, M. 166
D« 27
Index
489
D* 182
decomposition
Hahn — 36
Jordan — 36, 366
of a subharmonic element 411
defect indices 349
defining function 17
dense 11
derivative
Dini — 239
distributional — 49, 72
DiEUDONNfi, J. 105, 471
difference
operator 244
symmetric — 19
differential operator
analytically elliptic — 193
as infinitesimal generator 243
formally hypoelliptic — 193
hypoelliptic — 80, 189
parabolic — 188
strictly elliptic — 413
strongly elliptic — 176, 188, 286
uniformly strong — 188
weaker 184
diffusion 398, 409
equation 398, 419, 425
dimension 21
Dini derivative 239
Dirac distribution 48, 50, 152
directed set 103
fundamental 105
Dirichlet's problem 286
discrete topological space 34
dissipative
Markov process 395
operator 208, 250
part 395
set 447
distance function 4, 30
distribution (= generalized func-
tion) 47
convergence of — 123
Dirac - 48, 50, 152
direct ( = tensor) product of — 66
local structure of a — 100
product with a function 51
tempered — 149
with compact support 62, 64
distributional derivative 49, 72
distributive
distributive
identity 365
lattice 365
divergence, gerneralized 406
domain 1, 21
Doob, J. L. 388, 471
Dorroh, J. R. 453, 471
duality map 447
duality theorem of Tannaka 337
dual operator 194
inverse of the — 224
resolvent of the — 225
dual semi-group 272
dual space 48, 91, 111
strong topology 111
weak* topology 111
Dufresnoy, J. 166
Duhamel principle 431
Dunford, N. 128, 225, 233, 325, 389,
471
integral 225
theorem 128
Dynkin, E. B, 398, 408, 471
Eberlein, W. F, 141, 471
-Shmulyan theorem 141
Egorov's theorem 16
Ehrenpreis, L. 183, 471
eigenspace 209
eigenvalue 209
eigenvector 209
elliptic differential equation
analytically — 193
formally — 193
strictly — 419
strongly - 176, 188, 286
energy inequality 430
equation
diffusion — 398, 419, 425
of evolution 418, 430, 438
wave — 427
equi-bounded 85
semi-group 232
equi-continuous 85, 213
semi-group of class (Co) 234
group of class (Co) 251
equi-measure transformation 215,
380
Erdelyi, A, 173, .471
ergodic
decomposition of a Markov pro-
cess 393
490
Index
ergodic
hypothesis 389
of Boltzmann 380
ergodic theorem
individual — 383, 388
mean —213, 382
of the Hille type 215
essentially
bounded function 34
self-adjoint operator 315
evolution equation 418, 430, 438
non-linear — 445, 446, 454
exponential of a bounded operator
244
extension of an operator 2, 21
smallest closed  78
extremal
point 362
subset 362
factor
group 220
space 22, 59
Fatou, Lebesgue----lemma 17
Feller, W. 249, 398, 403, 406, 471
Feynman, R. P., integral 465
field, normed — 129
finite intersection property 6
finitely additive 71
measure 118
finitely valued function 16, 130
F-lattice 369
Foias, C. 472
formally hypoelliptic 193
forward and backward unique con-
tinuation property 425
Fourier, J. B. J.
