Introduction to -
Tensor Products of
Banach Spaces
With 31 Figures
?^9'f
uf
Springer

w
32-X «*£
_ * -Raymond A. Ryan
. j"^" #&3epartment of Mathematics
National University of Ireland, Galway
Joepgr"
Ireland
I ^rfs'tdikV •'
Bntz5h Library Cataloguing m Publication Dat*
Ryan, Raymond A
Introduction to tensor products of Banach spaces
(Springer monographs m mathematics)
I Banach spaces 2. Tensor products
I. Title
515 732
ISBN 1852334371
Library of Congress Cataloging-in-Pubhcanon Data
Ryan, Raymond A-
I introduction to tensor products of Banach spaces / Raymond A Ryan
p. cm -- (Springer monographs m mathematics)
Includes bibliographical references and index.
ISBN 1-85233-437-1 (alk. paper)
1 Banach spaces 2 Tensor products I Title II Series
QA322.2.R92 2002
515*732—dc21 2001049821
Mathematics Subject Classification (2000) 46-01,47-01,46A32,46B22,46B28,46M05,47B10
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the
Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or
by any means, with the prior permission in writing of the publishers, or m the case of reprographic reproduction in
accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction
outside those terms shonld be sent to the publishers
Springer Monographs in Mathematics ISSN 1439-7382
ISBN 1-85233-437-1 Springer-Verlag London Berlin Heidelberg
a member of BertelsmannSpringerScience+Business Media GmbH
http://www springer co uk
© Spnnger-Verlag London Limited 2002
Printed in Great Britain
The use of registered names, trademarks, etc m this publication does not imply, even in the absence of a specific
statement, that such names are exempt from the relevant laws and regulations and therefore free for general use.
The publisher makes no representation, express or implied, with regard to the accuracy of the information contained m
this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made
Typesetting Camera-ready by the author
Printed and bound at the Athenaeum Press Ltd., Gateshead, Tyne fit Wear
12/3830-543210 Printed on acid-free paper SPIN 10792170

Preface
This book is intended as an introduction to the theory of tensor products of
Banach spaces. The prerequisites for reading the book are a first course in
Functional Analysis and in Measure Theory, as far as the Radon-Nikodym
theorem. The book is entirely self-contained and two appendices give
additional material on Banach Spaces and Measure Theory that may be
unfamiliar to the beginner. No knowledge of tensor products is assumed.
Our viewpoint is that tensor products are a natural and productive way
to understand many of the themes of modern Banach space theory and that
"tensorial thinking" yields insights into many otherwise mysterious
phenomena. We hope to convince the reader of the validity of this belief.
We begin in Chapter 1 with a treatment of the purely algebraic theory of
tensor products of vector spaces. We emphasize the use of the tensor product
as a linearizing tool and we explain the use of tensor products in the duality
theory of spaces of operators in finite dimensions. The ideas developed here,
though simple, are fundamental for the rest of the book.
In Chapter 2 we introduce the projective tensor product. The space of
bounded bilinear forms on the product X x Y of two Banach spaces is the
dual of the projective tensor product, X®^Y. Projective tensor products with
l\ and L\ (fi) are particularly easy to interpret and the Bochner integral for
vector valued functions is introduced. The first hint of the influence of finite
dimensional structure appears here with the definition of the -Ci-spaces. We
give some elementary applications of the Rademacher functions, including the
Khinchine Inequality and its implications for the structure of tensor products
of Lp(fj) spaces. Finally, we introduce the class of nuclear operators.
The injective tensor product is studied in Chapter 3, along with the class
of integral operators. The attempt to represent the injective tensor product
Li(fj,)®£X as a space of vector valued functions leads to the definition of
the Pettis integral and among the applications we prove the Orlicz-Pettis
theorem, which provides a bridge between weak and norm summability. The
importance of the class of integral operators derives in part from the fact
that certain canonical operators, such as the injection of Loo(fi) into Li(fi),
where y. is a finite measure, are of this type. We study this idea in detail.
Chapter 4 is devoted to the approximation property and related matters.
The chapter begins with a list of open problems from the previous chapters,

viii Preface
all of which depend on the presence of the approximation property in some
form for their solution. We give a detailed account of the property and its
applications. We then turn to the question of when the projective, or injective,
tensor product of reflexive spaces is also reflexive. The chapter concludes with
an examination of the construction of tensor product bases for spaces that
have Schauder bases.
The Radon-Nikodym property, and its role in the theory of tensor
products, is the subject of Chapter 5. We give a complete account of the vector
measure theory needed for this purpose. Projective and injective tensor
products with the space of measures are characterized in terms of spaces of vector
measures. We develop the representation theory of operators on C(K) spaces
by means of vector measures, culminating in the phenomenon of the
coincidence of the classes of integral and nuclear operators in the presence of
the Radon-Nikodym property. The possession of this property by reflexive
Banach spaces and separable duals, among others, is established through a
reformulation of the property in terms of the representability of operators
on Li(/j,) spaces. We present several striking applications that illustrate the
power of the Radon-Nikodym property, finishing with the Principle of Local
Reflexivity.
Chapter 6 begins with a discussion of uniform crossnorms and the central
concept of a tensor norm. The Chevet-Saphar norms gp and dp are
introduced, leading to a study of the class of p-summing operators. Among the
results treated here are the Pietsch Domination Theorem for p-summing
operators. The highlight of the chapter is the Grothendieck inequality. After
proving this fundamental result, we give some of its applications to operators
between classical Banach spaces.
In Chapter 7 we embark on a thorough investigation of the basic
properties of tensor norms. We begin with Schatten's construction of a dual norm,
pointing out its shortcomings and the benefits to be obtained from adopting
the Grothendieck definition. The question of when the two duals coincide
leads in a natural way to the concept of accessibility for a tensor norm. After
examining the interconnections between accessibility, duality and the
possession of injective or projective properties, we study the injective and projective
associate norms that can be generated from a tensor norm. This machinery
enables us to identify the duals of the Chevet-Saphar tensor norms and this
in turn leads to the classes of p-integral operators. We then turn our attention
to the Hilbertian tensor norm, u>2, and the central role it plays in the theory
by virtue of the Grothendieck Inequality. After an account of the associated
classes of 2-factorable, or Hilbertian, and 2-dominated operators, we conclude
with Grothendieck's classification of the fourteen "natural" tensor norms.
In Chapter 8 we present a very brief introduction to the theory of Operator
Ideals, concentrating on the connection with the parallel theory of Tensor
Norms. Each tensor norm a generates two Banach ideals, the ideal of a-
nuclear operators and the ideal of a-integral operators, which coincide in

Preface ix
finite dimensions. We show that these ideals axe the smallest and largest
ideals respectively that are associated with a. We also look at the relationship
between a Banach ideal and the tensor norm associated with it.
Each chapter is accompanied by a set of exercises. They are, for the most
part, straightforward applications of the subject matter of the chapter in
question, and are designed to help the reader in coming to terms with the
concepts introduced there. However, we have resisted the temptation to use
the exercises as a device for introducing topics that could not be
accommodated in the text for reasons of space.
We make no claims whatever to originality, except perhaps in the
arrangement of the material. Our overriding objective has been to provide a
self-contained, accessible and compact introduction to the subject. We have
been highly selective in our choice of topics, preferring to spend time on a
small set of important themes rather than include everything one might like
to have seen. The expert will find that many of his or her favourite areas
have been omitted. We hope that the reader will go on to study the more
comprehensive works mentioned in Appendix A. Most of all, we hope that
the reader will enjoy this book.
It is a great pleasure to thank all those who have helped me in one way
or another. My colleagues in the Mathematics Department in Galway have
always been supportive. Departmental heads Ted Hurley and Martin Newell
provided crucial practical assistance. My students, Padraig Kirwan and Bog-
dan Grecu, learned much of this material with me and read many early
drafts. I am grateful to Raymundo Alencar, Richard Aron, Juan Bes,
Fernando Blasco, Joe Diestel, Klaus Floret, Mikael Lindstrom, Manuel Maestre,
Yannis Sarantopoulos, Andrew Tonge, Barry Turret and Ignacio Zalduendo
for many helpful conversations and for sharing their knowledge with me.
Sean Dineen provided vital encouragement and advice at critical moments.
Michael Mackey, Niall Madden, Gotz PfeifFer and Dirk Werner all gave
invaluable assistance with lEX-related matters. Jose Ansemil and Chris Boyd
gave generously of their time in reading drafts of the book and providing
many corrections, suggestions and ideas. Dirk Werner not only read and
corrected many drafts, but was also a knowledgeable and attentive sounding
board. Without his deep knowledge and experience, this would have been a
much smaller book.
The support of the Millennium Research Fund of the National University
of Ireland, Galway, is gratefully acknowledged. The book was typeset using
I£IeX and the diagrams were produced using Paul Taylor's Commutative
Diagrams package. Finally, I would like to express my appreciation to Karen
Borthwick and all the editorial staff at the London office of Springer Verlag
for their patient and professional support.
Galway, September 2001 Ray Ryan

Notation and Terminology
The letters X, Y, Z will denote Banach spaces over the scalar field K, which
may be either the real numbers, R, or the complex numbers, C. The letters
E, F, M, N will usually denote finite dimensional Banach spaces. The closed
unit ball of a Banach space X will be denoted by Bx-
The term operator will mean a bounded linear mapping. The space of
operators from X into Y will be denoted by £>(X, Y) and the dual space of
X by X*. Elements of a dual space X* will typically be denoted by tp or ip
and elements of a bidual X" by symbols such as a;** or y**. If tp is a linear
functional on X, we shall denote the value of <p at an element x of X either
by ip{x) or (x,<p).
We shall denote the space of bounded bilinear forms on the product
X x Y of two Banach spaces by 3(X x Y), with the norm given by
||B|| = sup{\B(x,y)\ : x € Bx,y e BY}.
C(K) and Lp{(i) will denote the usual sequence spaces, where 1 < p < oo.
We shall follow the traditional abuse of notation in using a symbol such as /
to denote both the equivalence class of a function in Lp{(i) and the function
itself.
The symbols cq, tp, will denote the usual sequence spaces and (^ will
denote the space K™ with the £p-norm. If X is a Banach space, the space
lp{X) consists of all sequences x = (xn) in X for which the scalar sequence
(||a;n||) belongs to lp, with the norm of a; defined to bewthe £p-norm of (||a;n||).
A similar remark applies to the space cq(X). The scalar sequence spaces
obtained when the indexing set N is replaced by a set J will be denoted by
cq(I) and tp(I); the context will remove any possible notational confusion
with the above-mentioned vector valued sequence spaces.

Contents
1 Tensor Products 1
1.1 Tensor Products of Vector Spaces 1
1.2 Tensor Products and Linearization 5
1.3 Tensors as Linear Mappings or Bilinear Forms 7
1.4 Tensor and Trace Duality 9
1.5 Examples and Applications 10
1.6 Exercises 12
2 The Projective Tensor Product :. 15
2.1 The Projective Norm 15
2.2 The Dual Space of X®*Y 22
2.3 Li (ii)®*X and the Bochner Integral 25
2.4 -Ci-spaces 30
2.5 Rademacher Techniques 32
2.6 Nuclear Bilinear Forms and Operators 39
2.7 Exercises 42
3 The Injective Tensor Product 45
3.1 The Injective Norm 45
3.2 C(K) and -C^-spaces 49
3.3 Li(ii)®eX and the Pettis Integral 51
3.4 The Dual Space of X®eY '' 57
3.5 Integral Operators : 62
3.6 Exercises 68
4 The Approximation Property 71
4.1 The Approximation Property 71
4.2 Reflexivity of Tensor Products 82
4.3 Tensor Product Bases 87
4.4 Exercises 91
5 The Radon-Nikodym Property 93
5.1 Vector Measures and the Radon-Nikodym Property 93
5.2 Tensor Products and Vector Measures 103
5.3 Operators on C(K) Spaces 108

xiv Contents
5.4 Operators on £i(/u) Spaces 114
5.5 The Principle of Local Reflexivity 122
5.6 Exercises 125
6 The Chevet-Saphar Tensor Products 127
6.1 Tensor Norms 127
6.2 The Chevet-Saphar Tensor Norms 133
6.3 p-summing Operators 140
6.4 Grothendieck's Inequality 152
6.5 Exercises 157
7 Tensor Norms 159
7.1 The Dual Norm 159
7.2 Injective and Projective Associates 165
7.3 The Chevet-Saphar Dual Norms and p-integral Operators ... 170
7.4 The Hilbertian Tensor Norm 176
7.5 Exercises 184
8 Operator Ideals 187
8.1 The Forms and Operators Associated with a Tensor Norm ... 187
8.2 Operator Ideals 194
8.3 Exercises 198
A Suggestions for Further Reading 201
B Summability in Banach Spaces 205
C Spaces of Measures 211
References 219
Index 223

1. Tensor Products
In this chapter we study tensor products from a purely algebraic viewpoint.
Our approach is to define tensors as functional that act on bilinear forms.
We„ explain^ how the tensor product can be seen as a "linearizing space"
for bilinear mappings. Tensors can also be viewed as bilinear forms, or as
linear mappings and we explore the connections between these ideas. In finite
dimensions, tensor products provide a means of understanding the duality of
spaces of linear mappings or bilinear forms, either through "tensor duality"
or the equivalent "trace duality". Finally, we look at some examples of tensor
products.
We work with vector spaces over the field K, which can be either the real
numbers, R, or the complex numbers, C. We denote the algebraic dual of the
vector space X by X". Thus the space X* consists of all the linear functionals
on X, that is, linear mappings from X into K. If x is an element of X and ip
is a linear functional on X, we shall denote the value of tp at x either by ip(x)
or by (x,ip). The vector space of linear mappings from X into Y is denoted
by L(X, Y) and of course L(X, K) is just the space X*.
1.1 Tensor Products of Vector Spaces
We recall that a mapping A from the cartesian product X x Y of vector
spaces into the vector space Z is bilinear if it is linear in each variable, that
is,
i4(a.ia:i + a2x2,y) =aiA(xi,y) + a2A(x2,y) and
A(x,PiVl + Q2y2) = PiA(x, yx) + 02A(x, y2)
for all a;,, a; 6 X, yity 6 "K and all scalars a„ /3„ i = 1, 2. We write B^XxYtZ)
for the vector space of bilinear mappings from the product Xjk_Y into Z\
when Z is the scalar field we denote the corresponding space of bilinear forms
simply by B(X x Y).
Now the tensor product, X ® Y, of the vector spaces X, Y can be
constructed as a space of linear functionals on B(X x Y), in the following way:
for x 6 X, y 6 Y, we denote by x ® y the functional given by evaluation at
the point (x,y). In other words,

2 1 Tensor Products
(a; ® y)(A) = (A, x ® y) = A(x, y)
for each bilinear form A on X x Y. The tensor product X ® Y is the subspace
of the dual B(X x Y) spanned by these elements. Thus, a typical tensor in
X ® Y has the form
n
« = ^A,a;, ®y,, (1.1)
i=i
where n is a natural number, A, 6 K, xx 6 X and yx 6 V. It is ^important
to realize that the representation of u is not unique - in general, there will
be many different ways to write a given tensor in the above form. We shall
deal with this question later. For now, we note a few elementary facts about
tensors. First, if u = X."=1 A, xx ® yx is a tensor and A a bilinear form, then
the action of u on A is given by
u(A) = (A,^\xx®yx\ =^\xA(xx,yx).
\ i=i ' i=i
Of course, the value of this expression is independent of the particular
representation chosen for u. Next, we observe that we can think of the mapping
(a;, y) »-). x ® y as a sort of multiplication on X x Y with values in the vector
space X ® Y. This "product" is itself bilinear, so we have, for example,
(i) (a-i + x2) ® y = xi ® y + x2 ® y,
(ii) x ® (yi + y2) = x ® yi + x ® y2,
(iii) A(a; ® y) = (Xx) ®y = x® (Ay),
(iv) 0®y = a;®0 = 0.
It is worth noting that the typical description of an element « of X ® Y given
in (1.1) can always be rewritten, using (iii) above, in the form
n
u = y^xx®yx.
i=i
Our first proposition shows how linearly independent sets and bases can be
transferred from the component spaces into the tensor product.
Proposition 1.1. Let X and Y be vector spaces.
(a) Let E and F be linearly independent subsets of X and Y respectively.
Then {x ® y : x 6 E, y 6 F} is a linearly independent subset of X ® Y'.
(b) If {e, : i 6 /} and {f3 : j 6 J} are bases for X, Y respectively then
{e, ® fj : (i,j) e I x J} is a basis for X ® Y.
Proof. We prove (a), from which (b) follows immediately. Suppose we have
« = Xir=i^>a;> ® Vi = 0, where xx 6 E and y, 6 F. Let <p, ip be linear

1.1 Tensor Products of Vector Spaces 3
functionals on X, Y respectively and consider the bilinear form defined by
A(x, y) — <p(x)il>{y). We have u(-A) = 0 and so
5^A,p(a:,)V>(y,) = ^V(i.)!/.) =0.
Since this holds for every ip 6 yK we can conclude that J^ILi \<p{x\)yi — 0
and so, by the linear independence of F, we have X,(p(xi) — 0 for every
ip 6 XK But, by the linear independence of E, each xi is nonzero and it
follows that A, = 0 for every i. D
It follows from this result that if X and Y are finite dimensional spaces,
then
dim(X ®Y)= dim(X) dim(y).
In particular, we see that, taking one of the component spaces to be the scalar
field,~the"space X ® K has the same dimension as X. In fact, regardless of
dimensional considerations, the tensor products X ® K and K ® X are each
canomcailly isomorphic to X;"we leave it as an exercise to the'reader to show
that the identifications x ® A •-)• Xx and A ® x h-> Xx respectively provide the
required isomorphisms.
For every non-zero tensor u 6 X ® Y there is a smallest natural number n
for which there is a representation of « containing n terms; let $^i=i x, ® yt
be such a representation. Then it is clear that the sets {xi,...,xn} and
{yii-•• iVn} are each linearly independent. Suppose, for example, that a-i
were a linear combination of X2,...,xn; then the term X\ ® yi could be
absorbed by the others, giving a representation of « with n — 1 terms. The
number n is known as the rank of u. Tensors of rank one are often referred
to as elementary tensors.
How can we tell if two tensors are the same? This question reduces to the
following: how is it possible to determine whether X."=1 xx ® yt is a
representation of the zero tensor? In principle, this can be determined by evaluating
"£"_, A(xt,yt) for every bilinear form A. Fortunately, there are easier ways
to proceed.
Proposition 1.2. The following are equivalent for v. = Ya=i a:»®V» 6 X®Y:
(i) « = 0;
(") EJLi fM^(yt) = 0 for every p6X',^e Y*;
(»i) T,"=i'P(xi)yi = 0/or every tpeXK
(iv) E,n=i Tp(yt)xi - 0 for every %l> 6 "K".
Proof. The proofs of (i) => (ii) => (iii) => (iv) are straightforward variations
of the proof of Proposition 1.1 and are left to the reader. To prove (iv) =>
(i), suppose that £;"=, x/}(yl)xl = 0 for every V 6 Yl. Let A 6 B(X x y).
Let E, F be the subspaces of X, Y spanned by {ii,... , a:n}, {yi,... , yn}
respectively and let B denote the restriction of A to E x F. Choosing bases

4 1 Tensor Products
for the finite dimensional spaces E, F and expanding the bilinear form B
relative to these bases yields a representation for B of the form
m
B(x,y) = Y^O,(x)uj}(y)
i=i
where 8j 6 & and Uj 6 F". We may extend the domains of Oj,u} to all
of X, Y respectively in the following manner: choose algebraic complements,
G,H for E,F respectively, so that X = E©Gand"K = F©jff. Then, if
x = a-i + a-2 6 X with x\ 6 E,x-i 6 G, let 0}(x) = 0,(a:i). The functionals Uj
are defined on Y in a similar way. We may now consider B as a bilinear form
on X x y by using the representation of B given above. Now A and B may
be different bilinear forms on X x Y, but they agree on E x F. Thus we have
n n
i=l i=l
n m m * n v
= E E *j (*H (».) = E M E wj (J'')1') = ° -
i=i j=i y=i ^i=i '
using (iv). Thus u(A) = 0 for every A 6 B(X x y). D
This proposition provides effective means of recognizing when two tensors
are the same and we will make extensive use of it. A subset, S, of a dual space,
X", is said to be separating if it contains enough functionals to distinguish
between the points of X; in other words, if tp(x) = 0 for every ip 6 S,
then x must be zero. It is easy to see that, in applying the proposition, the
functionals used may be restricted to lie in separating subsets of the relevant
duals. An important instance of this idea occurs when the component spaces
in the tensor product are dual spaces: in order to show that £)"=i f* ® V"i =0
in X* ® "V", it suffices to have J27=i V»(a")'0»(j') = 0 f°r every x 6 X, y 6 Y.
We have similar variations on statements (ii) and (iii) of the proposition.
Next, we look at the interaction between tensor products and some
commonly used vector space constructions. Suppose that E and F are subspaces
of the vector spaces X and Y respectively. Then E®F can be considered as
a subspace of the tensor product X®Y in a natural way - if u — J27=zi xi®yi
is an element of E ® F, we can think of « as an element of X ® Y by the
formula (A,u) = "jrj"=1 A{xityx), where A 6 B(X x Y). This gives us a
linear mapping of E ® F into X ® V. To see that this mapping is injective,
suppose that X."=1 xt ® y, = 0 in X ® V. We have seen in the proof of the
above proposition that for every bilinear form B on E x F there is a
bilinear form AoaX xY such that J^L, B(xi,y{) = £;"=1 .Afo.y,). Therefore
Er=i ^foi ft) = 0 and it follows that X)"=1 a:, ® y, = 0 in E ® F.
Next, we show that tensor products respect direct sums:
Proposition 1.3. If Y = F0 0Fx toen X ® "K = (X ® Fo)-0 (X ® Fi).

1.2 Tensor Products and Linearization 5
Proof. It is clear that X ® Y is spanned by the subspaces X ® Fo and
X ® Fi. It only remains to show that (X ® F0) fl (X ® Fi) = {0}. Let
« 6 (X ® Fo) D (X ® Fi), so that we have two representations of u:
n m
v. = ]P «i ® y i = 5Z wi ® zi '
with y, 6 Fo and z,- 6 Fi- Then for every ip 6 X1 we have £"=1 f{vi)yi =
Y,?=i f{w,)z,. But FoflFi = {0}, and so Y£=i V>MVi = 0 for every <p 6 X".
It follows that « = 0. □
It follows from this result that the quotient space (X® Y)/(X®Fo) can be
identified with X® (Y/Fo). Of course, the same assertions hold for subspaces
and quotients of the first component of a tensor product.
1.2 Tensor Products and Linearization
The primary purpose of tensor products is to linearize bilinear mappings. To
explain this, let A: X x Y —"> K be a bilinear form. We recall that each tensor
« 6 X ® Y acts as a linear functional on the space of bilinear forms and so
we may define a mapping A: X ®Y -tKby u€. X ®Y i-t (A, u) 6 K. Now
A is easily seen to be a linear functional on X ® Y. Furthermore, we have
A(x ®y) = [A, x®y) = A(x, y) and it follows that
M^2x*®y*j
]T A(xt, y.).
We have factored the bilinear form A as the composition of the special bilinear
mapping (a;, y) & X xY i-t x®y €. X ®Y and the linear functional A that
maps X ® Y into the scalar field. On the other hand, it is easy to see that if
"J is a linear functional on the tensor product X ® Y, then the composition
of "$ with the bilinear mapping (a;, y) •-). x ® y is a bilinear form, A, on
X x y for which A = *. Thus the bilinear forms on X x Y are in one-to-one
correspondence with the linear functionals on X ® Y. In summary, we have
B{X x Y) = (X ® y)'.
The same idea can be applied to bilinear mappings. If A: X x Y -»
Z is a bilinear mapping, we define a linear mapping A: X ®Y —> Z by
■^(!Cr=i a"i®jft) = S"=i ^(^ii !.»)• To show that this mapping is well defined,
suppose that J2"=i xt®yt = 0. Then, for each ip & 2$, the composition tpo A
is a bilinear functional onXxr" and so
f\y^Mxx,yt)\ ='^<poA(xi,yl) = (J2xl ®yt,ip°A) =0.

6 1 Tensor Products
It follows that Ya=i Mx*> Vi) ~ 0- Therefore A is well defined. Thus the
bilinear mapping A is associated with the linear mapping A and, as above,
this yields an identification:
B(X xY,Z) = L(X ®Y,Z).
The situation is illustrated in the diagram below:
XxY - Z
/A
X®Y
The special bilinear mapping (x,y) & X x Y i-t x ® y & X ® Y acts as a
"universal" bilinear mapping - every other bilinear mapping on XxY factors
through this one via a linear mapping on the tensor product. We summarize
these results:
Proposition 1.4. IFbr every bilinear mapping A: X xY -» Z there exists a
unique linear^mapping A: X ® Y —)• Z such that A(x, y) = A(x ® y) for all
x 6 X, y 6 YYThe correspondence A <—> A is an isomorphism between the
vector spaces B(X x Y, Z) and L(X ®Y,Z).
Now, we have chosen to define the tensor product X ® Y in a particular
way, as a space of linear functionals on B(X x Y). There are other ways
to construct the tensor product, some of which we shall meet in the next
section. The next result demonstrates that these approaches yield essentially
the same object.
Proposition 1.5 (Uniqueness of the Tensor Product). LetX andY be
vector spaces. Suppose there exists a vector space W and a bilinear mapping
B: X x Y -» W with the property that, for every vector space Z and every
bilinear mapping A from XxY into Z, there is a unique linear mapping
L: W —» Z such that A = L o B. Then there is an isomorphism J from
X ®Y into W such that J(x ®y) = B(x, y) for every x 6 X, y 6 Y.
Proof. First, we observe that the uniqueness of the linear mapping L for
each A implies that the image of^xVinlf under B must span W. Now,
applying the given property of W and B to the bilinear mapping (a:, y) 6
X x Y i-t x® y e X ®Y, we obtain a linear mapping L: W -> X ® Y
such that L(B(x, y)) = x ® y for all x, y. On the other hand, the bilinear
mapping B factors through X ® Y to give a linear mapping B: X ® Y -)• W
with the property that B(x ® y\ = B(x, y) for every x, y. Thus, we have
BoL(B(x,y)) = B(x,y) andLoB(x®y) = x®y for all x, y. Since the spaces
X ® Y and W are spanned by the elements x ® y and B(x, y) respectively, it
follows that J = B is the required isomorphism. D

1.3 Tensors as Linear Mappings or Bilinear Forms 7
We illustrate the principle that linear mappings on X ® Y are essentially
the same as bilinear mappings oa X x Y with some applications. First, we
show that the tensor products X®Y and Y®X are canonically isomorphic.
For each u = X."=1 x, ® y, 6 X ® Y, the transpose of u is the tensor
, n V "
To see that u i-> «' is a well defined linear mapping, we observe that this
mapping is simply the linearization of the bilinear mapping from X xY into
Y ® X that takes (a;, y) to y ® x. It is a simple matter to show that the
transpose yields an isomorphism of X ® Y with Y ®X.
We can use the same idea to define the tensor product of linear mappings.
Let S: X -» E and T: Y -» F be linear mappings. Then we may define a
bilinear mapping by (a;, y) 6 X x Y h-> (Sa:) ® (Ty) & E ® F. Linearization
gives a linear mapping S ®T: X ®Y -¥ E ® F such that
(S®T)(a.®y) = (S.r)®(ry)
for every x 6 X,y 6 Y. The tensor product mapping S®T can inherit some
properties from its components. For example, we leave it to the reader to
verify that if S and T are both infective (respectively, surjective) then S®T
is also injective (respectively, surjective). ~~
1.3 Tensors as Linear Mappings or Bilinear Forms
We have chosen to define the tensor product X ® Y as a space of linear
Junctionals on the space B(X x Y). There are other, equally natural approaches,
some of which we describe in this section. First we show that tensors can be
viewed as bilinear forms. With each pair x 6 X, y,& Y we can associate a
bilinear form, BXtV on X' x y", where BXty(ip,xj>) — <p(x)\l>{y). The mapping
(a;, y) 6 X x Y h-> BXiV 6 B(X* x y") is easily seen to be bilinear and so there
is a unique linear mapping from X ®Y into B(X* x y") that maps x ® y
to Bz,y. To see that this mapping is injective, suppose that "£"=1 BXiiVx = 0.
Then "£?=i p{xi)il>{y,) = 0 for every ip 6 X',^i 6 y" and so, by
Proposition 1.2, it follows that ">]]"=1 xt®yt = Q'mX®Y. Thus, we have a canonical
embedding:
X®YcB(XtxYi).
When the spaces in question are duals, there is a simpler embedding:
Xs ® y» C B(X x Y),
where the tensor X."_, ip,®^,'^ identified with the bilinear form that maps
(x,y) to X.r=i <Pi(x)il>t(y). In view of these canonical embeddings, we shall

8 1 Tensor Products
feel free to consider tensors as bilinear forms, should the occasion demand. If
it is necessary to preserve the formal distinction between a tensor « and the
bilinear form associated with it, we shall denote the bilinear form by Bn.
It is also possible to view tensors as linear mappings. Indeed, if B is a
bilinear form on X x Y, then we may associate two linear mappings with B
- the mapping Lb ■ X -» Yf, and the mapping Rb ■ Y -* X', defined by
(y,LB(x)) = (x,RB(y))=B(x,y).
Thus, each tensor « = "££=i a", ® jk generates two linear mappings, which we
write as Lu and Ru-
n n
Ln(>p) = Ylfix^ and RnW = Yl^(y%)Xi'
i=l >=1
and so we have two more identifications:
X®Y <zL{X*,Y) and X®Y CL(Yl,X).
Finally, when one of the component spaces is a dual space, we have a natural
embedding into a smaller space of linear mappings:
Xl®Y cL(X,Y) and X ® Y* C L(Y, X).
The elements of L(X, Y) that correspond to tensors in X' ® Y under this
identification are the linear mappings of finite rank, that is, mappings whose
range is a finite dimensional subspace of Y.
Consider now the linear functional on the tensor product X' ® X that
linearizes the bilinear functional (ip, x) •-). ip(x). This functional is known as
the trace and denoted by tr, or by tr* if it is necessary to specify the space
X in question. We have
tr (^2<Pi®xA =]Ty,(a:,)
and, of course, this value is independent of the representation. The trace
functional is of particular interest when the tensors are interpreted as linear
mappings. We collect together some of its most useful properties in the next
proposition. In particular, we see that if a linear mapping on a finite
dimensional space is represented by a matrix, then the trace coincides with the
usual definition of the trace of a square matrix. We recall that the transpose
of a linear mapping S: X -> Y is the linear mapping 5*: "K" -> Xi given by
(x,Stil>) = (Sx,xl>).
Proposition 1.6. Let X and Y be vector spaces.
(a) LetU: X -> X be a linear mapping of finite rank. The transpose, Ul,
xdso has finite rank and trxU = tr*. Ul.

1.4 Tensor and Trace Duality 9
(b) Let S: X -*Y and T:Y-tX be linear mappings, at least one of which
is of finite rank. Then ST and TS have finite rank and
try ST = trxTS.
(c) Suppose X is finite dimensional and V: X -» X is a linear mapping. If
V is represented by the matrix A relative to some basis for X, then tr* V
is the sum of the diagonal entries of A.
Proof, (a) If X3"=i Vx ® xi is a representation of U, then U* — X."=1 xx ® ipi
and so trx« Ul = J^=1(xit fx) = tr* U.
(b) Suppose that S = Ya=i P.®y. has finite rank. Then TS = ;£"=1 <pl®Tyl
and ST = Y^=i rV. ® V. and so
n n
tr^TS = £<ry„p,) = ^{yi,r'W> = try ST.
1=1 i=i
(c) Let {ei,...,en} be a basis for X with coordinate Junctionals {tpi,...,tpn},
so that each x & X has the expansion x = X."=1 y>,(a;)e,. If A = (a,,-) is the
matrix of V relative to this basis, then we have
n n n
V{x) = ][>,(a.)V(e,) = EEft(*K«-
J=l J=l1 = 1
Therefore V = £"i=1 ^®a.je. and so tr* V = £"j=i aijMe») = E"=i «u-
D
1.4 Tensor and Trace Duality
In this section we look again at the duality theory .we have developed for
tensor products. Recall that the dual space of the tensor product X ® Y is
the space B[X x Y) of bilinear forms onXxr" and that the tensor product
X' ® "V" is canonically embedded in the space B(X x Y). Now if X and V
are finite dimensional, then it is easy to see that B(X x Y) = X* ® y', and
so we have
(x ® y)" = x« ® y«,
with the duality relationship between the spaces X ®Y and X' ® y"
summarized by the relation
(a; ® y, <p ® ip) = p(z)V>(i,).
In this case, each of the tensor products X ® Y and X" ® y" is the dual
of the other, and we have a very pleasing duality, which we shall refer to as
tensor duality. It is interesting to look at this duality using spaces of linear

10 1 Tensor Products
mappings rather than tensor products. Continuing with our assumption of
finite dimensionality, consider the space L(X, Y) of linear mappings from X
into Y. We may identify this space with the tensor product X' ® Y, and so
its dual space is X ® y', which it is natural to interpret as L[Y, X). Thus we
have
L(X,Y)l=L(Y,X).
Let us examine the workings of the duality between these spaces. Suppose
S 6 L{X,Y) and T 6 L(Y,X). Using the identifications described above, S
and T correspond to tensors « = £"=i <Pi®Vi 6 X1 ® Y and v = J^Li a:,- ®
t/'j 6 X ® y' respectively. Then a straightforward computation shows that
(S,T) = (u,v) = try ST. Therefore the duality between the spaces L(X,Y)
and L(Y, X) is given by
(5,r)=try5r = trxr5.
We shall describe this relationship as trace duality.
We emphasize that tensor duality and trace duality are simply different
ways of looking at the same thing, and we are free to work with whichever
formalism best suits the context.
What can we say in infinite dimensions? If we drop the assumption that
X and Y are finite dimensional, then in our description of tensor duality we
must revert to our original identification of the dual space of X ® Y with
the space B(X x Y). This latter space will in general be strictly larger than
X' ® y'. And, in the language of trace duality, we must replace the space
L[X, Y) with the smaller space of linear mappings of finite rank, which we
shall denote by FL[X,Y). Thus in general we have
FL(X,Yf=Hy,Xn),
with the duality given by (S, T) = try ST = trxu TS, as in the finite
dimensional case.
1.5 Examples and Applications
Spaces of Matrices
Consider the tensor product K" ® K"*, where K" and K*" are endowed with
the standard bases, which we denote by {ei,..., en} and {/i,..., fm}
respectively. This tensor product can be identified with the vector space Mron(lK)
of m x n matrices with entries in the field K. The tensor e, ® f} is identified
with the matrix that has the entry 1 in the (t, j) position and zeros elsewhere.
Thus the tensor product basis {et®f3} corresponds to the standard basis for
Mmn(K).

1.5 Examples and Applications 11
Vector Valued Functions
We now describe a process whereby a vector space of scalar valued functions
can be converted into a space of vector valued functions. We begin with a very
general situation and then look at some examples. Let F(S) be the vector
space of all functions from a set S into the scalar field K, with the usual
pointwise definitions of addition and multiplication by scalars. Now let X be
a vector space and let F(S,X) be the vector space of all functions from S
into X. For each / 6 F[S) and x 6 X we may define a function from S into
X by s h-). f[s)x. Let us for the moment write this function as / • x. Consider
the bilinear form (/, a;) 6 F[S) x X h-> / -x 6 F(S, X). Linearizing, we obtain
a linear mapping
n n
]T /, ® Xi 6 F[S) ® X ,-> ]T /, • xx 6 F{S, X).
x=i »=i
Let us show that this mapping is injective. To this end, suppose that the
function X."=1 /t • ar, is zero. Then we have "£"=1 ft{s)xi = 0 for every a 6 S.
But the evaluation functionals, / h-> f(s), clearly form a separating subset of
the dual space of F[S) and so, by the remarks following Proposition 1.2, we
have "£r=i fi ® -c. = 0 in .F(S) ® X.
Thus, we have an embedding of F(S) ® X into F[S, X) - taking the
tensor product of the function space F[S) with the space X has the effect of
"adjoining" vector values to the functions in F(S). In the light of this
identification, we may dispense with the notation introduced above, so the tensor
X.r=i /t ® ^ may also be taken to represent the function a h-> J^JLi fi{s)xi-
The same ideas can be applied to any subspace, G(S), of F(S) to
produce a "vector valued version" of G[S). For example, let P(K) be the vector
space of polynomial functions on K and let P(K, X) be the space of
polynomial functions on K with values in the vector space X. Each element P
of P(K, X) has the form P(s) = YJk=o s*** where xk 6 X. Let pk be the
monomial functions, pjt(s) = s*. Then the polynomial P is given by the
tensor X.*=oP.c ® Xk- On the other hand, it is easy to see that every element of
the tensor product P(K) ® X yields a polynomial function from K into X.
Thus we have a representation of the space P(K,X) as a tensor product:
P(K,X) = P(K)®X.
For another example, let fi be the vector space of all scalar sequences
that are eventually zero. Thus, a typical element of SI has the form w =
(si,...,sn,0,0,...). Then the elements of the tensor product SI ® X can
be considered as sequences in X that are eventually zero. Let il{X) denote
the space of all sequences in X that are «ventually zero. If a; 6 il{X), then
x = (a-i,... ,a;n,0,0,...) = X."=i ei®x„ where ei is the scalar sequence that
has 1 in the n-th place and zero elsewhere. Thus we see that
il(X) = il®X.

12 1 Tensor Products
Vector Valued Measures
Let A be an algebra of subsets of a set S. We denote by FM[A) the vector
space of finitely additive scalar valued measures on A. Thus, an element,
fi, of FM(A) is a function from A into the scalar field such that /n(0) = 0
and /n(U"=1.4i) = X.?=i fi(At) for every finite, pairwise disjoint collection of
sets Ai,...,An that belong to A. For each vector space, X, we can define
in just the same way the vector space FM{A, X) of finitely additive X-
valued measures on A Proceeding just as above, we then have an embedding
of the tensor product FM [A) ® X into FM [A, X) under which the tensor
ft = J2?=i A*» ® x* corresponds to the vector valued measure
n
We leave it to the reader to verify that if X is finite dimensional, then
FM(A) ® X = FM(A,X).
Functions of Two Variables
We have already seen that the tensor product of dual spaces, X' ® V", can
be embedded in the space B[X x Y) of bilinear forms. This process can
be generalized, so that the tensor product of two function spaces can be
interpreted as a space of functions of two variables. Thus, if S and T are
two sets, then there is an injective linear mapping from F[S) ® F(T) into
F(S x T) which associates with the tensor Y%=i /»® 9*tne funct>on (s. 0 *-*
EIU/*(»>*(*)■
In some cases, a space of functions of two variables may be represented
exactly as a tensor product of function spaces of one variable. For example,
let P(K x K) be the vector space of polynomial functions in two variables.
Then we have
P(K x K) = P(K) ® P(K) •
1.6 Exercises
Exercise 1.1. Show that the tensor product X ® Y contains isomorphic
copies of both X and Y.
Exercise 1.2. Let X = Y = R2. Show that 2ei®ei-2e2®ei+ei®e2-e2®e2
is an elementary tensor in X ® Y.
Exercise 1.3. Show that X ® (Y ® Z) is isomorphic to (X ® Y) ® Z.
Exercise 1.4. Show that the tensor product mapping S ® T is injective
(respectively, surjective) if and only if both S and T are injective (respectively,
surjective).

1.6 Exercises 13
Exercise 1.5. Let E and F be subspaces of X and Y respectively. Show
that (X/E) ® (Y/F) is not, in general, the same as X ® Y/E ® F.
Exercise 1.6. Show that there is a linear mapping C: {X ® Yl)®{Y ® Z) ->
X ® Z such that C((a; ®ip)®{y® z)) = ip{y)\x ® z).
Exercise 1.7. Show that the rank of a tensor « 6 X ® Y is the same as the
rank of the linear mapping from X^ into V associated with u.
Exercise 1.8. The symmetric tensor product X ®„ X is the subspace of
X® X spanned by the tensors x®„y = [x®y + y® x)/2. Show that X®„X
is a universal space for symmetric bilinear mappings on X x X in the following
sense: B:XxX-tZ is a. symmetric bilinear mapping if and only if there
exists a unique linear mapping T: X®„X -> Z such that B(x,y) =T(x®,y)
for every x, y 6 X. (To say that B is symmetric means that B(x,y) = B(y,x)
for all x, y 6 X.)
Exercise 1.9. The alternating tensor product X ®a X is the subspace of
X ® X spanned by the tensors a:®0y=(a:®y-a:® y)/2. Show that X ® X
is the direct sum of the subspaces X ®„X and X ®a X.
Exercise 1.10. Show that if X and Y are infinite dimensional, then B(X x
Y) is strictly larger than X* ® Y*.
Exercise 1.11. Find conditions on the set S or the vector space X that will
ensure that F(S,X) = F{S) ® X.
Exercise 1.12. Let /: F(K) -> P(K) be the indefinite integral operator:
AELo °-ktk) = E*=o(fc +1)" W*+1- Describe the mapping I® I if P(K) ®
P(K) is identified with the vector space of polynomials in two variables.

2. The Projective Tensor Product
In this chapter we investigate the simplest way to norm the tensor product of
two Banach spaces. The projective tensor product linearizes bounded bilinear
mappings just as the algebraic tensor product linearizes bilinear mappings.
The projective tensor product derives its name from the fact that it behaves
well with respect to quotient space constructions. The projective tensor
product of (i with X gives a representation of the space of absolutely summable
sequences in X and projective tensor products with L\ [p) lead to a study of
the Bochner integral for Banach space valued functions. We also introduce
the class of Li-spaces, whose finite dimensional structure is like that of l\. We
study some techniques that make use of the Rademacher functions, including
the Khinchine inequality. Finally, interpreting the elements of a projective
tensor product as bilinear forms or operators leads to the introduction of the
concept of nuclearity.
2.1 The Projective Norm
Let X and Y be Banach spaces. How should we norm the tensor product
X ® y? Consider first the elementary tensors. It is natural to require that
ll«®»ll<IWIII»l|. .. (2-1)
Now let « be any element of X ® Y. If X!"=i a", ® yt is a representation of u,
then it follows from the triangle inequality that the norm must satisfy
Ni<£iMiiy.ii-
Since this holds for every representation of «, it follows that
IMI<irf{£>.||||y.||},
the infimum being taken over all representations of u. The right hand side
of this inequality is the biggest possible candidate for a "natural" norm on
X ® Y. This norm is known as the projective norm and is defined as follows:

16 2 The Projective Tensor Product
7r(«) = inf{^||a:l||||yl||:« = ^a;i®yl}. (2.2)
If it is necessary to specify the component spaces in the tensor product, we
shall denote this norm by itx,y(u). or 7r(«; X ® Y).
Proposition 2.1. Let X and Y be Banach spaces. Then w is a norm on
X ® Y and n[x ® y) = ||a;||||y|| for every x 6 X, y 6 Y.
Proof. First, we show that 7r(Au) = |A|7r(u). This is obvious when A is
zero, so suppose that A ^ 0. If u = X."=1 a:,- ® yi is a representation of «
then Au = Yli=\ (■^a:») ® &> and so we have i"(A«) < Y^=\ ll*3"*!...!*.. =
W X.r=i lla;»lll|j'»ll- Since this holds for every representation of «, it follows
that 7r(A«) < |A|7r(«). In the same way, we have 7r(«) = 7r(A_1A«) <
|A|-1tt(A«)> giving |A|tt(«) < tt(Au). Therefore tt(Au) = |A-7r(«).
Now, to prove that 7r satisfies the triangle inequality, let «, v 6 X ® Y and
let e > 0. It follows from the definition that we may choose representations
« = EIU *.®ifcand »=* EJli "j®*/ such that Si=i lkilflfy.il < »(u)+e/2
and S™=1 KHHzJ < n(v) + e/2. Then £?=i a* ® Vi + EJLi •», ® *, is a
representation of « + v and so
n m
*(« + t/> < J] IIx.IHIkII + E ll^-IHKII < *(«) + *(„) + e.
Since this holds for every e > 0", we have 7r(« + v) < w(u) + ir{v).
Now suppose that 7r(«) = 0. Thenrfor every e > 0 there is a representation
Er=ia:» ® y. °f « such that E"=1 lla;illl|j'il. < e- Hence, for every ip 6 X*,
t/1 6 Y* we have | ElLi f{xi)i>{y%)\ < elMUM- Since the value of the sum
X.r=i (P{xi)'lP{yi) is independent of the representation of «, it follows that
EILi V'(a;i)V'(j'») = 0- But the dual spaces X*, Y* are separating subsets
of the respective algebraic duals and so, by Proposition 1.2, it follows that
« = 0.
Finally, we must show that k{x ® y) = ||x|| ||y||. On the one hand, it is clear
that 7r(a;®y) < ||a;||||y||. Choose tp 6 BX', ip 6 By such that <p(x) = ||a;|f
and t/^y) = ||y|[. Consider the bounded bilinear form, B, on X x Y given by
B(w, z) = ip(w)ip(z). The linearization of B is a linear functional on X ® "V
and
I* (e *. ® ft) I < E !■*{*• ® »)i=E m*.Mw>i £ X>«ii ii wii>
1 m=i ' i .-=i i=i .=i
which implies that \B[u)\ < w(u) for every « € X ® V. Therefore B is a
bounded linear functional on the normed space {X ® Y, w) of norm at most
one. Hence ||a;||||y|| = B(x ®y)< ir{x ® y) and we are done. □

2.1 The Projective Norm 17
We shall denote by X ®„ Y the tensor product X 0 Y endowed with
the projective norm, w. Unless the spaces X and Y ate finite dimensional,
this space is not complete. We denote its completion by X®WY. The Banach
space X®„Y will be referred to as the projective tensor product of the Banach
spaces X, Y.
We have seen in the proof of Proposition 1.2 that, in order to establish
that a tensor «in X®„ Y is zero, it suffices to show that ^J^ <p{xi)il>(yi) = 0
for every tp 6 X*, t/i 6 Y*, where X."=1 a", ® yj is any representation of u.
The question of identifying the zero tensor in the completed tensor product
X®„Y is a more delicate matter, to which we will return in Chapter 4.
If A, B are subsets of X, Y respectively, let us write A ® B for the set
{x®y : x 6 A, y 6 B}. This notation, though convenient, w rather dangerous,
and will be used sparingly. We denote the convex huB of a set 5 by co(5)
and its closure by co(5).
Proposition 2.2. The closed unit ball of X®„Y is the closed convex hull of
the set Bx ® By.
Proof. The closed unit ball of X®*Y is the closure of the closed unit ball
of the uncompleted tensor product X ®„ Y, and so it suffices to prove the
proposition for the space X ®„ Y. Suppose that u is in the open unit ball of
X ®„ Y. Then, by the definition of the projective norm, there is a
representation of « of the form "£"=1 xx ® yt, where xt and jfc are all non-zero, and
ElU ll*.lllly.ll < 1- Let «. = IM"1*.. * = lly.ll-1* and A, = ||x,||||y,||.
Then u = Y,"=1 A.u;, ® z„ with wt 6 Bx, z, 6 BY, A* > 0 and YZ=i \ < 1-
Therefore « 6 co[Bx ® By)- It follows that the closed unit ball of X ® Y is
contained in co(Bx ® By). On the other hand, it is clear that Bx ® By is
contained in the closed unit ball of X ®„ Y, and so the same is true for the
closed convex hull of Bx ® By. □
Next, we consider the tensor product of operators. Let S 6 £>{X, W) and
T 6 &[Y, Z). We have seen in Chapter 1 that there is a unique linear mapping
S®T:X®Y -» W ®Z such that S ® T(x ® y) = {Sx) <8> (Ty) for every
x & X, y &Y. Let « 6 X ® Y and let X."=1 a;, ® yi be a representation of «.
Then
7r(5®r(«)) = T(^(5a;i)®(ry,)) < l|S|||w£>illll!fcll.
from which it follows that 7r(S®T(u)) < ||5||||r||7r(tt). Therefore S ®T
is bounded for the projective norms on X ® Y and W ® Z and \\S ® T\\ <
||S||||T||. On the other hand, it follows easily from S®T{x»y) = (Sx)®{Ty)
that ||S®T|| > ||5||||r||. Thus, we have \\S ®T\\ = [|5||jr||. The last step
in this construction is to take the unique bounded extension of the operator
S ® T to the completions of X ®„ Y and W ®„ Z. We shall denote this
operator by S ®* T. We summarize the properties of this process:

18 2 The Projective Tensor Product
Proposition 2.3. Let S: X -*• W and T-.Y-tZ be operators. Then there
is a unique operator S®*T: X®*Y -> W®TZ such that S ®* T{x ® y) =
{Sx) ® {Ty) for every xeX.yeY. Furthermore, \\S ®„ T\\ = \\S\\\\T\\.
The projective tensor product does not respect subspaces. In other words,
if W is a subspace of the Banach space X, so that W ® Y is an algebraic
subspace of X ® y, then the norm induced on W ® Y by X ®„ Y is not, in
general, the projective norm 7riy.y(.). To understand why this is so, consider
the definition of the norm 7r(«; W®Y), where u 6 W®Y. The infimum that
defines this norm is taken over the set of all representations of u in W ® Y.
If we enlarge the space W to X, then the set of representations of u becomes
bigger, and so we have
W <ZX=> ir(u; X ® Y) < ir(u; W ® Y).
We shall explore this phenomenon later. For now, let us note a positive result
for complemented subspaces.
Proposition 2.4. LetE, F be complemented subspaces ofX, Y respectively.
Then E ® F is complemented in X ®„Y and the norm on E ® F induced
by the projective norm of X ®„Y is equivalent to the projective norm we,f-
If E and F are complemented by projections of norm one, then E ®„ F is a
subspace of X ®„Y and is also complemented by a projection of norm one.
Proof. Let P, Q be projections from X, Y onto E, F respectively. It is easy
to see that P®Q is a projection of X®„Y onto E®F. Let u 6 E®F. We have
seen that itx,y(«) < 'tb,f(«)- Now let "£"=i xt ® yt be a representation of u
in X ® Y. Then u = P ® Q[u) and so X."=1 {Px%) ® (Qyt) is a representation
of u in E ® F. Therefore
»*,*■(«) < E ii^.inwv.11 $ \\pw\\Q\\ E iwiii^ii ■
1=1 1=1
Since this holds for every representation of u in X ® Y, it follows that
»x.y(«) < *e,f{u) < \\P\\\\Q\Wx,y{u) ■
If E and F are complemented by projections of norm one, then we have
i-£,,f(u) = 7rx,y(«) for every u&E®F. Also, \\P® Q\\ = \\P\\\\Q\\ = 1, and
hence P ® Q is a projection of norm one onto E® F. D
Taking completions, we see that these results also hold for the complete
projective tensor products.
We now prove a result that explains the choice of the name "projective"
for the norm w. We say that an operator Q: Z —► Y is a quotient operator
if Q is surjective and ||y|| = inf{||z|| :z£Z, Q[z) = y} for every y 6 "K, or,
equivalently. Q maps the open unit ball of Z onto the open unit ball of Y.
This means simply that Y is isometrically isomorphic to the quotient space
Z/kerQ.

2.1 The Projective Norm 19
Proposition 2.5. Let Q: W -» X and R: Z -» Y be quotient operators.
Then Q ®„ R: W®„Z -» X®„Y is a quotient operator.
Proof. It suffices to show that Q ® R: W ®„ Z -» X ®„ Y is a quotient
operator. To see that Q ® R is surjective, let X."=1 xx ® y, 6 X ®T Y. There
exist w, 6 W and zx & Z such that Q(wt) = a:, and R[z%) = y, for each i.
Then Q ® R(Y^=i «"t ® 2t) = X."=1 a:, ® y». Therefore Q®Ris surjective.
Now let « 6 X ®„ y. If Q ® R(v) = u, then w(u) < ||Q||||il||7r(i.) =
7r(u). Given e > 0, choose a representation "£»=.i a;, ® y, of u such that
X.r=i lla;»lll|j'»ll "^ 7r(u) + £- Now for each i, choose w, 6 VV and ;z, 6 Z
such that Q(u>,) = x„ R{zt) = y, and ||w,|| < (1 + 2~-ne)||a:,||, ||z,|| < (1 +
2~ne)l|y.||- Then Q ® -R(£"=1 «". ® *.) = « ^d, using the fact that n"=1(l +
a,) < exp(X)"=1 |a,|), we have
*(£>,«*) <e*(X>.llllifrll) <eie(n(u) + e).
Since this holds for every e > 0, it follows that 7r(u) = inf{7r(u) : v 6
W ®„ Z, Q ® R(v) = u}. □
In general, the projective norm, 7r(u), of a tensor u can be quite difficult to
compute. The definition of this norm requires that we examine every possible
representation of u, and this is clearly not a practical proposition. In the next
section we shall investigate the duality theory of projective tensor products
and this will yield an alternative approach. However, there is one space whose
tensor products are relatively easy to describe:
Example 2.6. The projective tensor product t\®„X.
Let X be a Banach space. We recall from Chapter 1 that the elements of
the tensor product l\ ® X can be viewed as X-valued sequences. Under this
identification, for each a = (an) 6 t\ and a. 6 X, the elementary tensor a®a;
corresponds to the sequence (anx) in X. Now we observe that this sequence is
absolutely summable, since "£n=i llan^|| < (X.„t=i I^QIM!* ^n other words,
a® a; belongs to the Banach space (i(X) of all absolutely summable sequences
in X, where the norm is given by
oo
||(«n)||l = X)||*»||.
n=l
Thus there exists a linear mapping J: li ®X -¥ d(X) satisfying J(a® x) =
(anx). Now, if Xllii fli®a"i is a representation of u 6 t\ ®X, where a, = (am)n
for each i, then
II / "* \ II °° II / "* \ II
U]Tama:,j = E ( E a"ia;--)
I \,=l /nlll n=|ll\,=| /nil
oo m m * oo v m
< E E ik-*.!! = E( E i°»i ) ii*.h = E iwiwi •
n=l t=l t=l Vi=l ' 1=1

20 2 The Projective Tensor Product
and, since this holds for every representation of u, it follows that
||J(«)||, <*(«).
To prove the reverse inequality, fix a representation YllLi a» ® xi °f u- Then
J(«) = («„), where u„ = Y^Li a»na"i- We claim that the series S^Li e» ® "n
converges to u in li ®„ X, where {e„} is the standard unit vector basis for
li. Let Etfc denote the projection of t\ onto the first k coordinates, so that
Ilfc(a) = X3n=i at»en- We have Iljfc(a) ->aas/c->oo and hence
(k \ i m km .
u- ]T}e„®unJ =7rf^a, ®a;, - y^ J~] en ® amxx J
n=l ' S=l n=l t=l '
=""(5Z (a* ®X{ ~ ]Ca,nen ® *•))= "■( ]C (a« ~ n*a«) ® *•)
m
<£lh-nfca,|||M,
t=i
from which it follows that 7r(« — X.n=i en ® «n) -> 0 as k -»• co. This proves
our claim. We now have
(OO v OO
^n®^) <£||tl»|| = |W«>lll-
n=l ' n=l
Therefore the linear mapping J: t\ ®„ X -¥ 1\{X) is an isometry. Now
1\{X) is complete and so J extends to a unique isometric operator from
the completed projective tensor product li®*X into t\{X). Furthermore,
this operator is surjective. To establish this, let v = [xn) 6 t\ (X). We shall
show that the series J^nLie" ® Xn CoavecSes ia ^i®ir-X^. It is then easy to see
that the sum of this series is mapped by J to «. Since we are working in the
Banach space 1\®„X, it is enough to show that this series is Cauchy. But this
follows immediately from ir("£n=.j en ® xn) < X.n=J ||a.n||- To summarize, we
have proved that there is a canonical isometric isomorphism
J: f,®.*->«,(*),
which allows us to identify these spaces.
The arguments we have given for the space t\ apply equally well to the
space l\ (I), where / is an arbitrary indexing set, to give an identification
ei(i)9.x = el(i,x)l
where li(I,X) is the Banach space of absolutely summable families in X
indexed by /, with the norm ||(a;,)||i = Yliei H3"*!!- ^° Prove this, we need
only recall that if (a.,) is an absolutely summable family, then there is a
countable subset, Jo. °f I such that x, = 0 if i & Iq-

2.1 The Projective Norm 21
We can draw some interesting conclusions from the identification of
ti(I)®jrX with t\(I,X). Recall that, in general, the projective tensor
product does not respect subspaces - if W is a subspace of X, then W®*Y need
not be a subspace of X®„Y. The situation is different if one of the spaces is
Proposition 2.7. Let X be a Banach space and let Y be a closed subspace
ofX. Then ti(I)®*Y is a subspace ofii(I)®^X.
Pro^>f. We have *,(J)®)ryr = h{I,Y) and ti(I)®xX = li{I,X) and it is
clear that t\(I,Y) is a subspace of ti(I,X). □
Our second application is a useful representation of the elements of a
complete projective tensor product, X®„Yt which complements nicely the
representation of the elements of the algebraic tensor product X ®Y. We
recall that every Banach space is a quotient of l\ (/) for some suitably chosen
indexing set /.
Proposition 2.8. LetX andY be Banach spaces. Letu 6 X®„Y ande > 0.
Then there exist bounded sequences (xn), (yn) in X, Y respectively such that
the series J^Li xn ® Vn converges to u and
oo
£.MWI<»(«> + e-
Proof. Choose an indexing set J and a quotient operator Q: ti(J) -¥ X.
Let J be the identity operator on Y. By Proposition 2.5, the tensor product
operator Q®^/: f](J)®,r' -»• X®*Y is a quotient operator. Therefore there
exists v 6 l\(J)®„Y such that Q ®„ I(v) = u and n(v) < n(u\ + e. Since
^i (J)®xY = (i (J, Y), we may identify v with an absolutely summable family,
(v3) in Y. Now there exists a countable subset Jo = {jn} of J such that v} = 0
if j # J0. Hence we may write v as Y^Li ejn ®«j«> witfa w(v) = E^Li IKJI-
Let a;n = Q{ein) and y„ = ty„. Then we have « = Q ®* I(v) = £,£1, a;„ ® yn
*"d EZi IMIIy.ll = *(») < »(«) +e. D
It follows immediately from this result that we have
{OO OO OO v
£Mllv»lh£ll*»llllv»ll<«>. u=£a:»®y»}.
n=l n=l n=l •*
the infimum being taken over all the representations of u 6 X®*Y as
described in the statement of the proposition. There are some minor
variations of this formula that are of use. For example, we may write u in the
form-X^l, A„a;„ ® yn where ||a;n|| = ||yn|| = 1 for every n, or even as
X.£li ^n^n ® yn where xn -»• 0 and yn -»• 0. We have corresponding
formulas for the projective norm:

22 2 The Projective Tensor Product
7r(u)
{oo oo oo <.
X) iA"i :u = YlXnXn ® y*> Yl iA"i < °°' ii^n= n^ii= 11
n=l n=l n=l ■*
{OO OO OO x
^lA"IH:rnllllJ'nll :u = YXnXn®yn> SlAnl <0°' a"»>!'n-*0|-
n=l n=l n=l ■*
2.2 The Dual Space of X®nY.
We have seen in Chapter 1 that the tensor product X ®Y linearizes bilinear
mappings on X x Y. We now add norms to this picture. Recall that a bilinear
mapping B: X x Y -* Z is said to be bounded if there exists a positive
constant, C, such that ||B(a;,y)|| < C||a"||||y|| for every x 6 X, y 6 Y, or
equivalently, if B is bounded on the product Bx x By of the unit balls. We
denote by H(X x Y, Z) the Banach space of bounded bilinear mappings from
X x Y into Z, where the norm is given by ||B|| = sup{||B(a:, y)|| : x 6 Bx, y 6
BY}- When Z is the field of scalars, we denote this space by "B(X x Y).
Theorem 2.9. Let B: XxY-tZbea bounded bilinear mapping. Then
there exists a unique operator B: X®rY -J- Z satisfying B(x ® y) = B(x, y)
for every x 6 X, y 6 Y. The correspondence B «—> B is an isometric
isomorphism between the Banach spaces "B(X x Y,Z) and L{X®„YtZ).
Proof. There exists a unique linear mapping B: X ®Y -J- Z such that
B(x ®y) — B(x, y) for every x,y. Let us show first that B is bounded for the
projective norm on X ® Y. For u = X."=1 xx ® yx 6 X ® Y we have
l|B(«>ll
£>(*„»,) <||B||£||*.||||».I|.
t=i
Since this holds for every representation of u, it follows that ||£(u)|| <
||B||ir(«). Therefore B is bounded and satisfies ||B|| < ||B||. On the other
hand, ||B_(.r,y)|| = \\B(x®y)\\ < ||^||||^||]]V|] gives ||B|| < ||B||. Therefore
ll^ll — ll^ll- Now the operator B: X ®„ Y -¥ Z has a unique extension to
an operator B: X®„Y -¥ Z with the same norm. The mapping B h-s- B
is clearly a linear isometry. It only remains to show that this mapping is
surjective. Let L 6 H(X®XY, Z). The bounded bilinear mapping defined by
B(x, y) - L(x ® y) satisfies B = L. □
Thus we have a canonical identification
V(X x Y, Z) = Jl(X®xY, Z).
If we take Z to be the scalar field, we obtain a canonical identification of
the dual space of the projective tensor product with the space of bounded
bilinear forms:

2.2 The Dual Space of X®„Y. 23
3(X x Y) = {X&vYY .
With this identification, the action of a bounded bilinear form B as a bounded
linear functional on X®„Y is given by
This duality yields a new formula for the projective norm:
tt(«) = sup{|(u, B)\:Be £(X x V), ||B|| < 1}. (2.3)
We now have two ways to compute a projective norm; the original
definition and the duality formula above. The next example shows how effective
this approach can be.
Example 2.10. The tensor diagonal in £2®*^-
It is not an easy matter to give a simple description of the space li®,,^-
However, this space contains a familiar subspace whose presence is rather
surprising. By the tensor diagonal in £2®*^ we mean the closed subspace,
Z>, that is generated by the tensors en ® en, where {en} is the standard unit
vector basis for ti- Let us compute the projective norm of an element u of
D of the form u = 5Zn=i o.nen ® en. By the definition of this norm, we have
7r(u) — !Cn=i \an\- Now consider the bilinear form on (2 x £2 defined by
k
B(x> y) = Yl sgn(a«)a;«3'« >
n=l
where sgn(a) is a scalar of absolute value one such that sgn(a)a = \a\. It is
easy to see that ||B|| = 1. Therefore, by the duality formula (2.3),
fc *,_
tt(«) > (v,B) = £anB(en,en) = ]£'|an|.
n=l n=l
Thus we have w(u) = X.n=1 la«l- ^ f°U°ws ^at D is isometrically isomorphic
to (i.
This subspace is even complemented in £2®*-^ by a projection of norm
one. The projection is the linearization of the bilinear mapping from t2 x ^2
into D given by (a;, y) i-i- Xl^Li xnyn^n. ® e„. We leave it as an easy exercise
to the reader that this operator has the stated properties.
The fact that £2®*^ contains a complemented isometric copy of li alerts
us to some negative aspects of the projective tensor product. For example,
although the space l2 is reflexive, &2®..4. >s not.
The duality theory that we have developed between projective tensor
products and spaces of bounded bilinear forms can also be formulated in

24 2 The Projective Tensor Product
terms of spaces of operators. With each bounded bilinear form B 6 "B(X x Y)
there is an associated operator Lb 6 £>(X,Y*), defined by (y,Lg(x)) =
B(x, y). It is straightforward to verify that the mapping B »-). Lb is an
isometric isomorphism between the spaces "B(X x Y) and L{X,Y*). Thus,
we have an identification:
(X&Y)* = L(X,Ym),
through which the action of an operator S: X -»• Y* as a linear functional
on X®„Y is given by
. n \ n
(£>,®J-,.S) = X>,.Sa-,).
\=i ' i=i
In the same way, we have another identification:
(*®iriT = £(y.**).
where the action of T 6 &(Y, X") on X®„Y is given by
(^z,®yl.r\ = £>l,ryl).
\t=i ' i=i
These identifications yield two variations on the duality formula for the
projective norm of an element of X®„Y:
Tr(u) = sup{|(«,5)| : S 6 L(XtV), \\S\\ < 1}
= suP{|(u,r)|:r6i:(y,x')1 ||r||<i}.
Of course, these are nothing more than restatements of (2.3). However, it is
sometimes more convenient to work with operators than with bilinear forms.
With the assistance of this duality theory, we now look again at the
question of when the projective tensor product respects subspaces. The following
proposition is a straightforward consequence of the duality formula (2.3) and
the Hahn-Banach Theorem.
Proposition 2.11. LetW and Z be subspaces ofX andY respectively. Then
W®TZ is a sub space of X®„Y if and only if every bounded bilinear form on
W x Z extends to a bounded bilinear form on X xY with the same norm.
If we fix the second component space and use the operator version of
duality, the existence of Hahn-Banach type extensions of bilinear forms translates
into an extension property of operators:
Corollary 2.12. Let W be a subspace of X. Then W®„Y is a subspace of
X®„Y if and only if every operator from W into Y* extends to an operator
of the same norm from X into Y*.

2.3 L1([i)®*X and the Bochner Integral 25
A Banach space Z is said to be injective if it has the property that, for
every Banach space X and every subspace W of X, every operator from W
into Z extends to an operator from X into Z of the same norm. Equivalently,
by considering the identity operator on Z, we see that Z is injective if and
only if Z is complemented by a norm one projection in any Banach space
that contains it as a subspace. It is not difficult to show that the space l.*, is
injective. This can also be seen from the fact that projective tensor products
with l\ respect subspaces. The corollary above shows that projective tensor
products with the Banach space Y respect subspaces if and only if Y* is an
injective space.
There is one important setting in which subspaces and projective tensor
products always work well together:
Theorem 2.13. Every bounded bilinear form on X xY has an extension to
a bounded bilinear form on X" x Y** with the same norm.
Proof. Let A be a bounded bilinear form on X x Y and let S be the
associated operator from X into V", so that A(x,y) = (y, 5a:) for every
x 6 X, y 6 Y. Consider the bounded bilinear form B on X" x Y** 'given
by B(x**,y") = (S*y",a.**), where S": Y** -)• X* is the adjoint of 5. If
x & X and y 6 V, then
B(x, y) = (S'y, x) = (y, Sx) = A(x, y),
and so B is an extension of A. Furthermore, ||-4|| = ||5|| = ||5*|| = ||B||. □
Corollary 2.14. X®*Y is a subspace of X**®,r"K**.
2.3 Xi(/i)(giwX and the Bochner Integral
In this section, we shall identify the projective tensor product L\ (/i)®,X with
a Banach space of integrable vector valued functions., In order to do this, it
is necessary first to put in place a suitable integration theory for functions
with values in a Banach space. Throughout this section, (f2, E, p) will denote
a measure space, by which we mean that E is a <r-algebra of subsets of the
set 0 and n is a positive measure on E, which we shall assume to be complete
(if E 6 E has measure zero, then every subset of E belongs to E). We shall
refer to the elements of E as measurable subsets of f2.
We begin with the definition of a measurable function. A function /: Q, -»•
X is said to be a simple function if / has only a finite number of distinct
values. Thus, there is a partition of Q, into disjoint subsets E\,...,En and
distinct nonzero elements a:i,....a:n of X such that /(w) = i, if w 6 Ex.
We may then write / as Y^!i=\ XExxt, where xe denotes the characteristic
function of the set E. We call this the canonical representation of /. We
say that / is a n-measurable simple function if / is simple and each of the

26 2 The Projective Tensor Product
sets Et is measurable. A function /: 0, —)• X is said to be a fi-measurable
function if there is a sequence (/n) of /n-measurable simple functions that
converges almost everywhere to /. We note that if / is /n-measurable, then
the scalar function ||/|| is also /n-measurable, since if (/n) converges /x-almost
everywhere to /, then (||/n||) converges /n-almost everywhere to ||/||.
There are other, equally natural, approaches to the definition of measur-
ability. A function /: fl —)• X is weakly fj,-measvrable if the scalar valued
function <pf is /z-measurable for every <p 6 X*. And we shall say that / is
Borel ^-measurable if /-1(0) is a measurable set for every open subset O of
X. The key ingredient that is needed to connect these concepts to the
definition of /Li-measurability is separability. The function /: Q, -»• X will be said
to be (i-essentially separably valued if there exists a measurable subset E of
fl whose complement has /n-measure zero, such that f(E) is contained in a
separable subspace Y of X. We also say that f(Q) is /n-essentially contained
in Y. We leave it as an exercise to the reader to show that if the measure fi
is (T-finite, then /: Q. -»• X is /n-measurable if and only if the function Xe/ is
/a-measurable for every E 6 E of finite /n-measure.
Proposition 2.15 (Pettis Measurability Theorem). Let (Q, E,/n) be a
a-finite measure space. The following are equivalent for a function f: £1 -)• X:
(i) / is fi-measurable,
(ii) / is weakly fx-measurable and n-essentially separably valued,
(iii) / is Borel n-measurable and fx-essentially separably valued.
Proof, (i) implies (ii): suppose that / is /n-measurable and let (/„) be a
sequence o£ ^measurable simple functions that converges /n-almost
everywhere to /. Then, for every <p 6 X", (<pfn) is a sequence of scalar valued
/Li-measurable simple functions that converges /n-almost everywhere to <pf>
and hence tpf is /n-measurable. Let Y be the closed subspace of X
generated by the ranges of the functions /„. Then Y is separable and f(Q) is
/n-essentially contained in Y.
(ii) implies (iii): suppose that / is a weakly /n-measurable function and
that f(Q.) is /n-essentially contained in a separable subspace Y of X. We
may assume without loss of generality that /(fi) is contained in Y> since we
may alter the values of / on a /x-null set without affecting either (ii) or (iii).
Since the embedding of Y into X is continuous, it suffices to prove that /
is Borel /n-measurable as a function from Q into Y. The fact that tpf is a
/Li-measurable function for every <p 6 X* implies that / is Borel /n-measurable
for the weak topology on Y. In other words, /_1([/) belongs to E for every
weakly open subset U of Y. Hence the same is true for weakly closed subsets
of Y. Now every closed ball in Y, being convex, is also weakly closed. And, by
the separability of V, every open subset U of Y is the union of a countable
family of closed balls. It follows that f~l{U) belongs to E. Therefore / is
Borel /Li-measurable.

2.3 Li(/l.)®„X and the Bochner Integral 27
(iii) implies (i): suppose that / is Borel /n-measurable and that f(Q.) is
essentially contained in a separable subspace Y of X. We may assume without
loss of generality that f(Q) C Y. By the remarks preceding the statement, it
suffices to consider the case where ft is finite. Let {yk} be a countable dense
subset of Y. Then, for each n, the union of the open balls B(yk, 1/n) is Y. By
the Borel measurability of /, the inverse images E£ = f~1(B(yic, 1/n)) are
measurable and cover 0, for every n. We convert this into a disjoint measurable
cover by setting F? = EJ1 and F£ = En \ (J5J* U • • • U Ejf_j) for k > 1. For
each n, let gn = Y,T=i XFfVk- Then we have ||f(w) - Sn(w)|| < 1/n for every
(j 6fJ. Therefore the sequence (gn) of countably valued functions converges
uniformly to / on £1.
We truncate the expressions defining the functions gn to get a sequence
of /Li-measurable simple functions that converges /n-almost everywhere to / as
follows. For each n, choose mn so that
\fc=m„-l-l '
and let hn = ^=1 XFfVk, so that ||/(w) - hn(u)\\ < 1/n for w 6 fi \ Cn,
where Cn = ljr=m„+i *?• Let C = f\n=i U2U <?*• Then C is a ^nuU set
and, if w £ C, then there exists n such that w £ Ck for any fc > n. Therefore
||/(w) - hn(w\\ < 1/fc < 1/n for every fc > n. □
This result can be applied to an arbitrary measure, provided the
function / is supported by a set of <r-finite measure. Also, we can extend to the
vector valued situation many of the standard properties of measurable scalar
functions. For example, on a <r-finite measure space, the pointwise almost
everywhere limit of a sequence of measurable functions is measurable. Egoroff 's
Theorem also holds for vector valued functions on a finite measure space; the
proof is exactly as for the scalar case.
We can now define our integral. Let / be a /n-measurable simple function,
with canonical representation Y^=iXAfXi and suppose that each of the sets
A, has finite /n-measure. The integral of / over a measurable subset E of Q.
is defined to be
f n
/ fdn = J2n(AlnE)xt.
Now a ^-measurable function /: 0. -¥ X is said to be Bochner mtegrable if
there exists a sequence (/n) of /n-measurable simple functions converging (i
almost everywhere to / and satisfying
Urn f\\f-fn\\dii = 0.
n-.cx.yn
The Bochner integral of / over a measurable subset E of Q. is

28 2 The Projective Tensor Product
/ fdfi= lim / fndfi.
The fact that this limit exists and is independent of the particular sequence
(/„) is proved exactly as in the case of scalar functions.
There is a very simple test for Bochner integrability:
Proposition 2.16 (Bochner's Theorem), if/: 0, -»• X is a (i-measvrable
function, then f is Bochner mtegrable if and only if the scalar function ||f||
is integrable.
Proof. Suppose that / is Bochner integrable. Let (fn) be a sequence of (i-
measurable simple functions converging /i-almost everywhere to / as in the
above definition. Then
/ 11/11 dfi = f ||(/ - /„) + /n|| dn<f ||/ - /„|| dfi + f ||/n|| dp
Jn Jn Jn Jn
for each n, and it follows that /n ||f|| dfj, is finite.
Conversely, suppose that ||f|| is integrable. Let (fn) be a sequence of /i-
measurable simple functions that converges /n-almost everywhere to /. Fix
5 > 0 and let
10 otherwise.
<(i + WM
Now (gn) is a sequence of /n-measurable simple functions that converges
^-almost everywhere to /. Furthermore, the sequence of scalar functions
(||/ — pn||) is dominated by the integrable function (2 + £)||/||. Therefore,
by the Dominated Convergence Theorem, we have fn \\f — gn\\ dfj. -J- 0. □
The Dominated Convergence Theorem is also valid for the Bochner
integral. The proof is a straightforward application of Bochner's Theorem.
Corollary 2.17. If f: Q, -t X is Bochner integrable then
if f*A< /"ii/hv
11./n II Jn
Proof. Let (/„) be a sequence of /n-measurable simple functions converging
^-almost everywhere to /, with limn fa fn dfi -»• /n / dfi- Then the sequence
(||/„||) converges in /n-mean to ||/||. Therefore
/ fdfi\ = lim / fndfi\\ = lim / fndfi
\\Jn I ||m->ooyn || n-.ex>||yn
lim f \\fn\\dii= f WfWdfi.
<
n—>o

2.3 Li(js)®„X and the Bochner Integral 29
Next, we show that operators respect the Bochner integral.
Proposition 2.18. Let f: Q, -)• X be Bochner integrable. If T 6 £>(X,Y)
then the function Tf: Q, -)• Y is Bochner integrable and
jjfdp = T^fjdp
Proof. Let (/„) be a sequence of /n-measurable simple functions such that
(/n) converges /n-almost everywhere to / and /n /„ dp -»• fnf dp. It is easy to
see that (Tfn) is a sequence of /n-measurable simple functions that converges
/n-almost everywhere to Tf and that /n Tfn dp = T(Jn fn dp) for every n.
Furthermore, we have /„ \\Tf - Tfn\\ dp < \\T\\ fn \\f - fn\\ dp. Therefore
fTfdp= lim [ Tfndp= lim t( f fndp)
-<J&/.A*)-T(i.'4
D
The Lebesgue-Bochner space L\ (p> X) is the Banach space of equivalence
classes of Bochner integrable functions /: Q. -»• X, with the norm
ll/lli=/ ll/ll<fo-
Jn
We follow the usual convention in using the symbol / for the equivalence
class containing /. The completeness of the space L-i (p, X) is proved exactly
as in the case of the space L\(p).
Example 2.19. The projective tensor product L\(n)®*X
We shall now see that the projective tensor product Li(p)®nX can be
identified with the space L\{p,X). We begin with the bounded bilinear mapping
of norm one from L\{p) x X into Li(p,X) that maps (/, a;) to the function
f ®x. Linearizing, we obtain an operator
J: Li(n)®*X -> L^X)
of norm one. Thus, we have ||Ju||i < 7r(u) for every u 6 L\{p)®„X. On the
other hand, let / 6 Li(p,X) be a Bochner integrable simple function with
canonical representation "£"=i XAtxx- Then / = Ju where u = Yl?=i XAt®xi
and
»(«) < Ell^.lllKII = f>(^)IM = ll-Hli •
i=l x=l
Therefore 5t(u) = |j Ju||i for such functions. Now let S be the normed subspace
of Xi(/z) -consisting of the integrable simple functions. Since S is dense in
•

30 2 The Projective Tensor Product
L\ (n), it follows that S® X is a dense subspace of L\ (/j)®,X The argument
we have given above shows that 7r(u) = ||Ju||i for every u 6 S®X. Therefore
J is an isometry. It only remains to show that J is surjective. But this follows
immediately from the definition of the Bochner integral, since we have seen
that the Bochner integrable simple functions are all in the range of J.
Taking /j, to be the counting measure on N, we obtain as a special case
the identification life^X = t\(X) established previously. Like £1( projective
tensor products with L\ (p) respect subspaces: if Y is a subspace of X, then
Li(n)®nY is a subspace of L|(/n)®wX. This follows immediately from the
fact that the L\ norm of a Bochner integrable function with values in Y is
unchanged if the range space is enlarged to X. Now if the measure ft is a-
finite, then the dual space of Li(fi) is Loo(fi). We can deduce from the results
of Section 2 that I>oo(i<) is an injective space. In fact, this is true for every
measure jj, but we shall not prove it here.
2.4 ,Ci-spaces
We have observed that projective tensor products do not, in general,
respect subspaces. This might suggest that it is not possible to deduce useful
information about tensor products involving the Banach space X from a
knowledge of tensor products with subspaces of X. In fact, even the finite
dimensional subspaces of X carry vital information. To see this, let us fix
Banach spaces X and V, and an element u of the tensor product X ® Y.
If M and N are subspaces of X and Y respectively such that u belongs to
M ® JV, then 7r(u; X ® Y) < 7r(u; M ® N). Now, given e > 0, there exists a
representation X."=1 x,®y, of u such that X."=1 ||a"i||||yi|| < t(«; X ® Y) + e.
Let M and N be the finite dimensional subspaces of X and Y spanned
by {q?i,..., xn} and {yi,...,yn} respectively. Then u lies in M ® N and
7r(u;M® JV) < ^Li .|z..|||ft|| < *(«.* ® Y) + e. Therefore
n(u;X ®Y) = inf{7r(u; M® N) : u 6 M®N, dimM, dimJV < co} .
(2.4)
The infimum is taken over all the pairs of finite dimensional subspaces M,
JV, for which u belongs to M® N. Thus the values of the projective norm are
completely determined by the behaviour of the norm on finite dimensional
spaces. The property described by the above equation is expressed by saying
that the projective norm is finitely generated.
We now show one way in which this property can be exploited. We have
already seen how well behaved the space L\ (p) is in the formation of
projective tensor products. Any space whose finite dimensional structure is like
that of L\ (fi) will inherit some of that good behaviour. We now proceed to
a description of such spaces. Let A > 1. A Banach space X is said to be an
£i, a-space if every finite dimensional subspace M of X is contained in a finite

2.4 jCi-spaces 31
dimensional subspace N, of dimension k, say, whose Banach-Mazur distance
from the space l\ is at most A. This means that there exists an isomorphism
r: N -> t\ such that ||T*M||T*—XM < A. If X is a -C.1>A-space for some A > 1
then X is said to be an -Ci-space. We shall see shortly that every Li(p) is a
.Ci-space, but first we show that this is a useful concept.
In order to state our main result about -Ci-spaces let us introduce some
terminology. We shall say that projective tensor products with X respect sub-
spaces isomorphically if, for every Banach space Y, and every subspace Z of
y, the projective norm on X ® Z is equivalent to the norm induced by the
projective norm on X ® Y.
Theorem 2.20. If X is a -Ci-space then projective tensor products with X
respect subspaces isomorphically.
Proof. Suppose that X is a Li^-space. Let Z be a subspace of Y and let
u € X ® Z. We shall show that '
tt(u; X ® Y) < tt(u; X ® Z) < Xw(u; X ® Y) .
Let e > 0. Then there exists a finite dimensional subspace Af of X such that
7r(«; M ® Y) < (1 4- e)w(u; X ® V). Since this inequality is unchanged if M
is replaced by a larger subspace of X, we may assume that there exists an
isomorphism T: M -»• t\ such that ||rH||'T,—*|| < A. Let J be the embedding
of M into X and let 5 = JT'1. Now « = ST ® /(«) where J denotes the
identity on Y, and so
7r(u;X®Z)=7r(Sr®J(u);X®Z)<||S||7r(r®/(u);*J®Z)
= ||5|k(r ® /(«); «{ ® y) < ||5||||r||ir(u; M®Y)<\(1 + e)n(u; X®Y).
Since this holds for every e > 0, we obtain 7r(u; X ® Z) < \ir(u; X ® Y). O
We leave it as an exercise for the reader to show that X is a £.i.i+e-space
for every e > 0 if and only if the following is true: for every e > 0, and every
finite dimensional subspace M of X, there is a finite dimensional subspace,
N, containing M, and an isomorphism T: N -¥ t* for some k, that satisfies
|||ra:|| - ||x||| < e||x|| for every x € N.
We show next that the space I>i(^t) is a /Ci,i+e-space for every e > 0.
We shall make use of some elementary facts about bases in finite dimensional
spaces. Let X be a finite dimensional normed space. Then X has an Averbach
basis, namely a basis {ei,..., en} of unit vectors such that the coordinate
funcionals {ej,...,e*} also have unit norm. Thus, if x = 2r=ia;»e»i ^hen
\xt\ < \\x\\ for every i. It follows that, if X is an n-dimensional subspace of
any Banach space Z, then there is a projection of Z onto X of norm at most
n. To define such a projection, let <pi,-■.,<pn be Hahn-Banach extensions of
the coordinate functionals ej,...,e* to Z, and take Tz = 2"=i ¥"i(z)ei f°r
ze Z.

32 2 The Projective Tensor Product
Proposition 2.21. L\(n) is a £>i,i+e-space for every e > 0.
Proof. Let M be a finite dimensional subspace of Li(fi) and let e > 0.
Choose an Auerbach basis, {/i,..., fn} f°r M. For each i, let g, be an
integrate simple function such that ||/, — s,||i < e/n2. Then it is easy to see that
the functions pi,... ,gn are linearly independent. Let M' be the subspace of
Li(fj.) spanned by pi,... ,gn. Let S: M -¥ M' be the isomorphism given by
Sfi = g, for each t. If / = £"""=, c,/, G M, then
l|/-S/Hi<Eklll/.-ff.||i<4Ek|<J||/||i. (*)
*—' n* *—' n
t=i t=i
Now since p, are measurable simple functions, there exist mutually disjoint
measurable sets E\,...,Ek, each having finite, positive measure, such that
each g, lies in the subspace N' of Li(fi) spanned by the characteristic
functions XBi i ■ ■ ■ i XBk ■ ^ is easv to see that N' is isometrically isomorphic to l\.
By the remarks preceding the proposition, there is a subspace G of N' such
that N' = M' ffi G, and the projection of N' onto M' with kernel G has norm
at most n. Let N = M (B G and define an isomorphism 7": JV -)• N' = t\
as follows. If / G JV, then f = h + k, where h £ M and k € G. We define
T/ = Sh + k. It is clear that T is an algebraic isomorphism. Furthermore,
using (*), we have
lll/lli - ||r/|U| < ||/ - r/||i = lift - Sfciu < -||fc||i < e||/||,,
since ||fc||i<n||/||i. / D
The dual spaces C(K)* are also .Ci^-spaces for every A > 1. We leave
the proof of this fact as an exercise for the reader. It can be shown that a
Banach space is an Li^-space for every A > 1 if and only if it is an Li(fi) for
some fj,. However, we shall not need to make use of this fact.
2.5 Rademacher Techniques
We have encountered two ways to compute the projective norm of a tensor
u in X ® Y. On the one hand, we have the original definition:
tt(u) = inf | ]T ||x,||||y,|| : « = ]T a:, ® y, | .
This is complemented by the duality formula:
tt(ix) =sup{|(„.B)| : B e B(Jf x Y), \\B\\ < 1}.

2.5 Rademacher Techniques 33
In order to use the first formula, we need to find the most efficient
representation of u. In this section, we show how this can be done in certain special
cases, using an averaging technique.
We illustrate this technique with a simple example. Consider the tensor
« = ei®ei+e2®e2inco®„.co. Using this representation of u, we see that
7r(u) < 2. However, we can also express « as
(ei + e2) ® (ei + e2) + (ei - e2) ® (ei - e2)
+ (-ei + e2) ® (-ei + e2) + (-ei - e2) ® (-ei - e2) ,
from which we obtain the estimate 7r(u) < 1. In fact, we have w(u) = 1, as
we can see by applying the unit norm bilinear form B(x, y) = X\y\ and using
the duality formula.
This idea can be extended to a sum of n terms. It is not difficult to verify
the identity
xi ® yi + ■ ■ ■ + xn ® yn =
— ]T (eia-i+■■■+£„a:n)® (dyi + ■■ ■ + e„y„). (2.5)
£.=±1
l<i<n
Although the number of terms in the representation has increased from n to
2n, this is compensated for by the factor 2_n. This simple averaging technique
has many applications. It can be formulated most elegantly in terms of the
Rademacher functions, which we now describe.
The Rademacher functions are a sequence of functions on the interval
[0,1], with values ±1. The first function in the sequence is the constant
function ri(t) = 1. The second is obtained by bisecting the interval [0,1] and
defining r2(t) to be 1 on [0,1/2) and -1 on [1/2, l]. Once again these
intervals are bisected and r^t) takes the constant values 1 and —1 alternately
on the intervals [0,1/4), [1/4,1/2), [1/2,3/4) and [3/4,1]. In general, the
function rn=i(t) takes the values 1 and —1 alternately on the 2n subinter-
vals [0, l/2n),..., [(2n - l)/2n, 1]. These functions are mutually orthogonal
on [0,1]:
j[W,«>«-{; n;i:
To see this, suppose that i < j. The function r,(<) is constant on each of
2* subintervals of [0,1] and r3(t) takes the values 1 and —1 equally often on
each of these subintervals. Therefore the integral is zero. On the other hand,
if i = j, then rt(t)r3(t) = rt(t)2 = 1 for every t and so the integral is 1. Using
the-same argument, we can establish a more general orthogonality statement.
If fci,..., kn are natural numbers, then
1
« = ei ® ei + e2 ® e2 = -

34 2 The Projective Tensor Product
/W>...rn(t)^t=(l ^.■■;.^-all even,
Jo 10 otherwise.
We now present the "averaging identity" in terms of the Rademacher
functions. We point out that integrals such as those in (2.6) above are actually
finite sums, since the interval [0,1] can be partitioned into a finite number of
subintervals in each of which the functions r, (t) that appear in the integral
are constant.
Lemma 2.22 (Rademacher averaging). Let E and F be vector spaces
and let Xi,...,xn £ E and yi,--.,yn £ F- Then
£ «. ® Ifc = /" {Y, r,(0«. j ® (V, r,(0lfc J dt.
Proof. Expand the integrand and use the orthogonality property of the
functions rt(t). □
The following example shows how Rademacher averaging can help us
compute projective norms.
Example 2.23. The tensor diagonal in lp®Tr£p, 1 < p < oo.
By the tensor diagonal we mean the closed subspace spanned by the tensors
en ® en. where en are the standard unit basis vectors. We shall denote this
subspace by Dp.
(a) The case p = oo. Consider the tensor u = Yl"=i a»e» ® e» =
!Ci=i (atet) ® et- Using Rademacher averaging, we have
« = J (j2 a'r'We,) ® (y, r,(t)et] dt.
We may interpret this integral as a Bochner integral and hence
7r(u) < sup
0<t<l
]Pa,r,(t)e, 5^r,(t)e,
oo t=l
= ll(a.)llo
since |r,(t)| = 1 for all values of i and t. Now choose an index j between 1 and
n such that IKa,)^ = \a3\ and consider the bilinear form on t^ x (^ given by
B(x,y) = sgn(aja;JyJ.Wehave||B|| = 1 and (u, B) = \a3\ = ||(a,)||oo, which,
by the duality formula, implies that 7r(u) > ||(o,)||oo- Therefore, 7r(u) =
||(ax)||oo- It follows that the mapping "5^=1 a,e, G Co >-> Yl?=i a»e» ® e» G ^oo
extends to an isometric isomorphism between these spaces. Therefore Doo
may be identified with cq.
(b) The case 2 < p < oo. Once again, we employ Rademacher averaging
to produce a more efficient representation of the tensor u = J2?=i a»e» ® e»>
but now we distribute the coefficients in a different way:

2.5 Rademacher Techniques
35
u = jf (Esgn(a,)|a,|,/2r,(t)e,) ® (£ W1/2r,(Oe,) <tt.
This yields
Ij n n
y]sgn(a,)|a,|,/2r,(t)e, yV|1/2r,(t)ei
/ n v 2/p
= EW?/2 =ll(a.)llp/2-
To prove the reverse inequality, consider the bilinear form on lp x lp defined
by B(x,y) = J£=l blXlylt where &, = sgnKJkl"/2-1. We have («.B) =
Z)i=i la»lp^2- In order to estimate the norm of B, we use Holder's inequality.
Thus
lE^Ifc|<H(6.)IUIWI/»l|y|l7.
where 0 < a,7,)9 < 1 and 1/a + 1/0 + 1/7 = 1. We take a=p/(p-2j, and
/3 = 1 = p. This gives ||B|| < (Y^=1 |a,|p/2)(p-2>/p. Therefore
x: lax^2=i(«5 s>i < 7r(«) f x: i«.r/2)
1=1 \=1 '
(p-2)/p
from which we obtain ||(a,)||p/2 < ir(u). Therefore n(u) = ||(a,)||p/2. It follows
that Dp is isometrically isomorphic to lp/2-
(c) The case 1 < p < 2. This case is argued exactly as in Example 2.10
and we find that Dp is isometrically isomorphic to l\ for these values of p.
To summarize, we have shown that the tensor diagonal Dp in lp®*lp
satisfies
Ux if 1 < p < 2,
tp/2 if 2 < p < 00 , >.
Co if p = 00.
Dp= {
The Rademacher functions belong to the space Lp[0, l] for every p. Our
next result will enable us to identify the closed subspace of Lp[0, l] generated
by them. To do this, we must compute Lp-norms of the form
j=i
where c} are scalars. The case p = 2 is easy, since the Rademacher functions
form an orthonormal sequence in Z.2[0,1]. Therefore
c.l2,
(2.7)
j=»

36
2 The Projective Tensor Product
where ||c||2 is the ^-norrn of the vector c = (ci,.. .,(^,,0,0,...). It is a
surprising fact that the Lp-norm of 2?=i cjrj ^ equivalent to the ^-norm of
the vector c for every value of p.
Theorem 2.24 (Khinchine's Inequality). Let r3 be the Rademacher
functions. For each p £ [l, oo) there are positive constants Ap and Bp such
that
AP\\c\\2 <
c,r,
i=i
< SP||C||2
for every finite sequence c— (ci,..., c„) of scalars.
Proof. We have already seen that H2y=i c_.r_.ll2 = Ilcll2 and so we may take
A.2 = B-i = 1. Now the norm of Lp[0,1] is an increasing function of p and
so it suffices to prove the Bp inequality for p > 2 and the Ap inequality for
p<2.
Step 1: The Bp inequality for p > 2 and K = R.
By the monotonicity of the Lp-norms, it is enough to prove the inequality
when p is an even positive integer. Accordingly, let p = 2m. Using the
Multinomial Theorem, we have s
2m
n iip /•! / n \ ^m
,=1 Up ■'0 \, = 1 /
«=!+• +fc„=2m
(2m)l
h\...kn\
Jo
The integral is zero unless all the k} are even, say k} = 2m}. Now fci + ■
kn = 2m if and only if mi -I 1- mn = m and so
c,r,
"3=1
Therefore
(2m)!
w*i+* mmb'Tnn=m
(2m,)!...(2mn)!
_-2mi
|C1
„2m„
<(2m)! X, K)r---(c2„)m<(2m)!
w* 14* ■ ■■ 4*mn =fn
E (ci)r--(c2„)m<(2m)!(^Cj2)
■+n»n=n> J = l
X>,J < (pO^Hclla,
i=i Up
when p is an even integer.
Step 2: The Bp inequality for p > 2 and K = C.
If c} = aj 4- ibj are complex numbers, then
c,r,
',-1
XXrJ + E&.
"j=i
+
P '
"J1!
J=l
<Bp(l|a||2+|l&||2)
<V2Bp(||a||i + ||b||22)1/2-V^Bp||cl|2

2.5 Rademacher Techniques 37
Step 3: The Ap inequality for 1 < p < 2.
By the monotonicity of the Lp-norms, it suffices to prove the inequality in
the case p = 1. Using Holder's inequality with the conjugate exponents 3/2
and 3 and the Bp inequality for p = 4, we have
-li n i2 -li n 2/3i n
= / Ec^\dt=L \52c>r> £c-
0.11 n I v2/!/.lin i4 v
. Ec4') u SH *)
,4/3
<ft
1/3
<
<
c,r,
",=i
2/3
*44/3|M|42/3-
It follows that
y=i
> B?\\42
and so we have proved the Ai inequality with Ai = Bt . □
We should point out that the proof we have given here does not by any
means give the best values for the constants Ap and Bp-
Khinchine's inequality shows that the closed subspace of Lp[0,1]
generated by the Rademacher functions is isomorphic to fa. F°r values of p greater
than 1 we can say a little more about this subspace:
Theorem 2.25. Let 1 < p < oo. The closed subspace Rp of Lp[0,1]
generated by the Rademacher functions is isomorphic to fa. If P > 1 then Rp is
complemented in Lp[0,1].
Proof. It follows from Khinchine's inequality that the mapping 5D?=i ci ei ^
52?=i c3r] extends to an isomorphism from fa onto Rp.
Now suppose that p > 2. Then Lp[0,1] C Z.2[0,1] and so, for every / in
Lp[0,1], the "Rademacher coefficients",
</>„)= [ f(t)rn(t)dt,
Jo
of / are square summable. We claim that the required projection of Lp[0,1]
onto Rp is given by the formula
oo
v = ]£(/>'■»)'•».
First, we use the Cauchy -criterion to show that this series converges in
I,PtO, 1]. We have

38 2 The Projective Tensor Product
m / m \ 1/2
E</.^)^ <^(Ei</'^>i2) >
]=n P S=n '
and, since ((/,r„)) G £2, it follows that ]L.nLi(/'rn)rn converges to an
element of Lp[0,1]. Therefore, the series defining Ppf converges in the space
LP[Q, 1]. To see that Pp is bounded, we use the Khinchine inequality again,
together with Bessel's inequality. For every n, we have
j=i p S=i ' y=i
<Bpl|/||2<Bp||/||P.
Letting n -»• 00 we get
||Pp/||p<Bp||/||p.
Now it is easy to see that Pp is a projection and so it only remains to identify
the range of Pp. The definition of Pp shows that PP(LP[Q, 1]) C Rp. On the
other hand, Pprn = rn for every n implies that Rp C Pp(Lp[Q, 1]). Therefore
Pp(Lp[0,l]) = Rp.
Now let 1 < p < 2 and let q be the conjugate index of p. In this case, we
define Pp = P*, where Pq is the projection of Lq[0,1] onto Rg constructed
above. Now Pp, being the adjoint of a bounded projection, is itself a bounded
projection. Thus we have only to show that the range of Pp is .Rp. First, we
show that Pprn = r„ for every n, from which it follows that Rp C PP(LP[Q, 1]).
Let g £ L,[0,1]. Then
(g,PPrn) = (Pq9,rn) = (y2(g,rk)rk,rn) = (g,rn)
and so Pprn = rn. We claim that the series 2»Li(/. rn)rn converges weakly
to Ppf for every / G LP[Q, 1]. Since the weak and norm closures of a subspace
are the same, this implies that Pp(Lp[0,1]) C Rp and the proof is complete.
To establish our claim, fix g € Lq[Q, 1]. Then
.E</,r,)r,) = E</,r,)te,r,)
(P,9,f) = (9,PPf)
as n -> 00. Therefore Ppf is the sum of the weakly convergent series
.C~=l(/.rn)rn- D
It can be shown that the subspace Ri of L[0,1] is not complemented (see
Exercise 6.11).

2.6 Nuclear Bilinear Forms and Operators 39
We can deduce some interesting structural information about projective
tensor products from this proposition. We know that the projective tensor
product respects complemented subspaces isomorphically and we have seen
in Example 2.10 that h&xk contains a complemented isometric copy of li.
Thus we have
Corollary 2.26. Let 1 < pi,P2 < oo. Then LP1[0, ljdi^LpjfO, 1] contains a
complemented subspace that is isomorphic to l\.
Once again, we see that the projective tensor product has some "bad"
properties - the tensor product of reflexive spaces fails dramatically to be reflexive.
2.6 Nuclear Bilinear Forms and Operators
We have seen in Chapter 1 that tensors can be viewed as bilinear forms by
means of the canonical linear mapping that associates with a tensor u =
£"=1 <pt ® i/h G X* ® Y* the bilinear form Bu £ B(X x Y) given by the
formula B„(a;, y) = Y^=\ V»(a;)V,»(y)- Now if X and Y are Banach•spaces,
the elements of the tensor product X* ® Y* define bounded bilinear forms
on X x Y and we obtain an injective operator of unit norm from X* ®„. Y*
into "B(X x Y). Taking the extension of this operator to the completion gives
an operator
J: X'®*Y* -> B(X x Y)
of unit norm. A bilinear form on X x Y is said to be nuclear if it lies in
the range of this operator. Bearing in mind the explicit description we have
found in Section 1 for the elements of a completed tensor product, we see
that a bilinear form B is nuclear if and only if there exist bounded sequences
(<pn) and (V>„) in X* and Y* respectively, satisfying YL™=\ I.VnllHlM < oo,
such that
oo
n=l
for every x £ X, y £ Y. We shall refer to an expression of the form J^nii V»»®
ipn as a nuclear representation of B. The nuclear norm of B is defined to be
( OO OO x
||B||„ = inf £ IKIIHiM : B(x,V) = £>»(*)•*»(») .
^=1 n=l >
the infunum being taken over all the nuclear representations of B. It is easy
to see that
11*11 < 11*11*
and it follows that the nuclear norm is indeed a norm. We shall denote by
"Bn(X x Y) the space of nuclear bilinear forms with the nuclear norm. We

40 2 The Projective Tensor Product
leave it as an exercise to the reader to verify that "Bn(X x Y) is a Banach
space.
It seems at first glance that "Bn(X x Y) is just X*®^* in disguise. A
closer examination reveals that this is not necessarily the case. The difficulty
is that the canonical mapping
J:X*®„Y* ->3w(Xxy)
might fail to be injective. Let u G X*®t,Y* and let 5D^=i fn ® "0n be a
representation of u. Now, if the corresponding bilinear form B is zero, then
Blx> V) = Z.rT=i ¥"n(a")-0n(v) = 0 for every a; G X, y G Y. But in order
to deduce that u = 0, we must have (u, A) = YlnLi Mfnt "0n) = 0 for
every A G H(X* x y*) = (X*®^y*)*. In general, these conditions are not
equivalent. The "obstruction" to the identity "BN(X x Y) = X*®nY* is the
kernel of J:
ker J = {u G X'®„Y* : Bu(x,y) = 0 Vz G X,y G Y} .
Looking again at the definition of the nuclear norm, we see that it is the
quotient norm of X*®*Y*/ker J and so the best we can say in general is
that
3N(X xY) = X*®wy*/ker J.
We shall investigate this problem in greater detail in Chapter 4.
For now, we shall be content to note one example in which the space of
nuclear bilinear forms coincides with the projective tensor product.
Example 2.27. Nuclear bilinear forms on cq.
We have c%®„Cq = li&rli = l\{l\). Now the space t\{l\) is isometrically
isomorphic to the space t\ (N x N) of summable families of scalars indexed by
N x N. Let {enm} be the standard unit vector basis for this space, so that a
typical element of <i(NxN) has the form a = (anm) = Yin
m flnTnenm) With
llall» = Ylnm \anm\- We now have an identification of cJ®»Co with ^i(NxN),
under which the elementary tensor en ® ej^ corresponds to the basis vector
enro. It follows that every u G <$$*<{ has the form u = Yln,m anmen ® ej,,
with 7r(u) = "F].im lanm|. Therefore the corresponding nuclear bilinear form
is given by
Bu(x, y) = ]T anmxnym .
ntm
Now, if Bu = 0, then by taking x = en and y = em we get anm = 0 for every
n,m and hence u = 0. So, in this case, the kernel of the canonical mapping
is trivial and we have
3N(co -x c0) = *,<5Mi = *i(N x N).

2.6 Nuclear Bilinear Forms and Operators 41
We can also look at nuclearity from the point of view of operators. There is
a canonical operator J: X'^Y -»• £>(X, Y) of unit norm that associates with
the tensor u = Y^Li Vn ® Vn the operator Lu: X -»• Y, defined by Lu(x) =
YlXLi ¥>n(a")y.i- The operators that arise in this way are called nuclear. The
nuclear operators form a Banach space, which we shall denote by N(X, Y),
or simply N(X) when Y = X, with the nuclear norm:
{OO OO v
Y,\\<P»\W\Vn\\:T(x) = 'E<P*(*)Vn\,
n=l n=l >
the infjirmm being taken over all the representations of T of the form T(x) =
Y^=\ 'Pn(x)yn, where (<pn) and (yn) are bounded sequences in X* and Y
respectively such that Y^=i I.Pn||||Vn|| < oo. As for nuclear bilinear forms,
we can identify the space N(X, Y) with a quotient space of the projective
tensor product:
,N(X,Y) = X*®lrY/keiJ.
We leave it as an exercise to the reader to verify that, if an operator T is
nuclear, then so is its adjoint. Furthermore, we have ||T*||n < ||T||.\r. The
converse of this statement, and the question of whether the nuclear norms of
T and T* coincide, is more difficult and will be dealt with in Chapter 4.
The proof of the following proposition is routine and will be left to the
reader.
Proposition 2.28. Let S: X -* Y be a nuclear operator and let T G
L(W, X) and R £ L(Y, Z). Then the operator RST is nuclear and
||E5r|U<||fl||||5|k||r||.
As is the case with bilinear forms, we have
imi < \\t\\n
for every nuclear operator. The next result shows that the nuclear norm can
be much bigger than the operator norm.
Proposition 2.29. Let X have dimension n. Then the identity operator on
X has nuclear norm n.
Proof. We note first that the tensor product X* ®„ X is complete and can
be identified with N(X). Choose an Auerbach basis (ei,..., en} for X with
associated coordinate functionals ej,...,e^. Thus we have ||e,|| = ||e*|| = 1
for each t. Then x = Yl?=ie*(x)e* ^or every x € X and so the identity
operator, J, is given by the tensor Yl?=i e* ® e»- Therefore

42 2 The Projective Tensor Product
by the definition of the projective norm. Now consider the trace functional,
trx on X* ®lrX. We have ((p®x, trx) = <p(x), and tr* has norm one. Hence
"*=(£, < ® e"tr*) ^ w (J2 < ®e-)= 11*11 * ■
Therefore \\I\\N = n. D
In infinite dimensions, the class of nuclear operators is quite small. For
example, we can see from the definition that every nuclear operator is the limit
in operator norm of a sequence of finite rank operators. Thus, in particular,
every nuclear operator is compact.
We have seen that the nuclear bilinear forms on Co x co have a simple
description. We now look at nuclear operators on cq.
Example 2.30. Nuclear operators on Co.
We recall that c^fa^X = <!®„X can be identified with the space (i(X).
Under this identification, the canonical mapping J: cJ®wX = l\ (X) -»• !N(co, X)
associates with the element x = (xn) of l\ (X) the nuclear operator given by
T(a) = Yl^=i an%n- It follows easily that J is injective. Hence we have
X(c0,X) = ei&wX = ll(LX),
and, for every nuclear operator T,
oo
||7-|U = £||ren||.
We conclude with some remarks about finite dimensional spaces. If X and
Y are finite dimensional, then every operator from X into Y is nuclear, and, of
course, the tensor product X*®^Y is complete. Thus, "X{X, Y) = X*®*Y. In
the language of trace duality, we see that the nuclear norm and the operator
norm are in duality; the spaces N(X, Y) and H(Y, X) are dual, and for every
Tel(X,Y),
\\T\\N = sup{| tr* ST\ : S g L(Y, X), \\S\\ < 1}.
2.7 Exercises
Exercise 2.1. Show that X®nY contains norm one complemented isometric
copies of X and Y.
Exercise 2.2. Show that X®nY is isometrically isomorphic to Y®t,X.

2.7 Exercises 43
Exercise 2.3. Show that the Banachspaces (X^Y)®^ and X®W("K®WZ)
are isometrically isomorphic.
Exercise 2.4. Show that X ®„ Y is complete if either X or Y is finite
dimensional.
Exercise 2.5. Show that X ®„ Y is never complete if X and Y are infinite
dimensional.
Exercise 2.6. Show that £i®„-£i is isometrically isomorphic to l\.
Exercise 2.7. Let X be a closed subspace of C[0, l] that is isometrically
isomorphic to l\. Show that X®„ cq is not a subspace of C[0,1] ®„ Co- (Hint:
every operator from a C(.K') space into an Lt (fi) space is weakly compact.)
Exercise 2.8. Show that L\ (/n)®^L| (v) is isometrically isomorphic to the
space L\(fi x v), where fix i/ is the product measure.
Exercise 2.9. In each of the following cases, determine whether the function
/: [0,1] -¥ X is Bochner integrable with respect to Lebesgue measure, m. If
/ is Bochner integrable, calculate the Bochner integral f0 f dm.
(a) X = loo anfI f(t) = (rn(0)i where rn(t) are the Rademacher functions.
(b) X = too and /(f) = (n-'Mt)).
(c)X = L1[0,l]and/(t)=X[0it].
(d)X = Loo[0,l]and/(t) = mi].
Exercise 2.10. Find a Banach space X and a function /: [0,1] -> X such
that ll/H is integrable with respect to Lebesgue measure, but / is not Bochner
integrable.
Exercise 2.11. Show that C(K)* is a .Ci.A-space for every A > 1.
Exercise 2.12. Show that the sequence of Rademacher functions is not an
orthonormal basis for ^2(0,1].
Exercise 2.13. Show that the normed spaces "Bn(X x Y) and "N(X, Y) are
complete.
Exercise 2.14. Find an example of a compact operator that is not nuclear.
Exercise 2.15. (a) Let T: X -> Y be a nuclear operator. Show that the
adjoint operator T*: Y* -+ X* nuclear and ||r*|U < \\T\\n-
(b) Let T be an operator whose adjoint T* is nuclear. Why is it not
possible to deduce that T is nuclear?
Exercise 2.16. An operator T: H -*• H on a Hilbert space is said to belong
to the trace class if Xl^Li K^me,,)! < 00 for every orthonormal sequence
{e„} in H.
(a) Show that every nuclear operator on H belongs to the trace class.
(b) Let T be defined on the real space ti by Tx = (—a-2, a-i, —a.4, a.3,...).
Show that T belongs to the trace class but is not nuclear.