absolutely convergent — series
294, 301
coefficient 86
expansion 74, 87
inverse Fourier transform 147
-Laplace transform 161
transform 146, 151, 153
fractional power of a closed operator
259
Fr^chet, M, 23, 39, 52, 275
-Kolmogorov theorem 275
-space (or F-) 52
Fredholm theory (Riesz-Schauder
theory) 274, 283
Freudenthal, H. 317, 364, 472
Friedmann, A. 191, 472
Friedrichs, K. 177, 208, 317, 430
472
theorem on semi-bounded opera-
tors 317
theorem on strongly elliptic dif-
ferential equations 177
F-space 52
Fubini-Tonelli theorem 18
Fujita, H. 431
Fukamiya, M. 360, 375, 472
function 1
almost periodic — 219, 330
Baire — 19
Bochner wz-integrable — 132
boundary — 163
continuous — 4
defining — 17
finitely valued — 16
generalized — 47
harmonic — 410
Heaviside — 50
holomorphic — 41
ideal - 47
integrable — 16, 17
locally integrable — 48
measurable — 15
positive definite — 346
potential — 410
rapidly decreasing — 146
slowly increasing — 150
testing — 47
uniformly continuous — 332
fundamental
directed set 105
solution 182
system of neighbourhoods 28
GArding, L. 175, 281, 287, 472
inequality 175
Gaussian probability density 235,
243, 268
Gfl-set 18
Gelfand, I. M. 128, 166, 281, 293,
295, 298, 356, 472
-Mazur theorem 128
-Raikov theorem 356
representation 298
general expansion theorem 323
generalized
derivative 49
divergence 406
Index
491
generalized
function 47
function with compact support 62
gradient 406
Laplacian 403
Leibniz’ formula 51
limit 104
nilpotent element 299, 302
spectral resolution 353
generator, infinitesimal — 231, 237
Glazman, I. M, 354, 473
gradient, generalized — 406
graph 77
Green's integral theorem 50
Grothendieck, A, 145, 279, 289,
293, 473
group 218, 332
of isotropy 400
-ring 355
group, equicontinuous — of class
(Co) 251
representation of a — 330
of unitary operators {Stone’s
theorem) 345
Hahn, H. 36, 70, 102, 105, 106
-Banach theorem 102, 105, 106
decomposition 36
Halmos, P. R. 95, 119, 323, 393, 409,
473
Hardy, G. H. 41
-Lebesgue class 41, 55, 163
harmonic 410
least — majorant 411
Hausdorff, F. 3, 11, 75, 473
axiom of separation 3
Baire- — theorem 11
Heaviside, O. 169
function 50
operational calculus 169
Hellinger, E, 323, 473
Helly, E. 109, 114, 473
Hermite polynomials 89
Hilbert, D.
-Schmidt integral operator 197,
277
-space 52, 81, 86
Hille, E. 215, 231, 232, 249, 255,
295, 398, 454, 461, 473
-Yosida theorem 248
(The result pertaining to (9) of
Corollary 1 of p. 248 is some-
-Yosida theorem 248
times called as the Hille-
Yosida-theorem)
Hirsch, F. 418, 474
Hoffmann, K. 363, 474
Holder's inequality 33
holomorphic 41
semi-group 254
strongly 128
weakly 128
homeomorphic 76
Hopf, E. 346, 382, 385, 389, 474
Hormander, L. 51, 79, 80, 182, 183,
188, 189, 193, 474
theorem 80, 191
//-theorem 392
Hunt, G. A. 398, 416, 474
hyperdissipative set 449, 450, 452,
453, 454, 457, 461, 465
hypermaximal operator 350
hypoelliptic 80, 189
formally 193
ideal 295, 372
function 47
maximal — 295
theory in normed rings 295
idempotent 83
identity
distributive — 365
modular — 365
resolution of the — 309
image 1
indecomposable Markov process 390
individual ergodic theorem 384, 388
(strict) inductive limit 28, 139
inequality
G&rding’s — 175
Holder’s - 33
infimum (= greatest lower bound) 2
infinitely divisible law 409
infinitesimal generator 231, 237
of non-linear evolution equation
445, 453
resolvent of the — 240
inner product 40
integrable function 16, 17
locally — 48
integral 16
Bochner — 132
Dunford — 226
Feynman — 438
492
Index
integral 16
Lebesgue — 19
Radon — 118
solution of the initial value
problem 465
integral operator 197, 277
interior 12
invariant measure
with respect to a Markov process
379, 393, 395
right invariant Haar measure 400
inverse 2, 21, 43, 77
involutive ring 300
isometric 113, 202, 203
isotropy 400
Ito, K. 398, 408, 474
It6, S. 419, 474
Jacobi matrix 349
Jacobs, K. 382, 474