3. The Injective Tensor Product
In this chapter we study the injective norm for tensor products. The injective
tensor product gives a representation of Banach spaces of continuous vector
valued functions and injective tensor products with I>i(/j) spaces provide
an introduction to the Pettis integral. The duality theory of injective tensor
products leads to the introduction of the important classes of integral bilinear
forms and operators.
3.1 The Injective Norm
We shall now take a different approach to norming the tensor product, X®Y,
of two Banach spaces. We recall from Chapter 1 that the elements of the
tensor product can be viewed as bilinear forms on the product X' x Y"* of
the algebraic duals. If X."=1 Xi®ytis any representation of the tensor u, the
associated bilinear form is given by
n
t=i
Now the restriction of Bu to the product X* x Y* of the dual spaces is
bounded and so we have a canonical algebraic embedding of X ® Y into
"B(X* x Y*). The injective norm on X ® Y is the norm induced by this
embedding. We shall denote the injective norm of u £ X ® Y by e(u), or by
£x,y(«)i or e(u;X® Y), if it is necessary to specify the component spaces in
the tensor product. Thus we have
(u) = sup i PT
e(u)=sup\\S2<p(xl)ij>(yl)
<peBx.,i/,eBy.\, (3.1)
where "£"=1 x, ® yt is any representation of u. We may also view the elements
of X ® Y as operators, either from X* into Y, or from Y* into X. Thus, the
operators Lu: X* -)• Y and R^: Y* -)• X are given by Luip = J^Li f(x,)y,
and .R,,^ = X."=1 V'(j't)'Ct) and these operators have the same nerms as the
bilinear form Bu. This gives two more formulas for the injective norm:

46 3 The Injective Tensor Product
{II " -I
»t=i ■*
{|| n -I
|^-0(».)-b. :rP£BY'\.
»t=i '
supj ^ '
Unless there is a danger of ambiguity, we shall from now on use the same
symbol for the tensor u and the bilinear form Bu, or either of the operators
associated with u as described above.
A simple but invaluable observation is that we may replace the balls Bx'
and By in any of these formulas by norming sets: a subset A of Bx' is said
to be a norming set if we have ||x|| = sup{|y>(x)| : <p £ A} for every x € X.
For example, the closed unit ball of X is a norming set in the bidual X",
and so, if u = Y?=i <Pi®ipi€X'®Y*, then
e(u) = supi\f2<Pt(x)rlh(y) '■ x € Bx,y € BY\ , (3.3)
^ 'm=l ■*
with similar variations on (3.2).
We denote by X ®c Y the tensor product X ® Y with the injective norm.
The completion, denoted by X®eY, is called the injective tensor product of X
and Y. Unlike the projective tensor product, there is no general representation
of the elements of the completed tensor product X®eY. However, since X®e
Y is a subspace of "B(X* x Y*), the completion, X®CY is simply the closure in
B(X* x Y*), or equivalently, in L(X',Y), or in L(Y\X). Thus the injective
tensor product X®eY may be considered as a subspace of the space "B(X* x
Y*) of bounded bilinear forms, or of either of the spaces of operators L(X*,Y)
or £j(Y*,X). And, in the special case in which one of the component spaces
is a dual, X*®CY is a subspace of L(X,Y). The operators from X into Y
that arise in this way, as limits in the operator norm of sequences of finite
rank operators, are known as approximable operators.
We summarize some useful embeddings of the injective tensor product
into spaces of bilinear forms or operators:
X®CY C -B(X* x Y'), L(X',Y), or L(Y',X) (3.4)
X*®CY C'BiX xY*), L(y*,X*)t or L(X,Y)
X'®eY* C "B(X x Y), £(*,*"*), or L(Y,X').
We now record some of the elementary properties of the injective norm.
The proofs are easy consequences of the definition and are left to the reader.
Proposition 3.1. Let X and Y be Banach spaces.
(a) e(u) < 7r(u) for every u G X ®Y.
(b) e(x ®y) = \\x\\\\y\\ for every xSX,y£Y.

3.1 The Injective Norm 47
(c) If <p £ X*, ip £ Y*, then ip®ip is a bounded linear functional on X®eY
and\\V9i»\\ = \\<p\\n\\.
We now consider the tensor product of operators.
Proposition 3.2. Let S: X -+W and T:Y-iZbe operators. Then there
exists a unique operator S®CT: X®eY -)• W®CZ such that (S®cT)(x®y) =
(Sx) ® (Ty) for every x€X,y€Y. Furthermore, \\S ®e T\\ = ||S||||T||.
Proof. Let S ®T: X ®Y -)• W ® Z be the tensor product operator. If
u = X.r=i x, ® y, € X ® y, then
ew,z(S®T(u)) = supj \j2ip(Sxt)ip(Tyt) : tp <E Bw.,r/, G Bz. \
= sup{\J2(S*ip)xl(T*ip)yl :<peBw.,r/, €BZ. \
<\\S*\\\\T*\\ex,y(u) = \\S\\\\T\\ex,Y(v)-
Therefore S ® T is bounded for the injective norms, and has norm at most
||5|| ||T||. On the other hand, for any e > 0, we may choose x € Bx and y G BY
such that \\Sx\\ > (1 - e)||5|| and ||ry|| > (1 - e)||r||. Then e(x ® y) < 1 and
e((5® T))(x® y) > (1 - e)2||5||||r||. It foUows that ||5® T\\ > \\S\\\\T\\ and
therefore \\S ® T\\ = ||5||||r||. Now the operator S®T has a unique extension,
with the same norm, to the completed tensor product. We define S ®e T to
be this extension.
D
It follows immediately from the formulas (3.2) that the injective tensor
product respects subspaces, so that, if X is a closed subspace of W and Y is
a closed subspace of Z, then X®CY is a closed subspace of W®CZ. However,
the injective tensor product does not, in general, respect quotients; if X is a
quotient space of W, then X®eY need not be a quotiept of W®eY. We refer
to the exercises for further details on this.
We now look at some examples.
Example 3.3. The injective tensor product cq®cX.
We shall show that co®cX can be identified with the Banach space co(X)
of sequences in X that converge to zero, where this space carries the norm
||(xn)|| = supn||xn||. Consider the canonical mapping J: Co ® X -)• co(X)
that takes the tensor u = X."=1a, ® x, to the X-valued sequence Ju =
("£r=i o.ikXt)k- To show that Ju is, in fact, an element of co(X), we observe
that
/J flits.
t=i
<
max |a,fc|y"||x,|| -► 0
l<t<n *—'

48 3 The Injective Tensor Product
as fc -»• co. Now we compute the norm of Ju. Using the duality between co
and l\, we have
|| Ju|| = sup
k
= sup sup
= sup sup
/f aikXi
n
Y, atk<p(x,)
n
t=i
Y o.tk<p(x,)
O
j2Ylbkatkp(Xt)
■■ sup sup
6GB/J <p£Bx.
sup sup
¥>eBx. 6eB4l
= e(u),
*=it=i
by (3.1). It follows that J extends to an isometry from co®eX into co(X).
But it is easy to see that J(co ® X) is dense in co(X) and therefore J is an
isometric isomorphism.
Example 3.4. The injective tensor product 1\®CX.
Let t\[X\ denote the space of all unconditionally summable sequences in
X, in other words, sequences (xn) such that the series J") xn converges in
X, regardless of the order of the terms. We shall make use of the Bounded
Multiplier Test, which states that a sequence (xn) belongs to ii[X] if and only
if the series "£ bnxn converges in X for every bounded sequence of scalars (bn).
(We refer the reader who is not familiar with this material to Appendix B for
further details.) Now, if (xn) is unconditionally summable, we may define a
linear mapping T: X* -^ li by the formula Tip = (<pxn), and an application
of the Closed Graph Theorem shows that T is bounded. Furthermore, we
see from the Bounded Multiplier Test that the adjoint operator T* takes its
values in X; we have T*b = £"". bnxn for every b = (bn) G l^. It follows that
T is continuous for the weak* topology on X* and the weak topology on lt.
Therefore, by Alaoglu's Theorem, T maps the weak* compact closed unit
ball of X* into a weakly compact subset of l\. But, by the Schur property
of £i (see Appendix C), every weakly compact set is compact in the norm
topology. We may conclude that the operator T: X* -»• £i is compact, as is
the adjoint T*: l^ -)• X. Now this association of each element of li[X] with
an operator suggests a natural norm for this space. Bearing in mind the fact
that an operator has the same norm as its adjoint, we have two formulas for
the norm of an element (xn) of li[X]:
IIC
{°° 1 fll °° 1
Yi \<P(xn)\: <P G Bx. \ = supi p^ bnxn : b e Btoo \
n=l ' lHn=l '
It is a straightforward exercise, which we leave to the reader, to use the second
of these formulas to show that £i [X] is complete in this norm. Furthermore,
if we fix an element (xn) of t\[X], then the compactness of the operator T
implies that sup{">]]fc>n |y(a:fc)| •" <P G BX'} -»• 0 as n -¥ co. In other words,

3.2 C{K) and .Coo-spaces 49
the sequence whose nth term is (zi,...,zn,0,0,...) converges to (xk) in
M*l-
Now let us show that ti[X] is isometrically isomorphic to the injective
tensor product t\®cX. We begin with the canonical mapping J: l\ ®X —y ti[X]
which maps the tensor u = Et=i atx, to the sequence Jv. = (Et=i aikxi)k-
We compute the norm of Ju:
oo I / n v oo / n ,
||Ju|| = sup V*L(y]a,fcX, J = sup sup J2 bk I 5") a*k>p{xx) )
veBx. £^| \rr[ J veBx. beBtoo k=1 \t=1 /
= sup sup Y"b(a,)p(x,)
v>gbx. 6eB/oo|t=1
e(u).
Therefore J extends to an isometry of (\®CX into l\[X\. But we have seen
that the finite sequences are dense in MX], an^ since J(l\ ®X) contains all
such sequences, it follows that J is an isometric isomorphism.
3.2 C{K) and .C^-spaces
We now investigate injective tensor products with the space C(K) of
continuous functions on a compact topological space K. We shall show that the
injective tensor product C(K)®CX can be identified with the Banach space
C(K, X) of continuous functions from K into X, where the norm on this
space is given by ||/||oo = sup{||/(t)|| : t G K}. Consider the canonical linear
mapping J: C(K) ® X ->• C(K,X) given by Ju(t) = ^JL, /.(*)»<. where
X."=i A ® Xi is any representation of u. We show first that the norm of the
function J v. coincides with the injective norm of the tensor u. We have
ll-HIc
sup
teK
£/•(')»
sup
teK
X>(/.)z.
where 6t € C(K)* is the evaluation functional, St(f) = f(t). Now the set
{St : t 6 K} of all evaluation functional clearly forms a norming set in
C(K)*, and so, using the identification of the injective norm with the operator
norm in &(C(K)*,X), we see that || Julia, = e(u).
It only remains to show that J(C(K) ® X) is dense in C(K, X). Let
/ G C(K,X) and let e > 0. Let {/(*i),- ■■, f(tn)} be an e-net for the
compact set f(K) and for each t, take Vt = [t € K : \\f(t) - f(tt)\\ <
e}. Then the open sets Vi,...,Vn cover K. Let {gi,---,gn} be a
partition of unity on K subordinate to this cover. Thus the gt 6 C(K) take
their values in the interval [0,1], the support of g, is contained in Vt and
E,"=i <?•(*) = 1 f^ every t g K. Let it = £,n=1 g{ ® f(U) <E C(K) ® X. Then
II J« - f\\ = IIE,=i ».(*)/(«.) - /(OH = IIE?=, ».(*)(/(«.) - /(0)ll- N°w>
filing t G AT, then, for each i, either ||/(t.) - /(i)|| < e or g,(t) = 0. Therefore

50 3 The Injective Tensor Product
|| Jv. — /|| < e. It follows that J is an isometric isomorphism, and C(K)®CX
can be identified with C(K,X).
This identification can be applied to obtain a representation of a space
of continuous functions of two variables as an injective tensor product of
two spaces of continuous functions. Let K\ and K2 be compact spaces. The
space C(Ki x K2) can be canonically identified with the space C{K\, C(K2));
the function / £ C(Ki x K2) corresponds to the function /1 defined by
/i(*i)(*2) = f(ti,t2)- Therefore we have
C(Ar,)®£C7(A:2) = C7(AT, x K2).
We recall that projective tensor products with the space L\(jt) respect
subspaces. We propose to prove an analogous result for C(K), namely that
injective tensor products with C(K) respect quotients. Our proof will shed
some light on the structure of the space C(K) itself. Before stating the result,
we make the following simple observation. In order to show that an operator
R: W -¥ Z of unit norm is a quotient operator, it suffices to establish that,
for every z £ Z and every e > 0, there exists w £ W such that \\Rw — z\\ < e
and|M|<(l + e)||z||.
Proposition 3.5. Let Q: Z -»• X be a quotient operator and let I be the
identity operator on C(K). Then I®CQ: C(K)®CZ -»• C(K)®eX is a
quotient operator.
Proof. Since C(K) ® X is dense in C(K)®CX, it suffices to prove that for
every u £ C(K) ® X and every S > 0, there exists v £ C(K)®CZ such that
/ ® Q(v) = u and e(v) < (1 4- 5)e(u). Let {n,...,xn} be a <5-net for the
compact subset Jv.(K) of X, where Ju is the continuous function associated
with u. For each i, let V, = {t £ K : \\Ju{t) - x,|| < 5}, so that {V, Vn}
is an open cover of K. Choose a partition of unity {pi,... ,gn} subordinate
to this cover, so that each g, is a continuous function with values in [0,1] and
support contained in V„ and X."=1 gt(t) = 1 for every t £ K. In addition,
we may assume that there exist points tt £ V, such that gt(tt) = 1 for each
i. Consider the element u' = XllLi gt ® xt of C(K)®eX and the associated
function Ju'. We have e(u — u') < S. Furthermore, || Ju'(t)|| < max ||x,|| and
also ||Ju'(i,)|| = ||x,|| for each i. It follows that e(u') = ||J«'||oo = max||x,||.
Now, since Q is a quotient operator, we may choose Z\,..., zn £ Z such that
Qz, = x, and ||z,|| < (l + <5)||x,|| for each i. Let v = X),n=i Pt®«. £ C(K)®CZ.
Then J ®e Q(v) = u' and e(v) = \\JvW,*, = max||z,|| < (1 + 5)e(u') <
(1 + S)(e(u) + S). Since 6 > 0 is arbitrary, this completes the proof. D
If we examine this proof to see why it works, we find that the cnicial
argument is a finite dimensional one. Indeed, the functions glt..., gn of the
partition of unity span a subspace of C(K) that is a copy of the n-dimensional
space ££, - the argument given in the proof shows that the norm in C(K) of
the linear combination J^ILi "^9* "s exactly IJalla, = max |o,|. The rest of the
proof essentially takes place in the tensor product £^ ®c X.

3.3 Li(n)®cX and the Pettis Integral 51
Any space with the same finite dimensional structure as C(K) will behave
in the same way. We say that a Banach space X is a £00,a-space for some
A > 1 if every finite dimensional subspace, M, of X is contained in a finite
dimensional subspace, N, of dimension n, say, whose Banach-Mazur distance
from P^o *s a* m°st A. Thus, there exists an isomorphism T: N —► £^o suc^
that ||T||||T-1|| < A. The space X is called a £„,-space if it is a £.oo,A-space
for some A. Arguing as in Section 2.4, but with the approximation by simple
functions replaced by partitions of unity, we find that the space C(K) is a
-C-x^i+e-space for every e > 0.
Let us agree on some terminology. We shall say that injective tensor
products with the space X respect quotients if, for every quotient operator
Q: Z —► Y, the tensor product operator I ®CQ: X®eZ —► X®eY is a
quotient operator. And we say that injective tensor products with X respect
quotients tsomorphically if, for every quotient operator Q: Z —)• Y, the
tensor product operator I ®eQ is surjective. This is equivalent to the statement
that the space X®eY is isomorphic to the quotient space associated with the
surjective operator J ®c Q.
Theorem 3.6.
(a) If X is a Hoc-space, then injective tensor products with X respect
quotients isomorphically.
(b) If X is a -Coo, i+e -space for every e > 0, then injective tensor products
unth X respect quotients.
Proof, (a) Let Q: Z -)• Y be a quotient operator. Let u € X®CY and let 5 >
0. Choose u' € X ® Y such that e(u — u') < S. Then, fixing a representation
of u', we obtain a finite dimensional subspace M of X, such that u' lies in
M®c Y, where its injective norm is the same. Now M is contained in a finite
dimensional subspace, N, of X, of dimension n, say, for which there exists an
isomorphism T: M -¥ £"£> with ||T||||T-1|| < A. We may assume, without loss
of generality, that ||r|| < 1 and ||r_1|| < A. Consider the element T ® I(u')
of e^®eY. We have e{T ®I{u')\tl0 ®Y) < e(«')- ^ce Co k the space
C(K) where K is the discrete space with n points, it follows that P^®^
is a quotient space of P^®^ and so there exists w £ Co®e^ such that
I®Q(w) = T®I(u') and e(w) < (l + 5)e(u'). Let v = T-1®I(w) £ X®CZ.
Then J® Q(v) = u' and e(v) < A(l + 5)e(u') < A(l + 5)(e(u) + 5). It follows
that for every S > 0 there exists z € X®CZ such that J ® Q(z) = u and
e(z) < X(l+S)e(u).
(b) follows immediately from the proof of (a). D
3.3 Lx{n)®eX and the Pettis Integral
Let (fl, E, (i) be a measure space. We have already seen that the
projective tensor product Li(ji)®„X can be identified with the space Li(fi,X) of

52 3 The Injective Tensor Product
Bochner integrable functions from Q, into X. We seek a similar representation
for the injective tensor product Li(p)®cX. This requires the development of
a new integral.
We shall assume throughout this section that the measure p is finite. We
recall that a function /: Q. -)• X is said to be weakly /n-measurable if the
scalar valued function ipf is /n-measurable for every tp S X*. The weakly
/Li-measurable function / is weakly integrable with respect to p if the function
ipf is integrable for every tp £ X*. We propose to show that we can define an
integral for such functions, but that this integral lies in the bidual, X". To
this end, consider the linear mapping S: X* -)• L\(p) given by Sip = ipf. To
see that S is bounded, we apply the Closed Graph Theorem. Suppose that
f(ipn) converges to ip in X* and (ipnf) converges to g S Li(p). The sequence
(ipnf) lias a subsequence (ipnkf) that converges /j-almost everywhere to g.
But (ipnk f) converges pointwise to ipf. Therefore g = ipf and we have
established that S is bounded. Now, since p is finite, the dual space of L\ (p) is
Looifi) and so the adjoint operator T = S*: L^p) -»• X" satisfies
(<P,Tg)= f g(tpf)dp
for g £ Loa(n). It now follows, by taking g = Xe, that if / is weakly integrable
then for every measurable subset E of Q. there is an element of X", which
we denote by fE f dp, such that
(tp,l fdp\ = I ipf dp
for every tp £ X*. The vector fEfdfi is called the Dunford integral of / over
E.
To see that we do need to go to the bidual to define this integral, consider
the function /: [0,1] -)• Co defined by f(t) = (fi(En)~1XB„(t))> where ft is
Lebesgue measure and (En) is any sequence of mutually disjoint Lebesgue
measurable sets of positive Lebesgue measure whose union is [0,1]. It is easy
to see that / is weakly /n-measurable, and if tp = (An) £ cj = llt then
/0 ipf dp = Yln ^n- Therefore the Dunford integral of / over [0,1] is the
element (1,1,...) of 1^.
We say that the weakly integrable function / is Pettis integrable if the
integral fEfdp belongs to X for every E € E. In this case, the integral
fEfdp is referred to as the Pettis integral of / over E. When / is Pettis
integrable, the operator T takes its values in X, since it is obvious that
Tg = fngfdp belongs to X for every measurable simple function, g, and
these functions are dense in L^p). The operator T is sometimes referred to
as the Dunford operator of /. We can say a little more about this operator.
Proposition 3.7. Let (fl, E,/n) be a finite measure space and let f: Q, -)• X
be Pettis integrable with respect to p. Then the operator T: L00(p) -)• X
defined by

3-3 Li(n)®cX and the Petdis Integral 53
Tg= f gfdn
Jn
is weakly compact.
Proof. We have already seen that the Dunford integral fn gf'dp, belongs to
X for every g £ L-x,^). Let ip £ X* and g £ Loo(/u)- Then, by the definition
of the Dunford integral,
(Tg,<p) = / g{<pf)dn= I ip(gf)d(i=(l gfdp,if>\.
Therefore T is continuous for the weak* topology on Loo(m) and the weak
topology on X and it follows that T is weakly compact. D
The definition of the operators S and T suggest a natural norm for the
Pettis integrable functions. The Pettis norm of the Pettis integrable function
/ is defined by
|/|i = sup^ / \tp}\ dp : >p e BX' \ = supi / gf d(i
■ g e BLoo{tl)
Of course, it is necessary to group functions into equivalence classes in order
for this formula to define a norm. We return to this question shortly.
Clearly, every Bochner integrable function is Pettis integrable, with the
same value for the integrals. Furthermore, it follows easily from the definitions
of the Pettis and Bochner norms that |/|i < ||/||i- Thus, the Pettis integral is
an extension of the Bochner integral. We now give an example of a Pettis
integrable function that is not Bochner integrable. Working again with Lebesgue
measure on [0,1] and X = Co, let f(t) = (nxi„(t)), where I„ is the interval
(l/(n +1), 1/n]. The Pettis integral of / is the vector (l/(n +1)) in c0. Also,
since Co is separable, / is Lebesgue measurable. However, ||/|j = ]C.H=i nXini
which is not Lebesgue integrable, and so the Bochner integral of f does not
exist.
The next result shows that the Pettis integral interacts well with
operators.
Proposition 3.8. Let f-.Q.-^Xbea Pettis integrable function and let
R: X —► y be an operator. Then the function Rf is Pettis integrable and
f^Rfdti = R^fjd^j.
Furthermore, |E/|, < \\R\\ |/|,.
Proof. Let r/>£Y*. Then ip(Rf) = R*ip(f) and so Rf is weakly integrable
with respect to ft. Also,
^J^Rfd^ = j^{Rf)dti=^jjdti,R*^i = (^R^jjd^j,^,

54 3 The Injective Tensor Product
and hence we have fn Rf dp = Rfnf d/j, £ Y. It follows easily from the
definition of the Pettis norm that \Rf\i < ||-R|||/|i. D
The Pettis integral has some unexpected, and rather unpleasant
properties. For example, let X be the space co(I), where the indexing set J is the
interval [0,1] and let /(f) = et for t € [0,1], where et are the usual unit
basis vectors. Now / is certainly weakly Lebesgue measurable, since every
ip £ X* is given by a vector (At) in l\ (I) which has only a countable number
of nonzero components. Thus <pf(t) = ^2nKnX{t„}(t) for some countable
subset {*„} of [0,1] and so <pf(t) = 0 almost everywhere for each tp.
Therefore the Pettis integral of / over every measurable subset of [0,1] vanishes,
although / is everywhere nonzero. This cannot happen if we insist on mea-
surability in the strong sense. We leave it to the reader to verify that if / is a
ft-measurable Pettis integrable function, then <pf(t) = 0 /z-almost everywhere
for every tp £ X* is equivalent to f(t) = 0 p-almost everywhere.
We may now define the vector space Pi (ft, X) of equivalence classes of
ft-measurable Pettis integrable functions in the usual way. The Pettis norm,
defined above, is a norm on this space. Unfortunately, the space P\(n,X)
need not be complete. We shall denote the completion of the space P\(n,X)
by Pi(n,X). A full understanding of this space entails a study of vector
measures, which we shall meet in Chapter 5.
We now direct our efforts to showing that the injective tensor product
Li(n)®eX can be identified with Pi(fi,X). The crux of the proof will be to
establish the density of the p-measurable simple functions in P\(n,X). The
following proposition will enable us to prove this.
Proposition 3.9. Let (fi, E, fi) be a finite measure space and let the function
f: Q, -> X be Pettis integrable with respect to ft. If(An) is a disjoint sequence
of measurable subsets ofil, then
oo
n=l
Proof. Suppose that the series J^nLi l/X/i„|i diverges. For each n, we may
choose ipn G Bx- such that fA \ipnf\dfi > (l/2)|/X/i„li- It follows that the
series X^jJLi/t \ipnf\d(i also diverges. Now, for each n, we may choose a
measurable subset Bn of An such that
/ fnfd/j, > - \<pnf\dfi,
Jb„ 4 Ja„
and so the series Xl^Li | fBn ipnf dp\ diverges.
Consider the operator U: X* -)• £| defined by
U<p
(1 ■"*•)„•

3.3 Li(im)®cX and the Pettis Integral 55
Now U is the composition of the operator S: X* -¥ L\ (fi) given by Sip = ipf,
with the operator from Li(fi) into t\ that takes the function g s Li(fi) to
(fB gdfi). By Proposition 3.7, S is weakly compact. Therefore U is also
weakly compact and so, by the Schur property of l\, it follows that U is
compact. Therefore the convergence of the series X.nii I Sb ff dft\ is uniform
m ip £ Bx' and so the series Y^=\ I Sb Vnf dfi\ must converge for every
sequence {<pn} m -Bx-, a contradiction. D
If we now assume in addition that / is /n-measurable, we have:
Corollary 3.10. Let (Q, E, p) be a finite measure space and let f: Q, -»• X
be a fi-measuTable Pettis integrable function. Then there exists a sequence of
fi-measurable simple functions that converges to f m the Pettis norm.
Proof. Since / is /n-measurable, the sets En = {u € 0, : ||/(w)|| > n] are
measurable. Let An = En\ En+1, so that (An) is a disjoint sequence of
measurable sets and En = \Jk>n Ak for every n. Then
l/X-sJi = sup / \<pf\dlj,<'^2\fxAk\i,
V>eBx. JEn k=n
and it follows from the proposition that \fXEn\i -4 0 as n -> oo.
Now, given e > 0, choose n so that \fxEn\i < e/2- Then / is bounded
on the complement of En and hence is Bochner integrable there. By the
definition of the Bochner integral, there exists a /n-measurable simple function
g, vanishing outside En, such that \\fxE% — g\\i < e/2, where ||.||i is the
Bochner norm. Therefore
1/ - g\i < \fxB' - g\i + \fxEnU < WfxB- - fflli + UxeJi < e.
a
We have seen that the Dunford operator T: L^p) -> X of the
Pettis integrable function /, given by Tg = fn gf dp, is weakly compact. The
behaviour of this operator improves along with that of /:
Proposition 3.11. Let p be a finite measure, let f be Pettis integrable with
respect to p and let T: L00(fi) -»• X be the weakly compact operator given by
Tg = fngfdp.
(a) If f t3 fi-measurable, then T is compact.
(b) If f is Bochner integrable, then T is nuclear.
Proof, (a) Let (/„) be a sequence of /n-measurable simple functions that
converges to / in the Pettis norm. Then the operators Tn: Loo(p) -¥ X
defined by Tng = fn gfn dfi are of finite rank and \\T - Tn\\ = '/-/„',-». 0.

56 3 The Injective Tensor Product
Therefore T, being the limit in operator norm of a sequence of finite rank
operators, is compact.
(b) Let / £ Li(n,X) = Li(/n)®^X. Then there are bounded sequences
(/„) and (xn) in Lx(p) and X respectively such that Yln ll/nllill^nll < °o
and / = X",n fn ® xn. Then Tg is given by the Bochner integral fn gf dfi for
g £ Loo(fj,) and we have
fY.ghxkdfi ^iipiiooEllMlill^ll-*0
'*>„
k>n
as n -¥ oo. Hence
T9 = Y,U 9fndii\xn
for every g € L00(fi). Therefore T is nuclear. □
The fact that the Dunford operator of a /n-measurable Pettis integrable
function is compact enables us to prove an important result that bridges
the gap between weak and norm summability. A sequence (xn) in a Banach
space X is said to be weakly subseries summable if the series "£n Xkn is
weakly convergent for every increasing sequence (fcn) of positive integers.
Equivalently, the series ^2n£nxn *s weakly convergent for every choice of
signs, en = ±1.
Proposition 3.12 (The Orlicz-Pettis Theorem). Every weakly sub-
series summable sequence in a Banach space is unconditionally summable
in norm.
Proof. Suppose there is a sequence (xn) in the Banach space X that is
weakly subseries summable but not unconditionally summable. Then,
replacing (xn) by a suitable sequence (enxn) with en = ±1, we may assume
that (xn) is weakly subseries summable, but that the series "£"]„ xn diverges.
Therefore there exist increasing sequences of positive integers, (m*) and (n*),
with rrifc < nfc < mfc+i for every fc, and 8 > 0, such that
J2 XJ
><5
for every fc. Let Zk = ^2^mk Xj- Then the sequence (zj) is weakly subseries
summable, but ||zj|| > S for every j. A simple application of the Closed Graph
Theorem shows that the weak sums of the form JV e}z}, where eT= ±1, lie in
a bounded subset of X. Thus we can define a bounded function /: [0; 1] -¥ X
by taking f(t) to be the weak sum of the series J"). r}(t)z}, where r3(t) are
the Rademacher functions. By the Pettis Measurability Theorem
(Proposition 2.15), / is measurable with respect to Lebesgue measure. Therefore /

3.4 The Dual Space of X®CY 57
is Bochner integrable and so the Dunford operator T: Z<oo[0,1] -» X is
compact. But this forces the set {z3} = {Tr3} to be relatively compact in X,
which is impossible, since (z}) converges weakly, but not in norm, to zero.
□
We conclude with the representation of the injective tensor product
Li(n)®cX as the completion of the space of /n-measurable Pettis integrable
functions.
Proposition 3.13. Let fi be a finite measure. The injective tensor product
L\(fi)®eX is isometncally isomorphic to the space P\(n,X).
Proof. Let J: Li(fi) ® X -)• P\(fi,X) be the canonical linear mapping and
let v. = XllLi fi®xt£ L, (fi) ® X. Then
|Ju|x = sup
veBx
dfi = sup sup
Y>eBx. geBLa>{ll
/ V) P/.V(-c.) dfi
= sup
sup
v>eBx. geBLooM t=,
X.^'/'XV'1')
= e(u).
Therefore J is an isometry. Now the /x-measurable simple functions lie in the
range of J and these functions are dense in A(/u, X). Therefore J extends to
an isometric isomorphism from Li(fj,)®cX into P\(fi,X). D
3.4 The Dual Space of X®EY
We now turn our attention to the dual space of the injective tensor product
X®CY. Since the injective norm on X ® Y is smaller than the projective
norm, every bounded linear functional on X®eY is the linearization of a
unique bounded bilinear form on X x Y. We wish to characterize the bilinear
forms that arise in this way.
Let us look again at the definition of the injective norm of an element
« = £?=! x3 ® yj oiX®Y:
e(u)=supj ^.pfc.Myt)
tpeBx.,ipeBY'
This formula encourages us to think of u as a continuous function on the
compact product space Bx- x By, where Bx- and By carry their weak*
topologies. Taking this point of view, we see that we have an embedding of
X ®eY into the Banach space C(Bx* x By), where this space carries the
usual supremum norm. This embedding extends to the completion:
X®CYCC(BX- xBy).

58 3 The Injective Tensor Product
This canonical embedding of the injective tensor product into a C(K)
space enables us to identify the dual. The Hahn-Banach Theorem allows us to
consider a bounded linear functional on X®eY as a bounded linear functional
on the space of continuous functions. Then the Riesz Representation Theorem
shows that action of this functional is given by integration with respect to
a suitable measure. To see how this works, suppose that B is a bounded
bilinear form on X xY whose linearization, B, is bounded for the injective
norm on the tensor product. Then B extends to a bounded linear functional
on C(Bx' x By) with the same norm. This functional is given by a regular
Borel measure, ft, on Bx> x By ■ Thus we have
(u,B)=f u(<p,1,)dp(ip,il>) (3.5)
J B}{ • X By •
for every u £ X®eY, and ||B|| = \\fi\\, where \\fi\\ is the variation norm of fi.
In particular, if we take u to be an elementary tensor, we see that
B(x,y)= f <p(x)rl>(y)dii(ip,iP) (3.6)
JBX. x9r.
for every x £ X, y £ Y.
Conversely, if p is a regular Borel measure on Bx- x By, we may define a
bounded linear functional on X®CY by (3-5) and the corresponding bilinear
form B satisfies (3.6). Furthermore, a simple computation shows that ||B|| <
IMI-
We have obtained a complete description of the dual space of X®CY. Our
findings are summarised in the following proposition.
Proposition 3.14. Let B be a bilinear form on X xY. Then B is a bounded
linear functional on X®eY if and only if there exists a regular Borel measure
fi on the compact space Bx- x By such that
B(x,y)= <p(x)ip(y) dfi(tp,ip)
J B jp* y,By
for every x £ X, y 6 V. -FWt/iermore, the norm of B is given by
\\B\\ = mi\M\,
where fi ranges over the set of all measures that correspond to B in this way,
and this mfimum is attained.
We say that a bilinear form B on X x Y is an integral bilinear form
if its linearization, B, is a bounded linear functional on the injective tensor
product X®CY. The integral norm of B is defined by
||B||/=inf||Ai||,

3.4 The Dual Space of X®eY 59
the infimum being taken over all the regular Borel measures on Bx. x By
that satisfy (3.6). The Banach space of integral bilinear forms, with this norm,
will be denoted by "Bj(X x Y). Thus, we have
(X®CY)* = 3i(XxY).
We note some elementary properties of the class of integral bilinear forms.
If B is an integral bilinear form on X x Y and S: W -)• X and T: Z -)• Y
are operators, then the bilinear form B' defined on W x Z by B'(w,z) =
B(Sw,Tz), is integral and ||B'||, < ||B||/||5||||r||.
Next, we observe that every nuclear bilinear form is integral. Indeed, if B
is nuclear, we may express it in the form
oo
B{x, y) = X. Xnfn(x)r/)n(y) ,
n=l
where (An) is a summable sequence of scalars, and (<pn), (ipn) are sequences
of distinct unit vectors in X*, Y*, respectively. We may interpret this sum
as an integral of the type given in (3.6), where fj, is the measure with'point
masses An at the points (tpn,ipn)- Therefore
VN(XxY)CBr(XxY).
Furthermore, we have \\fi\\ = J^nLi l^nl and the sequence (An) may be chosen
so that X^nLi |At»| is arbitrarily close to the nuclear norm of B. It follows from
the definition of the integral norm that
l|B||/ < W\n
for every integral bilinear form B.
Finally we recall that the injective tensor product respects subspaces; if
X, Y are subspaces of W, Z respectively, then X®CY \s a subspace of W®CZ.
It follows by the Hahn-Banach Theorem that every integral bilinear form on
X x Y can be extended to an integral bilinear form onW x Z with the same
integral norm.
We now look at some examples.
Example 3.15. Integral bilinear forms on Co x cq.
We have seen that co®eX = co(X). Therefore (co®eco)* = ti(ti) = £i®,.£i
and so, in this case we have "B/(co x c0) = "B^ (co x Co) with equality of the
nuclear and integral norms.
Example 3.16. Integral bilinear forms on C(K) spaces.
Let K-i and K^ be compact spaces. We have seen that C(K{)®CC{K2) =
C(Ki x K2) and it follows that a bilinear form B on C(Ki) x C(K2) is

60 3 The Injective Tensor Product
integral if and only if there exists a necessarily unique regular Borel measure
(i on K\ x K2 such that
B(f,g)= f f(s)g(t)dti(s,t)
JK1xK2
for every / £ C(Ki), g G C(K2). By the uniqueness of the measure ft, we
have ||B||/ = ||/u||. A special case of this is the integral bilinear form on
C(K) x C(K) given by
B(f,g)= f /(t)p(W),
Jk
where ft is a regular Borel measure on K. Again, we have ||B||/ = ||/u||.
This example is the prototype of all integral bilinear forms. It can viewed
in purely measure theoretic terms:
Example 3.17. The canonical integral Mmear form on L00(/l() x L00(/li).
Let (CI, E, ft) be a finite measure space. Consider the bilinear form B: Loo (fi) x
■^oo(^) -»• K defined by
B(f,g)= f fgd».
We shall show that B defines a bounded linear functional on the tensor
product L>oo{ii) ®c Loo(li)- Since the measurable simple functions are dense in
Loaili), it suffices to consider tensors of the form u = Y%=\ /• ® P«i where
/,, g, are measurable simple functions. Taking the canonical representations
of /, and p,, we obtain a representation of u as a finite sum of the form
u = X.,-fc CjkXA, ® XBk, where (A}) and (Bfc) are finite sequences of
mutually disjoint measurable subsets of H, each having nonzero measure. Let us
compute the injective norm of u. On the one hand, since the closed unit ball
of Li (ft) is a norming set in Loo(n)*, we have
e(u) =supl\^2cjk(xAjJ)(XBk,g)\ : f,9 e Lifji), ll/lli, llplli < l}
<sup{2>jfc|^ \f\dnJB lffl^:||/||i,IHIi<l}
< sup|cjfc|sup
< sup|cjfc|.
j,k
Now fix j and fc. Let / = fi(A])~1XAj and g = fi(Bk)~1XBk- Then / and g
are unit vectors in L\ (fj.) and so

3.4 The Dual Space of X®CY 61
e(u)>
Y^Cr>(XA<-'f)('X-B.,9)
\c,k\-
Therefore e(u) = sup^ fc \c]k\. For such a tensor u, we have
K«,B>|
< sup |cjfc| Y^V(A] fBfc) < sup |c,fc|A»(fi) = e(«)/^(fi),
j,*
and so B is bounded on Loo(n) ®c Loo(/L(). This computation also shows that
||B||/ < n($l)- On the other hand, if we take / = g = 1, then the elementary
tensor u = f ® g has unit injective norm and hence (u, B) = n(fl) < \\B\\j.
Therefore ||B||/ = fi(Q).
Our next result shows that every integral bilinear form can be factored
through the canonical integral bilinear form on a suitable L00(/l().
Theorem 3.18. LetX andY be Banach spaces. A bilinear form B onXxY
is integral if and only if there exists a finite measure space (fi, E, p] and
operators S: X -)• Loo(fi), R:Y -* L^fi) such that
i(x,y) = f(Sx)(Ry)dp
Jn
for every x € X, y eY. Furthermore,
11511, = inf!|5||||E||Mfi),
where the tnfimum is taken over all such factorizations of B, and this mfimum
is attained.
Proof. Suppose that B is integral. Then there exists a regular Borel measure,
v, on the compact space Bx- x By such that
Hx,y) = J
Br' X By
<p(x)ip(y) du
for every x € X, y £ Y and ||B||/ = \\v\\. By the Radon-Nikodym Theorem,
there is a Borel measurable function, h, on Bx* x By such that \h(t)\ = 1 for
every t G Bx' x By and du = hd\v\. Let ft = \v\ and Q = Bx* x By with
E the Borel u-algebra. Let S: X -* Loo(fi), R:Y-> I>c»(l<) be the operators
given by (Sx)(ip,-p) = tp(x) and (Ry)(ip,r/)) = h(ip,ip)tp(y). Then
B(x,y)= [(Sx)(Ry)dfi
and, since ||5||, ||A|| < 1, we have ||B||, = \\u\\ = rfCl) > \\S\\\\RMn).
Conversely, given such a factorization of the bilinear form B, it follows
that B is integral and ||B||/ < ||S||||.R||A*(fi). D

62 3 The Injective Tensor Product
We note that, by multiplying R or S by a constant, it is possible to assume
that the measure p in this factorization is a probability measure.
We conclude with a look at some of the finite dimensional aspects of
integral bilinear forms. If X and Y are finite dimensional, then X ®e Y =
-B(X' x y*) and thus -B,(X x Y) = X* ®„ Y* =S#x Y). Therefore,
every bilinear form is integral and the integral and nuclear norms are the
same. We shall see in Chapter 5 that this coincidence of the integral and
nuclear bilinear forms is a phenomenon that can also occur in certain very
special infinite dimensional spaces. However, in general, the spaces of integral
and of nuclear bilinear forms are different.
We can find the influence of finite dimensional considerations at work in
infinite dimensions. Indeed, a bilinear form B on the product X x Y of any
two Banach spaces is integral if and only if there exists a positive constant
C such that, for all finite dimensional subspaces M, N of X, Y respectively,
we have
\\BM,N\U<C,
where Bm,/V denotes the restriction of B to M x N. Furthermore, the integral
norm of B is the infimum of all such C. This is an immediate consequence of
the fact that the injective norm, like the projective norm, is finitely generated;
in other words, we have
e(u) =inf{eM|W(u) : M C X,N CY, u G M ® N}
for every u € X ® Y. The remarks above show that this property of the class
of integral bilinear forms is not shared with the nuclear bilinear forms.
3.5 Integral Operators
We now examine the operators that correspond to the integral bilinear forms.
Every operator T: X -> Y corresponds to a bilinear form, Bt : X x Y* -)• K,
defined by Br(x,ip) = (Tx,r/)). We say that T is an integral operator if the
bilinear form Bt is integral and we define the integral norm of T, denoted
by ||T||/, to be the integral norm of the bilinear form Bt- Clearly, we have
||T|| < ||T||/. The space of integral operators from X into Y with this norm
will be denoted by 3(X, Y). The correspondence T ■-> BT embeds 3(X, Y) as
a subspace of B/(X x Y*). We leave it to the reader to verify that D(X, Y)
is closed in -B/(X x V). Therefore 3(X,Y) is a Banach space.
There is a plentiful supply of integral operators. Every nuclear operator is
easily seen to be integral, with ||r||7 < ||T"||jy Furthermore, if T: X -»• Y is
integral and V: W -> X, U: Y -► Z are operators, then the operator UTV
is integral, with
FTV||,<||[/||||r||,||V||.
If fi is a finite measure, then the canonical injection

3.5 Integral Operators 63
J: Looi/j.) -¥ L,(/n)
is integral, with integral norm equal to fi(ii), since the corresponding bilinear
form is the canonical integral bilinear form on Loo(m) x £00 (/■*)■ This operator
is the prototypical integral operator:
Theorem 3.19. An operator T: X -¥ Y is integral if and only if there
exists a finite measure space (fl, E, (i) and a pair of operators S: X —)• Loo{\i),
R: L\{fi) -)• Y" such that JT = RIS, where J is the canonical
embedding ofY into Y" and I is the canonical mapping from Loo(/li) into L\{^i).
Furthermore,
||r||/ = inf||5||||Ji||Ai(n),
where the infimum is taken over all such factorizations ofT, and this infimum
is attained.
Proof. Suppose that T is integral. Applying Theorem 3.18 to the integral
bilinear form Bt, there exists a finite measure space (fl, E, (i) and operators
S: X -)• Loo(fi) and U: Y* -)• Loo(fi) such that
<r*,t/>)= j'(Rx)(Ui>)dp
for every ieX^cI", with ||r||, = ||BT||/ = ||S||||t/|.At(fi). We obtain
the required factorization by taking R to be the restriction of U* to Li(fi).
Then, bearing in mind that the closed unit ball of £.(a0 is a norming set in
L^M*, we have ||r||7 = ||5||P|Hfi).
Conversely, suppose that there exists a factorization of the form JT =
RIS. Let V: Y* -»• L^ifi) be the restriction of R* to Y*. Then the bilinear
form associated with T satisfies
BT(x,iP) = (Tx,iP) = (RISx,1>)
= (ISx, Vi>) = f (Sx)(Vt/>) dfi.
Jn
Therefore BT is integral, and ||BT||/ < ||-R||||V||/a(fi). □
T J
X ► Y - Y*
R
L°°(t>)
iiM
The factorization scheme given by the proposition is shown in the diagram
above. An examination of the proof shows that the space Loo(fi) can be

64 3 The Injective Tensor Product
replaced by a C(K) space, where the measure fi is now a positive regular
Borel measure on K and / is the canonical mapping of C(K) into Li(/i),
with the same statement regarding the integral norm of T. This factorization
is shown below:
— Y
C(K)
R
LM
The next result illustrates the power of these factorizations. We recall that
an operator is said to be completely continuous if it maps weakly convergent
sequences into norm convergent sequences.
Proposition 3.20.
pletely continuous.
Every integral operator is both weakly compact and com-
Proof. It suffices to prove these properties for either of the canonical
operators J: Loo(/L() -»• Li(fi) or /: C(K) -)• Li(fi). Now the operator /: L^fi) -)•
Li(fi) factors through the canonical injection Loo(n) -»• Lp(fi) for every p > 1:
^(a*)
Lifa)
Hence, by the reflexivity of Lp((i), the operator J is weakly compact.
To see that the canonical operator I: C(K) -)• L\{(i) is completely
continuous, suppose that the sequence (/„) converges weakly to zero in C(K).
Then the functions /„ are uniformly bounded and pointwise convergent to
zero and it follows from the Bounded Convergence Theorem that (/„)
converges to zero in Li(/u)-norm. □
In the factorization scheme for an integral operator T: X -»• Y, it is
essential that we enlarge the range space to the bidual of Y. In some cases, it
is possible to factor T directly. If Y is complemented in Y** by a projection of
norm one, we may compose JT with this projection to obtain a factorization
of the form shown below:

3.5 Integral Operators 65
— Y
R
Looiii) —j- L^fi)
and we have ||T||/ = inf ||.R||||S||/Lj(fl), where the infimum is taken over all
such factorizations. This applies in particular if Y is a dual space or an L\ (ft)
space.
If we insist on factoring the operator T rather than JT, we obtain a
smaller class of operators. We say that the operator T: X -»• Y is Pietsch
integral if T can be factored in this way as RIS, and the Pietsch integral norm.
of T is defined to be ||T"||P/ = inf ||.R||||S||fj(fi). The space of Pietsch integral
operators from X into Y is denoted by 73(X, Y). This space is complete in
the Pietsch integral norm. It is easy to see that
J4(X, Y) C 73(X, Y) C 1(X, Y),
and ||T||/ < ||T||p/ < ||T||jv for every nuclear operator T. In general, these
spaces of operators are distinct. From the remarks above, we see that the
spaces J(X, Y) and 73 (X, Y) are the same if Y is a dual space. Another
special case occurs when the domain is Co- We have seen that cq®cX = cq{X)
and hence
a(co, X) = T3(co, X) = ei9.X = X(co, X),
with equality of the integral, Pietsch integral and nuclear norms.
Proposition 3.21. Let T: X -)• Y be an operator and let J:Y -»• Y" be
the canonical inclusion. The following statements are equivalent:
(i) T is integral. .'■
(ii) JT is integral.
(iii) T* is integral.
(iv) T" is integral.
Furthermore, ifTis integral, then
lir||/ = ||jr||/ = nr||/ = ||r"||/.
Proof. The equivalence of (i) and (ii), and the equality of the integral norms
of T and JT, follow immediately from the remarks above concerning
operators into a dual space.
(i) implies (iii): suppose that T is integral. Let JT = RIS, where
R: Li(fi) -»• Y** and S: X -»• L^fi) are operators and /: L^fi) -»• L\(p)
is the canonical injection. Then T*J* = S*I*R*. Now J* is the canonical

66 3 The Injective Tensor Product
projection of y*** onto Y* and so we may restrict R* to obtain an
operator U from Y* into Loo(/L(). And the operator /*: Loo(fi) -» Looin)* factors
through the canonical injection of Loo(fi) into L\{(i). Therefore, we have a
factorization of T* as shown below.
Hence T* is integral and, since \\U\\ < ||-R*||, it follows that \\T*\\i < ||r||/.
The proof that (i) implies (iii) shows that (iii) implies (iv) and that ||T**||/ <
||T*||/ whenever T* is integral.
(iv) implies (i): Suppose that T** is integral. Then there exists a
factorization, T" = RIS, where R <E £(L,(/n),y") and S £ ^(X",^^)).
Restricting T** to X gives the operator JT: X -> Y". Hence we have a
factorization of JT as shown below.
Therefore T is integral and ||r||7 < ||T"||/.
We have shown that (i), (ii), (iii) and (iv) are equivalent and that, if T is
integral, then
»r||/< 112-11/<||ri/< urn/= 11^11/.
Therefore these norms are equal. □
We recall that the space of bounded bilinear forms "B(X x Y) is isometri-
cally isomorphic to the space -C(X, Y*). Under this identification, the bilinear
form B is associated with the operator T, defined by (Tx, y) = B(x, y).
Proposition 3.22. A bilinear form B on X x Y is integral if and only if
the associated operator T: X -> Y* is integral The mapping B h-> T is an
isometric isomorphism ofBj(X x Y) with 3(X, Y*).

3.5 Integral Operators 67
Proof. Suppose that the bilinear form B is integral. Then there exist a
probability space (ft, E,/li), and operators U: X -» Looiii) and V: Y -»
Loo(fi) such that B(x,y) = fn(Ux)(Vy) dp for every x £ X, y £ Y, and
\\B\\i = \\U\\\\V\\. Thus (Tx,y) = B(x,y) = (IUx,Vy) = <V*/[/x,y), where
I: Loo(n) —"> L\{fi) is the canonical injection. Therefore T = RIU, where R
denotes the restriction of V* to the canonical image of Li(p) in Loo(p)*, and
it follows that T is integral, with ||r||/ < ||B||/.
Conversely, suppose that T is integral. Let T = RIS be a factorization
of T, where R £ £.(£00(ji),*") and 5 G £(X,L,(/n)), with ||r||/ = ||il||||5||.
Then
B(x,y) = (Tx,y) = (RISx,y) = (ISx,R*y)
= f(Sx)(R*y)d»
Jn
for every x £ X, y £ Y. Therefore B is integral and ||B||/ < ||S||||fl*|| = ||r||/
It follows that B is integral if and only if T is integral and that \\B\\j is
equal to ||r||/. □
It follows from this result that the dual space of the injective tensor
product X®CY may be identified with the space J(X, Y*):
(X®CY)* = -B/(X x Y) = 3(X, Y*).
In finite dimensions, every operator is integral and the integral norm
coincides with the nuclear norm. Just as the integral bilinear forms can be
characterized by their behaviour on finite dimensional subspaces, the integral
operators are also "finitely determined". To make this precise, let B: X x Y* -> K
be the bilinear form associated with the operator T: X -> Y. We have seen
that B is integral if and only if there exists a positive constant, C, such that
for every pair of finite dimensional subspaces, M, N of X, Y* respectively,
the restriction, Bm,n oi B to M x N satisfies ||Bjvf,w||/ < C\ or equivalently,
IISm.jvII/V < C Furthermore, the integral norm of fl' is the infimum of all
such C. Now every finite dimensional subspace N of Y* is the dual space
of a quotient space X/F, where F is a subspace of X of finite codimension.
Denoting the injection of a subspace G by Ig and the quotient operator with
kernel G by Qg, we have Qn = Ip- Therefore the finite dimensional
characterization of integral bilinear forms translates to operators in the following
way:
Proposition 3.23. An operator T: X -> Y is integral if and only if there
exists a positive constant C such that for every finite dimensional subspace E
of X and every finite codimensional subspace FofY, the finite dimensional
operator
QfTIe- E -» X -» Y -» Y/F
satisfies \\QfTIe\\n < C Furthermore, we have \\T\\i = inf C, where the
infimum is taken over all such pairs E, F.

68 3 The Injective Tensor Product
This result suggests that the integral norm is the most natural extension
to infinite dimensions of the nuclear norm for finite dimensional operators.
We shall explore this idea further in a more general setting in due course.
3.6 Exercises
Exercise 3.1. Show that cq®cco is isometrically isomorphic to Co-
Exercise 3.2. Show that ^®e^2 is not a Hilbert space.
Exercise 3.3. Let Q: Z -> Y be a quotient operator and let / be the identity
operator on X. Show that the tensor product operator I ®e Q: X®eZ -»
X®CY is a quotient operator if the following "lifting" condition is satisfied:
for every operator S: X* -> Y and every e > 0, there exists an operator
T: X* -» Z such that QT = S and ||r|| < ||5|| + e.
Exercise 3.4. Let Q: l\ -> Co be a quotient operator and let / be the
identity operator on (\. Show that I ®c Q: li®eti -* co®cti >s not a quotient
operator.
Exercise 3.5. Identify the tensor diagonal in lp®clp for 1 < p < co.
Exercise 3.6. (a) Show that if X is infinite dimensional then too®eX ^
loo{X).
(b) Find a description of loo®cX as a space of sequences.
Exercise 3.7. Show that Loo(fi)®cX is isometrically isomorphic to the
space Koo(fi, X) of (equivalence classes of) /n-measurable essentially compact
functions with values in X, where the norm of / is the essential supremum
°f 11/11- (/ 's said t° t>e essentially compact if there exists a /n-null set whose
complement is mapped by / into a relatively compact subset of X.)
Exercise 3.8. Show that if X and Y have the Schur property, then so does
X®eY. (The Banach space X has the Schur property if every weakly
convergent sequence in X converges in norm.)
Exercise 3.9. Let X and Y be Loo-spaces. Show that X®eY is also a Loo-
space.
Exercise 3.10. Let K be a compact topological space and let Q: Z -> X
be a surjective operator. Show that every continuous function from K into
X can be lifted to Z\ in other words, if /: K -> X is continuous, then there
exists a continuous function g: K -> Z such that Qg = f.
Exercise 3.11. Let /: [0,1] -» L,[0,1] be the function f(t) = X[o,t]- Show
that / is Pettis integrable with respect to Lebesgue measure on [0,1] and
compute the Pettis integral of /.

3.6 Exercises 69
Exercise 3.12. Let / = (/„): [0,1] -» Co, where /„ are Lebesgue integrable
functions on [0,1] with disjoint supports. Let (i be Lebesgue measure on [0,1].
(a) Show that the function / is Pettis integrable with respect to ft if and
only if sup ||/n||i < oo. If / is Pettis integrable, show that
|/|, = sup H/4,.
n
(b) Show that / is Bochner integrable with respect to [i if and only if
YLn ll/nll < oo. If / is Bochner integrable, show that
oo
ll/ll. = £ll/»ll.-
n=l
Exercise 3.13. (a) Show that the bilinear form on (2 x d given by B(x, y) =
X.n xnVn is not integral.
(b) Show that the same bilinear form is integral on t\ x t\ and find its
integral norm.
Exercise 3.14. Show that every nuclear operator T is Pietsch integral and
that uriipj < ||r||A,.
Exercise 3.15. Show that the space 73(X, Y) is complete in the Pietsch
integral norm.
Exercise 3.16. Show that the canonical injection of £i into cq is integral
and find its integral norm.