John, F. 474
Jordan, C. decomposition 36, 366
Jordan, P. 39
Kakutani, S. 10, 31, 119, 127, 215,
286, 369, 375, 378, 382, 388, 474
Riesz-Markov- — theorem 119
Kantorovitch, L. 175, 475
Kato, T. 208, 215, 218, 251. 260,
266, 272, 315, 431, 436, 437, 444,
445, 447, 453, 457, 475
Kelley, J. L. 363, 475
Kellog, O. D. 286, 475
kernel
Aronszajn-Bergman — 96
Hilbert-Schmidt — 197, 277
reproducing — 95
Khintchine, A. 388, 475
Kisy^ski, J. 429, 475
Kodaira, K. 325, 475
Kolmogorov, A. 275, 398, 475
Chapman- — equation 379
extension theorem of measures
293
Komatsu, H. 273, 444, 476
Komura, Y. 445, 446, 448, 449, 450,
451, 452, 453, 476
Konishi, Y. 465, 476
Котаке, T. 437, 476
Kothe, G. 68, 105, 476
Krein, M. 10, 375, 476
-Milman theorem 362
Krein, S. 10, 375, 476
Krylov, N. 321, 393, 476
-Weinstein theorem 321
Ladyzhenskaya, O. A. 431,476
Laguerre polynomials 89
Laplace transform 162, 237
Laplacian, generalized — of Feller
403
lateral condition 406
lattice 2
B- - 369
distributive — 365
F- - 369
homomorphism 372
modular — 365
vector — 364
Lax, P. D. 92, 98, 430, 476
-Milgram theorem 92
negative norm 98, 155
Lebesgue, H. 41
-Fatou lemma 17
measure 19
-Nikodym theorem 93
Legendre polynomial 89
Leibniz’ formula 51
Leray, J. 100, 477
Lerch’s theorem 167
Lewy, H. 182, 477
lexicographic partial order 367
Liggett, T. 454, 457, 465, 477
limit
generalized — 104
point 3
Lindeberg's condition 398
linear
continuous — functional 43
functional 21
independence 20
operator 21
(= total) ordering 2
space 20
subspace (or, in short, subspace)
21
topological space 25
Lions, J. L. 430, 438, 477
Liouville’s theorem 129, 380
Lipschitz condition 135
Lipschitzian mapping 447, 449, 450,
461, 463
Littlewood, E. 360
locally
compact space 5
convex space 26
Index
493
locally
integrable 48
sequentially weakly compact
space 126, 141
lower bound 2
ZJ-space, abstract — 369
Lumer, G. 250, 477
Phillips- —theorem 250
Maak, W. 224, 477
Malgrange, B. 183, 193, 477
-Ehrenpreis theorem 183
mapping 1
continuous — 4
finitely-valued 130
homeomorphic — 76
isometric — 113
w-almost separably valued — 130
one-(to-)one — 2
separably valued — 130
strongly 23-measurable — 130
weakly S3-measurable — 130
Markov, A. 119
Markov process 379
canonical measure of a — 406
dissipative — 395
ergodic decomposition of a — 393
indecomposable — 390
metrically transitive — 390
spatially homogeneous — 398
temporally homogeneous — 379
Maruyama, G, 398, 477
maximal 2
ideal 295
ideal (in vector lattice) 373
set of — probability 394
symmetric operator 350
maximally dissipative 449, 451, 453
Mazur, S* 108, 109, 120, 477
Gelfand- — theorem 129
theorem 109, 120
McKean, H. 398, 408, 477
mean
convergence of £-th order 34
ergodic theorem 213, 215, 382,
388
value of an almost periodic
function 222
measure 15
absolutely continuous — 17, 93,
135, 375
Baire — 18
Borel - 18
measure 15
canonical — of a Markov process
406
complex — 35
contracted — 378
convergence in — 38
invariant — of a Markov process
380, 395
Lebesgue — 19
product — 18
regular Borel — 19, 393
signed — 35
singular — 378
slowly increasing — 150
space 15
topological — 18
measurable
function 15
set 15, 20
strongly — 130
weakly — 130
metric 4
space 3
metrically transitive Markov process
390
Meyer, Ph A. 364, 417, 478
Mikusinski, J. 166, 169, 478
Milgram, A. N. 92, 478
Lax- — theorem 92
Milman, D+ 126, 362, 478
Krein- — theorem 362
Mtmura, Y. 119, 286, 340, 345, 478
Minkowski, H, 24, 33
functional 24, 26
inequality 33
Minlos, R, A. 293, 478
Minty, G. 447, 478
mixing hypothesis 392
Miyadera, I, 249, 454, 463, 478
Mizohata, S. 419, 478
modular
identity 365
lattice 365
moment problem 106
momentum operator 198, 315, 349,
353
Moore-Smith limit 104
Mqrrey, С. B. 193, 478
multiplicity
of an eigenvalue 209
of the spectrum 321
multi-valued mapping 446
494
Index
Nagumo, M. 231, 295, 478
Nagy, B. von Sz. 345, 354, 479
Naimark, M. A. 295, 325, 352, 360,
479
Nakano, H. 323, 479
Narasinham, M. S. 437, 479
nearly orthogonal element 84
negative
norm 98, 155
variation 35, 366
neighbourhood 3
Nelson, E. 438, 479
Neumann, C. series 69
Neumann, J. von 39, 91, 93, 200,
203, 204, 215, 222, 295, 315, 332,
340, 345, 350, 479
-Riesz-Mimura theorem 340
theorem on
Cayley transform 202
real operators 316
spectral resolution 315
Nikodym, O. 93
Radon- — theorem 378
nilpotent 372
generalized — 299, 302
Nirenberg, L. 178, 193, 438, 479
non-dense 11
non-negative functional 393
norm 30, 43
of a bounded linear operator 43
negative — 98, 155
quasi---23, 31
semi- — 23
normal
operator 202
spectral resolution of a — —
306
space 7
normed
factor space 60
field 129
linear space 30
ring 295
semi-simple — — 298
nuclear
operator 279, 289
space 291
null space 21
O-convergence 366
Oharu, S, 463, 479
Okamoto, S, 169
one point compactihcation 5
one-to-one mapping 2
open
base 60
mapping theorem 75
set 3
sphere 4
operational calculus
Dunford’s — 226
on functions of a self-adjoint
operator 343
Heaviside's — 169
Mikusinski’s — 169
operator 21
accretive — 208
adjoint — 196
adjoint differential — 80
bounded linear — 43
closable — 77
closed linear — 77
generalized spectral resolution of
a closed symmetric — 353
compact - 274, 277, 325
comparison of — s (Hbrmander’s
theorem) 79
completely continuous — 274,
277
conjugate — 194
continuous linear — 43
coordinate — 198, 351
d- - 170
dissipative — 208, 250
dual — 194
inverse of the — — 224
resolvent of the — — 225
operator
essentially self-adjoint — 315,
350
exponential of a bounded — 244
extension of an — 2, 21
fractional powers of a closed —
259
Hilbert-Schmidt integral — 197,
277
hypermaximal (= self-adjoint)
- 350
idempotent — 83
identity — 44
integral — 197, 277
inverse — 21, 77
isometric — 202, 203
linear 21, 43
maximal symmetric 350
Index
495
operator
momentum — 198, 315, 349, 353
norm of a bounded — 43
normal — 202
spectral resolution of a — —
306
nuclear — 279, 289
positive — 317
pre-closed — 77
projection — 82
real — 316
self-adjoint — 83, 197, 343, 350
functions of a — — 338
spectral resolution of a — —
313
spectrum of a — — 319
semi-bounded — 317
symmetric — 197, 349, 353
of the trace class 281
unitary — 202
spectral resolution of a — —
306
ordered set
linearily — 2
partially — 2
semi-------2
totally — 2
Ornstein, D. S. 385, 389, 479
orthogonal 82
base 86
complement 82
nearly — 84
projection 82
orthogonalization (E. Schmidt) 88
orthonormal system 86
Paley, E. R. A. C. 161, 479
-Wieher theorem 161, 162, 163
parabolic differential equation 188
Parseval(-Desch6nes, M. A.)
relation 86, 148
theorem for the Fourier transform
154
partial order 2
partition of unity 60
Pazy, A. 453, 457, 458, 463, 480
Peetre, J. 193, 480
Peter, F. 330, 480
-Weyl-Neumann theory 330
Petrowsky, I. G. 193, 480
Pettis, B. J. 131, 480
phase space 380, 381
Phillips, R, S. 215, 224, 231, 249, 250,
260, 272, 295, 409, 425, 430, 480
-Lumer theorem 250
theorem 273
Pitt, H. R. 360, 480
Plancherel's theorem for the Fourier
transform 153
Poincare, H. 227
point spectrum 209
Poisson, S. D,
equation 410
integral representation 165
kernel 268
summation formula 149
polar set 136
polynomial
approximation theorem 8, 249
Hermite — 89
Laguerre — 89
Legendre — 89
Tchebyschev — 89
Pontrjagin, L. 338, 480
positive
definite function 346
functional 92
operator 317
variation 35, 366
Post, E. L.-Widder inversion for-
mula 261
potential 410
Poulsen, E. T. 437, 480
pre-closed operator 77
pre-Hilbert space 39
principle of the condensation of sin-
gularities 72
projection 82
product
direct (= tensor) — of distribu-
tions 66
inner (= scalar) — 40
measure 18
pseudo-resolvent 215
quasi-norm 23, 31
quasi-normed linear space 31
completion of a — 56
quasi-unit 375
Quinn, В. K. 465, 480
radical of a
normed ring 298
vector lattice 372
496
Index
Rado, T. 411, 480
Radon, J.