4. The Approximation Property
In this chapter we introduce the approximation property for Banach spaces.
The possession of this property leads to the resolution of several outstanding
issues concerning projective and injective tensor products. We then consider
the following question: when are the projective or injective tensor products of
reflexive spaces themselves reflexive? A satisfactory answer requires the use
of the approximation property. Finally, we study tensor products of Banach
spaces with Schauder bases.
4.1 The Approximation Property
There are several unanswered questions in the previous chapters.
1. The space N(.Y, Y) of nuclear operators is a quotient of the projective
tensor product X*®„Y. When do we have X(X, Y) — Xm®*Y1
2. If the operator T has a nuclear adjoint, is T itself nuclear?
3. The injective tensor product X*®eY can be identified with the space of
approximable operators from X into Y. Thus, X*®EY is a subspace of
the space X(X, Y) of compact operators. WJien do we have X(X, Y) =
X*®eY?
4. Every nuclear operator T is integral and ||T||; jC. ||r||w When do the
integral and nuclear norms coincide on Ji(X, Y)\
These questions all reduce to one that is very simple to state. Recall
that a finite rank tensor « = Yl?=ix' ® V» 6 X ®Y is zero if and only if
X.r=i ¥'(a;>)y> = 0 f°r every ip 6 X*. Now consider the problem of
identifying when an element of the completed tensor product X®*Y is zero. Let
Xl^Li a"n ® J/n be a representation of the tensor u e X®WY. Can we deduce
from
oo
Yi f(xn)yn = 0 for every ip e X" (4.1)
that u = 0? Since the dual space of X®„Y is "B(X x Y) = L(X, Y*), we have
u = 0 if and only if

72 4 The Approximation Property
f^(yn, Txn) = 0 for every T e £(X, V). (4.2)
n=l
It follows easily from (4.1) that ^2™=i{Vn,Txn) = 0 for every finite rank
operator T. Now, if X and Y are infinite dimensional, there will be
operators that are not of finite rank. Thus, there is a gap between the conditions
expressed in (4.1) and (4.2). This gap can be bridged if it is possible to
approximate an arbitrary operator by an operator of finite rank, at least on the
set {xn}. Now, we may assume without loss of generality that the sequence
(a:n) tends to zero. Then the set {xn} is relatively compact and so we arrive
at the following condition on the space X: every operator from X into Y*
can be approximated on a compact subset of X by a finite rank operator.
Of course, there is an equivalent formualtion of (4.1) that leads to a similar
statement concerning operators from Y into X*. Our first result shows that
the approximability of operators either from X or into X on compact sets is
equivalent to the approximability of the identity operator on X on compact
sets.
Proposition 4.1. The following are equivalent for a Banach space X:
(i) For every compact subset K of X and every e > 0 there exists a finite
rank operator S: X —> X such that \\x — Sx\\ < e for every x 6 K.
(ii) For every Banach space Y, every operator T: X -» Y, every compact
subset KofX and every e > 0, there exists a finite rank operator S: X -»
Y such that \\Tx — Sx\\ < e for every x 6 K.
(iii) For every Banach space Y, every operator T: Y -» X, every compact
subset K ofY and every e > 0, there exists a finite rank operator S: Y -»
X such that \\Ty — Sy\\ < e for every y 6 K.
Proof, (i) implies (ii): let T 6 &(X,Y), T ^ 0, let K C X be compact and
let e > 0. By (a), there exists a finite rank operator R: X -> X such that
||a; - Rx\\ < e/\\T\\ for every x 6 K. Then S = TR is a finite rank operator
from X into Y and ||Ta: — 5a;|| < e for every x & K.
(i) implies (iii): let T 6 £>(Y,X), let K C Y be compact and let e > 0.
Applying (i) to the compact subset T{K) of X, there exists a finite rank
operator R: X -+ X such that ||a: - Rx\\ < e for every x 6 T(K). Then
S = RT is a finite rank operator from Y into X and \\Ty — Sy\\ < e for every
yeK.
Since each of (ii) and (iii) clearly implies (i), the proof is complete. D
A Banach space X is said to have the approximation property if the
equivalent statements of this proposition hold for X. We shall see that this property
is precisely what is needed to deduce (4.2) from (4.1). First, we look at some
examples of spaces with the approximation property.
Example 4.2. C(K) has the approximation property.

4.1 The Approximation Property 73
Let J be a compact subset of C(K) and let e > 0. By the Arzela-Ascoli
Theorem, J is equicontinuous and so there is a finite open cover {U\,..., Un}
of K such that, for each »', if s,t 6 £/,, then \f(s) — f(t)\ < e for every
/ 6 J. Let {pi,... ,Sn} be a partition of unity on K subordinate to the cover
{Ui,...,Un}. Choose points t, 6 Ut and let S: C(K) -» C(K) be given by
Sf = Y,?=i f{tx)9x- Then T is a finite rank operator and ||/ - Tf\\ < e for
every / 6 J.
Before giving the next example, we give a sufficient condition for the
approximation property that does not require any knowledge of the compact
sets.
Proposition 4.3. Suppose that there exists a uniformly bounded net (Ta) of
finite rank operators on X such that Tax -» x for every x 6 X. Then X has
the approximation property.
Proof. Let (Ta) be a net of finite rank operators such that sup ||Ta|| — C<
oo and Tax -> x for every x 6 X. Let K be a compact subset of X and let
e > 0. Choose a J-net, {x\,..., xn} for K, where 5 = min{e/3,e/3C}. There
exists ao such that if a > a0. then ||a;, — Taa:,|| < e/3 for each i. Let x 6 K
and choose i such that [|a; — a;,-|[ < 5. Then
\\x - T^xW < \\x - xt\\ + \\x, - Taox,\\ + \\Taox, - Taox\\ < e.
a
We apply this to obtain some more examples of spaces with the
approximation property.
Example 4.4. Every Banach space with a Schauder basts has the
approximation property.
Let (en) be a Schauder basis for X and let e\ be the ^associated coordinate
functional^. For each n, let Pn be the finite rank operator on X given by
Pn.x — Z]r=ieI'(a;)a;>- Then the operators Pn are uniformly bounded and
Pnx -> x for every x 6 X. Therefore X has the approximation property.
It follows that the spaces cq and lp, for 1 < p < oo, all have the
approximation property. Indeed, for every indexing set /, the spaces Co(7) and lp(I)
(1 < p < oo) have the approximation property, since we may define a
uniformly bounded net of finite rank operators, (Pf), indexed by the set of finite
subsets of /, by PpX = X.,ej?e*(a:)e, and we have Ppx -)• x for every x. In
particular, we see that every Hilbert space has the approximation property.
We now show that the spaces Lp(n), where ft is any measure, have the
approximation property. In this case, we can use averaging operators to
approximate the identity operator.
Example 4.5. Lp(n) has the approximation property for every p 6 [1, oo).

74 4 The Approximation Property
Let 1 < p < oo. We construct a net of finite rank operators as follows. Let
A = {Ai,..., An} be a finite collection of disjoint measurable sets of finite
positive measure. Let Ta be the finite rank operator on Lp(fj,) defined by
Note that, since n{At) < oo, the integral of / over A, is defined. Indeed, we
have fAt \f\dfi < (/4i Iffdfiyir. It also follows that ||TU|| = 1 for every A.
If A and B are two such collections of measurable sets, we define A < B if
every set in A is the union of a subcollection of B. This construction yields a
directed set and we claim that for every / 6 Lp([i) the net [TaJ) converges
to /. In view of the uniform boundedness of the operators Ta, it suffices to
prove this for the simple functions in Lp(fi). But every such function can
be written as / = X."=1c,x.4J where A = {Ai An} consists of disjoint
measurable sets of finite positive measure and it is easy to see that Tsf — f
for B > A.
The spaces C(K)*, Loo(fi) and Lot)(/L()* also have the approximation
property. Thus, all the classical Banach spaces of sequences and functions
encountered thus far, along with their duals, have the approximation property. There
do exist spaces without this property; it is a deep result of Enflo that the
space £(£2,^2) does not have the approximation property.
We now show that the approximation property meets our needs. In the
proof of the next proposition we shall make use of the fact that every compact
set in a Banach space is contained in the closed convex hull of a sequence
that converges to zero.
Proposition 4.6. Let X be a Banach space. The following are equivalent:
(i) X has the approximation property.
(ii) Ifu = Xl^Li <Pn®xn 6 X*®XX, where (<pn), (xn) are bounded sequences
m X*, X respectively with Y^Li ll¥,n||||a"„|| < 00 and i/J^Li fn{x)xn =
0 for every x 6 X, then u = 0.
(iii) For every Banach space Y, ifu = Y^=\ xn®yn 6 X®„Y, where (xn),
(j/n) are bounded sequences in X, Y respectively with J^nLi INnllllynll <
00 and if J^Li ip(xn)yn = 0 for every ip & X*, then u — 0.
(iv) For every Banach space Y, ifu = Y,™=i xn®yne X®*Y, where {xn),
(j/n) are bounded sequences in X, Y respectively with Xl^Li INnllllVnll <
00 and if Y,™=i Hyn)xn = 0 for every ip 6 Y*, then u = 0.
Proof, (i) implies (iv): suppose that X has the approximation property. Let
« = X.n=i xn®Vn £ X®XY, where (a;n) and (yn) are bounded sequences in X
and Y such that Y,™=i INn||||y„|| < 00, and suppose that ^=1 ^l>{yn)xn = 0
for every ^ £ Y*. Assume, without loss of generality, that xn -» 0 and
£~=i ..V»|. < oo- Let T 6 H{X,Y*) = {X®„Y)* and let e > 0. Since X has

4.1 The Approximation Property 75
the approximation property, there exists a finite rank operator, S: X -» Y*,
such that ||Ta;n - 5a;n|| < e for every n. We have Sx = Y£=i Vi(x)^u where
tp, 6 X* and V>, 6 Y*. Now
oo m tn / oo v
(«, 5) = J2 ]T <p,(xn)r/>,(yn) = ]T <p, ( J2 iMife)*-) = 0 ■
n=l «=1 .= 1 Vi=l '
Therefore
OO OO
|<«,T)| < |(«,r- 5)| + \(u,S)\ < Y,\\Txn - Ss„|||M < e Ell^ll"
n=l n=l
Since e is arbitrary, it follows that («, T) = 0 for every T 6 C(X, Y*).
Therefore u = 0.
(iv) implies (iii): suppose that « = "£nLi xn ® !/» 6 ^®»^ satisfies the
conditions given in (c). Then, for every ip & X* and ^ 6 Y*t
(OO v OO / OO \
n=\ ' n=l \l=l '
Therefore X.!H=i ^{v^)xn = 0 f°r every rf> 6 V*, and so, by (iv), we have
u = 0.
(iii) implies (ii): if « = Y,Z=i P* ® x* satisfies £)£1, Vn{x)xn. = 0 for
every xeX, then "J^i ^(-c„)y„(a;) = 0 for every x e X, 9 e X*. Therefore
E^Li 0(a"n)Pn = 0 in X* for every 0 6 X* and it follows from (iii) that
EJJLi in®y>n = 0 in X®,X*. Since the mapping x ®<p «->• y> ® a: is an
isomorphism between X®,J' and X*®,X, we have « = 0.
(ii) implies (i): suppose that X does not have the approximation property.
Let E be the locally convex space of operators from X to itself, with the
topology of uniform convergence on compact sets. Thus, the topology on E
is generated by the continuous semi norms
pK{T)=SWp{\\Tx\\:xeK},
where K ranges over the compact subsets of X. Let F be the subspace of
E consisting of the finite rank operators. Then the statement that X does
not have the approximation property is equivalent to the statement that the
identity operator, J, does not belong to the closure of F. It follows from the
Hahn-Banach Theorem that there exists a continuous linear functional «""> on
E, such that $(7") = 0 for every T 6 F and $(/) = 1.
Since $ is continuous, there exists a compact subset, K, of X such that
|$(T)| < pk(T) for every T e E. Since K is contained in the closed convex
hull of a sequence (a;n) that converges to zero, we have
|$(r)| < sup ||ra;n|| for every T e E.
n

76 4 The Approximation Property
We may consider the sequence (Txn) as an element of the Banach space
co(X). Let Z be the subset of co(X) consisting of all such elements, as T
ranges over E. Then Z is a subspace of co(X) and we may view $ as a
bounded linear functional on Z. We use the Hahn-Banach Theorem again
to extend this functional to co(X). Therefore, since the dual space of co(X)
is ti{X*), there exists (<pn) 6 t\{X*) such that $(7") = Y^=i <Pn(Txn) for
every T 6 E. But then
OO
n=l
while $(T) = 0 for every finite rank operator T implies that Y^=i tPn(x)xn =
0 for every x 6 X. Therefore (ii) does not hold. D
An examination of this proof shows that, when applying one of the tests
(ii), (iii) or (iv), it suffices to use a separating set of vectors. For example, if
U is a separating subset of X*, then the condition given in (iii) is equivalent
to the requirement that J^Li <p(xn)yn = 0 for every ip& U.
We recall that X*®„X is a subspace of X*®^X**. Applying this fact,
together with the observation that X is a separating subset of the dual of
X*, we obtain:
Corollary 4.7. If X* has the approximation property, then so does X.
The converse of this result is false; there exist Banach spaces with the
approximation property whose duals do not have this property. For example,
the aforementioned example of Enflo, £.(£2,^2). is the dual of the projective
tensor product £2®*^. which has the approximation property (see Exercise
4.5).
We can now address the first of the questions posed at the beginning of
this section.
Corollary 4.8. Let X and Y be Banach spaces.
(a) If X* orY has the approximation property, then
X*®7rY = yt(X,Y).
(b) If X* orY* has the approximation property, then
r«,r = 2w(Xxy).
Proof. We prove (a), from which (b) follows easily. Recall that there is a
quotient operator J: X*®XY -> N(X, Y), which associates with the tensor
« = S^Li fn ® yn the nuclear operator (Ju)(x) = "£!Hi fn{x)yn- It follows
from the proposition that if X* or Y has the approximation property, then
the kernel of J is trivial and hence J is an isometric isomorphism. D

4.1 The Approximation Property 77
The next result shows that the trace functional can be defined for nuclear
operators on a Banach space X, provided X has the approximation property.
Corollary 4.9. Let X be a Banach space with the approximation property.
If u= Y^n°=\ fn®xn& X*®„X, where (ipn)> (xn) are bounded sequences in
X*, X respectively with Y,n°=i ll¥,n||||a"n|| < °° and */Z)^Li <Pn(x)xn = 0 for
every x 6 X, then J^nLi <Pn(xn) = 0.
Proof. Let tr be the trace functional on X*®„X. Thus tr is the linearization
of the bounded bilinear form on X* x X given by (<p,x) »-). ip(x). Prom the
proposition, we have u = 0 and hence tr(u) = "£n=i <pn{xn) =0. D
We can now answer the second of our questions. If T: X —> Y is a nuclear
operator and J^nLi <pn®yn is a nuclear representation of T, then the adjoint
operator is given by T*ip = J^Li ip(yn)<Pn- It follows that T* is nuclear, with
UT'IIn < ||T||;v. To prove the converse, we need the approximation property.
Proposition 4.10. Let X be a Banach space whose dual X* has the
approximation property. If the operator T: X -»• Y has a nuclear adjoint, then T is
nuclear and \\TI\n = \\T'\\N.
Proof. We have T* 6 7f(Y*,X*) = Y"®„X*, since X* has the
approximation property. We shall show that T* lies in the closed subspace Y®„X*
of y"®,J*. By the Hahn-Banach Theorem, it suffices to show that
every bounded linear functional on Y**®„X* that vanishes on Y®„X* also
vanishes on T*. Let S G (Y**®^X*Y = L(Y",X") and suppose that S
vanishes on Y®„X*, in other words, that Sy = 0 for every y 6 Y. Now
choose a nuclear representation J^nLi y" ® <pn for T*. The restriction of T**
to X is the operator T and so XlnLi <pn{x)y" belongs to Y for every x 6 X.
Therefore
oo
Y^fn(x)Sy" = 0 for every x 6 X.
n=l
It follows that the same statement holds if x is replaced by an arbitrary
element a;** of X". Hence, using the approximation property of X* and
Proposition 4.6(iii), we get
oo
£p»®Sy"=0 \nX*®xX".
n=l
Therefore, by Corollary 4.9,
oo
(r',5) = ^n,5y") = 0.
n=l
Thus T* belongs to Y®„X* and so T is nuclear. In addition, we have \\T\\N -
\\T*\\N, since Y®„X* is a subspace of Y"®XX*. a

78 4 The Approximation Property
We come to our third question: when do we have X(X, Y) = X*®CY1
The problem now is to approximate an arbitrary compact operator in the
operator norm by a finite rank operator. The approximation property allows
us to approximate any operator by a finite rank operator on a compact set. We
wish to move the compactness assumption from the domain of approximation
to the operator. We shall show that there is an equivalent statement of the
approximation property that has this form.
Let K be a convex, balanced, compact set in a Banach space X. The
subspace of X spanned by K is the union of all the positive scalar multiples
of K. This space is normed by the Minkowski functional of K:
\\x\\K = inf{A > 0 : x <E XK}
and the closed unit ball for this norm is K. The normed space defined in this
way will be denoted by Xk ■
Lemma 4.11. Let K be a convex, balanced, compact subset of the Banach
space X.
(a) The normed space Xk is complete.
(b) There exists a convex, balanced, compact subset L of X containing K
such that K is compact m the Banach space Xl ■
Proof, (a) Suppose that XK contains a Cauchy sequence (a:n) that does not
converge. We may assume without loss of generality that xn 6 K for every
n. Since the injection XK -> X is bounded, (a:n) converges in the norm of
X to some x G K. Then the sequence zn = x„ — x is Cauchy in XK but
does not converge there, while zn -»• 0 in X. Now, choosing a subsequence,
there exists S > 0 such that \\zn\\K > 6 for every n. On the other hand, there
is a positive integer N such that \\zn - zm\\K < 5 if n,m > N. Therefore
zn - zm £ SK for all n, m > N. Letting m-» oowe get zn 6 5K for every
n> N. Therefore \\zn\\K < 8 for every n > N, a contradiction.
(b) Since K is compact, there exists a sequence (xn) in X that converges to
zero such that K lies in the closed convex hull of {xn}. Choose an increasing
sequence (An) of positive scalars that diverges to infinity such that Ana:n
still converges to zero in X. Let L be the closed, convex, balanced hull of
{\ixn}- Then ||a.„|'L < A"1 and so the sequence (a:„) converges to zero in
Xl. Therefore K is compact in X^. D
We can now give a formulation of the approximation property in terms of
uniform approximation of compact operators. Recall that an operator is said
to be approximable if it is the limit in the operator norm of a sequence of
finite rank operators. We shall use the following property of adjoint operators
in the proof: let T: W -)• Z be an injective operator. Then T*(Z*) is dense
in W* for any locally convex topology with respect to which the dual space
of W* is W. In particular, this holds for the topology of uniform convergence
on the compact subsets of W.

4.1 The Approximation Property 79
Proposition 4.12. Let X be a Banach space.
(a) X has the approximation property if and only if for every Banach space
Y, every compact operator from Y into X is approximable.
(b) X* has the approximation property xf and only if for every Banach space
Y, every compact operator from X into Y is approximable.
Proof, (a) Suppose that X has the approximation property. Let T: Y -» X
be a compact operator and let e > 0. There exists a finite rank operator
R: X -» X such that ||a: — Rx\\ < e for every x in the relatively compact
subset T(BY) of X. Let S = RT. Then S has finite rank and \\T - S\\ < e.
Conversely, suppose that every compact operator with values in X is
approximable. Let K be a compact subset of X and let e > 0. By the lemma,
there exists a convex, balanced, compact set L containing K such that K is
compact in the Banach space Xl- Now, since the injection J: Xl —> X is
compact, there exists a finite rank operator R: Xl -» X such that \\I — R\\ <
e/2. Let R = YZ=i A ® «., where i/> 6 (XL)* and x, 6 X. Since I: XL -» X
is injective, I*(X*) is dense in (Xl)* for the topology of uniform convergence
on the compact subsets of Xl- Therefore, given 77 > 0, there exist tpi £ X*
such that ||y>,(a.) — Vl>(a;)l| < V f°r every x 6 K and every i. Let S: X -^ X
be the finite rank operator X."=1 <pt ® x,. If x 6 K, then
n n
\\Rx~Sx\\ < £\<p,(x) -<M*)IIWI < ^EM <e,
«=i «=i
taking 77 < e/(2X."=| ||a"i||)- Since ||a; - Rx\\ < e/2 for every x € K, we have
[[a; — Sx\\ < e for every x £ K. Therefore X has the approximation property,
(b) Suppose that X* has the approximation property. Let T: X -¥ Y
be compact and let e > 0. Since T* :Y* -» X* is compact, there exists a
finite rank operator R: X* -» X* such that \\ip — Rtp\\ < e for every <p in
the relatively compact set T*(By). Since T is compact, T** maps X** into
y and so the composition T**R* is a finite rank operator from X" into Y.
Let S be the restriction of this operator to X. We claim that this finite rank
operator approximates T uniformly. If a; 6 Bx, then
||ra; - 5a;|| = sup \x/,Tx - r/>T"R*x\ = sup |(r'i/,)a: - (RT'r/>)x\
< sup ||r>-fl(r»||<e.
il>eBY.
Therefore T is approximable.
Conversely, suppose that every compact operator with X as domain is
approximable. We use (a) to show that X* has the approximation property.
Let T: Y -» X* be a compact operator. Let Jx denote the canonical
embedding of X into X**. By hypothesis, the compact operator T*Jx' X -)• Y*
is approximable. It follows that the adjoint operator (T*Jx)*'- Y** -»• X* is

80 4 The Approximation Property
approximable. But T is the restriction of this operator to the canonical image
of Y in "K". Therefore T is also approximable. Hence X* has the
approximation property. □
We now have the answer to our third question.
Corollary 4.13. If either X* or Y has the approximation property then
X(X,Y) = X*®CY.
There is just one question remaining: under what conditions do the
nuclear and integral norms coincide on the space N(X, Y) of nuclear operators?
In order to give a satisfactory answer to this question, it is necessary to
strengthen the approximation property. We say that the Banach space X has
the bounded approximation property if there exists a positive constant A such
that, for every compact subset K of X and every e > 0, there exists a finite
rank operator S: X -» X such that ||5|| < A and ||a: — 5a:|| < e for every
x £ K. If this holds with A = 1 then X is said to have the metric
approximation property. A look back at the examples we have given shows that in every
case the space in question has the metric approximation property. Indeed,
the condition given in Proposition 4.3 actually implies the bounded
approximation property and, moreover, if every operator in the net of approximating
finite rank operators has norm at most one, then the space in question has
the metric approximation property. Thus all the spaces C[K), Co, (p, Lpiji),
along with their duals, have the metric approximation property. If X has a
Schauder basis (en), then the norm on X may be replaced by the equivalent
norm
||.r||'=sup||Pn(.r)||,
n
where Pn is the projection onto the first n coordinates, and X has the metric
approximation property in this norm.
The equivalent formulations of the approximation property given in
Proposition 4.1 also have metric versions. An inspection of the proof of that
proposition shows that the metric approximation property for X is equivalent
to the approximability of an operator T into or out of X on a compact set,
by a finite rank operator S that satisfies ||5|| < ||T||.
Theorem 4.14. The following are equivalent for a Banach space X:
(i) X has the metric approximation property.
(ii) For every Banach space Y, the canonical mapping X®„Y -» (X*®CY*)*
is an isometric embedding.
(iii) The canonical map X®XX* -» [X*®eX)* is an isometric embedding.
Proof, (i) implies (ii): the canonical mapping from X®„Y into (X*®CY*)*,
or 3/(X* x y*), has norm one. Thus we must show that ||u||j > ir(u) for
every u 6 X®*Y. Fix u 6 X®*Y and e > 0. Choose a representation
X..n=i xn ® yn of u with xn -» 0 and 0 < Xli^i ||yn|| < °o- Since the dual

4.JL X tie Approximation nuyai;
space of X®XY is £(X,y*), there exists T 6 L(X,Y*) with ||T|| = 1, such
that |(u,T)| > 7r(«) — e. Applying the metric approximation property, there
exists a finite rank operator S: X -J- V* such that ||5|| < 1 and
||r.rn-S.r4<e(f>n||)
for every n. Then \(u,T — S)\ < e and it follows that
|(«,5)|>7r(«)-2e.
Now S: X -¥ Ym, being a finite rank operator, belongs to X*®CY*, and
since the operator norm coincides with the injective norm, we have e(S) < 1.
Therefore,
Mi>\(u,S)\>w(u)-2e.
Since e > 0 is arbitrary, it follows that ||u||j = ir(«).
(ii) implies (iii) is trivial.
(iii) implies (i): suppose that X®„X* is a subspace of (X*®CX)*. Since
the closed unit ball of X*®CX is dense in the closed unit ball of the completed
tensor product X*®eX, it follows that
tt(«) = sup{|(u, S)\:SeX*® X, e(5) < 1}
for every « 6 X®*X*. On the other hand, we have
Tr(u) =sup{|(«,r)| :Tel(X,X), \\T\\ < 1}.
Therefore, the set F = {S 6 H{.X,X) : S has finite rank and ||5|| < 1} is
weak'-dense in the closed unit ball of -C(X, X), where these sets are taken as
subsets of (X®*X*)*. Thus, the identity operator, J, on X can be weak*-
approximated by elements of F, so that if x 6 X, ip 6 X* and e > 0, then
there exists S 6 F such that \(Sx, ip) — (x,ip)\ < e. It follows that for every
x 6 X, the point x belongs to the weak closure of the set Fa; = {Sx : S & F}.
But this set is convex, and so its weak and norm closures coincide. Thus, x
lies in the norm closure of the set Fa; for every x. Now consider the strong
operator topology on &(X, X), generated by the seminorms T h-> \\Tx\\, as x
ranges over X. We have shown that the identity belongs to the closure of F
for this topology. Therefore there exists a net (Sa) of finite rank operators,
each having norm at most one, such that Sax -> x for every x 6 X. Hence,
by the remarks preceding the proposition, X has the metric approximation
property. D
We recall from Proposition 3.21 that the integral norm of an integral
operator T: X -)• Y is unchanged if we consider T as an integral operator
from X into V*. Thus 3{X, Y) is a subspace of 3(X, Y**).
Corollary 4.15. If X* has the metric approximation property, so does X.

82 4 The Approximation Property
Proof. Applying part (ii) of the proposition to X*, with Y = X, we see
that the canonical mapping of X*®„X into (X"®CX*)* = 3(X*,X"') is
an isometric embedding. Now the elements of X*®*X, considered as integral
operators from X* into X***, take their values in X*. But 3(X*,X*) is a
subspace of 3(X*,X***). Therefore, the canonical mapping of X*®*X into
3(X*,X*) = (X®eX*)* is an isometric embedding. Hence, by statement (iii)
of the proposition, X has the metric approximation property. D
When a dual space has the metric approximation property, we have
another canonical embedding:
Proposition 4.16. If the dual space X* has the metric approximation
property, then the canonical mapping X*®„Y -)• (X®CY*)* is an isometric
embedding for every Banach space Y.
Proof. The canonical mapping X*®*Y -> (X"®CY*)* = 3(Y*,X*") is
an isometric embedding. But the elements of X*®„Y, considered as integral
operators from Y* into X***, take their values in X*. Therefore X*®„Y
is isometrically embedded into the subspace 3(Y*,X*) = (X®eY*)* of
(X"®CY*)* D
We can now answer our last question.
Corollary 4.17. If either X* or Y has the metric approximation property,
then N(X, Y) is a closed subspace of3(X, Y).
Proof. Since X* or Y has the approximation property, N(X, Y) = X*®*Y.
We have seen that, if either X* or Y has the metric approximation property,
then the canonical mapping of X*®*Y into (X®eY*)* = 3(X,Y**) is an
isometric embedding. Arguing as in the proof of the proposition, it follows
that the canonical mapping of N(X, Y) into 3(X, Y) is an isometric
embedding. D
There are spaces with the approximation property that fail to have the
metric approximation property. However, in the next chapter we shall prove
a striking result of Grothendieck to the effect that every reflexive space, and
every separable dual space, with the approximation property has the metric
approximation property.
4.2 Reflexivity of Tensor Products
We have seen that the projective tensor product of reflexive spaces can fail
to be reflexive - the projective tensor product of two Hilbert spaces, for
example, contains a copy of t\ and so cannot be reflexive. For the same
reason, LPl [0,1]®ILP:|[0,1] contains a complemented isomorphic copy of (i
for every pi, p2 and so cannot be reflexive. In this section, we find sufficient

4.2 Reflexivity of Tensor Products 83
conditions for the tensor products X®„Y and X®eY to be reflexive and we
give some examples of pairs of spaces that satisfy these conditions.
Since the dual space of X®XY is £>(X, Y*), the reflexivity of X®nY is
equivalent to that of -C(X, Y*). We shall begin with a sufficient condition for
the space of operators to be reflexive. First, we need a characterization of
weak convergence in the space of compact operators.
Lemma 4.18. Let X be a reflexive Banach space. A bounded sequence (T„)
in %(X,Y) converges weakly to T £ %(X,Y) if and only if the sequence
(Tnx) converges weakly to Tx for every x £ X.
Proof. This condition is clearly necessary. To establish sufficiency, consider
the compact space K = Bx x By, where Bx and By carry the weak
and weak* topologies respectively. Each compact operator T: X -* Y
defines a continuous function fa on K, given by /r(a". VO = {Tx,il>), and
||T|| = H/rlloo- Therefore X(X,Y) may be regarded as a closed subspace
of C(K). Now let (Tn) be a bounded sequence in X[X, Y) and suppose that
Tnx converges weakly to Tx for every x £ X. Then the sequence of functions
/t„ converges pointwise to the function fa- Furthermore, this sequence is
bounded in C(K). It follows from the Riesz Representation Theorem for'the
dual space C(K)* and the Bounded Convergence Theorem that fcn converges
weakly to fT in C(K). Therefore (Tn) converges weakly to T in X{X, Y). O
Theorem 4.19. Let X and Y be reflexive Banach spaces. If every operator
from X into Y is compact, then £>(X, Y) is reflexive.
Proof. Let (Tn) be a sequence in the closed unit ball of &(X,Y). We shall
prove that (Tn) has a weakly convergent subsequence.
Suppose to begin with that X is separable. Let {a:*.} be a countable, dense
subset of X. By the reflexivity of Y, the sequence (TnXk)n has a weakly
convergent subsequence for every k. Hence, using a diagonal argument, there
exists a subsequence (Tnj) of (Tn) such that the sequence {TniXk)j converges
for every k. It follows that the sequence (Tnjx) converges weakly for every
x £ X. Indeed, let x £ X and let e > 0. Choosing k such'that ||a; — Xk\\ < e/2,
we have
Tnrx - Tn,x = (rn„ - Tn,){x - xk) + (Tnrxk - Tn,xk)
for every r and s. Since
||(rnr-rn, )(*-**)II <*
for every r, s, it follows that the sequence (Tnjx) is weakly Cauchy in Y.
We may now define T: X -)• Y by taking Tx to be the weak limit of the
sequence (Tnjx) and it follows from the Uniform Boundedness Principle that
T £ L(X,Y) = X(X,Y). Applying the lemma, we see that (Tnj) converges
weakly to T.

84 4 The Approximation Property
We now drop the condition that X be separable. Each of the adjoint
operators T„: Y* —)• X* is compact and as a result, has a separable range.
Therefore, there exists a separable subspace Z of X*, such that T„(Y*) C Z
for every n. Let Q be the quotient operator from X onto X/Z^. Now X/Z1-
is a separable reflexive space, since its dual space is Z. Furthermore, each of
the operators Tn factors through this quotient space. To see this, let x € X
and suppose that Qx = 0, that is, x € Zx. Then, for every ^> € Y*, we have
(Tnx>,J>) = (x>T*Tl>) = 0.
It follows that Tnx = 0 for every n. Therefore, there exist operators
5„: X/Z-i- -¥ Y such that SnQ = Tn and ||5n|| < 1 for every n. Now Q(BX)
is dense in the closed unit ball of X/Z1- and so every operator from X/Z1-
into Y is compact. By the first part of the proof, (Sn) has a subsequence
(Sn ) that converges weakly to some operator S. Then, by the lemma, the
sequence (Tnj) converges weakly to T = SQ. O
The converse holds if we assume that one of the spaces has the
approximation property.
Theorem 4.20. Let X and Y be reflexive Banach spaces, one of which has
the approximation property. If L{X, Y) is reflexive, then every operator from
X to Y is compact.
Proof. Suppose there is an operator T: X -)• Y that is not compact. Then
there is a sequence (a;n) in the unit ball of X that converges weakly to
some x, for which the sequence (Txn) does not converge in norm. Taking
a suitable subsequence of the sequence (xn — x), we obtain a sequence (wn)
that converges weakly to zero and 5 > 0, such that ||Tu.n|| > 5 for
every n. Choose ^>n S Sy such that ^n(Tu;n) = ||Tu.n|| for every n. Now
(X®*Y*)* = L(X,Y) and so X®„Y* is reflexive. Therefore, the bounded
sequence (wn ® ^in) has a subsequence, (wnk ® V'n.,). that converges weakly
to some « G X®XY*. Since (wnk ® xl>nk,T) > 5 for every k, it follows that
(«, J1) > 5 also.
If ip G X* and y G Y, then
(Wn„ ®l/>nk>'P®y) = <P{Vnk)^n„ (y) -»• 0 ,
and hence («, tp ® y) = 0. Thus, u is zero when considered as a bilinear form
on X* x Y. However, since X or Y* has the approximation property, the
canonical mapping of X®„Y* into H(X* x Y) is injective. Therefore « = 0
in X®„Y*, which implies («,T) = 0, a contradiction. d
We now translate these results back into the language of tensor products.
We shall make use of a theorem of Grothendieck, to be proved in the next
chapter (Theorem 5.50), to the effect that every reflexive Banach space with
the approximation property has the metric approximation property.

4.2 Reflexivity of Tensor Products 85
Theorem 4.21. Let X and Y be reflexive Banach spaces, one of which has
the approximation property. Then the following are equivalent:
(i) X®XY is reflexive.
(ii) (x&jry = x'&jr.
(iii) Every operator from X to Y* is compact.
(iv) X*®CY* is reflexive.
Proof, (i) implies (ii): if X®„Y is reflexive, then so is {X®„Yy = &(X, Y*).
Therefore
(X&,Y)' = L(X,Y') = X(X,Y') = X*®CY*,
since either X* or Y* has the approximation property.
(ii) implies (iii) is trivial.
(iii) implies (iv): if every operator from X into Y* is compact, then
X*®CY* = X[X,Y*) = &(X,Y*) is reflexive by Theorem 4.19.
(iv) implies (i): If X*®CY* is reflexive, then, by Proposition 4.16, the
subspace X®„Y of the reflexive space (X*®CY*)* is also reflexive. O
We have seen that the projective tensor product Z»Pl[0, l]®*!,,^ [0,1] is
never reflexive. It follows immediately from this proposition that the same is
true for the injective tensor product:
Corollary 4.22. The injective tensor product LPI[Q,l]®eLp,[0,1] is not
reflexive for any Pi, P2-
After all these negative results, it is a pleasant suprise to find some non-
trivial examples of reflexive tensor products.
Theorem 4.23 (Pitt's theorem). Ifl<p<q<<x> then every operator
from lq into lp is compact.
Proof. Suppose that there is an operator T: tq -»• lp that is not compact.
We may assume that ||T|| = 1. Then, arguing as in theproof of Theorem 4.20,
there is a weakly null sequence (a:n) in tq and S > 0, such that S <*||a:n|| < 1
and ||Ta:n|| > S for every n. For each m, let Pm and Qm be the projections
on either tv or tq defined by
m
Pmx = ^T x*et a11*1 QmX = ])P x,et.
«=1 «>m
We begin by choosing a positive integer Si such that
l|Q.,*i||,<*/2.
Let z! = P5la:i. Then H^H, > 5/2 and ||r^||p > 5/2.
Since (a:n) converges weakly to zero, there is a positive integer m > 1
such that

8t> 4 The Approximation Property
ll^^nJ|,<<5/4.
Choose «2 > Si such that IIQaj^risll? < 5/A and let
■2*2 = -En 2 * ai^na ^/sa^na •
Then
II, "IK?*, *»,..* >S/2.
Since ||Ta:n.,||p xS and ||r|| = 1, we also have ||T^||P > 5/2.
Proceeding in this way, we obtain a sequence (zt) in lq and an
increasing sequence (sk) of positive integers, with the following properties:
5/2 < 11-ZfcH, < 1 and ||Tjzfc||p > 5/2, and the support of zk lies in the range
Sfc-i + 1 < i < sk.
We may now apply a similar technique to the sequence (Tz/c) in lp, since
this sequence also converges weakly to zero. Thus, we obtain an increasing
sequence (rm) of positive integers and a subsequence (ztm) of (zt), such that
WPr^Tz^ \\p < 5/2m+2 and \\QTmTzkm ||p < <S/2m+2
for every m. To simplify our notation, let us write ym = Zkm and wm =
Tzkm - um, where
Um^Pr^Tz^+Q^Tz^.
We now have
Tym = wm+um,
where the vectors wm are disjointly supported and
lkm||P>l|rym||p-||«m||p><5/4
for every m. Furthermore, we have
oo °° X X
Now let A be any element of the closed unit ball of tq that is not in lp.
Let a — X^m=i ^m-Vm- Since the vectors ym are disjointly supported, we have
oo oo oo
j=1 m=l m=l
and so the series X3m=i Amym converges to a in lq. Therefore the series
Ylm=i ^mTym converges to Ta in lp. However,

M
M
4.3 Tensor Product Bases
M
E Aro«,
m=l
1/P
87
= (£iAmHK.ii?) - EA-u"
\n=l ' m=l
r / M \ 1/P M
^ EN -£imIp
^m=l ' m=l
-1" *
2
for every M. Hence ||£™=1 AmT^mllp
-, ooasM-> oo, a contradiction. 0
W P < 9> then there is an injection of tv into £, that is not compact. This
gives a complete description of the reflexive tensor products of tv spaces.
Corollary 4.24. Let 1 < p, q < oo, with conjugate indices p', q1.
(a) lp®*tq is reflexive if and only if p > q' and if this holds, then
(tp®*tqY = lj/®elq' ■
(b) lp®ctq is reflexive if and only ifp'>q and if this holds, then
(lp®elqY = V®*V •
4.3 Tensor Product Bases
In this section we shall consider the formation of tensor products of Schauder
bases. We begin by recalling some elementary facts. The sequence (en) is a
Schauder basis for the Banach space X if every x £ X is the sum of a
convergent series of the form Y^=i xnen and the scalars xn are uniquely determined
by x. The coordinate functional e^ € X* are defined by e„(x) = xn and
two uniformly bounded sequences of projections, Pn and Qn, are defined by
PnX = E"=i SiCi md Qn* = x-Pnx = E^.i=i xte,- The number supn ||Pn||
is called the basis constant and the basis is said to be monotone if the basis
constant is one. The formula ||i||' = supn H-Pn^ll defines an equivalent norm
on X relative to which the basis (en) is monotone.
The following criterion will be useful: a sequence (en) in a Banach space
X is a Schauder basis for X if and only if {en} spans a dense subspace of X
and there exists a positive constant C, such that
Ea>e>
<C
Ea'e'

88 4 The Approximation Property
for every sequence of scalars (a,) and every m, n with m < n.
Now let X, Y be Banach spaces with Schauder bases (en), (fn)
respectively. The tensor products en ®'fm can be ordered as shown in the following
diagram.
e. ® h -s- t\ ® h t\ ® h ei ® h
4 4 4
e2 ® /i <- e2 ® /2 e2 ® /3 e2 ® /4
4 4
e3®/i <- e3®/2 <- e3®/3 e3 ®/4
4
■ ■• e4®/3 f- e4®/4
Thus, the ordering is t\ ® /i, ej ® /2, e2 ® /2, e2 ® /i, ei ® /3, e2 ® /3, ... .
This ordering is known as the square ordering.
We shall denote by P.J and P2 the projections onto the first n coordinates
in X and Y respectively. Thus, P\x = J^, ej(a:)e, and Pn2y = ^=i /,'(y)/,.
If u = "£« ajye, ® /j is a finite sum in X ® V, then the coefficients ay are
given by at] = e* ® /*(«)■ Therefore, the projection of « onto the first k
coordinates is given by
pkU= £ <«/;(«),
where /fc is the set consisting of the first k ordered pairs of indices (t, j) in
the square ordering. Now a glance at the diagram illustrating this ordering
shows that the projection P* can take one of three forms:
P1 ® P2 if jfc = n2
Pn ® pn + PLn* ® (/»+i ®/»+il, H n2 < k < n2 + n + 1,
Pn+i ® ^„2+i - «+i ® e„+i) ® P(n+1)*-k , if n2 + n + 1< fc < (n + l)2.
Let j4 and B be the basis constants of the bases for X and Y respectively.
Then ||e;®en|| < 1A and ||/,t®/n|| < 2B. Now both the injective and
projective norms satisfy \\S ® T\\ = ||5||||r|| where S and T are operators.
Therefore ||Pfc|| < 3AB in both cases. Now, it is easy to see that the tensor
products en ® fm span dense subspaces of both the projective and injective
tensor products. Thus, we have
Proposition 4.25. Let X, Y be Banach spaces with Schauder bases (en),
(/„) respectively. Then the sequence (en ® fm), with the square ordering, is
a Schauder basis for both X®*Y and X®CY.
We shall refer to the Schauder basis (en ® fm), with the square ordering,
as the tensor product basis.
Recall that the basis (en) for X is said to be shrinking if the sequence
(e^) of coordinate functionals form a Schauder basis for the dual space X*.
This is equivalent to the condition that, for every ip £ X*, the norm of the

4.3 Tensor Product Bases 89
restriction of ip to the subspace Q\X = (I — P£)X of X tends to zero as
n -¥ oo. Or, to put it another way, if (a;n) is a bounded sequence in X such
that P„xn = 0 for every n, then xn tends weakly to zero.
There is a simple test for weak convergence in an injective tensor product.
The idea is much the same as that used in the proof of Lemma 4.18. Recall
that X®eY may be regarded as a subspace of C(K), where K is the product
Bx> x By of the weak'-compact dual unit balls. Weak convergence of a
bounded sequence in a C(K) space is equivalent to pointwise convergence.
Therefore, a bounded sequence («n) in X®CY converges weakly to zero if
and only if un(<p,\l>) -)• 0 for every ip G X*, rj) 6 Y*. We may consider
the elements of X®eY either as compact operators from X* into Y, or as
compact operators from Y* into X. This enables us to give two equivalent
formulations of the criterion for weak convergence:
Proposition 4.26. Let X, Y be Banach spaces. The following are equivalent
for a bounded sequence (un) in X®eY:
(i) («n) converges weakly to zero m X®eY.
(ii) For every ip £ X*, the sequence (unip) converges weakly to zero in Y.
(iii) For every \j> G Y*, the sequence (unxj>) converges weakly to zero in'X.
We can now show that injective tensor products preserve the shrinking
property.
Proposition 4.27. Let X and Y be Banach spaces with shrinking Schauder
bases (en) and (/„) respectively. Then the tensor product basts for X®eY is
shrinking.
Proof. Let («n) be a bounded sequence in X®CY with the property that
Pnun = 0 for every n, where Pn are the projections onto the first n
coordinates for the tensor product basis (e,® f}). For each n, let fc(n) be the largest
positive integer such that k(n)2 < n. Then P^nj ® -Pfc(„)(un) = 0 and so
Un = I ® (J - Pfc2(n))M + (/ - Plln)) ® P2(„)(1*n) = < + < -
Now, for every ip € X*, the sequence (u'nip) in Y is bounded and has the
property that P%(u'ntp) = 0 if n > fc2. It follows from the shrinking property
of the basis (/.) that [u'nip) tends weakly to zero. Therefore (u'n) converges
weakly to zero in X®eY. Similarly, for every ip € Y*, the sequence (u^V)
tends weakly to zero in X and so the sequence (uj() converges weakly to zero
in X®CY. Therefore un = u'n + «(( converges weakly to zero. a
While the injective tensor product preserves the shrinking property of
bases, the projective tensor product preserves the dual property of bounded
completeness. Recall that the Schauder basis (en) for the Banach space X is
boundedly complete if X is the dual of a space with a shrinking basis and
(en) is the dual basis. Equivalently, X is the dual of the closed subspace of
X* spanned by the coordinate functional ej'..

90 4 The Approximation Property
Proposition 4.28. Let X and Y be Banach spaces with boundedly complete
Schauder bases (en) and (/„) respectively. Then the tensor product basis for
X®*Y is boundedly complete.
Proof. There are Banach spaces W, Z, with shrinking bases («n), {vn),
respectively, such that W* = X, Z* = Y, and «* = e„, u* = /„ for every
ri. Replacing the norm of X by an equivalent norm, we may assume that the
basis (en) is monotone. Then W* has the metric approximation property, and
so X®„Y = W&nZ* is a closed subspace of (W®CZ)*. But (en ® fm) =
« ® <J is a Schauder basis both for X®*Y and for {W®eZ)*. Therefore
X®*Y={W®eZ)\ a
The proof of this proposition gives some more information about the
nuclear and integral bilinear forms on spaces with shrinking bases:
Corollary 4.29. Let X and Y be Banach spaces with shrinking Schauder
bases, at least one of which is monotone. Then every integral bilinear form
on X x Y is nuclear and the integral and nuclear norms coincide.
We conclude with a negative result. A Schauder basis (en) for a Banach
space X is unconditional if, for every x £ X, the expansion of x relative to
this basis is unconditionally convergent. The following example shows that
the tensor product of unconditional bases need not be unconditional, even
for the most well behaved spaces.
Example 4.30. The standard tensor product bases for £2®*^ and ^®e^2
are not unconditional.
Let (en) be the standard orthonormal basis for l<i. We partition the set of
positive integers into consecutive blocks of length 2n: let I\ = {1,2}, I2 =
{3,4,5,6}, and so on. Let
x = ^2 -un, where un = 2_n/2 ]T e3.
n=i n j6/„
Since («n) is an orthonormal sequence, this series defines an element of (2-
Now consider the expansion of the tensor x ® x £ t2®*l2 relative to the
tensor product basis. We have
x ® x = Y^ xnxmen ® em .
This series converges for the square ordering. We claim that the convergence
is not unconditional. We shall establish this by constucting a bounded bilinear
form, B, on (2 x ^2 for which the series £"". xnxmB(en,em) is not absolutely
convergent.

4.4 Exercises 91
Consider the bilinear form on the 2-dimensional Hilbert space l\ given by
the matrix
*-(-M)-
We have ||-4i|| = 21/2. For n > 1, let An be the 2n x 2n matrix defined
recursively by
a _ ( An-i An~i\
An ~ \-An^ An^J ■
Then An defines a bilinear form on the 2n-dimensional Hilbert space £|" w>th
norm 2n/2. Now let B be the bilinear form on 1% given by taking the direct
sum of the unit norm bilinear forms 2~n/2An. Thus, B is given by the block
diagonal matrix
2-'A2
2~n/2An
\ '•■/
It is easy to see that B is bounded, and so B defines a bounded linear
functional On ^2®ir^2- But
°° °° 2n/2
«... n=l «,j6/n n=1
which diverges. Therefore the series "£ xtx3et ® e} is only conditionally
convergent.
This shows that the tensor product basis for £2®*^ is not unconditional.
It follows that the tensor product basis for the injective tensor product ^2®e^2
also fails to be unconditional, since (l2®c&2)* = ^2®ir^2'
4.4 Exercises
Exercise 4.1. Let X have the approximation property and let S: X —¥
Y and T: W -)• Z be injective operators. Show that the operator S ®„
T: X®„W -¥ Y®„Z is injective.
Exercise 4.2. Let A[D) be the Banach space of complex functions on the
closed unit disc, D, in the complex plane, that are continuous on D and
analytic in the interior of D, where the norm is given by ||/|| = sup{|/(;z)| :
z € D}. Show that A(D) has the approximation property.

92 4 The Approximation Property
Exercise 4.3. Show that if X* or Y* has the metric approximation
property, then the canonical mapping of X*®„Y* into (X®eY)* is an isometric
embedding.
Exercise 4.4. If X* or Y has the metric approximation property, then
~N(X, Y) is a closed subspace of 3(X, Y). What can be said if the metric
approximation property is replaced by the bounded approximation property?
Exercise 4.5. Let X and Y be Banach spaces with the bounded
(respectively, metric) approximation property. Show that both X®„Y and X®eY
have the bounded (respectively, metric) approximation property.
Exercise 4.6. Show that the space 130*130*13 is not reflexive.
Exercise 4.7. Let (en), (/„) be shrinking bases for X, Y respectively. Show
that the tensor product basis (en ® fm) for X®*Y is shrinking if and only if
every operator from X to Y* is compact.
Exercise 4.8. Let (en), (/„) be boundedly complete bases for X, Y
respectively. Find a necessary and sufficient condition on X and Y so that the
tensor product basis for X®eY is also boundedly complete.
Exercise 4.9. Show that every operator from cq into l^ is compact.
Exercise 4.10. Let X be a Banach space with an unconditional basis.
Taking the standard bases for li and cq, show that the tensor product bases in
<i®,X and cq®cX are unconditional.