integral 118
-Nikodym theorem 378
Raikov, D. A. 295, 356, 480
range 1, 21
closed — theorem 205
rapidly decreasing function 146
Rasmussen, S, 454, 481
Ray, D. 398, 481
Rayleigh's principle 321
real operator 316
reduction theory 349
reflexive space 91, 113, 126, 139, 140
semi- — 139
regular space 7
regularization 29
relative topology 3
Rellich-G&rding theorem 281
representation
Riesz' — theorem 90
of a semi-group 246
spectral — for a
normal operator 306
self-adjoint operator 313
unitary operator 306
of a vector lattice as
point functions 372
set functions 375
reproducing kernel 95
residual spectrum 209
residue class 220, 297
resolution of the identity 309
resolvent 209
equation 211
isolated singularities of a — 228
Laurent expansion of a — 212
pseudo- — 215
set 209
resonance theorem 69
restriction 2, 21
Rickart, С E, 295, 360, 481
Riemann, B.
-Lebesgue theorem 357
sum 104
Riesz, F. 84, 90, 92, 119, 215, 274,
283, 340, 345, 364, 378, 481
-Markov-Kakutani theorem 119
representation theorem 90
-Schauder theory 283
ring 294
involutive — 300
ring 294
normed — 295
semi-simpie - — 298
symmetric — 300
Ryll-Nardzewski, C 166, 481
ff-additive 15
Saks, S, 70, 95, 481
Sato, M. 166, 481
scalar product 40
Schatten, Rt 281, 481
Schauder, J, 274, 282, 481
Riesz- — theory 283
theorem 282
Schmidt, E. orthogonalization 88
Schur, I. 122, 334
lemma 334
Schwartz, J* 325, 389, 482
Schwartz, L. 23, 46, 100, 158, 165,
166, 178, 249, 482
u-complete 367
Segal, I. E, 360, 482
self-adjoint operator 197, 343, 350
essentially — 315, 350
functions of a — 339
spectral resolution of a — 313
spectrum of a — 319
semi-bounded operator 317
semi-group (of class (Co)) 232
contraction — 250
convergence of a sequence of —
269
differentiability of a — 237
dual - 272
equi-bounded — 232
equi-continuous — 234
holomorphic — 254
infinitesimal generator of a — 237
representation of a — 246
semi-norm 23
semi-reflexive space 139
semi-scalar product 250
semi-simple normed ring 298
separably valued mapping 130
separating points 9
separation, axiom of — 3
sesqui-linear functional 92
set 1
Baire — 18
Borel — 18
closed — 3
Index
497
set 1
convex — 24
G6- - 18
open — 3
ordered — 2
cF-finite 15
Shmulyan, V. L. 141, 482
signed measure 35
Silov.-G. 166, 295, 482
simple convergence topology 110
simple spectrum 322, 347
singular
element of a vector lattice 375
measure 378
slowly increasing
function 150
measure 150
Smith, P, A. 390, 482
Sobczyk, A. 105
Sobolev, S. 23, 55, 155, 174, 482
lemma 174
space 55
Sobolevski, P. E, 431, 437, 482
solution of a partial differential
equation
fundamental — 182
genuine — 178
weak - 177
space
Banach (or B-} — 23, 52
barrel — 138
bidual - 91, 113, 139
bornologic — 44
complete — II, 52, 105
discrete topological — 34
dual - 48, 91, 111
strong — — 111
weak* — — 111
eigen — 209
factor — 22, 59
Fr£chet (or F-) — 52
Hilbert — 52, 81, 86
space
homogeneous — 401
linear — 20
linear topological — 25
locally compact — 5
locally convex — 26
locally sequentially weakly com-
pact — 126, 141
measure — 15
metric — 4
space
normal — 7
normed linear — 30
nuclear — 291
null — 21
phase — 381
pre-Hilbert 39
product — 6
quasi-normed linear — 31
reflexive - 91, 113, 126, 139, 140
regular — 7
semi-reflexive — 139
sequentially complete — 105
Sobolew — 55
topological — 3
uniformly convex — 126
space-like unique continuation pro-
perty 419
spatially homogeneous Markov pro-
cess 398
spectral
mapping theorem 227
radius 211
resolution for a
closed symmetric operator 353
normal operator 306
self-adjoint operator 313
unitary operator 306
theory 209
spectrum 209, 299
continuous — 209, 323
multiplicity of the — 321
point — 209
residual — 209
of a self-adjoint operator 319
simple — 322, 347
sphere, open — 4
<7-ring 15
Steinhaus, H. 73
Stone, M, H. 9, 231, 253, 323, 325
345, 482	.