5. The Radon-Nikodym Property
In this chapter we introduce the Radon-Nikodym property for Banach spaces.
We begin with a study of vector measures, that is, measures with values in
a Banach space. Those spaces for which the classical Radon-Nikodym
Theorem extends to vector valued measures are said to have the Radon-Nikodym
property. The identification of injective and projective tensor products of
spaces of scalar measures in terms of spaces of vector measures sheds some
light on this property. We then examine the representability of various types
of operators on C(K) and Lj (p) spaces and we uncover some classes of
Banach spaces, such as the reflexive spaces and the separable dual spaces, that
possess the Radon-Nikodym property. We also relate the possession of this
property to the coincidence of the integral and nuclear operators. Finally,
we give some applications of the Radon-Nikodym property, including the
Principle of Local Reflexivity.
5.1 Vector Measures and the Radon-Nikodym Property
Let E be a u-algebra of subsets of a set Q. and let M(E) denote the Banach
space of scalar measures on E with the variation norm, \\n\\ = \fi\(Q,), where
|/n|(E) denotes the variation of p on the set E.
A vector measure on E is a countably additive function fj, on E with
values in a Banach space X. Thus, if (En) is a sequence of mutually disjoint
measurable subsets of Q, with union E, then the series J2n A*(-®n) is required to
converge to fi(E). Since the ordering of the terms is arbitrary, it follows that
this series must converge unconditionally. Furthermore, thanks to the Orlicz-
Pettis Theorem (Proposition 3.12), in order to establish countable additivity
for a function ft: E —)• X, it suffices to show that the scalar function tpfj, is
countable additive for every ip € X*. In other words, p: E -»• X is a vector
measure if and only if (pfi is a scalar measure for every ip € X*.
We now consider the definition of the variation for vector measures.
Proceeding exactly as in the scalar case, let
\n\i(E) = supj^ \\n(A,)\\ :{AU.. .,An} a partition of E
•

94 5 The Radon-Nikodym Property
for each E £ E, where a partition of E is a finite set of mutually disjoint
measurable sets whose union is E. It is easy to see that the variation is a
monotonic, non-negative, extended real-valued function. Now, the variation
of a scalar measure is a finite, positive measure. However, in the vector valued
case, the variation need not be finite valued.
The vector measure ft is said to have bounded variation if |/j.|j (E) is finite
for every E € E, or equivalently, if |/n|i (ii) is finite. We leave it as an exercise
to the reader to verify that if ft has bounded variation, then the variation
\fi\i is a finite, positive measure on E. Furthermore, it is not difficult to show
that the set of vector measures with values in X that have bounded variation
is a Banach space with respect to the variation norm:
IMIi = Mi(n)-
We illustrate these ideas with some examples.
Example 5.1. Some co-valued vector measures.
Let E be the u-algebra of all subsets of N. Fix an element a = (an) of cq and
let
It is easy to see that ft is a vector measure with variation |/n|i(E) =
SngE la»l- Therefore fi has bounded variation if and only if a belongs to
«,.
Example 5.2. An Lp[0, l]-valued vector measure.
Let E be the u-algebra of Lebesgue measurable subsets of [0,1] and let ft: E -»•
Lp[0,1] be given by
KE) = XB ■
If 1 < p < oo, then p is a vector measure. However, if p = oo, then jj, is only
finitely additive. When p = 1, the variation of ft on a measurable set E is
m(E), the Lebesgue measure of E. Thus ft has bounded variation. We leave
it to the reader to verify that the variation of ft is unbounded for 1 < p < oo.
There is an alternative approach to the definition of a variation for vector
measures that has the advantage of always being bounded. Let fj.be a. vector
measure on E with values in X. Then tpfi is a scalar measure for every ip £ X*
and so we may define a linear mapping T^: X* -)• M(E) by T^ip = tpfi. A
straightforward application of the Closed Graph Theorem shows that T^ is
bounded. Therefore, we may define the semivariation of (i on a measurable
set E by
\li\oo{E) = sap{\Vp\(E) :<peBx.},
and we see that

5.1 Vector Measures and the Radon-Nikodym Property 95
I4»(£)<Mi(£) and IaiU(B) < H~(n) = II^H
for every E £ E. It follows easily from the definition that the semivariation of
ft is a monotone, countably subadditive function on E, taking its values in a
bounded subset of R. However, the semivariation is not, in general, countably
additive.
The semivariation satisfies a useful inequality:
IMS).. < MM < 4sup{\\fi(F)\\ :FeZ,FcE} (5.1)
for every E G E. This is an immediate consequence of the corresponding
fact for scalar measures and the definition of the semivariation. It follows
that every vector measure is bounded, since we have ||/j(E)|| < \fi\oo(E) <
\fi\oo(ty = ||rj| for every E e E. However, we shall see shortly that a much
stronger statement holds.
We denote by M(E, X) the vector space of vector measures on E with
values in X. The semivariation norm is defined on this space by
Woo = Moo(n) = 11^11.
Proposition 5.3. The space M(E,X) of vector measures is complete in the
semivariation norm.
Proof. Let (fin) be a Cauchy sequence in M(E,X). Since
\\l*n(E) - llm(E)\\ < \fln ~ flm\oo(E) < \W ~ A<m||oo ,
the sequence (fin(E)) is Cauchy in X for every E £ E. Therefore, we may
define ft: E -»• X by fi(E) = limn/n„(E) for every E € E. Then <pfi(E) =
]imnipfin(E) for every E G E, tp G X*, and it follows from the Nikodym
Convergence Theorem (see Appendix C) that tpfi is a scalar measure for
every ip £ X*. Therefore ft belongs to M(E,X). Let T^: X* -»• M(E) be
the operator associated with ft, so that T^tp) = ipp. for tp £ X*. Since
ll^„ - T^JI = \\fin - fi^U, the sequence (T^J is Cauchy in £(X',M(E)).
It is easy to see that the operator 7), is the limit of this sequence. Therefore
IK-HlooHir^-rj-^o,
and so the sequence (fin) converges in semivariation norm to ft. D
Vector measures can be generated by means of indefinite integrals. Recall
that if the scalar function / is integrable with respect to a positive measure
A, then the formula fi(E) = JBf dX defines a measure with variation given by
|/n|i(E) = fB |/| dX. This process can be imitated in the vector valued case,
using the Pettis integral. If the function is assumed to possess stronger
properties, such as measurability or Bochner integrability, we see a corresponding
improvement in the behaviour of the indefinite integral:

96 5 The Radon-Nikodym Property
Proposition 5.4. Let X be a finite, positive measure on the a-algebra E.
(a) If f: Q. -t X is Pettis integrable with respect to X, then /j,(E) = fBf dX
defines a vector measure on E with semivariation given by
Moo(E) = suplf \vf\ dX:ipe Bx.
for every E £ E. In particular, the semivariation norm of jj, satisfies
||/i||oo = |/|i. where |/|i is the Pettis norm of f.
(b) If f is a X-measurable Pettis integrable function, then the vector measure
fi takes its values in a compact subset of X.
(c) If f is Bochner integrable with respect to X, then fi has bounded variation
and the variation is given by
Mi(E)= f WfWdii.
JE
In particular, the variation norm of ft satisfies ||jj||i = ||/||i, where ||/||i
is the Bochner norm of f.
Proof, (a) By the definition of the Pettis integral, ipn(E) = fEtpf dX for
<p G X* and E G E. Therefore ipfi is a measure for every ip G X* and so fi is
a vector measure. The formula for the semivariation of jj, follows immediately
from the fact that \tp/j,\(E) = /£ \ipf\dX.
(b) If / is A-measurable and Pettis integrable, then, by Proposition 3.11,
the operator T: Loo(A) -»• X, given by Tg = fngf dX, is compact. Since
fi(E) = Txb, it follows that the values fi(E) lie in a compact subset of X
(c) Suppose that / is Bochner integrable with respect to A. If {Ai,..., Am}
is a partition of a measurable set E, then
Eh^.)ii = E / fdx ^E/ ii/n<*A=/../ma.
Therefore fi has bounded variation and \fi\\{E) < fE \\f\\ dX for every E £ E.
To prove the reverse inequality we approximate / by a simple function. Fix
e > 0 and let g be a A-measurable simple function such that J"n \\f — g\\ dX <
e. Let v be the vector measure given by the indefinite integral of g. It follows
easily that |Mi(E) - |i/|i(S)| < e. Let g = YZ=\Xe,x, be the canonical
representation of g. Let {Ai,...,Am} be a partition of E £ E. We may
assume, by refining this partition, that each of the sets E D E, is a union of
a subset of {A3}. Then
m m II . n
EH^)II=E / 9dx =J2\M\HEnEt)= / \\g\\dx
and hence |i/|, (E) = /£ ||5|| dX. Hence

5.1 Vector Measures and the Radon-Nikodym Property 97
H, (E) <\uU(E) + e= f \\g\\ dX +e < f \\f\\dX + 2e
JE JE
for every e > 0. Therefore \fi\i(E) < fE \\f\\ dX and the proof is complete. 0
A deeper understanding of vector measures requires a knowledge of the
structure of the weakly compact subsets of the Banach space M(E) of scalar
measures. For the benefit of readers who may be unfamiliar with this material,
a complete account is given in Appendix C. In particular, we shall make use
of the following result.
Proposition 5.5. Let K be a bounded subset o/M(E). The following
statements are equivalent:
(i) K is relatively weakly compact.
(ii) The elements of K are uniformly countably additive.
(iii) There exists a finite, positive measure X on E such that the elements of
K are uniformly absolutely continuous with respect to X.
We recall that to say that the elements of K are uniformly countably
additive means that if (E„) is a sequence of mutually disjoint measurable sets,
then the convergence of the series X3nLi n(E„) is uniform in fi £ K. Equiv-
alently, if (F„) is a decreasing sequence of measurable sets with intersection
F, then fi(Fn) tends to fi(F) uniformly in fi G K. The statement that the
elements of K are uniformly absolutely continuous with respect to A means
that, for every e > 0 there exists 5 > 0 such that if E is a measurable set
with X(E) < S, then \fi(E)\ < e for every fi e K.
Now let ft: E -)• X be a vector measure and let K be the bounded subset
of M(E) consisting of the scalar measures ipfi, where tp £ Bx- ■ If (E„) is a
sequence of mutually disjoint measurable sets, then
k=n
<
i>(£*)
fc=n
for every <p £ Bx-- It follows that the measures ipfi, as <p ranges over Bx-,
are uniformly countably additive and hence K is relatively weakly compact
in M(E). In terms of the operator associated with fi, we have:
Proposition 5.6. Let fi: E —)• X be a vector measure. Then the operator
T,,,: X* —> M(E), given by T^tp) = ipfi, is weakly compact.
We have seen that the range of a vector measure is bounded. In fact, much
more is true:
Corollary 5.7 (Bartle-Dunford-Schwartz Theorem). Let fi be a
vector measure. Then the values of ft lie in a weakly compact set.

98 5 The Radon-Nikodym Property
Proof. Consider the weakly compact adjoint operator T*: M(E)* —> X**.
For each E 6 E, the characteristic function xb acts as a bounded linear
functional on the space M(E) by (A, xe) = HE) and has unit norm. It is
easy to see that n(E) = T*(xe). Therefore the subset K = {(i(E): E 6 E}
of X is contained in the relatively weakly compact subset T*(B) of X**,
where B is the closed unit ball of M(E)*, and it follows that K is relatively
weakly compact in X. O
The operator TJ,: X* -)• M(E) associated with a vector measure can be
thought of as an "adjoint" of ft, with measure-theoretic properties of fj, being
reflected by operator-theoretic properties of T^. Our next result is another
illustration of this principle.
Proposition 5.8. Let ft: E —)• X be a vector measure with the associated
operator T^: X* -¥ M(E) given by T^tp) = tpfj,. Then the following are
equivalent:
(i) (i has bounded variation.
(ii) Tp is a Pietsch integral operator.
(iii) Tp is an integral operator.
Furthermore, if (i has bounded variation, then
IMIi = HT„||/ = ||rj„.
Proof, (i) implies (ii): suppose that (i has bounded variation. Let A denote
the variation of /j,. Then A is a finite, positive measure on E. We shall show
that there is an operator V: X* -¥ Loo(X) of unit norm, such that T^
factors as UIV, where U: £i(A) -»• M(E) is the canonical embedding given by
Uf(E) = fEf dX and /: Loo (A) -»• Li(A) is the canonical injection:
T
v* f
M(E)
U
It then follows that TJ, is Pietsch integral, with Pietsch integral norm
satisfying
\\TJpi < ||A|| = IMI, ■
To define the operator V, let ip € X*. The measure ipfi is absolutely
continuous with respect to A since, if A(E) = 0, then fi{E) = 0 and hence ipn(E) = 0.
Therefore, by the Radon-Nikodym Theorem, there exists a unique element
fv of Lj (A) such that ipfi(E) = fE fv dfi for every E € E. The mapping
ip £ X* h-). fv £ L\ (A) defines an operator and we have

5.1 Vector Measures and the Kadon-iNiKOaym nupaij „„
T„(V)(E) = ME) = f Udfi = Ufv(E)
JE
for every E 6 E. Thus T^tp = Ufv for every ipSX*.
We claim that fv G £oo(A) for every ip € X*. To see this, let g be a
measurable, simple function with canonical representation $3iLi °tXEi and
let tpSX*. Then
I/.
9fvd\
X>W(E.) <IMl£WIIM£i)ll
1=1 t=l
<Ml£|c,|A(E,) = IMINIi-
It follows that /y, belongs to L^ (A) for every ip € X* and that the operator
V: y> G X* h-> /^ t= Loo (A) has norm at most one. Finally, we have T^ = UIV
as asserted.
(ii) implies (iii) is trivial, so we proceed to the proof of (iii) implies (i).
Suppose that the operator T^ is integral. Then T^ defines a bounded linear
functional on the injective tensor product X*®e'M(E)*. Let £ be a
measurable subset of Q. and let {A\,...,An} be a partition of E. For each »" let
tp, £ Bx* be such that ||j*(.A,)|| = (ji(A,),(p,). Then
£ Ha»(^)II = X>(^),*>.> = it<Tw,XA.)
t=l t=l t=l
'^^®XA{,T^\<e\yi>pl®XA.J\\Tfl\\I.
We compute the injective tensor norm:
e ( £ p, ® XA, ) = sup y2<pl(x)xA,
\,=1 J *&bx fr?
M(E)-
= sup sup
x&Bx ||^||=1
^,W"(^.) < sup £>(A)|<1
"ll=» ^
Therefore "£"=i ||/z(j4»)|| < ||2^||/ for every partition of E and so fi has
bounded variation. Furthermore, we have ||/n||i < ||Tj,||/. Therefore
iyi<lirAi||/<||rjw<Wi,
and-so these norms coincide. O
We now consider absolute continuity for vector measures. The definition
is the same as in the scalar case: the vector measure \i is said to be absolutely
continuous with respect to the finite, positive measure A if, for every e > 0
there exists 5 > 0 such that if E is a measurable set with A(E) < S, then
||/n(E)|| < e. As in the scalar case, we have:

100 5 The Radon-Nikodym Property
Proposition 5.9. Let p: E -»• X be a vector measure and let X be a finite,
positive measure on E Then \i xs absolutely continuous with respect to X if
and only if X(E) = 0 implies fi(E) = 0.
Proof. It is clear that this condition is implied by absolute continuity.
Conversely, suppose that p is not absolutely continuous with respect to A.
Then there exists e > 0 and a sequence (En) of measurable sets such that
X(En) < 2-" and \fi\oo(En) > \\fi(En)\\ > e for every n. Let Fn = \Jk>nEk
and let F be the intersection of the decreasing sequence (Fn). Then A(F) =
limn A(Fn) = 0. Since the set {\<pfi\ : f> G Bx-} is weakly compact, \ipp\(Fn)
converges to \ipfi\(F) uniformly for ip G Bx-- Therefore \n\oo(F) > e. Now,
by (5.1), F contains a measurable subset G with ||^(G)|| > |/li|oo(F)/4. Thus
X(G) = 0, but fi(G) ^ 0. O
We now construct an integral for scalar functions with respect to a vector
measure. It will be sufficient for the applications that we have in mind to
define the integral for bounded measurable functions. While it is possible to
define the integral first for simple functions and then pass to limits of
sequences of simple functions, we take a different approach that enables us to
make use of some of the machinery that we have developed. Let B(Q, E)
denote the Banach space of bounded scalar functions on Q, that are measurable
with respect to the a-algebra E, with the norm ||/||oo = sup{|/(w)| : <j G Q}.
We note that the E-measurable simple functions are dense in B(Q,, E). The
space B(fi,E) can be considered as a subspace of the dual space M(E)*,
where (A, f) = JQf dX for A G M(E). / G B(fi. E).
Now let fi: E -)• X be a vector measure and let T^: X* -> M(E) be the
associated operator. Restricting the adjoint of T^ to the space B(fi,E) yields
an operator S: B(fi,E) -► X**. We have Sxb = fi(E) for every E G E
and so, if g = X."=] c.^is, is the canonical representation of a E-measurable
simple function, then Sg = "£"_j c,/n(E,) G X. Since these functions are
dense in B(Q, E), it follows that S takes its values in X. Thus, we may define
the integral of a bounded, E-measurable function / over a measurable set
E by JE f dp = S(xEf) £ X. The following properties of this integral are
immediate.
Proposition 5.10. Let p: E —> X be a vector measure.
(a) If f is a bounded ^-measurable function on Q and E G E then
f fdp
JE
M°o(E).
In particular,
< ll/IUMIo
f fdp
Jn

5.1 Vector Measures and the itadon-iNucoayiu nupcuj
(b) If V: X -¥ Y is an operator, then Vfi is a vector measure and
L",Vi"v{L!d")
for every bounded H-measurable function f.
In order to prove a bounded convergence theorem for this integral, we
require a version of Egoroff's Theorem for vector measures.
Proposition 5.11 (Egoroff's Theorem for Vector Measures). Let
n:Y,—*X be a vector measure and let (/„) be a sequence of ^-measurable
functions that converges pointwise to f. Then for every e > 0 there exists a
measurable set E such that |/li|oo(-E) < £ &n& (fn) converges uniformly to f
onQ.\E.
Proof. Let A be a finite, positive measure with respect to which the elements
of the relatively weakly compact set {tpfi: ip 6 Bx-} are uniformly absolutely
continuous. Then there exists 5 > 0 such that X(E) < S implies \fi\oo(E) < e.
By the classical Egoroff Theorem, there exists a measurable set E such that
A(E) < 5 and (/„) converges uniformly on Q, \ E. Then E has the required
properties. Q
Proposition 5.12 (Bounded Convergence Theorem for Vector
Measures). Let ft: E -»• X be a vector measure and let (/„) be a uniformly
bounded sequence of H-measurable functions that converges pointwise to f.
Then
lim / f„dfi= / f dy,.
Proof. Given e > 0, let E be a measurable set such that |/n|oo(£) < e and
(/„) converges uniformly on the complement of E. Then
\lU-fn)dil
<suP{||/H- /„(w)||:u, e n\E}Moo(n\E)
+ \\f~fn\\oo\fi\oo(E)
for every n. The proposition follows immediately. Q
We turn now to the central theme of this chapter - the Radon-Nikodym
Theorem for vector measures. The natural extension to the vector case of the
classical Radon-Nikodym Theorem would assert that if the vector measure p
has bounded variation and is absolutely continuous with respect to a finite,
positive measure A, then ft is an indefinite Bochner integral with respect to
A. Unfortunately, this is false for many of the classical Banach spaces.
Example 5.13. The Radon-Nikodym Theorem fails in cq-

102 5 The Radon-Nikodym Property
Let E be the <r-algebra of Lebesgue measurable subsets of [0,1]. Since the
Rademacher functions rn are mutually orthogonal on [0,1], we may define
fi(E)= (frn(t)dt\ 6 Co
for E £ E. It is easy to see that ft is weakly countably additive and so, by the
remarks at the beginning of this section, (i is a vector measure. Furthermore,
we have ||/u(E)|| < m(E) for every E G E, where m denotes Lebesgue
measure. It follows that n has bounded variation and is absolutely continuous
with respect to m. Suppose that ft is given by the indefinite Bochner integral
of some function / = (/„): [0,1] -)• cq- Then
f fn(t)dt= f rn(t)dt
JE JE
for every E G E and every n. It follows that fn(t) = rn(t) almost everywhere
for every n. But this is impossible, since / takes its values in eg.
We shall see that if the Radon-Nikodym Theorem fails for the Banach
space X, then it fails for any Banach space that contains X as a subspace.
So, for example, the theorem fails for £«., C[0,1] and Loo[0,1], since each of
these spaces contains c0.
The Banach space X is said to have the Radon-Nikodym property if the
Radon-Nikodym Theorem is valid in X. In other words, if E is a u-algebra
of subsets of a set Q, and ft: E -)• X is a vector measure of bounded variation
that is absolutely continuous with respect to a finite, positive measure A,
then there exists a A-Bochner integrable function f:Q.-i-X such that
ti{E)= f fdX
JE
for every E G E.
Example 5.14. The space (i has the Radon-Nikodym property.
Let E be a u-algebra of subsets of a set Q. and let ft be a vector measure on
E with values in £]. We may write fi{E) = (fin(E)), where (/j,n) is a sequence
of scalar measures. A straightforward calculation shows that ft has bounded
variation if and only if the sequence (fin) is absolutely summable in M(E).
Indeed, we have
M.(£)=f>n|(£)
n=l
for every measurable set E. If ft is absolutely continuous with respect to A,
then so is each of the measures fin. Therefore there exist /„ G £i(A) such
that dfj,n = fn dX for every n. Since

5.2 Tensor Products and Vector Measures 103
£/l/n|dA=f>n||<0O,
n=l"'n n=l
the series J.^!=i l/n(w)l converges almost everywhere. Therefore, we may
define /: Q. -* i-i by f(J) = (/„(w)) and it follows from the Pettis Measurability
Theorem and Bochner's theorem that / is Bochner integrable with respect
to A. Finally, we have
»(E)=(Jjnd\] = jjd\
for every E 6 E.
We have seen in Proposition 5.4 that every vector measure that is given by
an indefinite Bochner integral takes its values in a compact set. This simple
observation yields another negative example.
Example 5.15. The space Li[0,1] does not have the Radon-Nikodym
property.
Let E be the u-algebra of Lebesgue measurable subsets of [0,1] and consider
the vector measure with values in Li[0,1] given by fi(E) = xb- We have
|^|, (E) = m(E) for every E G E and so p is absolutely continuous with
respect to m. However, the set {xe : E £ E} is clearly not relatively compact
in Li[0,1] and so ft cannot be given by an indefinite Bochner integral.
5.2 Tensor Products and Vector Measures
It is natural to consider the elements of the tensor product M(E) ® X as
vector measures - the tensor ft = X."=1 (it ® xt corresponds to the X-valued
measure defined by fi{E) = "£"","=1 fit(E)xt. We shall .-compute the injective
and projective norms of ft.
Lemma 5.16. Let X be a finite, positive measure on the a-algebra E and let
Ma(E) denote the closed subspace o/M(E) of measures that are absolutely
continuous with respect to A. Then Ma(E) is isometrically isomorphic to
Li(A) and is complemented in M(E) by a projection of norm one.
Proof. It follows from the Radon-Nikodym Theorem that M.x(E) is
isometrically isomorphic to Li(A). Now, by the Lebesgue Decomposition Theorem,
for each fj, € M(E) there is a unique pair of measures /ia, pa, such that (ia is
absolutely continuous with respect to A, fi, is singular with respect to A and
fi = fia + lis- The projection P: M(E) -)• Ma(E) is given by Pfi = fia. By
the uniqueness of the Lebesgue decomposition, we have \fj,\ = |/ia| + \pa\ and
so \\ii\\ = M(fl) > \iiam) = WpaW- Therefore ||P|| = 1. d

104 5 The Radon-N.kodym Property
We can now use the knowledge gained in previous chapters of the tensor
products L| (A) ®e X and L\ (A) ®x X to identify the projective and injective
norms on M(E) ® X.
Proposition 5.17. The projective and injective norms on M(E) ® X
coincide with the variation and semivartation norms respectively.
Proof. Fix a representation X."_j pt ® xx of p. 6 M(E)® X. Let A be a finite,
positive measure on E with respect to which the measures pi,...,p„ are all
absolutely continuous. Then, by the Radon-Nikodym Theorem, there exist
S, 6 ^i(A) such that dpt = gt d\ for each ». Let g = X."=1 9i®x< 6 Li(X)®X.
Then p is given by an indefinite Bochner integral:
Now p lies in the subspace Lj (A) ® X of M(E) ® X. We may compute the
injective norm of p in this subspace. Since the injective norm on Li(A) ® X
coincides with the Pettis norm, we have
e{p) = sup j / \ipg\ dX-.tpe Bx- \ = ||/i||oo ,
by Proposition 5.4. In the same way, since Lj (A) is complemented in M(E)
by a projection of unit norm, we may compute the projective norm of p in
Li(A) ® X. Therefore, again using Proposition 5.4, we have
»M = I \\9\\d\=\\ph.
Jn
a
Let 3C(E, X) denote the subspace of M(E, X) consisting of the compact
vector measures, that is, measures p for which the set {p(E) : E 6 E}
is relatively compact. It is easy to see that 3C(E, X) is a closed subspace of
M(E, X). Thus 3C(E, X), with the semivariation norm, is a Banach space. We
note that if p is a compact vector measure, then the operator T^: X* -> M(E)
associated with p is compact. To see this, let (<pn) t>e a sequence in Bx-- Let
(tpna) be a subnet that is weak'-convergent to some tp £ Bx-. This net, being
bounded, converges uniformly on every compact subset of X and
l|7>„a - 2>|| = \\<pnap - ipp\\ = \<pnap - ipp\(n)
< 4 sup \{<pnap - <pp)(E)\ = 4 sup \(tpnci - <p)(p(E))\.
JS6"C E6"G
It follows that (T^ipn^) converges in norm to T^ip.

5.2 Tensor Products and Vector Measures iuj
We are now in a position to identify the injective tensor product M(E)®eX
as a space of vector measures. In the proof of the next proposition, we shall
make use of the following fact: every separable subspace of M(E) is contained
in a subspace of the form Ma(E) for some finite, positive measure A. This
follows immediately from the fact that, for every countable subset {An} of
M(E), there exists a finite, positive measure A with respect to which all the
An are absolutely continuous.
Theorem 5.18. Let E be a a-algebra of subsets of a set Q and let X be a
Banach space. The injective tensor product M(E)®eX is isometrtcally
isomorphic to the Banach space 3C(E, X) of compact vector measures tvith the
semivariation norm.
Proof. The elements of M(E) ® X are clearly compact vector measures, and
hence M(E)®eX is a closed subspace of 3C(E, X). It only remains to show
that M(E) ® X is dense in DC(E, X). We have seen that if the vector measure
(i is compact, then the associated operator TJ,: X* -¥ M(E) is compact.
It follows that Tp maps X* into a separable subspace of M(E), and so, by
the remark preceding the proposition, T^X*) is contained in a subspace
of M(E) of the form I>i(A). Let e > 0. By the proof of the approximation
property for Li(A), there exists a partition of Q, into disjoint, measurable
subsets j4i, ..., An of positive A-measure, such that \\Uf — f\\ < e for each /
belonging to the relatively compact set T^Bx'), where U is the finite rank
operator on Li(fi) given by
Uf = YiW1(fA fdxjxA,-
Therefore, for every tp 6 B*-, we have
n
,=i MA) '
For each i, let A, be the scalar measure defined by A,(E) = X(A,)~1X(A, HE)
and let x, = fj,(A,) £ X. Then
tp/j, - tp ^2 A, ® x,
<e,
M(S)
for every <p 6 Bx' ■ Taking the supremum over ip gives
/J - yi fit ® x,
<e.
Therefore M(E) ® X is dense in DC(E, X).
a

106 5 The Radon-Nikodym Property
An examination of this proof shows that those compact measures that
are absolutely continuous with respect to a fixed finite, positive measure
A lie in the closure of Z<i(A) ® X in M(E)®eX. This yields an alternative
characterization of the injective tensor product L\(X)®CX to that obtained
in Chapter 3:
Corollary 5.19. Let X be a finite, positive measure on E. Then Li(X)®cX
is isometncally isomorphic to the space of compact vector measures on E
that are absolutely continuous with respect to X, where this space carries the
semivanation norm.
We turn now to the projective tensor product M(E)®,X. Since the
projective norm on M(E) ® X coincides with the variation norm, the elements
of the completed tensor product are vector measures of bounded variation.
However, not every vector measure of bounded variation arises in this way.
We shall say that a vector measure ft: E —> X has the Radon-Nikodym
property if fi has bounded variation and, for every finite, positive measure A on
E with respect to which ft is absolutely continuous, there exists a A-Bochner
integrable function / such that fj,(E) = fE f dp for every E 6 E. Thus, the
Banach space X has the Radon-Nikodym property if and only if every vector
measure with values in X has the Radon-Nikodym property.
In order to establish that a vector measure ft of bounded variation has the
Radon-Nikodym property, it suffices to show that (i is given by an indefinite
Bochner integral with respect to its variation. Indeed, suppose that this is
the case, so that
ti{E) = [ fd\p\lt
JE
for every E 6 E, where / is Bochner integrable with respect to the scalar
measure |/n|i. Suppose that p is absolutely continuous with respect to a finite,
positive measure A. Then |/n|i is also absolutely continuous with respect to A
and so there exists g 6 £i(A) such that \fi\i(E) = JEgdX for every E £ E.
It follows from the definition of the Bochner integral that the function gf is
Bochner integrable with respect to A and
H(E)= f gfdX
JE
for every E 6 E.
It is not difficult to find non-trivial examples of vector measures with the
Radon-Nikodym property. The following result shows that every indefinite
Bochner integral has this property.
Lemma 5.20. Let X be a finite, positive measure on the a-algebra of subsets
of the set ft and let f: ft —> X be Bochner integrable with respect to A. Then
the vector measure defined by

5.2 Tensor Products and Vector Measures 107
fdX
»(E) = f
Je
IE
has the Radon-Nikodym property.
Proof. By the Pettis Measurability Theorem, we may assume that X is
separable. Consider the function h: X —> X denned by
{ ' \0 ifa: = 0.
This function is Borel measurable and so, again using the Pettis
Measurability Theorem, the function g = h o f is |^|i-measurable. Furthermore, g
is bounded, and hence is Bochner integrable with respect to \p\i. Now, the
variation of ft is given by
|dA
Je
and so we have
Mi(£)= f
Je
li{E)= f fd\= [ g\\f\\d\= f gdfrU.
Je Je Je
It follows from the remarks preceding the lemma that ft has the Radon-
Nikodym property. Q
We shall denote by Mi(E, X) the space of measures with the Radon-
Nikodym property, with the norm given by the variation.
Lemma 5.21. The space 3Yti(E, X) is complete in the variation norm.
Proof. Let (/j,n) be a Cauchy sequence in Mi(E, X). Let A be a finite,
positive measure on E such that each of the scalar measures |/nn|i is absolutely
continuous with respect to A. Then there is a sequence■(•/„) in Li(X, X) such
that
ltn(E)= f fndX
Je
for every E 6 E and every n. Then (/„) is Cauchy in Li(X,X), since
[ \\fn-fm\\dX=\\fln-flm\\1
Jn
for all m,n. Let / be the limit of this sequence. Then the sequence (/n„)
converges in variation norm to the measure fi(E) — JEfdX and this
measure, being given by an indefinite Bochner integral, has the Radon-Nikodym
property. Q
We can now characterize the projective tensor product M(E)®^X.

108 5 The Radon-Nikodym Property
Theorem 5.22. Let E be a a-algebra of subsets of a set Q, and let X be
a Banach space. The projective tensor product M(S)®,X is isometrically
isomorphic to the Banach space Mi(E, X) of vector measures with the Radon-
Nikodym property, with the variation norm.
Proof. We have seen that the projective norm coincides with the
variation norm. Therefore it only remains to show that 3Yt(E) ® X is dense in
Mi(E,X). Let fi 6 Mi(E,X). Then there exists / 6 L,(|^|i,X) such that
li{E) = JBfd\fi\i for every E 6 E. Since Lj(|/n|,,X) = X,i(Ia*|i)®«-X", there
exist bounded sequences (gn) and (a;n) in Li(|/n|i) and X respectively, such
that 2^=1 llffnlllknll < oo and / = Y,™=i 9n ® xn, the series converging
in Lid/nli)®**. Let fin 6 M(E) be deBned by p„(E) = JEgnd\tJ,\i. Then
A* = E~=i ^" ® Xn e M(E)®, JT. O
Corollary 5.23. // X has the Radon-Nikodym property then M(E)(g.,r.X' is
the space of vector measures of bounded variation, with the variation norm.
5.3 Operators on C(K) Spaces
Let K be a compact topological space and let "BK be the u-algebra of Borel
sets. Every bounded linear functional on the space C(K) is given by
integration with respect to a uniquely determined regular measure on the u-algebra
3«■. It is natural to ask if there is an integral representation for operators.
In other words, if T: C(K) —> X is an operator, does there exist a vector
measure ft with values in X such that
Tf= [ fdfi
JK
for every / 6 C(K)1
Let us see how we might attempt to construct such a vector measure.
The dual space C(K)* is the Banach space of regular Borel measures on K
with the variation norm. The Banach space of bounded Borel measurable
functions on K, with the supremum norm, is a subspace of C(K)**\ if g is
such a function and ft 6 C(K)*, then (fi,g) = JKgdfi. In particular, we
may consider the characteristic functions of Borel subsets of K as elements
of C(K)**. Now let T: C(K) —> X be an operator. We may define a function
fix on the (T-algebra of Borel subsets of K with values in the bidual X** by
Pt(E) = T"Ub). (5.2)
Thus, for each ip 6 X*, we have
{<p,liT{E)) = T<p{E), (5.3)
where T*tp 6 C(K)* is interpreted as a regular Borel measure on K. It is
unfortunate that /lit takes its values in X** rather than X. However, this

&.<i uperau>r» uu«-. v«v,-_.p-~w
function has almost all the properties we require. To begin with, we see that
HT is weak*-countably additive: if (En) is a sequence of mutually disjoint
Borel subsets of K, then it follows from (5.3) that the series "£„ Ht(E„) is
weak'-convergent to pr(E). Furthermore, pr is weak?-regular in the sense
that the measure tppT is regular for every tp 6 -X"*. And finally, we have an
integral representation of sorts:
(Tf,<p)= [ fd<pnr (5.4)
for every / 6 C(K) and every tp £ X*. We shall refer to the function pt
as the representing measure for the operator T. It is clear from (5.2) that
fiT will take its values in X if T** maps C(K)** into X, in other words, if
T is a weakly compact operator. Before developing this idea, we clarify the
meaning of regularity for vector measures. Just as in the scalar case, a vector
measure p on the u-algebra of Borel subsets of if is said to be regular if, for
every Borel set E and every e > 0, there exist an open set O containing E
and a closed set C contained in E, such that \\p(0 \ C)|| < e. Our next result
shows that regularity is equivalent to "weak regularity".
Lemma 5.24. Let p be a vector measure on the a-algebra of Borel subsets
of the compact space K with values in the Banach space X. Then p is regular
if and only if the scalar measure tpp is regular for every tp e X*.
Proof. It is easy to see that the regularity of p implies that of tpp for every
tp 6 X*. Conversely, suppose that tpp is regular for every tp 6 X*. Since
the set {tpp : tp s Bx'} is weakly compact in the space of regular Borel
measures, it follows that there is a regular positive Borel measure A such that
the measures in this set are uniformly absolutely continuous with respect to
A (see Appendix C). Thus, given e > 0, there exists 5 > 0 such that if E is
a Borel set with A(£") < 5, then 1^(^)1 < e for every tp € BX', and hence
H/Li^i?)!! < e. The regularity of p now follows immediately from the regularity
of A. .- D
We can now present the fundamental result about the correspondence
between weakly compact operators on C(K) and vector measures on the
Borel (T-algebra of K.
Theorem 5.25. Let T: C(K) -)• X be a weakly compact operator. Then
there exists a unique regular vector measure p on the Borel subsets of K with
values in X such that
Tf= [ fdp
JK
for every f £ C(K). Conversely, if p is a regular vector measure on the Borel
subsets of K with values in X, then Tf = fK f dp defines a weakly compact
operator from C(K) into X. IfT and p are related m this way, then

v.
\\
110 5 The Radon-Nikodym Property
imi = iHi~,
the semwartation norm of p.
Proof. We have seen that p(E) = T**(xe) defines a weak'-counHably
additive, weak'-regular function from the Borel u-algebra "Br into X**, and
that
(Tf,<p)= f fd<pp
Jk
for every / G C(K) and ip £ X*. If T is weakly compact, then T** map
(C(K))" into X and hence fi{E) £ X for every Borel set E. It follows thait^
^p is weakly countably additive and so p is a vector measure. Furthermore,
the scalar measure tpp is regular for every <p £ X* and so p is regular
Lemma 5.24. Let / £ C{K). Since
(Tf, <p) = J f dipit = I j f dp, <p^
for every ip € X*, it follows that Tf = fK f dp.
Conversely, let p be a regular vector measure on the Borel subsets of K
with values in X and let T: C(K) -¥ X be the operator defined by Tf =
fK f dp. Then T*tp is the scalar measure ipp for tp £ X*. Since {tpp : tp £
Bx'} is a relatively weakly compact set of measures, it follows that T*, and
hence T, is a weakly compact operator.
Now
P7|| = ||/ fdp
\\Jk
for every / £ C(K), and so ||T"|| < \\p\\x
On the other hand,
||W|| = ||rvi| < ||r|| = ||r||
for every tp £ Bx-, and hence
< ||r||. Therefore ||r|| = \\p\\c
a
A Banach space X is said to have the Dunford-Pettis property if, for
every Banach space Y, every weakly compact operator from X into Y is
completely continuous. Now the weakly convergent sequences in C(K) are
precisely the uniformly bounded pointwise convergent sequences. Therefore,
the above proposition in tandem with the Bounded Convergence Theorem
for vector measures (Proposition 5.12) yields:
Corollary 5.26. The space C(K) has the Lhmford-Pettis property.
Properties of the operator T: C(K) -¥ X are reflected by properties of
the representing measure. We shall pursue this idea at some length. We begin
with compactness.
Proposition 5.27. The operator T: C(K) -¥ X is compact if and only if
the representing measure of T is a compact vector measure.

D.J upeiabuia uu i^-ny w^wv^w
Proof. Suppose that T is compact. Then so is T" and, since the
characteristic functions of Borel sets have unit norm in C(K)**, the set {p{E) : E £
3«■} = {T**xe} is relatively compact in X. Therefore (i is compact.
Conversely, suppose that fj, is a compact vector measure. By Theorem 5.18,
there exists a sequence (fi„) in M(3j() ® X that converges to /n in the semi-
variation norm. For each n, let Tn: C(K) -> X be the finite rank operator
defined by Tnf — fK f dfin. We have
l|r„/ - r/|| =
/ fd(Hn-fi)
J K
lllA*n-A*llc
and it follows that the sequence (T„) converges to T in the operator norm.
Therefore T is compact. D
We now consider the class of integral operators. We have seen in
Proposition 5.8 that the operator T: X* -¥ M(E) associated with a vector measure
ft is integral, or Pietsch integral, if and only if fj, has bounded variation. Our
next result is a variation on this theme.
Proposition 5.28. Let T: C(K) -i X be an operator with representing
measure p. The following are equivalent:
(i) T is a Pietsch integral operator.
(ii) T is an integral operator.
(iii) (i is a vector measure of bounded variation.
Furthermore, ifTts Pietsch integral, then
im|p/ = ||r|u = |Hli,
the variation norm of p.
Proof, (i) =>■ (ii) being trivial, we consider the implication (ii) => (iii).
Suppose that T is an integral operator. Then T*: X* -»• C(JfjQ* is integral, with
the same integral norm. Let J be the canonical inclusion of C{K)* into the
space M(Bjr) of Borel measures on K. Then IT* : X* -»• M(Bjr) is the
operator associated with the vector measure ft and so, by Proposition 5.8, ft has
bounded variation and
ii^iu = h/t^iu < ii2-ii/ = imi/.
To prove (iii) => (i), suppose that ft has bounded variation. Then the variation
|jlc| i is a finite, positive Borel measure on K. We shall show that T is Pietsch
integral by factoring it through the canonical injection J: C(K) -»• Li(|/n|i).
We define an operator R: Li(|/n|i) -¥ X as follows. Let / = Yl"=iatXAi be
a Borel measurable simple function, where the sets A\,...,An are mutually
disjoint. Let

112 5 The Radon-Nikodym Property
Rf= [ fdfi = ^2a^(Al).
Jk ,=i
Then
\\Rf\\ < £l«.HH^)ll < £ HMi^O = / l/l <*Mi.
t=i t=i Jk
and hence R extends uniquely to an operator of unit norm. This operator
satisfies
Rf= f fdii
Jk
for every / G C(K). Therefore T = RJ and it follows that T is Pietsch
integral, with Pietsch integral norm ||T||p/ < ||j*||i.
This proves the equivalence of (i), (ii) and (iii). Combining the inequalities
obtained in the course of the proof gives ||T||p/ = ||T||/ = H/allj. D
We note in passing that this result yields a representation of the dual of the
space C(K, X). Since this space is the injective tensor product C(K)®CX,
its dual is the space of integral operators from C{K) into X*. Therefore
C(K,X)* can be identified with the space of regular vector measures of
bounded variation on the Borel subsets of K with values in X*, this space
being endowed with the variation norm. A complete description of this
duality would require a treatment of integrals of the form JK f dp, where / takes
its values in X and the vector measure ft is X*-valued. This integral is known
as the Bartle integral.
We conclude this treatment of operators on C(K) with a characterization
of the class of nuclear operators. We shall see that the operator T: C(K) -)•
X is nuclear if and only if the representing measure of T has the Radon-
Nikodym property. This is essentially a reformulation of Theorem 5.22, which
can be interpreted as saying vector measure fj, £ 3Vt(E, X) has the Radon-
Nikodym property if and only if the associated operator T^: X* -> 3Vt(E)
is nuclear. First, we clarify the relationship between the space M(3k-) of
measures on the Borel u-algebra of K and the subspace C(K)* of regular
measures.
Lemma 5.29. Let 3^ be the a-algebra of Borel subsets of the compact space
K. Then C(K)* is complemented in M(3/f) by a projection of norm one.
Proof. Each measure ft £ M(3^-) defines a bounded linear functional ip on
C(K) by (/, ip) = fK f dfi. Let fj,' be the unique regular measure associated
with ip. Then Pfi = ft' defines a projection of M("Bk) onto C(K)*, and it is
easy to see that ||/n'|| < ||/n|| for every p. D
It follows that the nuclearity of an operator with values in C(K)*, and
the value of the nuclear norm, are unaffected if we consider the operator as
taking its values in the larger space "M(3j<-).

Proposition 5.30. Let T: C(K) -J- X be an operator with representing
measure ft. Then T is a nuclear operator if and only if ft has the Radon-
Nikodym property. Furthermore, ifT is nuclear, then \\T\\n = HmIIi-
Proof. Since C(K)* has the approximation property, the nuclearity of T
is equivalent to that of T*:X* -* C(K)*. By the lemma, this in turn is
equivalent to the nuclearity of the operator IT*: X* -J- M(3jc), where / is
the embedding of C(K)* into M(3k). Since IT* is the operator associated
with the vector measure ft, the result follows from Theorem 5.22. Finally, the
equality of norms follows from the preceding proposition. D
Corollary 5.31. Suppose that the Banach space X has the Radon-Nikodym
property. Then, for every compact space K, the spaces of integral, Pietsch
integral and nuclear operators from C(K) into X coincide, and the integral,
Pietsch integral and nuclear norms are the same.
The coincidence of the classes of integral and nuclear operators in the
presence of the Radon-Nikodym property is a fundamental fact, with far-
reaching consequences. The result we have proved above is the kernel of a
more extensive phenomenon, by virtue of the fact that all integral
operators factor through integral operators on C(K) spaces. Recall that an
operator T: Z -)• X is Pietsch integral if and only if there exists a probability
space (ft, E,/li) and operators S: Z -J- Loo(fi) and R: Li(fi) -»• X such that
T = RJS, where J: Loo(fi) -»• L1(fi) is the canonical injection, and that
||r||p, = inf ||fl||||5||. Let K be the closed unit ball of Z* with the weak*
topology. By the injectivity of the space L^p), the operator S extends to
an operator S': C(K) -»• Loo(/^) with the same norm. Let U: C(K) -)• X be
the composition RJS' and let J: Z -¥ C(K) be the canonical embedding:
R*
C(K)■— LooM -?~~ .Lifo)
Then U is a Pietsch integral operator and it is easy to see that the Pietsch
integral norm of T is given by ||T||p/ = inf ||t/||p/, where the infimum is
taken over all such factorizations T = UI. The following proposition is now
an immediate consequence of the above corollary.
Theorem 5.32. Suppose that the Banach space X has the Radon-Nikodym
property. Then, for every Banach space Z, the spaces of Pietsch integral and
nuclear operators from Z into X coincide and the Pietsch integral and nuclear
norms are the same.

114 5 The Radon-Nikodym Property
This result can be formulated in the language of tensor products:
Theorem 5.33. Let X and Y be Banach spaces such that X* has the Radon-
Nikodym property and either X* or Y* has the approximation property. Then
(X®CY)* = X*®„Y*.
Proof. Since X* is a dual space, the integral and Pietsch integral operators
from Y into X* are the same. The assumption of the approximation property
ensures that the space of nuclear operators from Y into X* can be identified
with the projective tensor product X*®„Y*. Therefore
(x®cy)* = 3(y,x*) = w(y.jr) = ■n(y,x*) = x'®,y.
D
We conclude with another result concerning the coincidence of integral
and nuclear operators, but now the Radon-Nikodym property relates to the
domain of the operator.
Theorem 5.34. Suppose that the dual of the Banach space X has both the
Radon-Nikodym property and the approximation property. Then, for every
Banach space Y, the spaces of integral and nuclear operators from X into Y
coincide and the integral and nuclear norms are the same.
Proof. Let T € 3(X,Y) C 3(X,Y**) = (X®CY*)*. Prom the preceding
proposition, we have (X®CY*)* = X*®*Y** = 74(X,Y**). Therefore T is
nuclear when considered as an operator with values in Y** and as such, its
nuclear norm is equal to the integral norm of T. Thus, if J is the canonical
embedding of Y into Y", then JT is nuclear and ||JT"||jv = ||T"||/. Now,
since X* has the approximation property, (JT)* is a nuclear operator from
Y*** into X* and its nuclear norm is the same as the nuclear norm of JT.
But the restriction of (JT)* to Y* is T*, and it follows that T* is nuclear
and ||r*||jv < ||(Jr)*||jv = \\JT\\n = ||T||,. Again using the approximation
property of X, the nuclearity of T* implies that T is also nuclear, with the
same nuclear norm. Therefore ||T||;y < ||T||/. Since the opposite inequality
holds in general, we have \\T\\N = ||r||/. D
5.4 Operators on Li(fx) Spaces
We have now seen some striking consequences of the Radon-Nikodym
property. However, we know very few spaces that have this property - apart from
the finite dimensional spaces, the only example we have is the space l1. In
this section we shall find many more examples, including all reflexive spaces
and all separable dual spaces. We do this by means of a reformulation of the
Radon-Nikodym property in terms of operators on Li(fj.) spaces.

Let 1/ be a vector measure on a u-algebra E of subsets of a set fi with
values in the Banach space X and suppose that v has bounded variation.
Then the variation \v\\ of v is a finite, positive measure on E. We have seen
in the proof of Proposition 5.28 that the integral J"n / dv can be defined for
every / G -t'i(|t/|i) and that it satisfies \\fQ f dv\\ < \\f\\i- Thus an operator
T: Li(|i/|i) -► X can be defined by
Tf= f fdu.
Jn
We now describe a property of the operator T that is equivalent to the
possession by v of the Radon-Nikodym property.
Let fibea finite, positive measure on E and let X be a Banach space. An
operator T: Li{p) -> X is representable if there exists a function g: Q, -I- X
that is /n-measurable and /n-essentially bounded, such that Tf is given by
the Bochner integral fQ fgdfi for every f € Li(fi). Of course, when X is the
scalar field, every operator is representable.
Proposition 5.35. The Banach space X has the Radon-Nikodym property
if and only if, for every finite, positive measure p, every operator from Li(fi)
into X is representable. '
Proof. Suppose that, for every finite, positive measure p, every operator
from Li(fji) into X is representable. Let v. E -J- X be a vector
measure of bounded variation. Then the operator T: Li(\v\i) -J- X given by
Tf = /n f dv is representable. Therefore there exists a \v\\-measurable, \v\\-
essentially bounded function g: Q, -»• X such that Tf = /n /</ d\v\\ for every
/ G Li(|i/|i). Then u(E) = T^e is given by the Bochner integral JEgd\v\1
for every E G E, and so i/ has the Radon-Nikodym property.
Conversely, suppose that X has the Radon-Nikodym property and let
T be an operator from Li(fi) into X. Then v(E) = T(xe) defines a vector
measure of bounded variation that is absolutely continuous with respect to
fi. Indeed, we have |i/|i(i?) < ||T||/n(i?) for every £eS. Thus there exists
g G Li(/j,,X) such that
u(E)= [ gdii
J E
for every EeE. Let us suppose for the moment that the function g is p-
essentially bounded. Then, by Bochner's Theorem, the function fg is Bochner
integrable with respect to ft for every / G Li(fi), and we may define an
operator from Li(fi) into X by / h-> J fg dp. However, this operator coincides
with T on the characteristic functions and, since these functions span a dense
subspace of L\(fi), it follows that
Tf= [ fgdfi
Jn
for every / g Lj(/lj). Therefore T is representable.

116 5 The Radon-Nikodym Property
It only remains to show that g is /n-essentially bounded. By the Pettis
Measurability Theorem, we may assume without loss of generality that X
is separable. Hence there exists a countable norming subset {<pn} of Bx--
Fixing n, we have
/ \Vng\ dfi = \>pnv\{E) < \vUE) < \\T\\»(E)
JE
for every E G E. It follows that there exists a /n-null set En such that
\>Png(f>)\ < \\T\\ for every u € ££. Then \<png(u)\ < \\T\\ for every n and
every ui outside the /n-null set \JkEk, anc^ so l.3(w)ll — ll^ll almost
everywhere. . □
Our next result indicates that the space l\ plays a central role in the
development of the Radon-Nikodym property.
Proposition 5.36 (Lewis—Stegall Theorem). An operator from Li(/j,)
into a Banach space X is representable if and only if it factors through t\.
Thus X has the Radon-Nikodym property if and only if, for every finite,
positive measure (i, every operator from L\((i) into X factors through t\.
Proof. Suppose that the operator T: L\{n) -> X factors through l\, so that
there exists operators S: Li(fi) -)• £j and R: t\ -)• X such that T = RS.
Since t\ has the Radon-Nikodym property, S is representable and it follows
easily that T is also representable.
Conversely, suppose that the operator T: L\(n) -J- X is representable, so
that there exists a /z-measurable, /z-essentially bounded function g such that
77= f fgdii
for every / £ Li(fi). We may assume without loss of generality that X is
separable and that g is bounded. By the proof of the Pettis Measurability
Theorem, there exists a sequence (/„) of /n-measurable functions, with values
in a countable subset of X, that converges uniformly to g. We may choose a
subsequence (/„,,) of this sequence such that \\g — /nJloo < 2-*-1 for every
fc. Let g\ = fni and, for fc > 1, let gk = fnk — /m,_i- Then g is the sum in
Li(fj.) of the absolutely convergent series £3*11 3* an^ so
r/ = E/ fa**?*
k=lJn
with this series converging absolutely in X for every / £ Li(fi). The values of
the functions gk lie in a countable subset of X and so, for each fc there exist
a sequence (xk})} of non-zero points in X and a sequence (Ek])} of mutually
disjoint measurable sets, such that

5.4 Operators on Li(p) Spaces 117
oo
3=1
For each / G L\{fi), let
fk3 = \\xk3\\j fdfi.
JEk,
We claim that the double sequence (fkj) is summable. Indeed,
oo oo - oo -
£i/*,i<££n**j/ i/i<fo = £/ i/Wm
oo
<n/iiiEw~-
Therefore, an operator S: Li(n) -»• ^(N2) can be defined by Sf = (fk]). If
we define the operator R: ^(N2) -¥ X by
Ra = Y^o.k]\\xk3\\~1xkj,
k,]
then T = RS. Since the space ^(N2) is isometrically isomorphic to t\, the
proof is complete. D
We note that this result can also be formulated in terms of vector
measures: a vector measure has the Radon-Nikodym property if and only if it
can be factored through an ti -valued measure of bounded variation.
The Lewis-Stegall Theorem can be used to infer structural properties of
representable operators from special properties possessed by t\. For
example, Ei has the Schur property, so that every weakly convergent sequence is
convergent in norm. Hence, we have:
Corollary 5.37. Every representable operator on Li(/j,) is completely
continuous.
We now begin to identify some classes of representable operators. We
begin with compact operators.
Proposition 5.38. Let ft be a finite, positive measure and let X be a Banach
space. Then every compact operator from Li(/j,) into X is representable.
Proof. Let T: Li(fi) -)• X be compact. Since Lco(^) has the approximation
property, T is the limit in operator norm of a sequence (T„) of finite rank
operators. Since finite rank operators are clearly representable, there exists
a sequence (gn) of /n-measurable, /n-essentially bounded functions such that
Tnf = Jr. fgndp for every / g Lj(^). Then ||9„ - gm\\oo = ||TB - rm|| for

118 5 The Radon-Nikodym Property
every n, m and it follows that (g„) converges in the essential supremum norm
to a /Li-measurable, /n-essentially bounded function g. An application of the
Dominated Convergence Theorem for Bochner integrals shows that
Tf= f fgdp
for every / G Li(fi). Therefore / is representable. D
We can extend this result to weakly compact operators with separable
range by means of the following device.
Lemma 5.39. Let K be a weakly compact subset of a separable Banach space
X. Then there is a norm, ||| • |||, on X with the following properties:
(a) lllzlH < ||a;|| for every x GX;
(b) K is compact in the normed space (X, ||| • |||);
(c) The weak topologies associated with the norms || • || and \\\ ■ \\\ agree onK.
Proof. Since X is separable, the unit sphere of X* contains a countable
norming subset, {^>n}- Let
oo
IIMI^E2""!^*)!-
n=l
It is easy to see that this defines a norm on X and that |||a;||| < ||a;|| for
every x £ X. Now the series defining |||a;||| converges uniformly on every
bounded subset of X. Therefore, if a sequence (a:*) converges weakly in X to
x, then |||a.fc — a;||| -)• 0. It follows from the Eberlein-Smulian Theorem that
the weakly compact set K is compact for the norm |||-|||. Finally, since the
|| • ||-weak topology is finer than the ||| • |||-weak topology, and K is compact
in the former and Hausdorff in the latter, it follows that these topologies
coincide on K. D
We can now establish the representability of every weakly compact
operator with separable range.
Proposition 5.40. Let p be a finite, positive measure and let X be a
separable Banach space. Then every weakly compact operator from Li(fi) into X
is representable.
Proof. Let T: Li(/n) -)• X be weakly compact and let K be the closure of
the image under T of the closed unit ball of Li(fi). Let Z be the completion of
the normed space (X, ||| • |||), where ||| • ||| is a norm on X with the properties
described in the lemma. Since X is separable, it follows from part (a) of the
lemma that Z is also separable. Let J denote the injection of X into Z. Then
JT is a compact operator and so there exists a /n-measurable, /n-essentially
bounded function g: Q, -)• Z, such that

5.4 Operators on L\ (ft) Spaces 119
JTf
= / fgdu
for every / £ L\{p). Applying this representation to the unit norm function
/ = fi(E)~1XE, where p(E) > 0, we obtain
l-E)Ladti£K
for every such E. It follows from this fact that g takes almost all its values in
the set K. Indeed, by the separability of Z, it suffices to prove that for every
open ball B = B(z,r) in Z that is disjoint from K, the set E = g~1(B) has
measure zero. Suppose this were false. Then
z-mL9dilhm\L{z~9)dil -m!^-^d^r>
a contradiction. Hence fi(E) = 0 and our assertion is proved. Therefore, by
the Pettis Measurability Theorem, g may be considered to be a /Lt-measurable
function from Q, into X. Furthermore, g is /n-essentially bounded as a function
with values in X and hence the Bochner integral fn fg dp exists in X for every
/ € Li(fi). Therefore T is representable. D
The technique we have employed here to convert a weakly compact set
into a compact set can also be adapted to deal with weak'-compact sets in
certain dual spaces. Suppose that X is the dual of a separable Banach space Y
and that X itself is also separable. Taking {yn} to be a countable dense subset
of the unit sphere of Y, we can define a norm on X by |||a;||| = X.n 2-n|a;(yn)|
as before and the statement of the lemma now applies to any weak*-compact
subset K of X, with the use of the Eberlein-Smulian Theorem replaced by
an appeal to the metrizability of the weak* topology on K. This yields the
following result.
Proposition 5.41. Let p be a finite, positive measure and let X be a
separable dual space. Then every operator from L\ (fi) into X is representable.
We now have our first example of a large class of infinite dimensional
spaces with the Radon-Nikodym property:
Corollary 5.42. Every separable dual space has the Radon-Nikodym
property.
We have seen that every weakly compact operator on L\ (fi) with values
in a separable Banach space is representable. We shall extend this result to
all Banach spaces by proving the following remarkable fact: every weakly
compact operator on Li(fi) takes its values in a separable space. We begin
with an observation about weak compactness in L\{p), where p is a finite,
positive measure. The canonical injection of Loo(fi) into Lj (p) factors through

120 5 The Radon-Nikodym Property
the canonical injection of Lp(fi) into L\(n) for every p and hence is weakly
compact. This enables us to identify an interesting class of weakly compact
sets:
Proposition 5.43. Let [i be. a finite, positive measure. Then every bounded
subset of Loo(fi) w relatively weakly compact in Li(fi).
In particular, we see that the set {xb} of characteristic functions of
measurable sets is relatively weakly compact in L\((i).
Theorem 5.44. Let p be a finite, positive measure and let X be a Banach
space. Then every weakly compact operator from Li(fi) into X takes its values
in a separable subspace of X and ts, therefore, representable.
Proof. Let T: L\(fi) -)• X be a weakly compact operator. We claim that
the set {Txb '• E £ E} is relatively compact in X. Since the characteristic
functions span a dense subspace of L1(fi) and every compact set is separable,
it then follows that T maps Li(fi) into a separable subset of X.
To establish this claim, let (En) be a sequence of measurable sets. Let Eo
be the u-algebra generated by these sets and let /no be the restriction of ft to
Eo- Since Eo is generated by a countable family of sets, the subspace I>i(i<o)
of Li(fj.) is separable. Let To denote the restriction of T to this subspace.
Then To is weakly compact and takes its values in a separable subspace of
X and so, by Proposition 5.40, T0 is representable. By Corollary 5.37, T0
is completely continuous and hence maps the relatively weakly compact set
{xe„} into a relatively compact subset of X. Thus (Txe„) is a relatively
compact sequence in X and our claim is proved. D
The fact that weakly compact operators on Li(/n) are representable has
many applications. First, since every operator with values in a reflexive space
is weakly compact, we have:
Corollary 5.45. Every reflexive Banach space has the Radon-Nikodym
property.
Next, we recall that the Banach space X has the Dunford-Pettis property
if, for every Banach space Y, every weakly compact operator from X into
Y is completely continuous. Since every representable operator is completely
continuous, we have:
Corollary 5.46. The space Li(fi) has the Dunford-Pettis property.
The next result combines the representability of weakly compact operators
with the Lewis-Stegall Theorem and the fact that the space t\ has the Radon-
Nikodym property to yield a striking and unexpected consequence.
Theorem 5.47. Let S: X -)• Y be an integral operator and T-.Y-^Za
weakly compact operator. Then the composition TS is nuclear.

5.4 Operators on ii(fi) Spaces 121
Proof. Since S is integral, there exist a finite, positive measure fj, and
operators B £ L(X, Looifi)) and A € -C(Li(/a), Y**) such that JS = AIB, where J
is the canonical embedding of Y into Y** and J is the canonical injection from
Loo(^) into Li(fi). Now T** is a weakly compact operator from Y** into Z.
Therefore, the composition T**A: Li(/j,) -J- Z is weakly compact and hence
factors through I\. These factorizations are shown in the diagram below:
L°°(f*)
MaO
U
The operator J is Pietsch integral and hence so is VI. Since t\ has the Radon-
Nikodym property, this operator is nuclear by Theorem 5.32 and hence TS =
T"JS= U(VI) Bis also nuclear. D
Next, we show that the Radon-Nikodym property is "separably
determined" .
Proposition 5.48. If every separable subspace of a Banach space X has the
Radon-Nikodym property, then X itself has the Radon-Nikodym property.
Proof. Let T be an operator from L1(fi) into X. We show that T takes its
values in a separable subspace of X by proving that the set {Txe : E £ E} is
relatively compact. Proceeding as in the proof of Theorem 5.44, we see that
every sequence (£^n) of measurable sets lies in a countably generated sub-<r-
algebra E0 of E and the restriction of T to L\{(1q) then takes its values in a
separable subspace of X. Therefore (Txe„) is a relatively compact sequence
in X. a
The space £].(/), where / is an uncountable indexing set, is an example of
a space that satisfies the hypothesis of this proposition. Since each element of
ii(I) is supported by a countable subset of /, it follows that every separable
subspace is contained in a subspace that is isometrically isomorphic to t\.
Proposition 5.49. If the Banach space X has the Radon-Nikodym property,
then so does every closed subspace of X.
Proof. Let T be an operator from Li(fi) into a closed subspace Y of X. Thus
there exists a /z-measurable, /n-essentially bounded function g, with values in
X, such that Tf = /„ fgdfi for every / G Lj(/n). Let Q: X -»• X/Y be the
quotient operator. Then
L
f(Qg)d» = QTf = 0

122 5 The Radon-Nikodym Property
for every / G Li(n). The function Qg, being /n-measurable, is essentially
separably valued and hence Qg = 0 almost everywhere. Therefore g is fi-
essentially V-valued and so T is representable. d
We finish this section with another striking application of the Radon-
Nikodym property.
Theorem 5.50. Let X be a Banach space that has the Radon-Nikodym
property and is complemented in its bidual by a projection of norm one. If X has
the approximation property, then X has the metric approximation property.
Proof. We recall that X has the metric approximation property if and only
if the canonical mapping from X'®,X into (X®eX*)* is an isometric
embedding. Now the possession by X of the approximation property implies
that the projective tensor product X*®*X can be identified with the space
"N(X, X) of nuclear operators on X. Since X has the Radon-Nikodym
property, the nuclear and Pietsch integral operators on X coincide and the nuclear
norm is equal to the Pietsch integral norm. The condition that X is
complemented in X** by a projection of unit norm implies that the spaces of Pietsch
integral and integral operators on X are the same. To summarize, we have
X*Q,X = 3(X,X).
On the other hand, the dual space (X®eX*)* can be identified with the space
3(X,X"). Therefore, with these identifications, the canonical mapping from
X*®„X into (X®CX*)* is nothing more than the canonical embedding of
3(X,X)mto3(X,X"). a
Corollary 5.51. IfX is a reflexive space, or a separable dual space, and has
the approximation property, then X has the metric approximation property.
5.5 The Principle of Local Reflexivity
In this section, we prove a fundamental fact about the bidual of a Banach
space. This result, known as the Principle of Local Reflexivity, shows that the
finite dimensional, or "local", structure of X** mimics that of X very closely.
This principle is rooted in the fact that finite dimensional spaces have the
Radon-Nikodym property. Our first proposition makes essential use of this
observation.
Proposition 5.52. Let E be a finite dimensional Banach space and let X
be any Banach space. Then (L(E,X))" = -C^X**).
Proof. Every operator from E into X is approximate and hence H(E, X) =
E* ®c X. Since E* has both the Radon-Nikodym property and the
approximation property, it follows from Theorem 5.33 that

5.5 The Principle of Local Reflexivity 123
H{E, X)* = (E* ®e X)* = E®*X*.
Therefore, we have
£(£■, xy = (e ®„ x*y = l(e, x").
a
As the dual space of £.(£", X) is the tensor product E ®c X*, the
action of T £ L(E,X") as an element of the bidual L(E,X)" is completely
determined by the equation
(x®<p,T) = (<p,Tx),
where x £ E and tp £ X*.
We can say a little more when the finite dimensional space E is contained
in X**. First, we introduce some notation. If E is a subspace of X**, we
shall denote by £.o(iS, X), the closed subspace of £.(£", X) consisting of those
operators that act as the identity on E fl X. In other words, T belongs to
L0(E,X) ifTx = x for every x£ EDX. Similarly, L0(E,X**) will denote
the subspace of L(E,X") consisting of the operators acting as the identity
on £ fl X.
Proposition 5.53. Let X be a Banach space and let E be a finite
dimensional subspace ofX*\ Then (L0(E,X)y* = L0(E,X**).
Proof. We wish to show that (L0(E,X)y is the subspace £,.(£.-X"**)
of L(E,X"). Suppose first that the operator T: E -)• X" belongs to
(L0(E,X)y*. Then, by Goldstine's Theorem, there is a bounded net, (Ta),
in &o(E, X) that converges in the weak* topology to T. Thus
(Tax, <p) = (Ta, x®<p)^(x®<p,T) = (tp, Tx)
for every x £ E and ip £ X*. Since Tax = x for every'--! £ E D X and every
a, it follows that Tx = x for every x£EC\X. Therefore T £ L0(E, X**).
Conversely, suppose T £ &0(E,X"). Let F be a finite dimensional sub-
space of E such that E is the direct sum of E fl X and F. Choosing a
basis {/i..... /m} for F, we can write each x £ E as « + "£yli x} f3, where
u £ EDX. Therefore Tx = u+YlT=i xi^fi- ^e^ (zi<*)a. • • ■. {zma)a be nets in
X that converge in the weak* topology of X** to Tfo,..., Tfm respectively.
Consider the net of operators (Ta), where
m
Then each Ta belongs to &0(E,X) and it is easy to see that the net (Ta) is
weak* convergent to T in L(E,X)". Therefore T £ (L0(E, X))**. □

124 5 The Radon-Nikodym Property
In the application of this proposition, we shall make use of Helly's Lemma,
in the following form: let Z be a Banach space, z** an element of Z**, M a
finite dimensional subspace of Z* and e > 0. Then there exists z € Z with
11*11 < ..***..+£. such that (•s.P) = (PiO for every V e M.
Theorem 5.54 (The Principle of Local Reflexivity). Let X be a
Banach space and let E and F be finite dimensional subspaces of X** and X*
respectively. Then, for every e > 0, there exists an operator T: E -»• X with
the following properties:
(a) (1 -6)11***11 < ||r.r"|| < (1 + e)||a.**|| for every x" <E E.
(b) (Tx**,<p) = (<p,x") for every x" eE,tp€F.
(c) Tx = x for every x G E fl X.
Proof. Since E is finite dimensional, we may assume, by enlarging the sub-
space F if necessary, that for every a:** £ E there exists <px~ in the unit
sphere of F such that
(l-e)||***|| <<¥>,...***)■
We now apply Helly's lemma to the bidual of the Banach space Z = SL>q{E, X),
namely £0(£;, X**). Let z" be the embedding of E into X** and M the finite
dimensional subspace of L{E, X)* spanned by the functionals a:** ® tp, as a:**
and ip range over E and F respectively. Therefore, there exists an operator
T in L0(E,X) such that ||T"|| < 1 + e/2 and
(Tx",tp) = (tp, x") for every a:" G E, <p £ F, (5.5)
and so T satisfies (b). Applying (5.5) with ip = <px», we obtain
(i-e)||***||<||r*»||
for every a:** £ E. Therefore, T satisfies (a). Finally, (c) is simply the
definition of L0(E,X). a
The Principle of Local Reflexivity has fundamental consequences and we
shall make extensive use of it. Indeed, many of the results in the preceding
chapters that relate to the bidual take on a new meaning in the light of this
result. For now, we illustrate its power with a couple of applications.
We have already encountered the classes of -Ci -spaces and /Coo-spaces. It
is a consequence of the principle of local reflexivity that X is an Lx-space
(respectively, an Loo-space) if and only if the bidual X" is an Li-space
(respectively, an Loo-space). We leave the details as an exercise for the reader.
For our second application, we prove a Schauder Theorem for
approximate operators.
Proposition 5.55. Let X and Y be Banach spaces. An operator T: X -»• Y
is approximable if and only if the adjoint operator T* is approximate.

5.6 Exercises 125
Proof. If T is approximable, it is easy to see that T* is approximable.
Suppose that T* is approximable. Let J denote the canonical embedding of Y
into Y". Then JT: X -¥ Y**, being the restriction of the approximable
operator T" to X, is approximable. Let e > 0. Then there exists a finite
rank operator S: X -► Y" such that ||JT-S|| < e. Now T is compact,
since its adjoint is, and so we may choose an e-net, {Txi,.. .,Txn} for the
relatively compact subset TBx of Y. Let E be the finite dimensional sub-
space of V* spanned by {Txu ... ,Txn} U S(X). By the Principle of Local
Reflexivity, there exists an operator R: E -¥ Y such that ||iZ|| < 1 + e and
T is the identity on EDY. We claim that the finite rank operator RS from
X into Y approximates T. To see this, fix x £ Bx and choose xt so that
||ra; - Tar, || < e. Since Txt € E D Y, we have RTx, = Tx, and hence
Tx - RSx = (Tx - Tx,) + (RTx, - RTx) + (RTx - RSx).
It follows that ||r - RS\\ < (3 + 2e)e. Therefore T is approximable. a
5.6 Exercises «
Exercise 5.1. Show that the space of vector measures of bounded variation
is complete in the variation norm.
Exercise 5.2. Show that the semivariation of a vector measure /iona set
E is given by
Ia*Io
(£^) = sup< ,^alfi(Al) : \a,\ = 1,{j4i,. ..,-4n} a partition of E\ .
Exercise 5.3. Show that the semivariation of a vector measure is countably
additive if and only if it coincides with the variation.
Exercise 5.4. Let E be the u-algebra of Lebesgue measurable subsets of
[0,1] and let fi: E -)• Lp[0,1] be the vector measure given by fi(E) = xe,
where 1 < p < co. Show that ft has bounded variation if and only if p = 1.
Exercise 5.5. Find an example of a function ft: E -J- X* that is weak*-
countably additive, but is not a vector measure (weak*-countable additivity
means that the scalar function E i-> (x,p(E)) is a measure for every x G X).
Exercise 5.6. Let p: E -»• X be a vector measure and T: X -»• Y an
operator. Show that Tfi is a vector measure that has bounded variation if (i has
bounded variation, with ||7yi||i < ||r||Hji||i <""! ||r^||oo < ||T||||ji||oo-
Exercise 5.7. Show that the space of measures M(E) has the metric
approximation property.

126 5 The Radon-Nikodym Property
Exercise 5.8. Show that M(E) is an -Ci.A-space for every A > 1.
Exercise 5.9. Show that a vector measure ft: E -)• X has the Radon-
Nikodym property if and only if the associated operator T^: X* -»• M(E)
is nuclear.
Exercise 5.10. Show that the £i-sum (X.n-X„.)i of a sequence of Banach
spaces with the Radon-Nikodym property also has this property. (The space
(]T) Xn)i consists of all sequences x = (a.n). where xn £ Xn for each n and
W--=E»KII<oo.)
Exercise 5.11. Show that the space C(K)** is injective. (Hint: show that
projective tensor products with C(K)* respect subspaces.)
Exercise 5.12. Show that the operator T: Li[0,1] -»• C[0,1] defined by
Tf(t)= f f(s)ds
Jo
is not representable.
Exercise 5.13. Find an operator from Li[0,1] into cq that is not
representable.
Exercise 5.14. Let 5: X -»• Y be a weakly compact operator and T: Y -»•
Z an integral operator and suppose that X* has the approximation property.
Show that the composition TS is nuclear.
Exercise 5.15. Show that the composition of two integral operators is a
nuclear operator.
Exercise 5.16. Use the principle of local reflexivity to show that X ®„ Y is
a subspace of X** ®„ Y** for every pair of Banach spaces X, Y.

6. The Chevet-Saphar Tensor Products
In this chapter we begin a general study of tensor norms. Taking the
projective and injective norms as our models, we formulate the appropriate
definition of a tensor norm. We then investigate the Chevet-Saphar tensor norms,
gp and dp. The dual spaces of the corresponding tensor products lead us to
the definition of the p-summing operators. We conclude with the fundamental
Grothendieck Inequality and some of its applications.
*
6.1 Tensor Norms
What properties should a tensor product norm possess? In this section, we
will attempt to provide a satisfactory answer to this question, taking the
projective and injective norms as models.
Let X and Y be Banach spaces. We say that a norm, a, on X ® Y is a
reasonable crossnorm if it has the following properties:
1. a(x® y) < ||a"||||y|| for every x £ X and y £ Y.
2. For every ip £ X* and rf> e Y*, the linear functional (p ® r/> on X ® Y is
bounded, and ||p®Vll < IMIIMI-
Of course, the projective and injective norms satisfy these conditions. Our
next result shows that these norms lie at opposite e'nds of the spectrum of
reasonable crossnorms.
Proposition 6.1. Let X and Y be Banach spaces.
(a) A norm a on X ®Y is a reasonable crossnorm if and only if
e(«) < a(u) < 7r(«)
for every u £ X ® Y.
(b) If a is a reasonable crossnorm on X®Y then a(x®y) = ||a:||||y|| for every
x £ X and y £Y. Furthermore, for every ip £ X* and ip £ Y*, the norm
of the linear functional ip®ip on (X®Y,a) satisfies \\<p ® V>|| = IMHMI-
Proof, (a) Suppose that a is a reasonable crossnorm on X ® Y. Then, for
every representation "£?=i Xj®y3 of u £ X ® Y,

128 6 The Chevet-Saphar Tensor Products
n n
a(u) < Yla(xi ®y,)<Yl \M\hA\
3=1 3=1
and it follows that a(«) < 7r(«). Also, since \\p ® rj>\\ < 1 whenever ip G B^.,
V £ By , we have
e(«) = sup{|(u,ip® V>) : V £ Bx~,il> £ By.}
< sup{|(«,«) : v e (X® Y,a)\ \\v\\ < 1} = a(u).
Conversely, suppose that a is a norm on X ® Y that lies between the
injective and projective norms. Then a(x® y) < ir(x ® y) = ||a"||||y|| for every
x G X and y G Y. Furthermore, since a(u) < 1 implies e(«) < 1,
||p ® VII < Bup{|(u,<p ® V)| : e(u) < 1} = j|p|||MI
for every <p G X* and ip eY*. Therefore a is a reasonable crossnorm.
(b) If a is a reasonable crossnorm on X ® Y, then a(x ® y) = ||a"||||y||
follows immediately from the fact that e(x ® y) < a(x ® y) < w(x ® y). Let
ip G X* and ip G Y*. Then, using the fact that n > a,
||p® V|| > sup{|(u,p®^)| : tt(«) < 1} = \\<p\\U\\
and it follows that \\tp ® Vll = IMIIMI- a
We have seen that the injective and projective tensor products behave well
with respect to the formation of tensor product of operators: if S: X —)• W
and T: Y -¥ Z are operators, then the operator S ®T: X ®Y ^ W ® Z is
bounded and ||5 ® T|| < ||S||||T||. This is clearly a desirable attribute and so
we add it to our requirements for a tensor product norm. A uniform cross-
norm is an assignment to each pair, X, Y of Banach spaces of a reasonable
crossnorm on X ® Y for which the above property holds.
If a is a uniform crossnorm and v. € X®Y, we shall denote the a-norm of
« by a(u;X®Y) or a^^u), or simply a(u) if there is no risk of ambiguity.
The tensor product X ® Y with this norm will be denoted by X ®a Y and
its completion by X®aY. If S: X -)• W and T: Y -»• Z are operators, then
the operator S ®T: X ®aY ^ W ®a Z satisfies \\S®T\\ < ||S||||r||. It has
a unique extension to an operator from X®aY into W®aZ with the same
norm. We shall denote this operator by S ®a T.
If a and @ are uniform crossnorms, we shall interpret relations such as
a < P to mean that a(u; X ® Y) < p(u; X ® Y) for every pair of Banach
spaces X, Y and every u £ X ® Y.
We make some elementary observations about uniform crossnorms. First,
we consider the norm induced on the tensor product of subspaces. Let E,
F be subspaces of X, Y respectively. Then the tensor product E ® F is an
algebraic subspace of X ® Y. Applying the uniform property to the tensor
product of the inclusion operators E -)• X and F -* Y, we see that

6.1 Tensor Norms 129
ax,y(«) < aB,F(u) (6-1)
for every u £ E ® F- Prom our experience of the projective norm, we know
that this inequality can be strict. Thus E ®a F need not be a subspace
of X ®a Y. On the other hand, we have seen that the injective norm does
respect subspaces. We say that a uniform crossnorm is injective if it shares
this property. Thus, a is injective if, whenever E, F are subspaces of X,
Y respectively, then the norm induced on E ® F by the norm of X ®a Y
coincides with the norm of E ®a F.
Next, we consider quotient spaces. Let Q: X -J- W and R: Y -J- Z be
quotient operators. Then, by the uniform property, the tensor product
operator Q®aR: X®aY -)• W®aZ has unit norm. We shall say that the uniform
crossnorm a is projective if this operator is always a quotient operator. We
have seen that the projective norm has this property: if W, Z are quotients
of X, Y respectively, then W ®„ Z is a quotient of X ®„ Y.
There is one property of the injective and projective norms that we do
not generalize. These norms are symmetric - interchanging the factor spaces
does not alter the norm. If a is a uniform crossnorm, the transpose of a is
the uniform crossnorm a1 defined as follows: for each pair of Banach spaces
X, Y and each « e X ® Y,
at(u;X®Y) = a(ut;Y®X),
where the transpose, «', of « = "£"=i x3 ® y3 is given by «' = X."=i y3 ® x}.
We say that a is symmetric if a1 = a. We shall encounter many examples of
non-symmetric norms.
There is just one more ingredient needed for a satisfactory theory of tensor
product norms. We recall that, although the projective norm is not injective,
the projective norm of a tensor « s X ® Y is determined by the values of the
projective norms of u in all the finite dimensional tensor products M ® N
that contain «. We have made essential use of this fact in our discussion
of -Ci-spaces and Loo-spaces in previous chapters. And the Principle of
Local Reflexivity demonstrates in a dramatic way the importance of the finite
dimensional structure of a Banach space.
We say that a uniform crossnorm a is finitely generated if, for every pair
X, Y of Banach spaces and every u £ X ®Y, we have
a(u;X ® Y) = inf{a(u; M ® N) :ueM®N, dimM, dimJV < co}.
(6.2)
Thus, the behaviour of a is completely determined by its values on tensor
products of finite dimensional spaces. Occasionally, it will be useful to say
that a uniform crossnorm is finitely generated on a certain class of Banach
spaces (or even on a single pair of spaces) if (6.2) holds for every pair of
spaces X, Y belonging to the given class.

130 6 The Chevet-Saphar Tensor Products
The reader may find it difficult to find an example of a uniform crossnorm
that is not finitely generated. In the light of our next result, this is not at all
surprising.
Proposition 6.2. Let X and Y be Banach spaces with the metric
approximation property. Then every uniform crossnorm is finitely generated on
X®Y.
Proof. Let a be a uniform crossnorm. It follows from (6.1) that
a(u;X®Y) < inf {a(u; M ® N) :u£M®N, dimM, dhaN < co}
for every u € X ® Y. To prove the reverse inequality, let u £ X ® Y and let
e > 0. Fix a representation "£" j x3 ®y3 of «. Using the metric approximation
property, there exist finite rank operators S, T on X, Y respectively, such
that ||S||, ||r|| < 1 and \\Sxj - x,\\, \\Ty3 -y,\\< e' for 1 < j < n, where e'
is chosed so that (X." j II3"., II + l|j'_.ll)e' < e- Now choose finite dimensional
subspaces M, N of X, Y respectively so that M contains {a-i,..., xn} and
the range of S, while N contains {yi,..., yn} and the range of T. Then both
« and S ® T(«) belong to M ® N and
ajf,»(« - S ® T(u)) = aMM C%2{x, ®y3- (Sx3) ® (Ty3)U
< f^Wx, - Sx,\\\\v,\\ + \\Sx,\\\\Vi ~ TVA\) < e.
If we consider S, T as operators from X, Y into M, N respectively, then, by
the uniform property of a, we have
aMMS9T(u)) < ||5||||r||ox,y(«) < ax,Y(u).
Putting these inequalities together, we obtain
«*.,«(«) < a*f,w(« - 5 ® T(u)) + aM,w(S ® r(«)) < ax,Y(u) + e.
We have shown that, for every e > 0 there exist finite dimensional subspaces
M, N of X, Y respectively, such that ctM^(u) < ax,Y(u) + e. Therefore
ax,y(«) = inf a.jvf,.v(u) and the proof is complete. O
We define a tensor norm to be a finitely generated uniform crossnorm. At
this point, we have only two examples of tensor norms - the projective and
the injective norm. We note in passing that the property of finite generation
is trivial for e and indeed for every injective uniform crossnorm.

6.1 Tensor Norms 131
Our next result gives two simple applications of finite generation. The first
illustrates how finite dimensional information can be used to infer general
facts and the second shows how to construct a tensor norm from a uniform
crossnorm on finite dimensional spaces- The proofs are straightforward and
are left to the reader.
Proposition 6.3.
(a) Let a, @ be tensor norms. Then a < P if and only if <xe,f < Pe,f for
every pair E, F of finite dimensional normed spaces.
(b) Let 7 be a uniform crossnorm defined on the class of finite dimensional
normed spaces. Then the formula
7(«; X ® Y) = inf {y(u; M ® N) : u £ M ® N, dim M, dim N < co}
defines a tensor norm that coincides with 7 on tensor products of finite
dimensional normed spaces.
There is an important special case in which every tensor norm behaves in
an injective fashion: t
Proposition 6.4. Let a be a tensor norm and let X, Y be Banach spaces.
Then X ®aY is a subspace of X** ®a Y**.
Proof. By the uniform property of a, we have ax",Y" < ax,Y on X®Y. To
prove the reverse inequality, let u £ X®Y and let e > 0. Fix a representation
X." j xi ® Vi °f v- Since a is finitely generated, there exist finite dimensional
subspaces M, N of X**, Y** respectively such that
ajK,w(ii) < ax",y(«) + £■
Now the value of a.M,.v(u) does not increase if M and N are enlarged. Hence
we may assume that {a-i,..., xn} C M and {yi,..., yn} C N. By the
Principle of Local Reflexivity, there exist finite dimensional'siibspaces E, F of X,
Y respectively and isomorphisms S: M -¥ E and T: N -»• F, each of norm
at most 1 -I- e, such that Sx} = x3 and Ty} = jy for each j. It follows that
S ® T(u) = u and hence
<*x,y («) < ob,f(ii) = oe,f(S ® T(u))
< (1 + e)2aM>N(u) < (1 + e)2ax»,Y"(«) + e(l + e)2 ■
Since e > 0 is arbitrary, we have a.x,y(u) < ox",y(u)- O
We conclude this section with some remarks about the dual of a tensor
product. Let a be a reasonable crossnorm on X ® Y. Since a is dominated
by the projective norm, every bounded linear functional on the Banach space
X®aY is given by a unique bounded bilinear form on X x Y. On the other
hand, the injective norm is the smallest reasonable crossnorm, and so every

132 6 The Chevet-Saphar Tensor Products
integral bilinear form on X x Y is also bounded for the norm a. Thus, the
dual space of X®aY is a space of bilinear forms that lies somewhere between
these extremes:
•B/(X x Y) C {X®aYY C £(X x Y).
Of course, the elements of this dual space may also be viewed as operators,
either from X into Y* or from Y into X*.
We recall from Chapter 2 that every bounded bilinear form onXxV
has an extension to a bounded bilinear form on X** x Y** with the same
norm. In fact, there are two "natural" ways to construct this extension. If
B £ "B(X x Y) corresponds to the operator T : X -)• Y*, so that B(x,y) =
(y,Tx), we may define
•B(x",y") = (T*y",x").
We shall refer to this as the canonical left extension of B. On the other hand,
we have B(x,y) = (x,Sy), where S G L(Y,X*). Thus we may define
B*(x",y") = (S-x",y")-
We call B* the canonical right extension of B.
Theorem 6.5. Let a be a tensor norm, let X and Y be Banach spaces and
let B € (X®aY)*. Then the canonical left and right extensions of B are
bounded linear functional^ on X**®aY** with the same norm as B.
Proof. We give the proof for the canonical left extension. Let T: X -> Y* be
the operator associated to B. Let u G X** ® Y" and choose a representation
EJLi XT ® v" for u-Then <U>*B> = EJUCfy;*.*;*}-We aPPlv the Prm-
ciple of Local Reflexivity to X**; take Mi and Ni to be the subspaces of X**
and X* spanned by {a;**,.. .,a;**} and {x\,... ,£*} respectively and take
e > 0. There exists an subspace Ei of X and an isomorphism R: Mj -J- Ei
of norm at most 1 + e such that (Rx**,x*) = (a;*, a;**) for every a;** G Mi
and x* £ Ni. Therefore,
j=i j=i y=i
We apply the Principle of Local Reflexivity now to Y**, taking M% and N2
to be the spaces spanned by the yf* and y* respectively. This yields an
isomorphism, 5, from M2 into a subspace E2 of Y of norm at most 1 + e,
and we have
JT(TRx?,y;-) = JT(sy;;TRx;n = /£ art* ® sy;; b) .

6.2 The Chevet-Saphar Tensor Norms 133
Now, by the uniform property of a, the tensor product operator R®S: M\®a
M2 -"» Ei ®a E2 has norm at most (1 + e)2. Thus,
k«, *b)\ < ax,Y ( e **r ® «»;*) iisii ^ «*.* (E •**" ® ■%**) 11*11
<(i+e)2a*#1,j1#1(,E*r®i';*)ii-Bii-
Since a is finitely generated, we may choose Mi and M2 so that
'..«.(E*r®w;*)
<*M,
is arbitrarily close to aX" y(Z]?=i *" ® Vj*) an^ '* follows that ||*B|| =
IIBII- ' Q
6.2 The Chevet-Saphar Tensor Norms
We now construct some new tensor norms. Let us begin with another look at
the definition of the injective norm on the tensor product X ® Y. We recall
that this norm may be defined by embedding the tensor product in the space
of operators H(X*,Y). Thus, if X."=1 Xj ® Vj is any representation of the
tensor u £ X ® Y, then
e(«) = sup I E f(xj) Vj '■ f e *** \ ■
Since e is the smallest tensor norm, we look for a norm that dominates
this expression. If we fix p G (l.oo) with conjugate index p', then applying
Holder's Inequality yields
(« \ i/p' / n \ '
Ei^-)ip (Eiiwii")
y=i ' V=i '
i/p' /_tl \ ]/p
e(«) < sup
A brief calculation shows that this expression is dominated by the projective
norm, 7r(«), and so we are on the right track. However, the formula above
now depends on the representation chosen for «. Borrowing an idea from the
definition of the projective norm, we arrive at the following candidate for a
tensor norm:
, n » i/p' , n \ Vp 1
jmp ^Ei^)ipj (Ew) }- (6-3)
{/ n x 1/p' / n_ \ 1/P '
ip£Bx.

134 6 The Chevet-Saphar Tensor Products
the infimnm being taken over the set of all representations of the tensor «.
This formula is rather cumbersome and it will be helpful if we pause to
organize our notation.
For 1 < p < oo, we shall denote by EP(X) the space of all p-summable
sequences in the Banach space X, that is, sequences (a;n) with the property
that the series "£nLi lla"»»llp converges. A norm is defined on this space by
IIWIIp = (f>n||P)
Up
The obvious modification is made in the case p = oo, so that ^(X) is simply
the space of bounded sequences in X with the norm IK-c^Hoo = supn ||a;n||.
We leave it to the reader to verify that lp(X) is a Banach space.
There are also weak versions of these spaces. We denote by £jt{X) the
space of weakly p-summable sequences in X, that is, sequences (a;n) with
the property that the scalar sequence (<pxn) belongs to tp for every <p £ X*.
An application of the Closed Graph Theorem shows that we may associate
with each weakly p-summable sequence (a;n) an operator S: X* -» £p, where
Sip = (<pxn). Now the adjoint operator sends the unit vectors en to the points
xn. It follows that 5* maps tp> into X when 1 < p < oo and when p = 1, S*
maps Co into X. We define a norm on the space ££{X) by taking the norm
of the operator S or the norm of its adjoint:
i/p
11(^)11- = Bupj(f; 1^1") P:^Bx-j
= supi ]T A„a;n : A £ Btp, \ when 1
(6.4)
< p < oo,
and
ii(*oiir
= sup] ^Ana:n
n=l
: AG Bc
■}
when p = 1.
Be,
In applying these formulas, we may replace the closed unit balls Bx"
or Beg by norming sets.
A moment's thought reveals that i^[X) may be identified with the space
£.(£p<, X) when 1 < p < oo, or with £(co, X) when p = 1. It follows that these
sequence spaces are complete. Since the weakly bounded and norm bounded
sequences are the same, the space F^,{X) coincides with i^X) and the weak
and strong norms are the same in this case.
We return now to our putative tensor norm. We may write (6.3)
using the compact notation developed above, with the understanding that

6.2 The Chevet-S*pli»r Tensor Norms 135
every finite sequence (xlt... ,xn) may be interpreted as the infinite
sequence (xi,...,xn,0,0,...) when it appears in a formula for an £p-norm.
The Chevet-Saphar norms are defined for 1 < p < oo as follows:
dp(«) = inf \\\(x,)\\2 ||(y,)||p : u = f>, ® y\ , (6.5)
taking the infimnm over all representations of u 6 X&Y. We may interchange
the roles of the weak and strong norms to define a parallel sequence of norms:
gp(u) = inf {||(*,)||p ||(y,)||J? : « = £>; ® y\ . (6.6)
It may be helpful to the reader to know that the notation for these norms
comes from the French words "gauche" and "droite*. Clearly, we have
for every p. It follows that every property of dp can be interpreted as a
property of gp and vice versa.
We shall find the following variations on (6.5) useful:
dp(«) = inf {||(A,)||P IK*,)!!? ||(y,)||oo :« = £>,*, ® y,} .
1 i=i > (6.7)
Of course, there is a similar variation on the formula defining gp. To prove
(6.7), let us denote the expression given there by 5p(v) for the moment. We
note first that every representation of u of the form Xl?=i ^ixi ® Vj can
be written as £?=i ^ ® (A,y,) and hence dp{u) < ||(a-,)||p/||(A,-y,)||p <
ll(Ay)||p \\[x,)\\% ||(yi)||oo, from which it follows that dp(u) < 5p{u). On the
other hand, let "£?=1 Xj ® y, be a representation qf tt. We can write « as
X,"=1 XjXj ® zj, where A,- = ||y,|| and ||z,-|| < 1 for.every j. Then 5p(u) <
ll(Vi)llp 11(^)11? and hence Jp(«) < dp{u).
Proposition 6.6. Lei 1 < p,q < co.
(a) dp and gp are tensor norms.
(b) Ifp<Q then dp>dq and gp > gq.
(c) d, = 3, = tt.
Proof, (a) We show first that dp is a reasonable crossnorm on every tensor
product X ® Y. We deal with the case 1 < p < co and we leave it to the
reader to make the appropriate small modifications in the other cases. Let
«i, «2 6 X ® Y and let e > 0. Choose representations «, = X." , xt] ® y,,
such that ||(.r„),||™. < (dp(«,) + £)>/p and ||(y„-);||p < (dp(«,) + e),/p' for
i = 1, 2. Then J") a:,,- ® y,,- is a representation of «x + «2 with

136 6 The Chevet-Saphar Tensor Products
ll(«tf)ll? Il(v<i)llp < (<W«i) + <*pM + 2e),/p (dp(«0 + dp(«2) + 2£),/p'
= dp(«i) + dp(«2)-l-2e
and it follows that dp(«i 4- u2) < rfp(«i) + dp(u2)- It is easy to see that
dp(Au) = |A|dp(«) for every A G Kand u € X®Y. Therefore dp is a seminoma.
But it is clear that e(«) < dp(u) < t(u) for every «, and hence dp is a
reasonable crossnorm.
The uniform property for dp follows easily from the definition. Finally,
the fact that the weak and strong lp-norms are unchanged if the range space
is enlarged shows that dp is finitely generated. Therefore dp is a tensor norm.
(b) Suppose that 1 < p < q < co. Let « G X ® Y and let e > 0.
It will suffice to show that d,(«) < dp(u) + e. By (6.7), we may choose a
representation of « of the form X.?=i \}Xj ® yj such that
ll(Ai)||p ll(xy)!|-, IK^JHco < dp(«) + e.
We rewrite this representation as
n.
so that
*,(«) < ll(A™)||, IKa;-^*,)!!? IKw^Hco • (6-8)
Consider the weak ^>-norm
ii(a;-p/^)h»,= sup (jz\^i-v,qW\v3A
!/«'
Fixing y>, we apply Holder's Inequality with the conjugate exponents p'/(p' —
q1) and p'/V. to get
/ n \ "■/«' / « \ >/p-i/« / « \ i/p'
Taking the supremum over tp £ Bx' and using (6.8), we have
<*»(«) < ll(A,)||p||(*,)||? ||{»,)||oo < dp(u) +e
and we are done. We leave it to the reader to dispose of the case q = co. The
case p = 1 will follow from (c).
(c)If X."=i xi ® W is a representation of «, then 7r(«) < "£"=i Ikjllll^ll <
IK^jJIloo ||(i,y)||i- Therefore 7r(«) < di(«). The reverse inequality follows from
(a). D

6.2 The Chevet-Saphar Tensor Norms 137
We say that a tensor norm a is right projective if, for every Banach space
X and every quotient operator Q: Z —> y, the tensor product operator
I®a Q- X®aZ -» X®ay is a quotient operator. There is a similar definition
for left projective tensor norms.
Proposition 6.7. The tensor norm dp is right projective for every p.
Proof. Let Q: Z -» Y be a quotient operator. We show that the operator
I ® Q: X ®a Z -» X®ay isa quotient operator for every X. We have
||/ ® Q\\ = 1 and so it suffices to prove that I ® Q maps the open unit
ball of X ®a Z onto the open unit ball of X ®a Y. Let u £ X ®a Y with
dp(«) < 1. Choose e > 0 so that (1 + e)dp(u) < 1. Now let YT,=\ xi ® Vi be
a representation of u such that 11(^)11^ ||(yj)||p < 1- Since Q is a quotient
operator, we may choose z3 G Z such that Qz3 = y,- and ||zj|| < (1 + e)||yj||
for every j. Let v = J")" x, ® z3 £ X ® Z. Then J ® Q(y) = u and dp(y) <
(l+e)ll(«,)ll?ll(»,)llp<l. - D
Of course, there is a corresponding result for the norms gp - these norms
are all left projective. When p = 1, then di = gi = w are both left and right
projective.
Bearing in mind the coincidence of the projective norm on 11 ® X with
the norm of li{X), it is natural to ask if there is an analogous description of
the Chevet-Saphar norms on lv ® X. However, when p > 1, there are two
norms, dp and gpt in play and the outcome is not quite as pleasing:
Example 6.8. The Chevet-Saphar norms gp and dp on Ep® X.
Let 1 < p < co. We may embed lp ® X algebraically as a subspace of EP[X)
in the usual way. If « = "£"=i a3 ® xj € Ep ® X, where a3- = (ajm) m> we may
write « as X.m=i ero ® uro, where «ro = X.?=i ajmXj. Thus, « is identified
with the element («ro) of tp{X). Now the unit basis vectors ero in lp satisfy
||(ero)||», = landso
<*p(«)<IHp- ''
On the other hand, with the help of Holder's Inequality, we have
(°o I, n ||P\1//p ( °° ( l*^> I\P\1//P
EEaJ»*J =(^( SUP Z30jm1p(*,) ) I
m=llly=, II / \ro=1 VeBx.|J = , \J J
(oo ,, n sl/p ,n \l/p'\P\l/P
E((EKJ') ,-(X>'*>>0 ))
= EEivr sup (x;w«,)ip
= (Ell(aJm)ror) " sup ff^l^*,)!"') " =HK)IIpII(*,)II£-

138 6 The Chevet-Saphar Tensor Products
This holds for every representation of u and so, by the definition of gp,
IMIp < pp(«) •
Putting these results together, we see that
«W«) < IHIp < ft>(«) (6-9)
for every « G ip ® X. In general, this is the best that can be said about these
norms. We refer to the exercises for some related results.
We turn our attention now to deriving a description of the elements of
the completed tensor product X&j Y similar to that proved in Chapter 2
for the projective tensor product.
Lemma 6.9. Let X and Y be Banach spaces.
(a) Let 1 < p < co. If (a;n) G (%(X) and (yn) G lP[Y), then the series
E~=i xn ® Vn. converges in X®dpY.
(b) If (xn) G ii"{X) and the sequence (yn) converges to zero in Y, then the
series Y^=\ xn ® Vn converges in X&a^Y.
Proof. We prove (a); the proof of (b) is virtually the same and is left to
the reader. Let us denote by (a^)™ the finite sequence (a;n,... ,a;ro), where
n< m. We have
4p(£>i®y,) <^((*XKp((j-,)n) <#((*,))(£>x)
»/p
for n < m. It follows that the series YlTLi xi ® Vi satisfies the Cauchy
condition. D
We show now that every element of X®d Y is the sum of a series of this
form.
Proposition 6.10. Let 1 < p < co and let X, Y be Banach spaces. If u £
X®drY ande > 0, then there exist sequences (xn) G £%,{X) and (yn) G lp{Y)
(or co{Y) when p = coj such that the series Xl^Li xn ® Vn converges to «
and
^(«)<ll(^)||?||(yn)||p<dp(«) + e.
Proof. The result has already been established for the projective norm
(Proposition 2.8) and so we may assume that 1 < p < co. Let r\ > 0.
Since X ® Y is a dense subspace of X®d Y, we may apply a standard
argument to obtain a sequence (ttm) in X ® Y such that « = X.m=i «ro with
dp(«i) < dp(u) +1) and dp(um) < j]2/4tm when m > 2.
We choose a representation of «i of the form "£ ' i X\3 ® y\} such that

6.2 The Chevet-Saphar Tensor Norms 139
\K*i]),\\2<dp{») + v and ll(yw)yllp<i-
For each m > 2, choose a representation of «ro of the form Ylj=i xmj ® Vmj
such that
ll(«mi),||? < >7/2ro and Hfo*,)^ < 7?/2ro .
Concatenating the finite sequences (xmj)j and [ym])j, we obtain infinite
sequences (a;n) and (y„). Thus, there is a strictly increasing sequence of
positive integers 1 = Ii < fc < ..., so that the blocks in the sequences (a;n)
and (yn) corresponding to lm < n < Jro+1 coincide with the finite sequences
{xmj)j and (ym])j respectively.
We claim that the series Xl^Li xn®Vn converges to u and has the required
properties. We note first that
, oo fcm . l/p . oo >. l/p
ii(»-)iip = (EEw) < I1+•»"E2_pro) •
Sn=l ]=\ ' ^ m=2 '
«
for 1 < p < oo. When p = oo, we have [yn) £ co{Y) and we may assume that
r\ is small enough so that sup||yn|| < 1.
To compute the weak £p.-norm of (a;n), fix <p € Bx-- Then
/ °° \l/p' / oo fcm .l/p' / °° / \P'\l/p'
(Ei^ip = (EEi^n ^(EiiK^Aiip') ) ■
\i=l ' \n=l j=\ ' V'm=l V ' '
Taking the supremum over ip, we obtain
noon? < ((dp(«)+7,)"'+/ f; 2-"'-*)
\ m=2 /
I/P'
It follows from the Lemma that the series Y^=\xn ® Vn converges in
X®dTY and it is clear that its sum is u. It is easy to see that dp(u) <
\\(xn)\\p' ll(yn)||p- Finally, using the estimates we have obtained for ||(aOI$
and ll(yn)llP. if V is sufficiently small, then \\{xn)\\% ||(y„)||p < dp(«) + £■ D
We now consider how the Chevet-Saphar tensor products can be
interpreted as spaces of operators. We recall that the elements of the projective
tensor product X*®KY can be viewed as nuclear operators from X into Y.
In the same way, there is a canonical linear mapping from X* ®Sr Y into
£>(X, Y) for each p that associates with the tensor "£?=i f3 ® y3 the operator
x h-). "£"_, ipj(x)yj. An application of Holder's Inequality shows that this
mapping is bounded. Therefore it extends to an operator Jp: X*®3pY ->
£>(X, Y). An operator from X into Y is called p-nuclear (or sometimes left
p-nuclear) if it lies in the range of Jp. Thus, T is a p-nuclear operator if there
exist sequences (ipn) e tp{X*) (or co(X') when p = oo) and (yn) e tp[Y),

140 6 The Chevet-Saphar Tensor Products
such that Tx = Y,™=i Vn(a;)y„ for every xeX.We denote by Np{X,Y) the
space of p-nuclear operators and we define the p-nuclear norm on this space
by
||r||WF=iiif||(^)||p||(yn)|p,
where the infimum is taken over all the representations of T as described
above. We leave it to the reader to verify that Np(X, Y) is complete in this
norm. Replacing the norm gp by dp in this construction leads to the class of
right p-nuclear operators. Since g-i = d\ = 7r, the (left) 1-nuclear and right
1-nuclear operators coincide with the nuclear operators.
It is clear from these definitions that the space of p-nuclear operators is
a quotient of the tensor product X*®3pY:
■Np{X,Y) = X*®SpY/keiJp.
We shall see later that tip{X,Y) = X*®9pY if either X* or Y has the
approximation property.
6.3 p-summing Operators
The dual space of a tensor product X®aY can be interpreted as a space of
bilinear forms on X x Y, a space of operators from X into Y*, or a space of
operators from Y into X*. A judicious choice often leads to the discovery of an
exciting new type of bilinear form or operator. In this section, we shall study
the dual of X®dpY as a space of operators from X into Y*. The operators
obtained in this way are known as the p'-summing operators.
Let T G &(X,Y*) be a bounded linear functional on X®dpY. We may
assume that 1 < p < oo, since d\ is the projective norm. The action of T
on a tensor u = "£"=i x3 ® y} is given by («,T) = ^2"=1(y},Tx}) and the
boundedness of T means that
<C||(*,)||J5||(y,)||p, (6.10)
where C is a positive constant, the minimum value of which is the norm of
T in (X®,ipY)*. The condition given in (6.10) can be simplified as follows.
Proposition 6.11. Let 1 < p < oo and let X and Y be Banach spaces. An
operator T: X -> Y* defines a bounded linear functional on X®d Y if and
only if there exists a positive constant, C, such that
Eii^iK^c' ^p f>*y
for every finite sequence (a.|,...,a:n) in X. Furthermore, the norm ofT in
{X®ap Y)* is the smallest value of C for which thts condition holds.

6.3 p-summing Operators 141
Proof. Suppose that T satisfies this condition. Let « = "£"=1 x} ® y} £
X®Y. Then
|<«,T)| = f^(y3,Tx3) < f>,|||M < (Enroll"') " (E WIPY"
by Holder's Inequality, and it follows that T is bounded on X ®dr Y.
Conversely, suppose T G {X®d„Y)*, so that T satisfies (6.10) for some
C > 0. Let a-i,..., xn G X and let e > 0. For each j, choose y, €Y such that
(y3,Tx3) = \\Txy and ||„,|| < (1 + e)\\Tx,\\'-K Then
E iit^hp' = X>.t*j> < c[i+e)\\[x,)\\? (eiit^iK)
3=1 3=1 \=l '
1/P
since p(p' - 1) = p'. It follows that "£"=, WTx,^' < &>'{1 + e)*'\\{x,)\\% for
every e > 0. Therefore T and C satisfy the stated condition.
The assertion about the norm of T follows immediately from the proof.
. D
At this point, it will be helpful if we interchange the conjugate indices p
and p\ Thus, we have the following characterization of T £ (X®4 ,Y)* for
1 < p < 00:
Enroll" <c sup f>*,|». (611)
3=1 *-eB*- 3=1
Now consider the action of T on infinite sequences in X. Suppose that the
sequence (a;n) is weakly p^summable. Then, from (6.11), we have
El|r-r„llp<c sup El^lp^CP(U(a;")llp)P
n=l VeBx- n=l
for every k. It follows that the sequence (Txn) is strongly p-summable in Y*,
and that
||(r«B)||p<C||(a;n)||J'. (6.12)
Conversely, if the operator T: X -» Y* sends weakly p-summable sequences
in X to strongly p-summable sequences in Y* and (6.12) is satisfied, then of
course, T belongs to {X®d ,Y)*. We have uncovered a new class of operators.
The next proposition gives some equivalent descriptions of this class. In order
to define these operators, it is not necessary that the range space be a dual.
Proposition 6.12. Let 1 < p < 00 and let X and Y be Banach spaces. The
following are equivalent for an operator T: X -¥ Y:

142 6 The Chevet-Saphar Tensor Products
(i) There exists a positive constant, C, such that
itiTx,,*,) < C||(s;)IP(<MIIp'
for all finite sequences (a-i,..., xn) and (V>i. • ■ •. "0n) of elements of X
and Y* respectively.
(ii) There exists a positive constant, C, such that
JT,\\{Tx,W<C sup £>*,!'
for every finite sequence (x\,..., xn) in X.
(iii) For every weakly p-summable sequence (xn) in X, the sequence [Txn)
is strongly p-summable in Y.
(iv) There exists a positive constant, C, such that
\\{Txn)\\p < C7||(*„)||-
for every weakly p-summable sequence (xn) in X.
Furthermore, the minimum value of the constant C is the same in each case.
Proof. The discussion above shows that (i), (ii) and (iv) are equivalent and
that each implies (iii). We prove that (iii) implies (iv).
If T satisfies (iii), then we may define a mapping S: P£{X) -» ip[Y) by
S((in)) = (Txn). It is easy to see that S is linear and it follows from the
Closed Graph Theorem that S is bounded. Hence T satisfies (iv). D
An operator T: X -» Y is said to be p-summing (or absolutely p-
summing) if it satisfies the equivalent conditions of this proposition. The
space of p-summing operators will be denoted by "Pp^X, Y). The p-summing
norm of T e 7P[X,Y), denoted by irp(T), is defined to be the minimum
value of the constant C that appears in the equivalent statements above. It is
easy to see that ||T|| < wp[T) for every p. The 1-summing operators are often
referred to simply as summing operators (or absolutely summing operators.)
We leave it to the reader to prove, with the help of the above proposition,
that the space of p-summing operators is complete in the p-summing norm.
Returning to our discussion of the dual space of the Chevet-Saphar tensor
products, we can now see that
{X®dpYy=?p,{X,Y*)
where 1 < p < oo. And, of course, there is a corresponding statement for the
transposed norm gp:
(X®gpYy=?p,{Y,X*).