theorem 253, 345
-Weierstrass theorem 9
strictly elliptic differential operator
413
strong convergence 30, 32
of operators 69
strong dual space 111
strong topology of operators 111
strongly elliptic differential operator
176, 286
strongly measurable mapping 130
498
Index
subadditive 23
subharmonic 411
subspace 21
support (= carrier) of a
distribution 62
function 26
supremum (least upper bound) 2
symmetric
difference 19
operator 197, 349, 353
ring 300
Szego, G. 90, 482
Tanabe, H. 432, 437, 438, 482
Tannaka, T. 336, 483
duality theorem 336
Tauberian theorem
special — 360
Wiener’s — 357
Taylor, A. 231, 483
Taylor’s expansion 129
Tchebyschev polynomials 89
tempered distribution 149
temporally homogeneous
equation of evolution 419
Markov process 379
temporally inhomogeneous equation
of evolution 419, 430
tensor product of distributions 67
testing function 47
6-formula 149
theorem
Ascoli-ArzelS, — 85
Baire - 12
Baire-Hausdorff —11
Banach — on
a.e. convergence of linear
operators 370
generalized limits 104
Bochner — 133, 134
on positive definite functions
346
Cauchy's integral — 128
theorem
Chacon-Ornstein— 385
closed graph — 79
closed range — 205
Crandall-Liggett — 454, 457, 470
Dunford — 128
Eberlein-Shmulyan — 141
Egorov — 16
ergodic — of Hille type 215
Feller — 404
Fourier's inversion — 147
Fr6chet-Kolmogorov — 275
Friedrichs — on
semi-bounded operators 317
strongly elliptic differential
equations 177
Fubini-Tonelli — 18
Gelfand-Mazur — 129
Gelfand-Raikov — 356
general expansion — 323
H- - 392
Hahn-Banach — 102, 105
Hilbert-Schmidt expansion — 326
Hille-Yosida — 248
Hormander — 80, 191
individual ergodic — 384, 388
Kolmogorov's extension — of
measures 293
Komura — 452
Krein-Milman — 362
Krylov-Weinstein — 321
Lax-Milgram — 92
Lebesgue-Fatou — 17
Lebesgue-Nikodym — 93
Lerch — 167
Liouville — 129, 380
Malgrange-Ehrenpreis — 184
Mazur — on
existence of continuous linear
functionals 108, 109
weak limits 120
mean ergodic — 213, 215, 382,
388
Milman — 127
Naimark — 352
von Neumann — on
Cayley transform 202
real operators 316
spectral representation 315
Neumann-Riesz-Mimura — 340
open mapping — 75
Paley-Wiener — 161, 162, 163
Peter- WeyLNeumann — 330
Pettis - 131
theorem
Phillips - 224, 225, 273
Phillips-Lu mer — 250
Blanche rei — 153
Radon-Nikodym 375, 37H
Rellich-GArding 281
Index
499
theorem
representation — for equicontin-
uous groups of class (Co) 251
semi-groups of class (Co) 246
resonance — 69
Riemann-Lebesgue — 357
Riesz-Markov-Kakutani — 119
Riesz' representation — 90
Schauder — 282
Schwartz - 100, 158, 178 (Weyl-
Schwartz —)
special Tauberian — 360
spectral-mapping — 227
Stone - 253, 345
Stone-Weierstrass — 9
Tannaka's duality — 337
Tauberian — of Wiener 357
Titchmarsh — 166
Trotter-Kato — 269
Tychonov 6
uniform boundedness — 68
Urysohn — 7
Vitali-Hahn-Saks — 70
Weierstrass — on