6.3 p-summing Operators 143
In the latter case, the action of a p-summing operator S: Y -» X* on the
tensor product is given by (£,"=, Xj ®y,,S) = Y."=i (xi ■ sVi )•
The space 7P{X, Y) is not necessarily a dual space. However, it follows
immediately from Proposition 6.12 that the p-summing property of an
operator, and the value of its p-summing norm, are unchanged if the range space
is enlarged. Thus, 7P{X,Y) may be embedded in 7P{X,Y") and we have
VP{X,Y) C ?P(X,Y") = {X®dr,Y*Y ,
where 1 < p < oo. Thus, the elements of VP(X, Y) act, in a canonical way,
as bounded linear functionals on X®d ,Y*\ we have
n % n
Y,x,®1,„t\=Yi(Tx„1,,).
3=1 ' i=i
We now collect together some of the elementary properties of p-summing
operators.
Proposition 6.13. f
(a) LetS £ 7P(X,Y) and letT: W -» X and R: Y -» Z be operators. Then
the composition RST is p-summing and
np{RST)<\\R\\np{S)\\T\\.
(b) Let T G 7p{X^Y). Then T" is also p-summing and has the same p-
summing norm as T.
(c) Let 1 < p < q < oo. Then ?P{X,Y) C ?q{X,Y) and wq{T) < wp{T) for
every Teyp\x,Y).
Proof, (a) By multiplying R and T by appropriate constants, we may assume
that ||r|| = 1, so that T*(BX>) C Bw.het wu.. .,wn SW. Then
£ \\RSTw,\\' < ||A|| £ \\STw3\\p < \\R\\*P{S)P ^up JT \vTw,r
3=1 ;=l feBx' i=l
= \\R\K{sf sup £|(rvH|'
n
<\\R\\wp{S)p sup ][>«;, |p,
and it follows from Proposition 6.12 that RST is p-summing and that
H-Rsrii < \\r\\*p{s)\f\\.
(b) By the remarks preceding the proposition, we may consider T as an
element of 7P{X,Y") = (X®dr,Y*)*. Then T" is the canonical extension

144 6 The Chevet-Saphar Tensor Products
of T to a bounded linear functional on X**®d ,Y*** and the result follows
from Theorem 6.5.
(c) follows immediately from the fact that if p < q then p' > q' and so
dp- <dq> onX®Y**. D
The fact that the norms dp are finitely generated translates into a dual
statement for the class of p-summing operators. We have seen that an
operator T; X —> Y is p-summing if and only if T defines a bounded linear
functional on the tensor product X ®d , Y*. The finite generation of dp>
means that
dp.(«;X ® Y*) = inf{dp-(«; M ® N) : u <E M ® N}
the infunum being taken over all pairs M, N of finite dimensional subspaces
of X, Y* respectively for which u € M ® N. The finite dimensional sub-
spaces of y* are in one to one correspondence with the subspaces of Y of
finite codimension. Thus, arguing, in exactly the same way as we did for
integral operators in Section 3.5, we obtain the following finite dimensional
characterization:
Proposition 6.14. Let 1 < p < oo and let X and Y be Banach spaces.
An operator T: X -> Y is p-summing if and only if there exists a positive
constant C such that for every finite dimensional subspace E of X and every
/mtte codimensional subspace F of Y, the finite dimensional operator
QFTIE: E -» X -» Y -» Y/F
satisfies itp(QfTIe) < C. Furthermore, we have 7rp(T) = inf C, where the
infimum is taken over all such pairs E, F.
We now look at some examples of p-summing operators.
Example 6.15. The canonical p-summing operator C(K) -> Lp[fi).
Let (J be a regular, positive Borel measure on the compact space K and
let J: C(K) -> Lp(fi) be the canonical mapping that associates with each
continuous function its equivalence class in Lv{ji). Suppose that the sequence
(/„) is weakly p-summable in C{K). Then the series Y^=\ l/n(*)lp converges
for every t G K and it follows that the sum of this series is a continuous
function on K. Then, by the Lebesgue Monotone Convergence Theorem,
oo oo .
£llJ/nl|P=£/ \fn{t)\"d^t) <00.
Therefore J is p-summing. We invite the reader to compute the p-summing
norm of J.
There is also a measure theoretic version of this example:

6.3 p-summing perators
Example 6.16. The canonical injection LooC/*) -» Lp[fi) is p-summing.
Let fi be a finite, positive measure on a <r-algebra of subsets of a set fi and
let Jp: Loo(/t) -» £P(/*) be the canonical injection. Let /i,...,/„ G £«>(/*)•
By (6.4),
ii(/,)r=suPf £>,/,
1 .7=1
■ (A,) e a
,}.
It follows that, for every Ai,..., An, we have
n
E*i/jM ^ IWUIMdll? M-almost everywhere.
3 = 1
Therefore, there is a /i-null set E so that this is valid for all rational values of
X} and all w £ E and hence it holds for all X} and all w £ E. It follows that
»/p
(£l/>)lP) P<IK/i)H;
for almost every w. Integrating, we obtain
EH-VjIIp = E /"l/jl"**^Mn)(ll(A)ll?)p.
Therefore Jp is p-summing and 7rp(Jp) < /*(n)1/p. But /i(n)1/" = ||JP|| <
ttp( Jp) and so ttp( Jp) = /*(n)1/p.
Example 6.17. The canonical injection l\ -» I2 is 1-summing.
Let X\,...,xn € l\, where x} = [x3k)k f°r each 3- Let r^ be the Rademacher
functions. Applying the Khinchine Inequality in Lt = Li[0,1], we have
n n f 00 \ 1/2 n ,00
EiM2 = E(EM2) ^VEE
=v/,i:|f;^rfc(o
Sjfcrfc
j=i 'fc=i
For each t, the sequence (rfc(t)) is a unit vector in tx and so
n 00 n
EE*wo < s"p E 1^1 = ii(^)iir•
J = l fc = l *-eB'~ J = l
Therefore, £" ll^lb < VllMIr-

146 6 The Chevet-Saphar Tensor Products
In fact, we shall see shortly that every operator from t\ into I2 is 1-
summing.
We now present a powerful characterization of the class of p-summing
operators. This result will enable us to prove some of the deeper properties
of these operators.
Theorem 6.18 (Pietsch Domination Theorem). Let 1 < p < co, let X
and Y be Banach spaces and let K be a weak?-closed norming subset of Bx',
endowed with the weak? topology. An operator T: X —» Y is p-summing if
and only if there exists a regular Borel probability measure, \i, on K and a
positive constant, C, such that
lira.il> < c f IHP <*/%>)
Jk
for every x £ X. Furthermore, the p-summing norm of T is the minimum
value ofC.
Proof. Suppose that T satisfies the above estimate. Then
for all a-i,..., xn £ X, and it follows that T is p-summing with irp(T) < C.
Conversely, suppose that T is a p-summing operator. For each finite subset
A of X, consider the function f^ on K given by
/^) = EHT*Hp-Mr)p£lHp-
x€A x€A
Then f^ is weak* continuous and the p-summing property of T implies that
fA (<p) < 0 for at least one ip £ K. Let L be the subset of C[K) consisting of
all the functions f*, as A ranges over the collection of finite subsets of X. It
is easy to see that Af4 = fv for all positive A, where A' = {A1/,pa;: x £ A},
and that fA + fB = fAuB if A and B are disjoint. It follows that L is a convex
subset of C{K).
Let P = {/ £ C{K) : f{<p) > 0 for every <p £ K). Then P is a convex,
open subset of C[K) and is disjoint from L. Therefore, by the Hahn-Banach
Theorem, there exists a regular Borel measure y, and a constant 7, such
that (/yi.ji) < 7 < (f,fi) for every f& £ L and f £ P. Since the zero
function belongs to L, we have 7 > 0 and so (i is a positive measure. Now,
by multiplying 7 and y, by appropriate constants, we may assume that fi
is a probability measure. We now have 0 < 7 < (f,fi) for every positive
function /. But the value of (/, y) can be arbitrarily small and hence 7 must
be zero. Thus, (f4,|t) < 0 for every f^ £ L. Now, if we take A to be a
singleton set {a;}, we have fA{f) = \\Tx\\p - wp{T)p\ipx\p and, bearing in
mind that y[K) = 1, we obtain

. p-summing pera ors
\\TxW><wp{T)p [ \ipxfdp.
Jk
Finally, it is clear from the proof that tp(T) is the minimum value of C.
D
It is possible to interpret the Pietsch Domination Theorem as a
factorization scheme. Let T: X -» Y be a p-summing operator and let K, (i and
C be as above. We shall attempt to factor T through the canonical
operator Jp: C(K) -> Lp(fi). Let V denote the embedding of X into C(K), so
that (Vx)(<p) = tp(x) for x £ X and tp £ K. Applying Jp takes us into
the subspace JPV(X) of Lp(fi). We can now define U: JPV(X) -> Y by
U(JpVx) = Tx and the domination condition tells us that U is bounded,
since ||[/(JpVa;)|| < C|| JpVa;||p for every x € X. Furthermore, we see that we
choose K and fi so that ||V|| < irp(T). This factorization is shown below:
As it stands, this is not very satisfactory. Ideally, we would like to factor T
through the canonical operator Jp. Thus, we need to extend U: JpV(X) -> Y
to Lp(fi). In general, this will not be possible unless Y is an injective space.
However, we may embed any Banach space Y as a subspace of an injective
space. For example, let ^oo(-By-) be the £«,-space with indexing set By (or
any weak'-dense subset of this ball). Then Y may be embedded in this space
by identifying y SY with (^iy) G ^oo(-By). Alternatively, we may compose
the canonical embedding of Y into C(By) with the embedding of the latter
space into its bidual to obtain an embedding of Y into the injective space
C(By')"- We can now extend the operator U to Lp((i) while preserving
norm, giving a more attractive diagram:

148 6 The Chevet-Saphar Tensor Products
T
Y ►M^y)
U
C(K) ^— Lp{y)
In addition, we have ||V||||.7p||||t/|| = np(T). We now show that thep-summing
operators are characterized by factorizations of this general form.
Theorem 6.19. Let 1 < p < oo and let X and Y be Banach spaces. An
operator T: X -t Y is p-summing if and only if there exists a compact space
K and a regular Borel probability measure \i on K, together with operators
V: X -» C(K) and U: Lp(fi) -» ^(By.), such that IT = UJPV, where I
is the canonical embedding ofY into loaiBy)- Furthermore,
7rp(r) = inf||[/||||V||,
where this infimum is taken over all such factorizations, and this infimum is
attained
Proof. We have already seen that every p-summing operator can be factored
in this way, with irp(T) = inf ||[/||||V||. Conversely, suppose that T can be
factored in the manner described. Since Jp is p-sumrning, it follows from
Proposition 6.13 that IT is p-summing. Therefore T is p-summing and has
the same p-summing norm as T. Also,
*P(T) = *P(IT) = *P(UJPV) < ||[/|kP(Jp)||V|| = \\U\\\\V\\.
This concludes the proof. O
To give an example of the power of this factorization, we recall from
Chapter 3 that the operator Jp: C{K) —> Lp{(i) is both weakly compact and
completely continuous and hence so also is the composition UJPV = IT. Now
these properties are unaltered if the range space is enlarged. Therefore, we
have:
Corollary 6.20. Let 1 < p < oo. Every p-summing operator is both weakly
compact and completely continuous.
This has some striking implications for infinite dimensional spaces:
Corollary 6.21. Let 1 < p < oo and let X be an infinite dimensional
Banach space. Then the identity operator on X is not p-summing.
Proof. The composition of a completely continuous operator with a weakly
compact operator is compact. O

6.3 p-summing Operators 149
In particular, taking p = 1, we see that every infinite dimensional Banach
space X contains a sequence (a;n) such that the series "£"]„ <p(xn) is
absolutely convergent for every tp £ X*, but "£"]„ ||a;n|| diverges. This is in marked
contrast with the situation in finite dimensions, where unconditional and
absolute convergence for series are the same. We have thus proved a weak form
of the Dvoretzky-Rogers Theorem, which asserts that in every infinite
dimensional Banach space there is an unconditionally convergent series that is
not absolutely convergent.
When the domain of a p-summing operator is a C(K) space, there is a
particularly simple factorization:
Proposition 6.22. Let 1 < p < oo. An operator T: C(K) -» Y is p-
summing if and only if there exists a regular Borel probability measure (i
on K and an operator U: Lp(fi) -» Y such that T = UJp. Furthermore, the
p-summing norm of T is the minimum value of \\U\\ over all such
factorizations:
C(K) ► Y
Proof. Since K is a weak'-compact norming subset of C(K)*, it follows from
the Pietsch Domination Theorem that there is a regular Borel probability
measure n on K, such that ||Ta:||p < 7rp(T) fK \x\p dp for every x € C(K).
Thus T is bounded for the Lp{p) norm on C(K) and it follows by the density
of the continuous functions in Lp(fi) that there is an operator U: Lp(fi) -> Y
such that \\U\\ < irp(T). On the other hand, T = [/^implies that wp(T) <
\\U\\ and so these norms are equal.
Conversely, if T = UJP is any factorization of T, then np(T) < \\U\\. O
The 2-summing operators have some special features that are unique to
this case. We begin by looking again at the argument that produced the
factorization scheme for p-summing operators. Recall the central idea: if
T: X -> Y is p-summing and K is a weak*-compact norming subset of Bx-,
then the Pietsch Domination Theorem provides a regular Borel probability
measure, fi, such that ||Ta:|| < 7rp(T)||JpVa:||p, where V is the canonical
embedding of X into C(K) and Jp: C(K) -> Lp(fi) is the canonical mapping.
We then define a bounded operator U on the subspace JPV(X) of Lp(fi)
by U(JpVx) = Tar. Our problem then was to extend the domain of V to
all of Lp(fi) and this forced us to embed Y in an injective space. But when
p = 2, the space ^(/n) is a Hilbert space and there is an easier alternative.

150 6 The Chevet-Saphar Tensor Products
We may assume that the operator U is defined on the closure of J2V(X).
This subspace has an orthogonal complement in L2{y) and so we may extend
U trivially by composing with the orthogonal projection of L2(fi) onto the
closure of J2V(X). Thus, we have:
Proposition 6.23. Let X and Y be Banach spaces. An operator T: X -» Y
is 2-summing if and only if there exists a compact space K and a regrrtar
Borel probability measure \i on K, together with operators V: X —> C(K)
andU: L2(fi) -» Y, suchthatT = UJ2V. Furthermore, w2(T) = inf ||t/||||V||,
where the infimum is taken over all such factorizations, and this infimum is
attained:
C(K)
J2
U
L2(fi)
As was the case with integral operators, our factorizations through the
canonical operators C(K) -> Lp{n) have an alternative formulation in terms
of the canonical operators Z»oo(/*) -> Lp{n). The advantage of the latter
approach is that Loa(fi) is an injective space. We use this idea in the following
result.
Proposition 6.24. Let X and Y be Banach spaces and suppose that X is
a subspace of the Banach space Z. Then every 2-summing operator from X
into Y has an extension to a 2-summing operator from Z into Y with the
same 2-summing norm.
Proof. Consider a factorization of T of the type described in the proposition.
The canonical mapping of C(K) into L2(fi) factors through the canonical
mapping J2: Loo(fi) -> L2(p). This gives the following factorization of T:
L°°{v)
A
U
i*M
and 7Tp(T) = ||l/||||V||. Since the space L00(fi) is injective, the operator
V: X -> Z<oo(/*) can be extended to an operator S: Z -> Loo(fi) with the
same norm. The operator UJPS is a 2-summing extension of T with the same
2-summing norm. O

6.3 p-summing Operators 151
This extension property of 2-summing operators has an interesting
consequence for the tensor norm i^. A tensor norm a is said to be left injective if,
whenever X is a subspace of Z, then X®a Y is a subspace of Z ®a Y for every
Banach space Y. This is easily seen to be equivalent to the condition that
every bounded linear functional on X®aY has an extension to a bounded
linear functional on Z®aY with the same norm. Since (X®d2Y)* = ^(X.y*).
we have:
Corollary 6.25. The tensor norm &% is left injective.
We conclude with an examination of the class of 2-summing operators on
Hilbert spaces. Let H\ and H2 be Hilbert spaces. An operator T. Hi —j* H2 is
called a Hilbert-Schmidt operator if there exists an orthonormal basis (e,),-ej
for Hi such that "£,e/ ||Te,||2 is finite. It is not difficult to show that this is
then satisfied for every orthonormal basis for Hi and that the value of the
sum X.,e/ ||Te,||2 is independent of the basis. The Hilbert-Schmidt norm of
T is defined by
'2(T)=(£||re,||2Y2.
Proposition 6.26. Let H\ and H2 be Hilbert spaces. An operator T: Hi -»
H2 is 2-summing if and only if it is a Hilbert-Schmidt operator. Furthermore,
the 2-summing and Hilbert-Schmidt norms coincide.
Proof. Suppose that T is 2-summing. Let (e,),e/ be an orthonormal basis
for Hi and let {ii,..., in} be any finite subset of the indexing set J. It is easy
to see that ||(e,-Jt||'£ = 1. Therefore
£||re,J|2<,r2(r)2.
7c=l
Since this holds for every finite subset of I, it followsi''that T is a Hilbert-
Schmidt operator and that o-2(T) < ^(T).
Conversely, suppose that T is a Hilbert-Schmidt operator. Fix
orthonormal bases (e,),e/ and(f,)jej for Hi and H2 respectively. Let a-i,. ..,xn € H\.
For each k, we have a:* = X.,(2".c.et)e« an^ so tne expansion 0f Tar* in the
basis (/,) is given by Txk = £, E.(*fc.e.)(Te„/,)/,. Therefore
EnTa;*ii2 = EE E^>e-)(Te-->^) * EEEk**.*)!2!^.,/,).2
k k ] 1 k } i
= Ei(a;*.e.)i2EEi(Te"^)i2^(ii(a:*)ii2)2'72(T)2-
Hence T is 2-summing and 7r2(T) < a2(T). 0

152 6 The Chevet-Saphar Tensor Products
6.4 Grothendieck's Inequality
In this section, we prove a fundamental result, known as Grothendieck's
Inequality, which will enable us to uncover some of the deeper facts about
p-summing operators. The finite dimensional structure of the spaces involved
is crucial and this will provide a further justification of the inclusion of the
property of finite generation in the definition of a tensor norm.
Theorem 6.27 (Grothendieck's Inequality). There w a positive
constant, K, with the following property: for any n, let (ay) be annxn matrix
andletxi,...,xn and j/i,..., y„ be in the closed unit ball of a Hilbert space H.
Then
]Cat/(a-ii Vj) - KsuT?\ ]Ca,J',,tJ : la*l' I**' - 11 '
Proof. We give the proof for the case of real scalars. The complex case is
dealt with by decomposition into real and imaginary parts. We may assume
that the vectors xt and y< lie in an n-dimensional Hilbert space. Since all
Hilbert spaces of a given dimension are isomorphic, we are free to work in
any n-dimensional space. We take H to be the subspace of L2.0.1] spanned
by the first n Rademacher functions n,...,rn. Thus x,(t) = Yl"=i xvri(t)
and W(0 = T,%i1h,r3(t) for t € [0,lj.
We begin by truncating the functions xt(t) and y,(t). For any function x,
we let
frnin{.r(t),l} if x(t)>0,
X{ ' (max{a;(t),-1} if a;(t)<0.
Writing ||a|| for supl "2,,, a,]S,t}\ : \s,\, \t,\ < 1 L we have
Yla'A^lb) = "!Ca,W xl{t)T,{t)dt\< / Val3xl{t)T,{t)
Tl Jo I Jo Y~1
dt < llall
We now estimate the Li norm of the functions of the form x — x. In general,
if x(t) ^ x(t), then |a;(t) -x(t)\ = |a;(t)| - 1. Using the fact that (A - 2)2 > 0,
we have A — 1 < A2/4 for every real number A and so
\x(t)-x(t)\<x(t)2/4
(6.13)
for all such t. On the other hand, if x(t) = x(t), then (6.13) is also true.
Therefore (6.13) holds for every function x and every t. Now suppose that
x = X.y=i bjrj with ||x|| < 1. Then, using the orthogonality properties of
the Rademacher functions established in Section 2.5 (see page 34),

6.4 Grothendieck's Inequality 153
16^ \x(t)-x(t)\2dt < J x(t)*dt
-L
53 hbkhbmr]{t)rk{t)ri{t)rm{t) dt
j,fc,£,m=l
53 b,bkbibm + ^ b,-bkbibm + 53 6i6fc6I6ro - 2 5^6*
££
j=m,
<3.
Therefore ||a;, - xl\\, \\yi - yl\\ < y/S/4 for each i. Let us write
IIWII = sup{|53a^^yy) : |M,||yJ < l}.
Then
>.j
+
and it follows that
•■J
53a«i(a;>'%~V3')
•J
<||a|| + |||a|||^/4 + |||o|||>/3/4,
„ 2
2-v/3
HI
and we are done.
The smallest constant that satisfies GrothendieckVInequality is known as
the Grothendieck constant We shall denote this constant by Kq, or by Kq(R)
or Kq(C) if it is necessary to distinguish between the real and complex cases.
The proof given here does not give a good estimate for this constant. It is
known that tt/2 < KG(R) < 1.782 and 1.338 < KG(Q < 1.405.
We now present some striking applications of Grothendieck's Inequality.
Theorem 6.28. Let X be an Li-space and let H be a HUbert space. Then
the norms dp for 1 < p < co are all equivalent on X ® H.
Proof. Since doo < dv < d\ — n for every p, it will suffice to show that
w < Cdoo for some constant C. Furthermore, since these norms are finitely
generated, it will be enough to prove the result for X = £?, with a constant
that does not depend on n. We shall show that w < Kadoo on Pf ® H.

154 6 The Chevet-Saphar Tensor Products
Let u G ii ® H with doo(u) < 1. Then we may choose a representation
Efcli x*®yt of w such that ||(a;t)||y < 1 and ||(y,)||oo < 1- Let x{ = Y,"=i <*3e,
for each j, where (e}) is the standard unit vector basis for C{. We have
■ I n | n y n ■. i
sup Y* sxxx = sup sup E t3 ( E stfltj ) ey
l*il<illfcT l"il<i|^<»lJ=i V,=i / I
supjr^ofya,*, :|«,|,|t,|<l|<l
To compute tt(u), let T <E (et®*H)* = £(£?, if') with ||r|| < 1. For each j,
let Tey = z3 £ H, so that ||zj || < 1 and (a;, Te3) is given by the inner product
(x,z3). Then
n n n n n
f=i t=i j=i t=i i=i
It follows from Grothendieck's Inequality that |(u,T)| < Kq and hence
w(u) < KG. Therefore w(u) < Kad^u) for every ue^gif. 0
Taking duals, we have:
Corollary 6.29. Let X be an -Ci -space and let H be a Hilbert space. Then
every operator T: X —» H is 1-summing.
In particular, we see that
£(<i,<a)=iPi(<i,<2).
Our second application will require a little preparation.
Lemma 6.30. Let (at]) be an n x n matrix and let a-i,... ,xn £ Q satisfy
\\(x,m < I- Then
EfEE^"^ ) ^-^0 supj Ea"',,<J :|s.l.l<yl < l|-
Proof. It follows from ||(a.,)||;f < 1 that Y,i \xij\2 < 1 for <&<& 3- Thus the
vectors Uj = (artJ), lie in the closed unit ball of £%• F°r each k, we may choose
a unit vector vj, in ^J so that
E <*;.*«.. = /E^*^''"*) '
Therefore

6.4 Grothendieck's Inequality 155
k ^ t j ' k 3 k » j '
= X) XX*^-w*) - K° sup| Yla»Js,<j : H'i*ii ^x [
by Grothendieck's Inequality. Q
The class of -Cp-spaces for 1 < p < oo is defined in exactly the same
way as the £i and £«,-spaces. Thus, a Banach space X is an £PiA-space
for some A > 1 if every finite dimensional subspace of X lies in a finite
dimensional subspace that is isomorphic, via an operator T, with some ££,
where ||T||||T-1|| < A. A Banach space is said to be an Lp-space if it is an
-Cp.A-space for some A.
Theorem 6.31. Let X be an Loo-space and Y an Lp-space where-! < p < 2.
Then every operator T: X -*Y is 2-summing.
Proof. Thanks to the finitely dimensional characterization of the class of
2-summing operators given by Proposition 6.12 (a), it is enough to prove
that if T is an operator from X = ££ into Y = t£, then n2(T) < C\\T\\,
where the constant C is independent of the dimension m. We shall prove that
n2(T)<KG\\T\\.
We begin with the matrix representation of T relative to the standard
bases (ej) and (/*) for ZQ and ££ respectively. Let Te, = Ylk ajkfk for
each j. Then
k 3
for every x £ t^ and so
\\Tx\\
/ P\ VP
(Y,Y,x>a>'< j v-
Let n,... ,xn £ ££ satisfy ||(a.,)||;? < 1. We claim that
1/2
(eii^-h2) ^k°
imi,
which is equivalent to
px 2/px 1/2
/ / p\ */p\ v*
\\T\\.
We now use the Lemma to establish a lower bound for ||T||. If we view T as
a bilinear form on P£o x £?, we see that

156 6 The Chevet-Saphar Tensor Products
||r|| = sup||^ojfcj,*fc : IMU ||z||p. < l} .
Now fix a vector z in the closed unit ball of F$. Then IKtjt-ZjOllp' ^ 1 f°r every
t in the closed unit ball of P£, and it follows that
Bup{|^oyik*fc«,tfc : \Sj\,\tk\ < l} < ||T||.
Applying the Lemma to the matrix (a,kZk), we obtain
YllYl Y,x*3aikZk J <KGsupl Y^aifrsJ, : |*,UI*j| < lj
<^g||T||.
In other words,
k V t j '
WW
for every z in the closed unit ball of £"?. Taking the supremum over z, we
have
/ / 2\ p/2\ 1/P
{E{Y,Y,xva»< ) ) <Ko\\n-
If we write this as
/ |2\p/2
ElEE*-^* ) £(*eimi)F
and observe that 2/p > 1, then an application of Minkowski's Inequality
yields
2\p/2
/ / px 2/px p/2 , 2n p/2
(E(EE«^ ) ) <E(EEw ) <(^imi)p.
Therefore
EEEvJ <Ka\\T\\,
V , \ fc j I / /
and the proof is complete. Q
Corollary 6.32. Let X be an Loo-space and Y an £jq-space where 2 < q <
co. Then the norms dp for 1 < p < 2 are all equivalent on X ® Y.

6.5 Exercises 157
Proof. It follows from the proof of the proposition that n < Kq da, on l^Ql?
for every to. It follows from the finitely generated nature of these tensor norms
that, if X is an -Coo.A-space and Y an Lqtll-space, then 7r < X/iKq d? on X&Y.
a
There are some instances of this result worthy of note: on the tensor
products too ® £«, and C(K) ® C(K), we have w < Kg d,2- Indeed, we also
have w < Kq </2- In fact, a sharper statement can be made. We shall return
to this question in the next chapter.
6.5 Exercises
Exercise 6.1. Let X and Y be Banach spaces and 1 < p < co. Does the
formula ||u|| = inf{||(ar^)||j,||(j/,)||p' : u = Y^-iX, ® j/J define a reasonable
crossnorm on X ® Y1
Exercise 6.2. (a) Let B G H(X x Y) and let (a;a), (yp) be bounded nets
that converge in the weak* topology to a;** € X**, y" € Y** respectively.
Show that the canonical left and right extensions of B are given by
*B(x",y") =]im\imB(xa,yp)
and
B*(x",y") = limlimB(a;aiy^).
(b) Show that *B = B* on X** x Y and on X x Y".
(c) Show that *B = B* on X" x Y" if and only if the operator T: X -»
Y* associated with B is weakly compact.
Exercise 6.3. Show that dp = gp on tp ® tp.
Exercise 6.4. Let (CI, E, (i) be a finite measure space. For 1 < p < co,
let Lp(n, X) be the space of equivalence classes of ^-measurable functions
/: Q. -» X for which the function ||/()||p is integrable, with the norm ||/||P =
(/nll/IW/p- Show that
dP(f) < ll/Hp < gP(f)
for every / belonging to the subspace Lp(fi) ® X of Lp(fi, X).
Exercise 6.5. Let 1 < p < co and let A G £«,. Show that the diagonal
operator D\: Co —"> Co given by D\X — (Ana;n) is p-summing if and only if
Xetp.
Exercise 6.6. Let /j, be a positive regular Borel measure on the compact
space K and let g £ Lp(fi). The multiplication operator Mg: C(K) -» Lp(y)
is defined by Mgf = gf. Show that Mg is p-summing and calculate irp{Mg).

158 6 The Chevet-Saphar Tensor Products
Exercise 6.7. Show that every integral operator T is 1-summing and that
'ri(r) < \\T\\j. Find an example of a 1-summing operator that is not integral.
Exercise 6.8. Show that Ti (co, Y) = N(co, Y) for every Banach space Y,
with equality of the 1-summing and nuclear norms.
Exercise 6.9. Let (fl, E,/t) be a finite measure space and let T: X -> Y
be a 1-summing operator. If /: Q, -» X is a /i-measurable Pettis integrable
function, show that the function Tf: Q. -» Y is Bochner integrable.
Exercise 6.10. Show that the space VP(X, Y) is complete in the p-summing
norm.
Exercise 6.11. Show that the closed subspace of Li(|t) spanned by the
Rademacher functions is not complemented.
Exercise 6.12. Show that every Lp((i) is an £p-space.
Exercise 6.13. Show that every £,2-space is isomorphic to a Hilbert space
(Hint: there is a quotient operator from some ^i(/) onto the £.2-space X.
This operator is 2-summing; now consider the factorization given by
Proposition 6.23.)
Exercise 6.14. Let X be an £p-space, where 1 < p < 2. Show that every
weakly summable sequence in X is strongly 2-summable (Hint: if (a:n) is a
weakly summable sequence in X, consider the .operator T: cq -> X given by
Ten = xn.)

7. Tensor Norms
In this chapter, we study the basic properties of tensor norms. We begin with
the dual norm and this leads naturally to the vital concept of accessibility,
which can be thought of as an analogue for tensor norms of the approximation
property for spaces. We then meet the various injective and projective norms
that can be associated with a tensor norm. Next, we turn our attention to
the identification of the duals of the Chevet-Saphar tensor norms in terms of
the classes of p-integral operators. In the final section, we meet the Hilbertian
tensor norm, which plays a central role in the theory. Grothendieck's
Inequality can now be interpreted as the statement that the Hilbertian tensor norm
and the largest injective norm are equivalent. Two new classes of operators
emerge: the Hilbertian and the 2-dominated operators. We conclude with
Grothendieck's classification of the natural tensor norms.
7.1 The Dual Norm
In finite dimensions, the definition of the dual norm is clear. If E and F are
finite dimensional normed spaces and a is a tensor norm, then E ® F is,
algebraically, the dual space of E* ®a F* and we may define a' to be the
dual norm:
E ®a. F = (E* ®a F*)* - c-
In other words, if u S E ® F, then
a'(u)=sup{\(u,v)\:veE*®F*, a(v) <1}.
To spell this out, recall that the duality between E® F and E* ® F* works
in the following way: if u = ">]]"_, x, ® y, £ E ® F and v = X.JL, f]®^] €
E* ® F*, then (u,v) = "£, ■ ^(x,)^(yl). Alternatively, we can use trace
duality: if we view u and v as operators from E* into F and F into E*
respectively, then (u, v) = tr vu.
It is tempting to use the same approach in infinite dimensions. If X and Y
are Banach spaces, consider the dual space (X*®aY*)*. While it is no longer
the case that this dual coincides with X ® Y, there is a canonical algebraic
embedding of X ® Y into it. Thus, the dual norm of (X* ®a Y*)* induces a

160 7 Tensor Norms
norm on X ® Y. We shall refer to this norm as the Schatten dual norm and
we denote it by a". To summarize, we have an embedding
X®a. Yc(X*®aY*)*
and, just as in finite dimensions, if u £ X ® Y, then
a3(u) = sup{|(u, v)| : v <E X* ® Y*, a(v) < 1} .
Let us apply this definition to the projective and injective norms. X®e Y is a
subspace of (X* ®„ Y*)* and so it3 = e. But we have seen in Chapter 4 that
the norm induced on X ® Y by (X* ®e V*)* is not, in general, the projective
norm. Indeed, by Proposition 4.14, the canonical embedding of X ®* Y into
(X* ®e y*)* is an isometric embedding for every Banach space Y if and
only if X has the metric approximation property. The existence of a space
without the metric approximation property shows that there is at least one
tensor product on which e3 < n, or, to put it another way, (ir3)s < it. This
is certainly not what we expect.
The flaw in the Schatten dual is not difficult to detect. While a" is easily
seen to be a uniform crossnorm, it fails to possess the crucial property of
finite generation. The remedy is simple; we use the extension process given
in Proposition 6.3 to extend the definition of the dual from finite dimensions.
Thus, we define the dual norm, a', to be the unique tensor norm that agrees
with the dual norm on tensor products of finite dimensional spaces. So, for
u £ X ® Y, we have
a'(u) = inf{a'B,F(u) :u£E®F, dim£\dimF< co} , (7.1)
the infimum being taken over all pairs of finite dimensional subspaces of X
and Y whose tensor product contains u. With this definition, all the
properties we would hope for follow easily from the finite dimensional case. So,
for example, we have n' = e and e' = w. And, of course, (a1)' = a for every
tensor norm a.
The relationship between the dual norm and the Schatten dual is of some
interest. While the definition of a' is the "correct" one, the norm a3 is easier
to understand and to compute. We have a3 < a' in general and these norms
coincide in finite dimensions. It follows that a3 = a' on X ® Y if and only
if a3 is finitely generated on X ® Y. We have shown in Proposition 6.2 that
every uniform crossnorm is finitely generated on the tensor product of two
spaces with the metric approximation property. Thus we have
Proposition 7.1. Let X and Y be Banach spaces with the metric
approximation property. Then a3 = a' on X ®Y.
This result does not explain the fact that w3 — tt' = e. This coincidence
can be explained by the possession by the injective norm of a property that
is dual to finite generation.

7.1 The Dual Norm 161
Let E and F be closed subspaces of the Banach spaces X and Y and
let QB: X -> X/E and QF: Y -> "K/F be the quotient operators. If a is a
uniform crossnorm then
a((QB ® Qf)u; (X/E) ® (Y/F)) < a(u)
for every u £ X®Y. A uniform crossnorm a is said to be cofinitely generated
if, for every pair of Banach spaces X, Y and every u £ X ® Y, we have
a(u) = sup{a((QB ® QF)u; (X/E) ® (Y/F)): dim X/E, dim Y/F < 00}.
In other words, the a-norm of a tensor u is determined by projecting u
onto finite dimensional tensor products and computing the norm there. It is
not difficult to show that the injective tensor norm e is cofinitely generated,
although this will also follow from our next result. It may come as a surprise
to find that the projective norm is not, in spite of its projective behaviour.
Proposition 7.2. Let a be a tensor norm.
(a) The Schatten dual norm a" is a cofinitely generated uniform crossnorm.
(b) a3 = a' if and only if the dual norm a' is cofinitely generated. *
Proof, (a) Let u G X ® Y and let e > 0. By the definition of the norm a",
there exists v e X* ® Y* such that a(v) < 1 and (u,v) > a"(u) - e. Since
a is a tensor norm, there exist finite dimensional subspaces M and N of X*
and Y* respectively such that v G M®N and a(v; M® N) < 1. There exist
subspaces E and F of X and Y respectively such that M = (X/E)* and
N = (Y/F)*. Then
<**((Qe ® Qf)u; (X/E) ® (Y/F)) > (u, v) > a*(u) - e.
It follows that a3 is cofinitely generated.
(b) If a" = a' then it follows from (a) that a' is cofinitely generated.
Conversely, suppose that a' is cofinitely generated. 'TJien, since a" and a'
coincide on tensor products of finite dimensional spaces, we have a" = a'.
a
Corollary 7.3. The projective norm w is not cofinitely generated.
We obtain another viewpoint on these phenomena if we interchange the
roles of a and a'. The canonical embedding of X ® Y into (X* ®a. Y*)*
induces the Schatten dual norm (a1)3 on X ® Y. On the other hand, we have
the-norm a" — a on this space. Then (a')s < a and, in general, these norms
are distinct. To keep our notation managable, we shall write
a = (a'Y
and we shall refer to a as the associated norm of a. Thus, we have an isometric
embedding:

162 7 Tensor Norms
X®sYc(X*®a.Y*)*. (7.2)
When the spaces in question are duals, the norm a can also be defined by
a simpler embedding. It follows from Theorem 6.5 that the same norm is
induced on X* ® Y* by (X** ®a. Y**)* and X ®a> Y. Thus we have another
isometric embedding:
X" ®& Y" C (X ®a, Y)* . (7.3)
The tensor norm a is said to be totally accessible if a = a. By
Proposition 7.2, this is equivalent to a being cofinitely generated. Our previous
results can now be phrased as follows: the injective norm e is totally
accessible, and the projective norm w is not.
Though the projective norm is not totally accessible, all is not lost. By
Theorem 4.14, we do have an isometric embedding of X®„Y into (X*®CY*)*
if just one of the spaces X, Y has the metric approximation property. We
shall see that this property is shared by a great many tensor norms and so we
give it a name. The tensor norm a is said to be accessible if the norms a and
a are equal on X ® Y whenever either X or Y has the metric approximation
property. We leave it as an exercise to the reader that this is equivalent
to the condition that a = a on X ® Y whenever either X or Y is finite
dimensional. Accessibility can be thought of an approximation property for
tensor norms. Indeed, Grothendieck, who coined this terminology, applied the
term "accessible" both to spaces and to tensor norms. All the tensor norms
we encounter in this book are accessible. The existence of a non-accessible
tensor norm was established in 1991 by Pisier.
We can summarize our findings as follows. In order that the norm induced
on X ® Y by the embedding into (X* ®ai Y")* coincides with a, one of the
following conditions suffices:
(a) Both X and Y have the metric approximation property.
(b) a is totally accessible.
(c) a is accessible and either X or Y has the metric approximation property.
We collect some useful results concerning accessibility:
Proposition 7.4. (a) Every injective tensor norm is totally accessible.
(b) Every projective tensor norm is accessible.
(c) A tensor norm a is accessible if and only if the dual norm a1 w accessible.
(d) If the tensor norm a is totally accessible then a' is accessible.
Proof, (a) Since a <a holds in general, we must prove the reverse inequality.
Let X and Y be Banach spaces and let Jx • X -> C(Bx-) and JY: Y ->
C(By) be the canonical embeddings. By the uniform property of a, the
tensor product operator
Jx ® Jy ■ X ®a Y -> C(BX.) ®a C(BY-)

7.1 The Dual Norm 163
has norm at most one. Let u £ X ® Y. Then
a((Jx ® JY)u; C(Bx.) ® C(By.)) < &(u;X ® Y).
Now every C(K) space has the metric approximation property and soa = a
on C(BX.) ® C(By.)- Therefore
a((Jx ® JY)u;C(Bx.) ® C(By.)) < a(u;X® Y).
But, by the injectivity of a, the a-norm of u in X ® Y is the same as the
a-norm of (Jx ® Jy)u. Hence a(u;X ® V) < d(u;X ® V) and we are done.
(b) If a is projective then a' is injective (see Proposition 7.5 below) and
the result follows from (a) and (d).
(c) To show that a' is accessible, it will suffice to prove that E ®a, Y is a
subspace of (E*®a Y*)* for every finite dimensional space E and every Y. By
the finite dimensionality of E, the dual space of E®a> Y coincides algebraically
with E*®Y*. Hence, by the accessibility of a, we have (E®a.Y)* = E*®aY*.
Therefore E®a.Y C (E ®a, Y)** = (E* ®a Y*)* and the proof is complete,
(d) follows from (c). O
We note that the dual of a totally accessible norm need not be totally
accessible. The injective norm provides an example.
We have seen the importance of injectivity and projectivity. Our next
result shows that, as one would expect, these are dual properties.
Proposition 7.5. A tensor norm a is projective if and only if the dual norm
a' is injective.
Proof. Suppose that a is projective. Let X and Y be closed subspaces of
W and Z respectively. Let u G X ® Y and let e > 0. If E and F are finite
dimensional subspaces of X and Y respectively such that u £ E ® F, then
a'(u;X ® Y) < a'(u;E® F). The dual space of E ®a> F is E* ®a F* and
hence there exists v e E* ® F* such that a(v; E* ® F?-) < 1 and
a'(u;E ® F) < (u,v)+e.
Let P: W* -¥ E* and Q: Z* -> F* be the canonical quotient operators.
Then P ®Q: W* ®a Y* -> E* ®a F* is a quotient operator and so there
exists w € W* ®a Z* such that a(w; W ® Z*) < 1 and (P ® Q)w = v. Then
(u, v) = (u, w) and so, by the definition of a3, we have
a'(u;X ®Y) < a'(u;E®F) <(u,v) + e = (u.to)-l-e
< a"(u; W ® Z) + e < a'(u; W ® Z) + e
Therefore a'(u; X ®Y) <a'(u;W® Z) and it follows that a' is injective.
Conversely, suppose that a1 is injective. Let P: W -> X and Q: Z -> Y be
quotient operators. We treat first the case in which all the spaces in question

164 7 Tensor Norms
are finite dimensional. By the injectivity of a', the space X* ®a. Y* is a
subspace of W* ®a. Z*. Hence X ®a Y = (X* ®a. V*)* is a quotient space
of W ®a Z = (W ®a» Z*)*.
We now consider the general case. Let u 6 X ® Y and let e > 0. It will
suffice to show that if a'(u;X ® Y) = 1, then there exists v £ W ® Z such
that (P ® Q)u = u and 0/(1;; W ® Z) < (1 + e)4. We begin by choosing finite
dimensional subspaces E and F of X and V respectively such that
a'(u; E ® F) < (1 + e)a'(u; X ® "K).
Using a compactness argument, there exists a finite dimensional subspace G
of W such that P maps G onto E and, for every x € E, there exists w € G
such that Pw = a; and ||a;|| < ||w|| < (1 + e)||a;||. Let E0 be the space E
with the norm given by the Minkowski functional of the set P(Bq). Then
P: G -¥ Eo is a quotient operator and the identity mapping from E to Eq
has norm at most (1 + e). We carry out an analogous construction for the
space F, obtaining a "close" renorming Fo of F which is a quotient of some
finite dimensional subspace H of Z. By the uniform property of a, we have
a'(u;Eo®F0) < (1 + e)2a'(u;E ® F).
Now i?o ®a' Fo is a quotient of G ®a' H and so there exists v £ G ®H such
that (P ® Q)v = u and
a'(v; G®H)<(1 + e)a'(u; E0 ® F0).
Combining these inequalities, we have
a'(v;W®Z) < a'(v;G®H) < (l + e)V(«;E®F) < (1 + e)V(u;X®Y).
This concludes the proof. O
An inspection of this proof shows that, in order to establish the projec-
tivity or injectivity of a tensor norm, it suffices to show that the relevant
property holds for tensor products of finite dimensional spaces.
We now turn our attention to the order structure of the set of tensor
norms. Not only is there a biggest and a smallest tensor norm; we see now
that every non-empty set of tensor norms has a greatest lower bound and a
least upper bound.
Proposition 7.6. The set of tensor norms is a complete lattice.
Proof. Let {a, : i £ 1} be a non-empty set of tensor norms. We define the
least upper bound first in finite dimensions. If E and F are finite dimensional,
let
a(u;E®F) = sup at(u;E®F)
16/

7.2 Infective and Projective Associates 165
for u £ E ® F. It is easy to see that a is a uniform crossnorm on the class of
tensor products of finite dimensional spaces and that a is the smallest such
norm that satisfies a,- < a for every i £ I. The unique extension of a to a
tensor norm as defined in Proposition 6.3 is the least upper bound of the set
{at : i £ /}.
To construct the greatest lower bound, consider the set {a't : i £ I}. Let
P be the least upper bound of this set, so that /3 is the smallest tensor norm
that satisfies a't < @ for every i 6 I- Then @' is the biggest tensor norm that
satisfies ft < a\ for every i £ I and hence /?' is the greatest lower bound. Q
A word of warning is in order here. It is tempting, but rather misleading,
to write sup( a, and inf, a, for the least upper bound and greatest lower
bound.
7.2 Injective and Projective Associates
We wish to determine the duals of the Chevet-Saphar tensor norms dp and
gp. We have seen that the accessibility, or total accessibility of a tensor" norm
greatly simplifies the determination of the dual. Now dp and gp are neither
injective nor projective for 1 < p < oo and so the results of the previous
section cannot be applied. However, dp is right projective and, even better,
d2 is also left injective. We take this observation as our starting point.
We recall that a tensor norm a is right projective if, whenever P: X -» Y
is a quotient operator, then the tensor product operator
I®P: Z®aX -» Z®aY
is a quotient operator for every Banach space Z. There are analogous
definitions for left projectivity and for left and right injectivity. By the remark
following the proof of Proposition 7.5, in order to show that a tensor norm
has any one of these properties, it suffices to establish''the property for tensor
products of finite dimensional spaces. Proceding as in the proof of
Proposition 7.5, we see that a is right (respectively, left) injective if and only if the
dual norm a1 is right (respectively, left) projective.
The notion of accessibility can also be developed in a one-sided fashion.
Recall that a tensor norm a is called accessible if the canonical embedding
of X ® Y into (X* ®a, y*)* induces the norm a oa X ®Y provided X otY
has the metric approximation property. In other words, we have (a')s = o; in
the presence of the metric approximation property. Similarly, accessibility of
a1 guarantees a" = a' under the same conditions.
A tensor norm a is said to be left accessible if the canonical embedding of
X®Y into (X* ®a> Y*)* induces the norm aon X®Y whenever the Banach
space Y has the metric approximation property. Similarly, a is said to be
right accessible if X ®a Y is a subspace of (X* ®a' Y*)* whenever X has the

166 7 Tensor Norms
metric approximation property. In each of these definitions, the assumption
of the metric approximation property can be replaced by the assumption of
finite dimensionality. Clearly, a tensor norm a is accessible if and only if it is
both left and right accessible. The following proposition is proved in exactly
the same way as Proposition 7.4.
Proposition 7.7. Let a be a tensor norm.
(a) If a is left injective or left projective, then a is left accessible.
(b) a is left accessible if and only if a1 is left acccessible.
One may, of course, replace "left" by "right" in the statement of this
result. It follows that dp is right accessible and di is accessible. We shall see
that much more is true for these norms.
We now describe a process by means of which an arbitrary tensor norm
can be modified so that the new norm has any desired property of left or
right injectivity or projectivity. We begin with right injectivity.
Proposition 7.8. For every tensor norm a there exists a biggest right
injective tensor norm P such that P < a.
Proof. Let "R. be the set of tensor norms that are right injective and bounded
above by a. Then 1Z is non-empty, since it contains the injective norm. Let P
be the least upper bound of this set. In order to show that /3 is right injective,
it suffices to consider finite dimensional spaces. Suppose that X, Y and Z
are finite dimensional and Y is a subspace of Z. Let u £ X ® Y. Then, by
the definition of the least upper bound, we have
Px,y(u) = sup 7x,y(w) = sup -yx,z(u) = Px,z(u) ■
-ten -ten
Therefore P is right injective. O
The tensor norm P is known as the right injective associate of a. We shall
use the notation a\ for this norm. The same construction can be carried out
on the left to obtain the left injective associate, which we denote by /a. Thus
/a is the biggest left injective tensor norm that is bounded above by a. If
a is already left or right injective, then the appropriate injective associate
norm coincides with a.
These constructions can be combined, giving the norms /(a\) and (/<*)\.
We leave it as an easy exercise to show that these norms are equal. Therefore,
we can define
/o\ = /(o\) = (/o)\.
The norm /a\ is the biggest injective norm that is bounded above by a. It
is known as the injective associate of a. The injective associate of a is the
biggest injective norm that is bounded above by a. In particular, /7r\ is the
biggest injective tensor norm.
Our definition of the injective associates, while elegant, can be difficult to
apply. The following proposition gives a more practical characterization.

7.2 Injective and Projective Associates 167
Proposition 7.9. Let a be a tensor norm.
(a) Let X and Y be Banach spaces. The norm induced on X ®Y by the
canonical embedding of X ®Y into X®a C(By) is a\.
(b) For every Banach space X and every compact space K, the norms a and
a\ coincide on X ® C(K).
(c) If Z is an injective space then the norms a and a\ coincide on X ® Z
for every Banach space X.
Proof, (a) For every pair X, Y of Banach spaces, let @ be the norm on X®Y
induced by the canonical embedding X ® Y -» X ®a C(By). It is easy to
see that @ is a uniform crossjaorm that satisfies @ < ct, and and is finitely
generated on the left, in the sense that
Px,y (u) = inf{0B)y(u) :ueE®Y, dim £ < co} .
Since a is a tensor norm, X ®a C(By) is a subspace of X ®a C(By )** by
Proposition 6.4. Therefore, @ is the norm induced by the canonical embedding
X® Y -> X®a C(By )**. It follows that f) is right injective. Indeed, suppose
that Y is a subspace of the Banach space Z. We have X ®p Y C ,X ®a
C(By)** and X ®fi Z C X®a C(BZ-)". But the injective space C(By)"
is complemented in C(Bz')** by a projection of unit norm and so X ®a
C(By)** is a subspace of X ®a C(BZ-)**. Therefore X ®/3 Y is a subspace
of X ®p Z. It follows from the right injectivity of f3 that this norm is finitely
generated on the right. Hence (3 is a right injective tensor norm. Since @ is
bounded above by a, we have @ < a\.
On the other hand, let 7 be any right injective tensor norm that is bounded
above by a. Then 7 < a on X ® C(By) and it follows from the definition
of P that 7 < a on every tensor product X ® Y. Therefore P is the biggest
right injective tensor norm that is bounded above by a.
(b) Let Y = C(K) for some compact space K. Then Y is complemented
in the space C(By) by the unit norm projection /•->/, where f(t) = f(5t).
It follows that X®a Y is a subspace of X®a C(By) &nd so, by the definition
of P = a\ in the proof of (a), a\ = a on X ® Y.
(c) follows from (a). O
This proposition provides a practical way to approach the injective
associates. We have
X ®a\Y C X ®a C(By) ■ (7.4)
Alternatively, we may embed Y in any injective space; for example, we may
take the canonical embedding of Y into the space too(By), to get
X®a\YcX®aeoo(By). (7.5)
Similar descriptions apply to the left injective associate /a and the injective
associate /a\.

168 7 5tensor Norms
We now turn our attention to the construction of the associated projective
norms.
Proposition 7.10. For every tensor norm a there exists a smallest right
projective tensor norm P such that @ > a.
Proof. Let @ = (a'\)'. Since @ > a if and only if /?' < a' and a tensor norm
is right projective if and only if its dual is right injective, it follows that @
has the required properties. O
We shall denote this tensor norm by a/ and we shall refer to it as the
right projective associate of a. In the same way, we have the left projective
associate, \a and the projective associate, \ct/, where
W = \(o/) = (\o)/.
In particular, the norm \e/ is the smallest projective tensor norm. Great
care must be taken when combining injective and projective associates. For
example, it is not true in general that \(<*\) is the same as (\a)Y Indeed, we
shall see that the case a = gp provides a counterexample.
There is a projective analogue of Proposition 7.9 that provides a useful
insight into the uses of the projective associate norms. The kernel of this
result lies, as always in these matters, in a finite dimensional statement:
Lemma 7.11. Let a be a tensor norm. If E is finite dimensional then a/ =
a on E ®£? for every n.
Proof. Using the definition of a/, the fact that P^ is a projective space, and
the definition of the dual norm in finite dimensions, we have the following
chain of isometric isomorphisms:
E®a/e? = E ®(a.\>' G = (E* ®«'\ O* = (e* ®a- e20y = E®ae^.
a
It follows that
x®a/t1(i) = x®ae1(i)
for every Banach space X and every indexing set J. More than this is true:
ti (I) can be replaced by any space that is a .Ci^-space for every A > 1.
Proposition 7.12. Let a be a tensor norm.
(a) Let X and Y be Banach spaces and let P: £(I) -> Y be a quotient
operator. Then the norm a/ on X ®Y coincides with the quotient norm
defined by the operator I ®P: X ®a£\(I) -> X ® Y.
(b) If Y is a Li^x-space for every A > 1, then the norms a and a/ coincide
on X ®Y for every Banach space X.

7.2 Injective and Projective Associates 169
(c) For every Banach space X and every finite, positive measure (i, the norms
a and a/ coincide on X ® L\ (p).
Proof, (a) Since I ®P: X ®a/ ti (I) -> X ®a/ Y is a quotient operator and
ti (I) is a £.i.A-space for every A > 1, the result follows from (b).
(b) follows from the Lemma.
(c) follows from (b), since Li(fi) is a £.|.vspace for every A > 1. Q
Thus far, we do not have many examples of totally accessible tensor
norms. Our next result will broaden our knowledge considerably.
Proposition 7.13. If a is a left accessible tensor norm then its right
injective associate a\ is totally accessible.
Proof. Let X and Y be Banach spaces. Let C(K) contain Y as a subspace.
Then X ®a\ Y is a subspace of X ®a C(K). Furthermore, since a is left
accessible and C(K) has the metric approximation property, X ®a C{K) is
a subspace of (X* ®a. C(K)*y.
On the other hand, C(K)* is a .Ci.vspace for every A > 1 and so X* ®ai/
Y* is a quotient space of X* ®a. c{K)*. But X* ®a,/ Y* = X* ®ay Y*.
Therefore X* ®ay Y* is a quotient of X* ®a> C(K)* and it follows'that
(X* ®ay Y*)* is a subspace of (X* ®a» C(K)*y.
The following commutative diagram summarizes these facts:
X®aXY * (X*®ayY*y
x ®a c(K) ► (x* ®a. c{Kyy
We have shown that all the solid arrows are isometric embeddings. Therefore
the canonical mapping X ®a\ Y -> (X* ®ay Y*)* is also an isometric
embedding. This completes the proof. ..- O
A moment's thought will reveal that there is no corresponding result for
the projective associate.
We can now generate many accessible norms:
Corollary 7.14. Let a be a tensor norm. Then the tensor norms /a\, (\a)\
and /(a/) are all totally accessible.
Now gp, being left projective, is left accessible. Therefore, we have:
Corollary 7.15. Let 1 < p < oo. The tensor norms gp\ and /dp are totally
accessible.
The case p = 2 is special since, by Corollary 6.25, (£2 is already left
injective. Therefore:
Corollary 7.18. The tensor norms d> and Q2 are totally accessible.

170
7 Tensor Norms
7.3 The Chevet-Saphar Dual Norms and p-integral
Operators
We are now ready to find the duals of the Chevet-Saphar tensor norms. The
identification of the dual norms will lead to a new class of operators.
Proposition 7.17. Let 1 < p < co and let X be a Banach space. Then
Pi®grX = OyC.x) for every n.
Proof. Each tensor u = "jfj™ , z} ® x} G £? ® "K corresponds to the operator
u: t^ -t X given by u(y) = 51™=i(zj>y)x}- Since ££, is finite dimensional,
every operator from this space is p-summing and so the spaces £J ®9p X and
•Pp^exn-^O coincide algebraically.
We show first that wp(u) < gp(u). To see this, take a representation u
as above. We may assume that z} ^ 0 for every j. Now let (j/*) be a finite
sequence in ££,. Then
fc fc }
For each fc, let tpk € Bx. satisfy
E<zJ'J'*)a'j =Vfc(E^,5"fc)a;j) =E(zi'5'*)(a;J'^)-
Writing z} = ||2,p, with p,|| = 1, we have, with the obvious modifications
for the case p = 1,
Eh^)iip=E(E^'^)^'^))p
^e(ei^-^)ip)(ei^>^)ip')
<EEi^>^)ip(ii(^)iip')p
=Ei^npEi^'^)ip(ii(^)iip')p
<(ll(^)llpll(^)llp'll(yfc)llp3)P-
This holds for every representation of u and every finite sequence (j/*) and
so, by the definitions of these norms, irp(u) < gp(u).
To prove the reverse inequality, we apply the Pietsch Domination Theorem
(Theorem 6.18) to the operator u, taking K to be the set {ei,..., en} of unit
basis vectors in ££. Therefore there exist nonnegative real numbers fix,..., fin
such that "jTJJl, A*» = 1 and
p/p'

7.3 p-integral Operators 171
My)llp<Tp(«)p£>.|y.r
t=i
for every y G ££,. For each t, let z* = fi] e, and Xi = f)tu(ex), where
A = K1/P if /*. "J4 0 and Pi = 0 if & = 0. Then u(y) = £"=, V»u(e.) =
53JL,(z,,y)a;, for every y G £"£,. We estimate gp(u) using the representation
ET-i * ® *.: we have ||(*,)||p = (£,w)1/p = 1 ™*
||(*.)II?=««p{ E^ :£l«.lP<l}
= sup{||U(^a,Ae,)|:^|a,|"<lJ
< 7rp(U)sup{]T>,|a,|'-tf' = £l<*.lP ^ -1} =M")-
* » »
Therefore 9p(u) < ||(*.)||P||(«.)||J5 < *p(«0- , 0
There is also a dual result in finite dimensions:
Corollary 7.18. Let 1 < p < co and let E be finite dimensional. Then
E ®g, C = {E* ®dp, <?)* = W.O /<"- e«ery n.
Proof. The transpose u »-). u' is an isometric isomorphism between X ®9p Y
and y ®d,, X. Hence E ®9p ££, is isometrically isomorphic with P^ ®dp E.
But
C ®«ip ^ = 3V(C £*)* = (<? ®V S*)*
by the above proposition. Applying the transpose again to the last space in
this sequence gives the space (E* ®d , fj)* = 7P{E*, ££,). Hence i5®Jp^ =
At this point, it will be helpful to interchange the'roles of the conjugate
indices p and p' and view this result as a statement about the dual of the
tensor norm dp in a special case. Thus, we have
E ®d'r C = {E* ®dr <?)* = E ®9r C
for every n and every finite dimensional space E. We now extend this
statement to infinite dimensions. Every C(K) space is a -Coo.A-space for every
A > 1. Therefore, since d' and gp are tensor norms, we have:
Proposition 7.19. Let 1 < p < co and let X be a Banach space. Then the
norms d'p and g? coincide on X ® C(K) for every K.

172 7 Tensor Norms
Now the norms dp are right projective and so the dual norms dl are right
injective. Proposition 7.9 tells us that the right injective associate normpp\ on
any tensor product X®Y is induced by the embedding X®Y -+ X®8rC(K),
where C(K) contains Y as a subspace. Putting these facts together with the
above result, we obtain an identification of the dual norm of dp-.
Theorem 7.20. Let 1 < p < co. The dual of the tensor norm dp is the right
injective associate norm gp>\.
Transposition gives a corresponding statement for gp, namely, g^ = /dp>
and, taking duals, gp = (/dp')'. Pursuing this a little further, we see that
0p = (AV)' = \(4') = \(3p\)-
On the other hand, gp is left projective, and so (\</p)\ = gp\- Summarizing,
we have
\(ft,\)=9P and (\9P)\ = 9p\. (7.6)
The case p = 2 is particularly pleasant. We have seen that gi is right
injective. Therefore
4=32 and ff2 = d2- (7.7)
The identification of the duals of the Chevet-Saphar norms enables us
to establish their accessibility. We have shown in Corollary 7.15 that gp\ is
totally accessible. Therefore, by Proposition 7.4:
Proposition 7.21. The tensor norms gp and dp are accessible for every p.
We now address the question of describing the dual space of X ®d> Y
in terms of a space of operators. It will be convenient to work with the
space X ®g> Y; the dual space for the norm d'p can be obtained simply by
transposing X and Y.
Let J be the canonical embedding of X into C(Bx-)- Then, by
Proposition 7.9, X®Sp Y = X®/d , Y is a subspace of C(BX.) ®dp, Y. If T € Z (X, Y*)
is a bounded linear functional on X ®gi Y then, by the Hahn-Banach
Theorem, T may be extended to a bounded linear functional on C(Bx-) ®d , Y
with the same norm. Denoting the extension by 5, we have the following
factorization of T:
C(BX.)