the existence of nowhere differ-
entiable functions 72
polynomial approximation 8,
249
trigonometric approximation
11
Weyl-Schwartz — 178
Wiener — 301
Wiener's Tauberian — 357
Titchmarsh, E. C. 166, 325, 483
theorem 166
topology 3
bounded convergence — 110
relative — 3
simple convergence — 110
strong — 111
uniform — of operators 111
weak — 111
weak* — 111
total
ordering 2
variation 35, 36, 38, 70, 118, 366
totally bounded 13
trace class, operators of the — 281
transition probability 379
translation operator 158, 243
transposed matrix 194
Treves, F. 193, 483
triangle inequality 30
trigonometric approximation theo-
rem (Weierstrass) 11
Trotter, H. F, 272, 483
-Kato theorem 269
Tvchonov, At theorem 6
uniform
boundedness theorem 68
strength 188
topology of operators Ill
uniformly
continuous function 332
convex R-space 126
unimodular group 219
unit 294, 372
unit sphere (or disk) 43
unitary
character 355
equivalence 323
invariant 323
matrix representation 330
operator 202, 253, 345
spectral resolution of a — —
306
unity, partition of — 60
upper bound 2
Urysohn, P. theorem 7
variation
negative 35, 36, 37, 366
positive 35, 36, 366
total 35, 36, 37, 38, 70, 71, 118,
119, 366
vector lattice 364
with unit 372
Vilenkin, N. Y. 281, 293, 483
Visik, I. M. 431,483
Vitali, G. 70, 483
-Hahn-Saks theorem 70
Watanabe, J. 266, 483
Watanabe, T. 417
wave equation 421
weak
convergence 120
in L1 (S, m) 121
topology 112
weak*
convergence 125
dual space 111
topology 111
500
Index
weakly measurable mapping 130
Wecken, F. J. 323, 483
Webb, G. 483
Weierstrass, C. 8, 9, 72, 249
Stone- — theorem 9
theorem on
the existence of nowhere diffe-
rentiable functions 72
polynomial approximation 8,
249
trigonometric approximation
11
Weil, A. 219, 483
Weinstein, A. 321
Weyl, H. 80, 178, 325, 330, 484
lemma 80
Widder, D. V, 261
Post- — inversion formula 261
Wiener, N. 23, 161, 162, 163, 165,
295, 301, 357, 484
Tauberian theorem 357
theorem 301
Yamabe, H. 425, 484
Yamada, A, 418, 484
Yosida, K. 90, 95, 169, 178, 215, 231,
249, 259, 260, 266, 268, 272, 286,
295, 305, 321, 323, 325, 338, 370,
375, 378, 383, 384, 392, 393, 398,
403, 411, 425, 430, 431, 436, 437,
461, 484
Hille- — theorem 249
individual ergodic theorem 384,
388
mean ergodic theorem 213, 382
Yushkevitch, A. A. 398, 485
Zorn, M. lemma 3
Zygmund, A. -class 392
Notation of Spaces
X2(G) 41, 53, 96
A (5,93) 35, 369
(c) 35, 115
Ы 34, 114
C(S) 9, 32, 294, 300, 369
C*(&) 26, 40
Cg (12) 26, 41, 57
©к(й) 28, 138, 291
Ф(Р) 28, 46, 293
®ft(I2) 27, 38
= ®°° (Q) 27, 52, 62, 293
exp(M) 269
Я-Л2 41, 55
H*(Q) 41, 57
Hq (Q) 41, 57
Hk(Q) 57
Hk0(Q) 57, 98, 279, 281
Яо *(*2) 99
(/*) 35, 117
L1 355
L(X, Y) 110, 294
L*(S), LP(S, 93, ж) 32, 53,115, 275,
369
£”($), L°°(S, 93, m) 34, 53, 118
im) 35
M(S, 93, m) 38, 45, 54, 117, 369
OM(Rn) 150
(5) 54
146, 293
(5,93, wi) 15, 70
№*'*>(&} 55
Wk(&) 55, 58, 155