7.3 p-integral Operators 173
Now S, belonging as it does to the dual space of C(Bx-) ®<* , Y, is a p-
summing operator from C(Bx') into Y* and the p-summing norm of S is
equal to the norm of the linear functional T. By Proposition 6.22 there exists a
regular Borel probability measure \i on Bx- and an operator U: Lp{(i) -» Y*
such that S = UJP and 7rp(S) = \\U\\. We now have a factorization of T:
Y*
C(BX.)
U
Lp(fi)
Furthermore, the norm of T as an element of the dual space (X ®9. Y)*
is equal to \\U\\. When p = 1 this is simply the description of an integral
operator and so the following terminology is appropriate. We shall say that
an operator T: X -» Z is a p-integral operator if the corresponding bilinear
form on X x Z* defines a bounded linear form on X <gy Z*. The p-integral
norm of T, which we denote by ip(T) is defined to be the norm of this
functional. We denote the space of p-integral operators, equipped with this
norm, by DP(X, Z).
Theorem 7.22. Let 1 < p < oo and let X and Y be Banach spaces. An
operator T: X -> Y is p-integral if and only if there exists a compact space
K, a regular Borel probability measure p on K and operators S: X -> C(K)
and R: Lp(fi) -> Y** such that JT = RIpS, where J is the canonical
embedding of Y into Y** and Ip is the canonical mapping of C(K) into Lp(fi).
Furthermore, the p-integral norm ofTis given by
•p(r)=inf||5||||fl||,
where the infimum is taken over all such factorizations, and this infimum is
attained: \
S
C(K)
R
Lp(f)
Proof. Suppose that T is a p-integral operator. Then the argument given
above shows that the operator JT has a factorization of the form given in
the statement of the proposition, with ip(T) = ||i2||||S||-

174 7 Tensor Norms
Conversely, suppose that JT has such a factorization. We may factor the
canonical mapping of C(K) into Lp(n) through the canonical mapping from
Loo(fi) in Lp(fi), which we also denote by Ip. We now have the following
diagram:
Si
Loo(li)
R
Lv(y)
where ||Si|| = ||S||. By the injectivity of L^p), the operator Si extends to
an operator S2: C(Bx') —"> £oo(/*) with the same norm as S. This yields the
following factorization of JT:
C(BX.)
R
Lp(ii)
where J: X -» C{BX-) is the canonical embedding and V = IpX2. Then V is
ap-summing operator with irp(v) < \\S2W = ||S||- Therefore the operator RV
defines a bounded linear functional on C{BX') ®d , Y* w»th norm at most
wp(RV) < \\R\\irp(V) < ||fl||||V||. Taking the restriction of this functional to
the subspace X®g> Y* of C(Bx')®d , Y*, we see that T belongs to the dual
space (X ®gi Y*)* and its norm is at most ||iZ||||S||. O
We now have a representation of the dual space of X ®g' Y:
(X®g,YY = 3p(X,Y-).
Furthermore, we have an isometric embedding of 3p(X,Y) into 3p(X,Y**)
and this give the canonical embedding:
3p(X,Y)C(X®g,pY*y.
The p-integral operators share many of the properties of the integral (or 1-
integral) operators. We summarize some of them in the following proposition.
The proof is left to the reader.
Proposition 7.23. Let T: X -*Y be an operator.
(a) If J is the canonical embedding of Y into Y** then T is p-integral if and
only if either of the operators JT, T** is p-integral. The p-integral norms
of T, JT and T" are the same.

7.3 p-integral Operators 175
(b) If T is a p-integral operator then for all operators R: W -» X and
S:Y-*Z, the operator STR is p-integral and
ip(STR)<\\S\\ip(T)\\R\\.
(c) T is p-integral if and only if there exists a positive constant C such that for
every finite dimensional sub space E of X and every finite codimensional
subspace F of Y, the finite dimensional operator
QFTIE: E -» X -» Y -» Y/F
satisfies ip{QFTlE) < C. Furthermore, ip(T) = inf C, where the infimum
is taken over all such pairs E, F.
It is interesting to compare the classes of p-integral and p-summing
operators. The factorization schemes that characterize these classes tell the whole
story. First, for the p-integral operators:
T
X » Y Y**
C(K) ► Lp{y)
And for the p-summing operators:
X ^— Y <■ EooiBy)
C(K) ► L» {
The only difference is that in the second case, the canonical embedding of Y
into Y** is replaced by the embedding of Y into the associated injective space
too(By) (or any injective space that contains Y). It follows immediately that
every p-integral operator is p-surnming and that
In general, these classes are not the same. For example, we have seen that
the injection of l\ into l<i is 1-summing, but, bearing in mind that li has the
Radon-Nikodym property, one can see that this operator is not 1-integral.
There are other significant differences between the p-integral and p-summing
operators. The p-surnming nature of an operator is not altered, nor is its

176 7 Tensor Norms
p-summing norm, if the range space is enlarged. This is clearly not the case
for a p-integral operator.
We conclude with an examination of the consequences of the accessibility
of the Chevet-Saphar norms.
Proposition 7.24. Let 1 < p < oo and let X and Y be Banach spaces. If
X* or Y has the metric approximation property, then the space of p-nuclear
operators from X into Y is isometricaily isomorphic to X*®SrY and the
p-nuclear and p-integral norms coincide on this space.
Proof. By the accessibility of gp, the canonical mapping from X* ®3p Y into
(X ®gi Y*)* = 3P(X, y**) is an embedding. Hence this mapping extends to
an embedding of the completed tensor product X*®gpY into 0P{X, Y**). But
each element of X*®grY takes its values in Y. The result-now follows from
the fact that 3P(X, Y) is a subspace of 3p(X, Y"). 0
7.4 The Hilbertian Tensor Norm
In this section, we find a concrete representation of the biggest injective
tensor norm, /7r\. We take the following approach to this problem. By
Proposition 7.9, the norm-/7r\ is induced" on X ® Y by embedding X and Y into the
projective tensor product of suitable C(K) spaces. For example, we have
X®h\YcC{Bx.)®vC(BY.).
Thus, if we can find an injective tensor norm, a, that is equivalent to the
projective norm on tensor products of C(K) spaces, then this norm will be
equivalent to /7r\ in general. Purthermorel by the finitely generated nature of
tensor norms and the fact that every C(K) space is a .Goo.A-space for every
A > 1, it will suffice to show that a is equivalent to w on £^o ® ^x> f°r every n,
provided, of course, the equivalence is uniform in n. In other words, we seek
a relation of the form Ca < w < Da on ££, ® ££,, with C and D independent
of n.
One of the consequences of Grothendieck's Inequality (Theorem 6.31)
states that 7r < KGd2 and w < KGg2 on ££, ® ££,. However, d2 is only
injective on the left, while g2 is only right injective. A little hopeful guesswork
suggests the following definition. We define the norm w2 on the tensor product
X ® Y of any two Banach spaces by
t»2(u) = inf{||(*,)||?||(y,)||? : u = £>, ® Vj\ .
We leave it as an exercise to-the reader to verify that w2 is a tensor norm.
For reasons that will soon be apparent, w2 is known as the Hilbertian tensor
norm. We now show that this norm satisfies one of our requirements.

7.4 The Hilbertian Tensor Norm 177
Theorem 7.25. The tensor norms w and vi2 satisfy u.2 < n < Kq v>2 on
Plo ® Plo for every n.
Proof. We prove the real case first. Let u € Plo ® Plo- To compute the
projective norm of u, let B be a bilinear form on ££, x ££, of unit norm. Thus
B(s,t) = Y^ijQijSit}' where
sup< praysiti = |*<|, |t,| < 1J = 1 •
I*4 !CfcLi xk ® V* be a representation of u and for each t, let «j = (a:*,)* and
vi = (yfci)fc- Then
\(u,B)\
k tj
^^OtjXkiyk, = ^aij^xkiykj = ^^(t^T.,)
«J fc
•J
Now ||u,-||2 = (Efckfci|2),/2 = (Efcl(^,ei)|2),/2 < ll(*OII? and similarly,
IKIh < IKl/OII™ f°r eacn *• Therefore, by Grothendieck's Inequality,
\{u,B)\<KG\\(xk)\\^\\(yk)\\^.
It follows from the definition of u.2 that \{u, B)\ < Kq u.2(u) and hence
w(u) < Kaw2(u).
For the complex case, consider the tensor u' — J2k Xk ® yH> wnere VH =
(yifcj)- It is easy to see that u' is well defined and that n(u') = n(u) and
W2(u') = u/2(w). Finally, imitating the proof for the real case gives n(u') <
Kq u>2(u). where Kq is now the complex Grothendieck constant. O
From the argument outlined above, we see that:
Corollary 7.26. u.2 < ir < Kq u>2 on every C(K) space.
Our next task is to show that u.2 is injective. Since W2 is finitely generated,
it will be enough to consider finite dimensional spaces.' If E and F are finite
dimensional and $3t=i xk®yk is a representation of u £ E®F, we may define
operators S: E* -> £J and R: (% -> F by Sip = (<pxk) and R\ = "££_, Afcyfc.
Then ||(a:fc)||2 = ||5|| and \\(yk)\\^ = IW K we view u as a operator from
E* into F, then u is the composition .RS.
On the other hand, suppose the operator u: E* -» F is factored through
some £j, so that there exist operators S: E* -> ££ and Jl: £5 -> F such
that u = .RS. Then there exist xk £ E and yk £ F such that Sip = (pa:*)
and RX = 53t=i -^V* and it follows that J2k=i Xk ® J/fc is a representation
of u. It is easy to see that the space £5 can be replaced by any larger Hilbert
space without altering the norms of the operators R and S, or the fact that
u = RS. Thus, we have proved the following striking description of the tensor
norm 102 in finite dimensions.

178 7 Tensor Norms
Proposition 7.27. Let E and F be finite dimensional Banach spaces and
let u G E® F. Then W2(u) = inf ||fl||||S||. where the infimum is taken over
all the factorizations of the operator u: E* -> F through a Hilbert space.
The "Hilbertian" nature of W2 exemplified by this result enables us to
prove its injectivity.
Theorem 7.28. The tensor norm w2 is injective.
Proof. It suffices to prove that w2 is injective for finite dimensional spaces.
Suppose that the Banach spaces E and F are subspaces of the finite
dimensional spaces Af and N respectively. Then
w2(u; Af ® N) < w2(u; E ® F)
for every u € E ® F. To prove the reverse inequality, let R\S\ be a
factorization of u through a finite dimensional Hilbert space H\, where u is considered
as an operator from Af * into N. Let P denote the quotient operator from Af *
onto E*. Consider the projection Q of Hi onto the orthogonal complement
of 5i(kerP). We may define an operator S from E* into Q(Hi) by setting
S(Pip) = QS^ip). Then ||5|| < ||Si||. Let H be the closure of S(E*) and let
R be the restriction of Rt to H, so that \\R\\ < \\Ri ||. Then RS is a
factorization of u through the Hilbert space H, and ||fl||||5|| < ||fli||||Si||. It follows
that w2(u;E®F) <w2(u;M ® N). □
Combining these facts about the Hilbertian tensor norm, we have:
Theorem 7.29. w2 < /ir\ < Kqw2.
Taking duals, we get a description of the biggest projective tensor norm:
Corollary 7.30. K^1 w'2 < \e/ < v/2.
It will be of great interest to determine the classes of operators generated
by w2 and w'2. Although we do not yet have an explicit formula for the norm
w'2, we already have enough information at hand to determine the dual space
of X ®wt Y. We use the following general principle, which is nothing more
than the definition of the property of finite generation of a tensor norm: if a
is a tensor norm, then an operator T £ &(X,Y*) defines a bounded linear
functional on X ®a Y if and only if there exists a constant C such that, for
every finite dimensional subspace E of X and every finite dimensional sub-
space F of Y, the finite dimensional operator QjtTIe'- E -> F*, considered
as an element of (E ®a F)* = E* ®a. F*, satisfies a'(QFTlE) < C:
T
X = »Y*
Ie
Qf
E
*F

7.4 The Hilbertian Tensor Norm 179
Furthermore, the norm of T in (X ®a Y)* is the infitrmm of the set of real
numbers C that satisfy this condition.
Thanks to this principle, the proof will reduce to the problem of patching
together a family of finite dimensional Hilbertian factorizations of an operator
to construct a factorization of the operator itself. For this, we need some
topological machinery. Let K be a compact topological space and let (a:,),ej
be a family of elements of K, indexed by the set J. If U is an ultrafilter on J,
then the limit limu xt of (a;,) with respect to U exists in K.
Theorem 7.31. Let X and Y be Banach spaces. An operator T: X -> Y*
defines a bounded linear functional on X ®„,/ Y if and only if there exists
a Hilbert space H and operators S: X -» H and R: H -» Y* such that
T = RS. Furthermore, the norm ofT as an element of (X ®w>a Y)* is the
infimum o/||iZ||||5||, ranging over all such factorizations ofT.
Proof. Suppose that T has such a factorization. Let u € X ® Y and let E,
F be finite dimensional subspaces of X, Y respectively such that u £ E®F.
Since (E ®w>3 F)* = E* ®W3 F*, it follows from Proposition 7.27 that
|(u,r)|<||i*||||S|K(u;E®F).
Therefore, taking the infirmim over E and F, we have
|(u,r)|<||i*||||S|K(u;X®y).
Conversely, let T £ (X ®wi Y)*. Then there exists a constant C such that
for every pair, E, F of finite dimensional subspaces of X, Y respectively,
there is a factorization RefSef °f T through a Hilbert space Hef> with
HflBFllllSBFll < C. We may assume that \\Ref\\ < 1 and ||5£,F|| < C for
every E, F:
T
X ► Y* ,.
\Sfj? Ref/
Hef
Let H be the ^-direct sum of the Hilbert spaces Hef- We propose to factor
T through a subspace of H. The factoring operators will be defined at each
point by taking the limit with respect to a suitable ultrafilter.
We take as indexing set the set / of all pairs (E, F), where E and F are
finite dimensional subspaces of X and Y respectively. For each (E,F) € I,
let T(E,F) be the set of all pairs (M,N) for which E C M and F C N.
These sets form a base for a filter; let U be any ultrafilter on J that contains
this filter.

180 7 Tensor Norms
Fixing x € X, consider the family in H denned by
\Sefx if x £ E,
VEF=\0 if x*E.
By the uniform boundedness of the operators Sef, the family (vef) lies in
a weakly compact subset of H and so it has a weak limit with respect to the
ultrafilter U. We define 5a; to be this limit. We leave it as a routine exercise
to the reader to verify that S is bounded, with ||5|| < C, and that HefSx =
Sefx whenever x G E, where Uef denotes the orthogonal projection of H
onto Hef-
We now define an operator R from the subspace S(X) of H into Y*. Let
v G S(X), so that v is the weak limit of a family (vef) as defined above,
for some x £ X. Then the family (Rvef) >s bounded, and so we may define
Rv to be its weak* limit in Y* with respect to II. Once again, we leave it to
the reader to verify that R is a well defined operator with ||-R|| < 1, and that
RSx — Tx for every x G X. The factorization is completed by taking ff0 to
be the closure of S(X) in H and extending R uniquely to Hq:
Y*
Finally, the assertion about the norm of T follows immediately from the
proof. O
An operator T: X -» Y is said to be a HUbertian operator (or a 2-
factorable operator) if there exists a Hilbert space H, together with operators
S: X -» H and R: H -» Y such that T = RS. The HUbertian norm of T is
defined by
||r||„=inf||fl||||5||,
the infimum being taken over all such factorizations of T. The observations
made about the p-integral operators in Proposition 7.23 also apply to the
Hilbertian operators, with the p-integral norm being replaced by the Hilber-
tian norm. In particular, we note that an operator T: X -> Y is Hilbertian
if and only if the operator JT: X -> Y** is Hilbertian and, furthermore, the
Hilbertian norms of T and JT are the same. We denote the Banach space of
Hilbertian operators from X into Y, with the Hilbertian norm, by L^(X, Y).
We may also define the class of Hilbertian bilinear forms. We say that a
bilinear form B £ "B(X x Y) is a Hilbertian bilinear form if the corresponding
operator T: X -> Y* is Hilbertian, and we define the Hilbertian norm, \\B\\ot

7.4 e ertian r orm
to be ||r||3{. We denote the Banach space of Hilbertian bilinear forms, with
the Hilbertian norm, by "Bx(X x Y).
We may now summarize the above proposition with the following
canonical identifications:
(X 9^ YY = 2X(X xY)= Loi{X, Y*).
We now turn our attention to the dual space of the Hilbertian tensor
product X ®u^ Y. Once again, Hilbert spaces play a central role.
Theorem 7.32 (Kwapien Domination Theorem). Let X and Y be
Banach spaces. The following are equivalent for a bilinear form B on X x Y:
(i) B defines a bounded linear functional on X ®W3 Y.
(ii) There exists a regular Borel probability measure \i on the product space
Bx' x By of the weak*-compact unit balls and a constant C such that
\B(x,y)\<c(f \<p(x)\2dA (7 My)\2df*)
\Jbx.xBy- / \JBx-xBy* /
f
for every x e X, y &Y.
(iii) There exists a Banach space Z and a pair of 2-summing operators
R: X -» Z and S.Y -» Z* such that
B(x,y) = (Rx,Sy)
for every x 6 X,y &Y.
Furthermore, the norm of T as a linear functional on X <8)„,a Y is given by
the infimum of C or the infimum of ^(fy^iS)•
Proof, (i) implies (ii): if B belongs to (X®Wa Y)* then, by the definition of
the norm u>2, there exists a constant C such that
EB(«„»,)|<C||(*,)||J||(y,)||?
Kn \1/2/ n \1/2 1
£l^(^)l2J (£My;)l2J : f e*x; * eby.I
for every pair of finite sequences (a:,), (y,) in X, Y respectively. Applying
the inequality ab < (a2 + b2)/2 where a and 6 are nonnegative real numbers
gives

182 7 Tensor Norms
for all finite sequences (a:,), (y:). We now use the same idea as in the proof
of the Pietsch Domination Theorem (Theorem 6.18) to find the measure fi.
Thus, working in the space C(BX' x By), consider the set L consisting of
the functions of the form
fAM)= £ *(*.»)-§ £ (l^)l2 + IV-(y)l2).
(x,v)eA (x,v)eA
where A is a finite subset of X x Y. Then L is a convex set and every function
in L takes at least one nonpositive value. Thus L is disjoint from the open
positive cone P of C(BX- x By) and so there exists a regular Borel measure
fi that separates L and P. Arguing exactly as in the proof of the Pietsch
Domination Theorem, we may assume that \i is a probability measure and
that (fA,li) < 0 for every Ja 6 L. Taking A to be a singleton set {(x,y)},
we get
\B(x,y)\<%([ \<p(x)\2dft+ f |V(y)|2^)
* \JBx.xBY- Jbx-x.By- /
(7.8)
for every x, y. In order to obtain the estimate given in (ii), we use the fact
that, if a and b are nonnegative real numbers, then
a& = mf{(t2a2-M-V)/2:t>0}.
Replacing (x,y) by (tx,t~ly) in (7.8) and taking the infimum over t, we
obtain (ii).
(ii) implies (iii): suppose that B satisfies (ii). Let Ix '• X -» Lvifi) and
Iy.Y-t L2(fi) be defined by Ix^f, *l>) = f{x) and Iwif, *l>) = V'(y)- These
operators, factoring as they do through the canonical mapping of C{BX- x
By) into I>2(l0. are 2-summing, with 2-summing norms at most 1. Let B'
be the bilinear form on Ix(X) x Iy{Y) defined by B'(IxX,IYy) = B(x,y).
Then B' has norm at most C and, since Z<2(/*) is a Hilbert space, B' may be
extended trivially to a bilinear form on Lvifi) x Li{n) with the same norm. Let
U: L2(fi) -¥ L2(fi) be the associated operator that satisfies (Ixx, U(Iyy)) =
B(x,y). Taking Z = L2(fi), with the operators S = Ix and R = Uly gives
the desired conclusion.
(iii) implies (i): suppose that B satisfies (iii). Then, if $3™ , xj ® Vj ls any
representation of u 6 X ® Y, we have
<tJt\\B*,f) (Ell^ll2) <**(R)*2(S)M*imMvM?
and it follows that B defines a bounded linear functional on X ®wa Y with
norm at most ^2{R)^2{S). O

7.4 The Hilbertian Tensor Norm 183
We remark that the Banach space Z that appears in statement (iii) of
this theorem may be taken to be a Hilbert space. This should not surprise us,
as every 2-summing operator can be factored through a 2-summing operator
into a Hilbert space.
A bilinear form B on X x Y is said to be 2-dominated if it satisfies the
equivalent conditions of this proposition. The 2-dominated norm of B is the
smallest constant that satisfies
< 82(B) ||(*,)||?||(y,)||3
for all finite sequences (a:,), (y}) in X, Y respectively. "We shall denote the
Banach space of 2-dominated bilinear forms with this norm by 1)2(X x Y).
We say that an operator T: X -» Y is a 2-dominated operator if the
corresponding bilinear form B on X x Y* is 2-dominated. It is not difficult to see
that this is equivalent to the existence of a constant C such that
< cu^mw^m
for all finite sequences (re,), (rj)}) in X, Y* respectively. The 2-dominated norm
of T, denoted by S2(T), is defined to the infimum of these C or equivalently,
the 2-dominated norm of the bilinear form B. We denote the Banach space
of 2-dominated operators by "D2(X, Y).
The following characterization of 2-dominated operators is just a
reworking of Proposition 7.32; we leave the proof as an exercise to the reader.
Theorem 7.33. Let X and Y be Banach spaces. An operator T: X -> Y
is 2-dominated if and only if there exists a Hilbert space H, a 2-summmg
operatorS: X -» H and an operator R: H -» Y whose adjoint R*: Y* -» H*
is 2-summing, such that T = RS. The 2-dominated norjn ofTis given by
S2(T) = Mw2(R')n2(S) \
where the infimum is taken over all such factorizations ofT.
We have a satisfactory description of the dual of a Hilbertian tensor
product:
(X ®W7 YY = V2(X x Y) = T>2(X, Y*).
Suppose we begin with the projective norm and generate new tensor
norms by taking the dual norm, forming left or right injective or
projective associates, and transposing. A tensor norm is said to be a natural tensor
norm if it can be obtained from tt by a sequence of these operations, applied
in any order. It is a remarkable theorem of Grothendieck that there are, up
to equivalence, only fourteen natural tensor norms. They are shown in the

184 7 Tensor Norms
following diagram. An arrow from a to P indicates that there is a constant C
such that a < Cp and a ~ P means that a and P are equivalent. The reader
will find it entertaining and instructive to verify all the relations expressed
in this diagram:
\(M = \g'o
/T=<4
(M)/~d2~/(W)
e/ = da
(e/)\ = 4»\
(t\)/ = 0
t\ = <c
\(M)~p2~(W)\
\e=Po
/(V) = /Po
7.5 Exercises
Exercise 7.1. Show that the Schatten dual, a3, of a tensor norm a is a
uniform crossnorm.
Exercise 7.2. Prove directly that the injective norm e is cofinitely
generated.

Exercise 7.3. Show that if X and Y have the metric approximation
property, then every uniform crossnorm is cofinitely generated on X ® Y.
Exercise 7.4. Let a and j3 be tensor norms. Show that a < 0 if and only if
a' > p'.
Exercise 7.5. Show that a tensor norm a is accessible if and only if a = a
on X ® Y whenever either X or Y is finite dimensional.
Exercise 7.6. Let A be a non-empty set of tensor norms with least upper
bound p. Explain why f3(u; X ® Y) is not in general given by the formula
suPae^ a(u> % ® ^) f°r ^ pairs of Banach spaces X, Y and all « 6 X ® Y.
Exercise 7.7. Let C be the collection of all cofinitely generated tensor
norms. What is the least upper bound of C?
Exercise 7.8. Show that /(a\) = (/a)\ and \(a/) = (\a)/ for every tensor
norm a.
Exercise 7.9. Let a be a left projective tensor norm. Show that a\ need
not be left projective.
Exercise 7.10. Show that, if an operator T is p-nuclear, then T is p-integral
and that tp(r)<||T||Np.
Exercise 7.11. Show that every 2-summing operator T is 2-integral, and
that 7r2(r)=t2(r).
Exercise 7.12. Show that an operator T: X -> Y is co-integral if and only
if T has the following property: for every Banach space Z that contains X
as a subspace, the operator JT: X -¥ Y" can be extended to an operator
from Z into Y". What is the norm tcx,(T)?
Exercise 7.13. Show that e/ = doo-
Exercise 7.14. For 1 < p < oo, let wp be defined on X ® Y by u>p(u) =
"tf IK^jJIlp ll(yj)llp'. where the infimum is taken over all representations of «.
(a) Show that wp is a tensor norm.
(b) Show that for every p there is a constant Cp such that wp < CPW2-
(Hint: every representation "£") Xj ® yy of « can be rewritten as
^1(Er'W«,)®(ErjWw)dt-
Use this and the Khinchine Inequality to estimate u.p(u).)
Exercise 7.15. Show that w2 < w'2-
Exercise 7.16. Show that \«.2 is equivalent to </2.

8. Operator Ideals
In this chapter we study the spaces of bilinear forms and operators that can
be generated from a tensor norm. For each tensor norm a, we have the a-
integral and the a-nuclear forms or operators. We introduce the concept of a
Banach operator ideal and we develop just enough of the theory to explain
the relationship between tensor norms and operator ideals. In particular, we
see that the a-integral and a-nuclear classes constitute the maximal and
minimal ideals respectively.
8.1 The Forms and Operators Associated with
a Tensor Norm
Let a be a tensor norm. If E and F are finite dimensional Banach spaces,
then E* ®a F* is a space of bilinear forms on E x F. This space can also be
represented as a dual:
E'®a F* = (E ®a> F)\
These two ways of associating a class of bilinear forms with a tensor norm
lead to different spaces in infinite dimensions.
If X and Y are Banach spaces, we say that a bilinear form B oa XxY is
an a-integral bilinear form (or that B is of type a) if J? belongs to the dual
space (X ®a> Y)*. The norm of B in this dual space is denoted by ||B||a and
will be referred to as the a-integral norm of B. We denote the Banach space
of a-integral bilinear forms by "Ba(X x Y). Thus
"Ba(X x Y) = {X ®a> Y)* .
We have plenty of examples to hand. If a is the projective norm w, we have
1ic(X x Y) = (X ®e V)*, which is the space of integral bilinear forms on
X x Y. This example explains the terminology we have chosen. On the other
hand, the e-integral bilinear forms on X x Y are simply the bounded bilinear
forms, since (X ®„ V)* = "B(X x Y). These represent the two extremes;
clearly, the integral bilinear forms are a-integral for every tensor norm a,
and, of course, every a-integral form is bounded.

188 8 Operator Ideals
We have seen that the Hilbertian norm plays a central role in the theory
of tensor norms. Theorem 7.31 shows that the ^-integral bilinear forms are
given by
VW3(XxY) = Vx(XxY),
the Hilbertian forms. And the forms associated with the dual norm are the
2-dominated bilinear forms:
^(X x Y) = V2(X x Y).
The norm induced on the subspace X* ® Y* of "Ba(X x Y) by the a-
integral norm is the associated norm a = (a1)" defined in Section 7.1. We
recall that a < a, so that the canonical mapping of X ®a Y into "Ba(X x Y)
has unit norm. We have seen that there are three situations in which the
norms a and a coincide on X* ® Y*:
(a) Both X* and Y* have the metric approximation property;
(b) a is totally accessible;
(c) a is accessible and X* or Y* has the metric approximation property.
In these cases, the completed tensor product X*®aY* is a subspace of the
space "Ba(X x Y). We shall return to this question shortly.
We now collect together some properties of a-integral bilinear forms that
have been established in Chapters 6 and 7.
Proposition 8.1. Let a be a tensor norm.
(a) A bilinear form B on X x Y is a-integral if and only if there exists a
constant C such that
\\Bo(IExIF)\\a<C
for every pair of finite dimensional subspaces E, F of X, Y respectively.
(b) If S: W -> X and T: Z -t Y are operators and B e "Ba(X x Y), then
the bilinear form B o{S xT) on W x Z is a-integral, and
l|Bo(Sxr)||a<||S||||r||||B||a.
(c) If B is an a-integral bilinear form on X xY, then the canonical left and
right extensions of B to X" x Y** are both a-mtegral.
Statements (a) and (b) are nothing more than the tensorial properties of
a', while (c) is Theorem 6.5. We recall that the canonical extensions of the
bilinear form B on X x Y are defined as follows. Let T: X -> Y* be the
corresponding operator, so that B(x, y) = (y, Tx) for every (a;, y) 6 X x Y.
Then the canonical left extension of B is defined by
•B(x",y") = (T*y",x")-

8.1 Associated Bilinear Forms and Operators 189
On the other hand, we may also associate an operator S: Y —> X* with
B, satisfying B(x,y) = (x,Sy). The canonical right extension of B is then
defined by
B*(x",y") = (S'x",y").
It is not difficult to see that *B = B* on X" x Y and on X x Y**. Exercise 6.2
gives another viewpoint on these extension processes.
These observations are helpful in analysing the operator versions of the
a-integral bilinear forms. Let T: X —> Y be an operator and let B be the
corresponding bilinear form on X x Y*, so that
B(x,iP) = (Tx,ip)
for every x 6 X and ip &Y*. The operator from X into Y** that corresponds
to this bilinear form is JyT, where Jy is the canonical embedding of Y into
y**. Conversely, starting with a bilinear form B on X x Y, we obtain an
operator T from X into Y* which in turn is associated with the bilinear form
*B = B* on X x Y". We shall say that the operator T: X -> Y is an a-
integral operator if the associated bilinear form B on X x V* is an a-integral
bilinear form. The a-integral norm of T, denoted by ||T||a, is defined to* be
the a-integral norm of B. The facts outlined above prove the following result.
Proposition 8.2. The following are equivalent for an operator T: X -» Y:
(i) T is an a-integral operator.
(ii) JyT: X —> T** is an a-integral operator.
(iii) T" is an a-integral operator.
Furthermore, the a-integral norms ofT, JyT and T** are the same.
In particular, we have the canonical embeddings
£,„(*.Y) C (X ®a> Y*)* C La{X, Y*') ■
The examples considered above translate immediately into the setting
of operators. Thus, the a-integral operators in the cases a = e and 7r are
the integral and the bounded operators respectively. The p-integral and p-
summing operators are also a-integral:
LSr(X,Y) = 3p(X,Y)
and
L„\(X,Y)=Vp(XtY).
And, of course, our previous identifications of the classes of u>2- and w'2-
integral bilinear forms may also be couched in terms of operators:
LW2(X,Y) = lx(X,Y)

190 8 Operator Ideals
and
HW'2(X,Y) = V2(X,Y).
Each property of a-integral bilinear forms has a counterpart for the a-
integral operators. We leave the proof of the following proposition as an
exercise.
Proposition 8.3. Let a be a tensor norm.
(a) An operator T: X —» V is a-integral if and only if there exists a constant
C such that, for every finite dimensional subspace E of X and every
finite codimensional subspace F of Y, the finite dimensional operator
QfTIb- E -» Y/F satisfies
\\QFTIE\\a<C
(b) Let T: X -» Y be an a-integral operator and let S: W -» X and R: Y -»
Z be operators. Then the operator RTS is a-integral and
||flrs||a<||fl||||r||a||S||.
The following result provides a useful test for an operator to be a-integral.
Theorem 8.4. An operator T: X -» Y is a-integral if and only if the tensor
product operator T ® I: X ®a< Y* -» Y ®„ Y* is bounded, where I is the
identity operator on Y*. Furthermore, the a-integral norm ofT satisfies
\\T\\a = \\T®I\\ = sup{|tr((r®/)«)| : a'(u) < 1} .
Proof. Let B be the associated bilinear form on X x Y*. Then, for u =
£"=1 xi ® $} 6 X ® Y*, we have
n n
(u,B) = Y,{Tx„1>j) = tr(£ray ® *,)
where tr is the trace functional on Y ®„ Y*. Therefore
(u,B)=tr((T®I)u)
for every u eX ® Y*.
It follows that, if T ® J is bounded, then
\{u,B)\ < |tr((r®/)«)| < 7r((r®/)«) < ||r®/||o'(u),
and so T is a-integral, with
\\T\\a < sup{| tr((r ® /)«) | : a'(u) < 1} < ||r ® /||.

8.1 Associated Bilinear Forms and Operators 191
Conversely, suppose that T is a-integral. Let « = X3"=i xj ® tyj 6 X ® Y*
and let v = (T ® /)«. To compute the projective norm of u, we take any
Se(Y®„Y*)*=JL(Y\Y*). Then
n
<», S) = £<r.r„ ^,) = ((/® S)u, B).
J=l
Hence,
\(v,S)\<\((I®S)u,B)\\<\\S\\a'(u)\\T\\a.
Taking the supremum over the operators S of unit norm, we have
w((T®I)u)<a'(u)\\T\\a,
by the uniform property of a'. Therefore T ® J is bounded and
l|r®/||<imu.
This concludes the proof. O
If a is a tensor norm we define a to be the tensor norm (a') . This is
sometimes referred to as the adjoint or contragredient norm of a.
Theorem 8.5 (Grothendieck Composition Theorem). Let a be a
tensor norm, let T: X -» Y be an a-integral operator and let S: Y -» Z be an
a-integral operator. If either a is accessible or Y has the metric
approximation property, then the composition ST is an integral operator, and
Iisr||, < ||symu.
Proof. Thanks to the finitely generated nature of the class of integral
operators, we may assume that X and Z are finite dimensional. Thus S and T are
finite rank operators and we have T 6 X* ® Y, S 6 Y* ® Z and S* e Z* ® Y.
Now, in finite dimensions, the integral and nuclear norms are the same and
so x
HSril, = 7r(Sr) = sup{|(W, ST)\: W 6 X ® Z\ e(W) < 1}.
Let W = Y^k xk ® lk 6 X ® Z* = L(X', Z*). Then e(W) < 1 simply means
that the operator norm of W is at most 1. Now,
(W, ST) = J2(STxk,ik) = ]C<T.rfc, S*7fc)
k k
= (jrxk<»S*yk,T) = (S'W,T)
k
and it follows that
\(W,ST)\ < a'(S*W)a(T) < a'(S') \\W\\a(T) < a'(S*)a(T).

192 8 Operator Ideals
The adjoint of the operator S is the same as the transpose of S considered
as a tensor, and hence we have
|(W,ST)|<4(5)o(r).
Finally, since X and Z are finite dimensional, the assumption that a is
accessible or that Y has the metric approximation property ensures that
a(S) = ||S||a and a(T) = \\T\\a (note that if a is accessible then so is
a = (a1)'). Therefore, taking the supremum over W,
lisrii, < iisymu.
This concludes the proof. O
This powerful result spawns a multitude of interesting composition
theorems in special cases. If a Banach space with the Radon-Nikodym property
can be interposed at a suitable point, the composition ST is even nuclear.
We present some examples.
Corollary 8.6.
(a) The composition of a pair of 2-summing operators is nuclear.
(b) The composition of a HUbertian operator and a 2-dominated operator, in
either order, is nuclear.
(c) Let 1 < p < oo. Then the composition of a p-summmg operator and a
p'-integral operator, in either order, is integral.
Proof. We prove (a); the proofs of the other assertions are left as an exercise
for the reader. The 2-summing operators are the a-integral operators where
a = d'2 = 02 and we have a = a in this case. Therefore the composition
of a pair of 2-summing operators is at least integral. However, by
Proposition 6.23, every 2-summing operator factors through the 2-summing operator
C(K) -» L2(|t) for some compact space K and some regular Borel
probability measure fi on K. Thus, if S: X -> Y and T: Y -> Z are 2-summing, we
have factorizations of the following form:
C(Jir,) L2(/*i) ► C(K2) L2(^2)
The operator from C{K{) into L2(/;i2) obtained by composing the operators
along the bottom of the diagram is the composition of two 2-summing
operators and hence is integral. But L2(/n) has the Radon-Nikodym property
and so this operator is nuclear. O
* Z

8.1 Associated Bilinear Forms and Operators 193
We now consider the spaces of "nuclear" forms and operators associated
with a tensor norm. Proceeding by analogy with the case a = w, we begin
with the canonical mapping of X* ® Y* into "Ba (X x Y). If we endow X* ® Y*
with the tensor norm a, then this mapping is bounded, with norm at most
one. Hence, it extends to an operator
Ja:X'®aY*-t'Ba(XxY).
The bilinear forms that lie in the range of this operator are known as a-
nuclear bilinear forms. The a-nuclear norm of an a-nuclear bilinear form B
is defined by
Na(B) = inf {a(u) : « 6 X*®aY*, Jau = B) .
We shall denote the space of a-nuclear bilinear forms, with the a-nuclear
norm, by yfa(X x Y). It is not difficult to see that this space is complete;
indeed 7Sa(X x Y) is a quotient space of X*®aY':
J^a(XxY)=X*®aY*/keiJa.
In the same way, there is a canonical mapping, which we also denote by
Ja, from X*®aY into La(X,Y). The operators that lie in the range of this
mapping are known as a-nuclear operators. The a-nuclear norm, Na(T), of
an a-nuclear operator is defined in the same way as for bilinear forms. Again,
the space Na(X, Y) of a-nuclear operators is the quotient space of X*®aY
by the kernel of Ja.
We have already encountered several examples of this construction. In
the case a = w, the a-nuclear operators are just the nuclear operators. When
a is the injective norm, we obtain the approximable operators. And if a is
the Chevet-Saphar norm gp, the corresponding class consists of thep-nuclear
operators. We have seen in Chapter 4 that N(X, Y) = X*®„Y if either X*
or Y has the approximation property. We now show that this is a special case
of a general phenomenon. ^
Proposition 8.7. Let a be a tensor norm. If either X* or Y has the
approximation property, then Na(X, Y) = X*®aY-
Proof. We give the proof for the case in which Y has the approximation
property. Supppose that Jau = 0 for some « 6 X*®aY- In order to establish
that « = 0, we must show that («,T) = 0 for every T 6 (X*®aY)* =
La'(X*,Y'). By Theorem 8.4, the operator T®I: X*®aY -» Y*®*Y is
bounded. If Y has the approximation property, then Y*®nY is the space
of nuclear operators on Y. Now Jau = 0 is easily seen to imply that the
nuclear operator (T®/)(«) is zero. Therefore, the trace functional on Y*®XY
vanishes on (T ® J) («). Hence, by the proof of Theorem 8.4, we have («, T) =
tr((r ®/)(«)) =0. O

194 8 Operator Ideals
The question of when N^X,Y) is a closed subspace of La{X,Y) has
already been answered. From the discussion at the beginning of this section,
we have:
Proposition 8.8. Let a be a tensor norm and let X and Y be Banach spaces.
The a-integral norm coincides with the a-nuclear norm on Na(X,Y) if one
of the following conditions is satisfied:
(a) Both X* and Y have the metric approximation property;
(b) a is accessible and either X* or Y has the metric approximation property;
(c) a is totally accessible.
In finite dimensions, the classes of a-nuclear and a-integral operators
are the same. In a sense that will be made clear in the next section, the
a-nuclear and a-integral operators respectively constitute the smallest and
biggest extension of this class of operators to infinite dimensions.
8.2 Operator Ideals
The theory of Banach operator ideals provides another means to study classes
of operators on Banach spaces, without explicit reference to tensor norms. In
this section, we give a brief introduction to Banach ideals.
A Banach ideal consists of an assignment to each pair of Banach spaces X,
Y of a vector space A(X, Y) of bounded operators from X into Y, together
with a norm, A, on this space, with the following properties:
(11) The one-dimensional operator ip® y belongs to A(X, Y) for every ip 6 X*
and y 6 Y, and A(<p ®y)< \\<p\\\\y\\.
(12) If S 6 A(X, Y) and if T 6 £(W, X) and R 6 £>(Y, Z), then the
composition RST belongs to A(W, Z) and A(RST) < ||-R||.4(S)||r||.
(13) [A(X, Y), A] is a Banach space.
We shall use the notation [.4, A] to refer to a Banach ideal, or simply A
if the norm is understood. Almost all the classes of operators we have
encountered constitute Banach ideals. Clearly, the largest Banach ideal consists
of all the bounded operators on Banach spaces, with the operator norm. At
the other end of the scale, we have the ideal [N, N] of nuclear operators: for
each pair of Banach spaces X, Y, N(X, Y) is the space of nuclear operators
and N(T) is the nuclear norm of T. Between these extremes, we have the
various Banach ideals of p-nuclear, p-integral and p-summing operators, and
the ideals of Hilbertian and 2-dominated operators. In addition to these, the
classes of approximable, compact, weakly compact and completely
continuous operators, and so on, each define a Banach ideal, where the ideal norm
in each case is the operator norm.
Every tensor norm generates two Banach ideals. Given a tensor norm a,
we have on the one hand the ideal [Na.-Wa]. where N^X, Y) is the space

8.2 Operator Ideals 195
of a-nuclear operators and Na the a-nuclear norm. On the other hand, we
may form the ideal [£.„, || • ||a], consisting of the a-integral operators with
the a-integral norm.
To go in the opposite direction, suppose [A, A] is a Banach ideal. We
may generate a tensor norm in the following way. First, if E and F are finite
dimensional spaces, consider the space A(E*, F). It follows from property (II)
that this space contains all the operators from E* into F. Thus A(E*, F) =
E®F and so we may define a norm on the tensor product E® F by a(u) =
A{u). Thus
E®aF=[A(E*,F),A] (8.1)
or, equivalently,
[A(E,F),A} = E*®aF. (8.2)
It is a simple matter to show that properties (II) and (12) imply that a is
a uniform crossnorm on the class of finite dimensional spaces. Indeed, finite
dimensional Banach ideals are just finite dimensional uniform crossnorms in
disguise. The passage to infinite dimensions is more delicate. We may apply
the process of finite generation to obtain a tensor norm:
a(u) = inf{a(u;E® F) : E C X,F cY,u 6 E® F}
where u belongs to the tensor product of the Banach spaces X, Y and the
infimum extends over all pairs of finite dimensional subspaces E, F for which
u 6 E® F. We shall refer to a as the tensor norm associated with the Banach
ideal A. The basic properties of the associated tensor norm are as follows.
Theorem 8.9. Let [A, A] be a Banach ideal.
(a) There is a unique tensor norm a such that
[A(E,F),A] = E*®aF „,
for every pair of finite dimensional spaces E, F- *
(b) For every pair of Banach spaces X, Y, every a-nuclear operator T from
X into Y belongs to A(X, Y) and furthermore,
A(T)<Na(T).
(c) For every pair of Banach spaces X, Y, every operator T in A(X, Y) is
an a-integral operator and furthermore,
\\T\\a<A(T).
Proof. We have already proved (a). To establish (b), let T 6 X* ® Y, so
that T is a finite rank operator from X into Y. Let M be a finite codimen-
sional subspace of X and N a finite dimensional subspace of Y such that

196 8 Operator Ideals
T 6 (X/M)* ® N. Then the operator T factors through a finite dimensional
operator Tmn : X/M -> N as shown in the diagram below:
X/M
Since the quotient operator Qm and the embedding In each have unit norm,
it follows from property (12) that
A(T) < A(TMN) = a(T; (X/M)* ® N),
by the definition of a. Taking the infimum over M and N, we get
A(T)<a(T)
for every T 6 X* ® Y. It follows immediately from the definition of the
a-nuclear norm that every a-nuclear operator T belongs to A(X,Y), with
A(T) < Na(T).
To prove (c), let T 6 A(X, Y). Let E be a finite dimensional subspace of
X and let F be a finite codimensional subspace of Y. Now let Tef denote
the composition QfTIe, as shown in the diagram below:
By property (12), we have A(TEF) < A(T). Now TEF 6 E' ® (Y/F) and
-4(7^) = a(TEF) = \\TEF\\a. Therefore ||r£;F||a < A(T) for all E, F. Taking
the supremum and applying Proposition 8.3 yields the desired conclusion.
It is clear from this theorem that the ideals of integral and nuclear
operators associated with a tensor norm are the "maximal" and "minimal''
Banach ideals respectively. We would like to develop this idea a little further.
In particular, it would be helpful to have intrinsic characterizations of these
ideals. A workable duality theory of Banach ideals is essential to achieve this
objective.
In finite dimensions we have trace duality: if E and F are finite
dimensional spaces, then the spaces of operators £>(E, F) and -C(F, E) are in duality.
If T 6 L(E, F) and S 6 £(F, E), we have

8.2 Operator Ideals 197
(S, T) = tiB ST = tr F TS. (8.3)
Now, if [A, A] is a Banach ideal, we may use trace duality to define a dual
norm for the ideal norm A; If S is an operator from E into F, we define
A'(S) = sup{|(S,r)|: T 6 L(F,E), A(T) < 1}. (8.4)
As we have seen in Chapter 1, this is essentially the same as the tensor duality
between F* ® F and F ® F*. The above operators S, T correspond to tensors
v. 6 E* ® F and v 6 E ® F*. Now, if a is the tensor norm associated with a
Banach ideal [A, A], then
(A(E, F))* = (F* ®aF)* = E ®a. F* = £a(F. E). (8.5)
We now proceed exactly as with the definition of the dual of a tensor
norm, using finite generation to define the dual ideal. Thus, for each pair X,
Y of Banach spaces and each operator T: X -» Y, we define
A*(T) = supiA'iQ'fTlE) : dim£, dimy/F < oo} , (8.6)
the supremum being taken over all pairs E, F, where E is a finite dimensional
subspace of X and F is a finite codimensional subspace of Y:
eSiHjly/f
Is
X = » Y
We define A* (X, Y) to be the set of operators T for which A*(T) is finite.
A careful study of this process reveals that A* (X, Y) is precisely the space
Ji&{X, Y) of a-integral operators and A* is the a-integral norm. In particular,
this'shows that [A*, A*] is indeed a Banach ideal. This Meal is known as the
adjoint ideal of [A, A], We summarize our findings: >
Proposition 8.10. Let [A, A] be a Banach ideal and let a be the associated
tensor norm. The adjoint ideal [A*, A*] coincides with the Banach ideal of
a-integral operators and the adjoint norm A* is the a-integral norm.
We define a partial ordering on the class of Banach ideals as follows:
[A, A] < [B,B] means that A(X,Y) is contained in "B(X,Y) for all Banach
spaces X and Y and B(T) < A(T) for every T 6 A(X, Y).
We may now define a Banach ideal [A, A] to be maximal if the only Banach
ideal [3, B] that satisfies [A, A] < [B, B] and B(T) = A(T) for every finite
rank operator is [A, A] itself. Similarly, [A, A] is minimal if the only Banach
ideal [£, B] for which [£, B] < [A, A] and B(T) = A(T) for every finite rank
operator is [.A, .A].
Qf

198 8 Operator Ideals
Theorem 8.9 demonstrates that the minimal ideals are precisely the ideals
of a-nuclear operators, where a is a tensor norm. We concentrate our
attention on the maximal ideals. These are the exact counterpart of the tensor
norms. Let us make a definition that will help us to express this idea. We
shall say that a Banach ideal [A, A] is finitely generated if, for every pair of
Banach spaces X, Y and every T 6 A(X, Y), we have
A(T) = sup{A(QFTIE) : dim £, dim IVF < oo},
where the supremum is taken, as usual, over all pairs E, F, where E is a
finite dimensional subspace of X and F is a finite codimensional subspace of
Y. This is usually taken as the definition of maximality. However, in the light
of our final theorem, there is no danger of confusion:
Theorem 8.11. The following are equivalent for a Banach ideal [A, A]:
(i) [A, A] is maximal.
(ii) [A",A"] = [A,A].
(iii) There exists a Banach ideal [B,B] such that [A, A] = [B*,B*].
(iv) [A, A] is finitely generated.
(v) [A, A] is the ideal of a-integral operators, where a is the tensor norm
associated with [A, A].
Proof, (i) implies (ii) follows from the fact that, if E and F are finite
dimensional, then A" (E, F) = A(E, F) and the norms A** and A are the same on
these spaces.
(ii) implies (iii) is trivial.
(iii) implies (iv) follows immediately from the definition of the adjoint
ideal.
(iv) implies (v) follows from Proposition 8.3.
(v) implies (i) also follows from Proposition 8.3. O
8.3 Exercises
Exercise 8.1. Show that an operator T is a'-integral if and only if the
adjoint operator T* is a-integral.
Exercise 8.2. Let a be a tensor norm. Show that a bilinear form B on X x Y
is \a/-integral if and only if the following is true: for every pair of embeddings
S: X -» C(Ki) and T: Y -» C(K2), there exists an a-integral bilinear form
A on C{K{) x C(K2) such that B = Ao (S x T). Show that the \a/-integral
norm of B is the infimum of ||j4||a.
Exercise 8.3. Formulate and prove a characterization of the /a\-integral
bilinear forms analogous to that given in the previous exercise for \a/-integral
forms.

8.3 Exercises 199
Exercise 8.4. Let a be a tensor norm. Show that an operator T: X -¥ Y
is a:\-integral if and only if for every (respectively, some) embedding U of Y
into a C(K) space, the composition UT is an a-integral operator.
Exercise 8.5. Formulate and prove characterizations of the a/-integral, /a-
integral and \a-integral operators analogous to that given in the previous
exercise.
Exercise 8.6. Show that there is no norm under which the finite rank
operators form a Banach ideal.
Exercise 8.7. Let [A, A] and [B, B] be Banach ideals and suppose that
A(X, Y) is contained in 1i(X, Y) for all Banach spaces X, Y. Show that
there exists a constant C such that B(T) < CA(T) for every X and Y and
every TeA(X,Y).
Exercise 8.8. Let [A, A] and [B,B] be Banach ideals. For each pair of
Banach spaces X, Y, let A o "B(X, Y) consist of all operators T: X -> Y
for which there exist a Banach space Z and operators S 6 "B(X, Z) and
R 6 A(Z, Y) such that T = RS. Show that A o "B(X,Y) is a vector space.
For each T 6 A o £(X, Y), let A o B(T) = inf A(R)B(S), where the infimum
is taken over all factorizations of T as described above. Show that A o B is
a quasinorm on A o "B(X, Y) (a quasinorm q on a vector space W has all
the properties of a norm, except that the triangle inequality is replaced by
q(u + v) < C(g(«) + q(vj), where C > 1 is a constant).
Exercise 8.9. Let [A, A] be a Banach ideal. The conjugate ideal, [AA, AA],
is defined as follows: an operator T 6 &(X, Y) belongs to A*(X, Y) if there
exists a constant C such that for every finite rank operator S: Y —> X, we
have | tr ST| < CA(T). A norm is defined for these operators by taking AA (T)
to be the infimum of C. Show that A*(T) < A*(T) for every T 6 A*(X, Y),
with equality if X and Y both have the metric approximation property.
Exercise 8.10. Show that the ideal of compact operators is neither minimal
nor maximal.
Exercise 8.11. Let X, Y and Z be Banach spaces. Show that there is an
operator from (X ®d2 Y*) ®„ (Y ®d2 Z) into X ®» Z with the property that
(a; ® (p) ® (y ® z) h-> tp(y)x ® 2.

A. Suggestions for Further Reading
In this appendix, we present some suggestions for further reading. Our choice
is highly personal; the reader may consult the books cited below for a fuller
set of references.
The first book dealing with normed tensor products was the 1950
monograph of R. Schatten, "A Theory of Cross-Spaces" [46], which summarized
the state of knowledge at that time. The ground-breaking works of A.
Grothendieck, "Produits Tensoriels Tbpologiques et Espaces Nucleaires" [21],
appearing in 1955, followed by "Resume de la Theorie Metrique des
Produits Tensoriels Topologiques" [19, 22] and a supplementary paper on the
Dvoretzky-Rogers Theorem [20, 23] in 1956, introduced a new cycle of ideas,
the ramifications of which are evident in Banach space theory today. The
fundamental importance of finite dimensional, or "local", considerations, the
classes of integral and summing operators, Grothendieck's Inequality, the
approximation property, the central role played by vector measures and the
Radon-Nikodym property, to name but a few, are all to be found, in one
form or another, in his work. However, these developments were not fully
appreciated at the time and virtually the only response for over a dozen
years was the paper of Amemiya and Shiga [1] in 1957, which explained
some of the results in the first part of the Resume and provided some of the
proofs that had been omitted by Grothendieck. It was not until 1968 that
J. Lindenstrauss and A. Pelczynski, in the paper " Absolutely Summing
Operators in Lp- Spaces and Applications" [34], brought some of the results in
Grothendieck's Resume to the attention of a wider audience. Most
importantly, they formulated Grothendieck's Inequality in the terms in which it
is usually presented today. However, the explanation was couched in terms
of operators rather than tensor products. In the following year, the paper
"The .Cp-Spaces" [35], by Lindenstrauss and Rosenthal, appeared. This
paper demonstrated the importance of local techniques and introduced the all-
important Principle of Local Reflexivity, which had, in a sense, been implicit
in the Resume. These papers led to an upsurge of activity, but now many
researchers found it easier to carry out their work using the parallel
language of operator ideals. Notable exceptions were the works of Chevet [3, 4],
Saphar [44, 43, 45] and Lapreste [31, 32], giving detailed treatments of the
tensor norms related to the classes of p-surnming, p-integral and p-nuclear

202 A Suggestions for Further Reading
operators; the 1970 paper by Kwapien and Pelczynski [30] on unconditional
structure in matrix spaces; the paper of Lewis in 1973 [33] on weak
compactness in injective tensor products; the solution in 1973 of the approximation
problem by Enflo [13] and the construction by Pisier [41] in 1983 of a Banach
space P for which the injective and projective tensor products P ®eP and
P ®„ P are isomorphic. Also noteworthy is the 1973 paper on Banach
operator ideals by Gordon, Lewis and Retherford [18] in which tensor product
methods play a crucial role.
The excellent survey by Gilbert and Leih [17] develops the material of the
Resume in tandem with the theory of operator ideals. A. Pietsch's book,
"Operator Ideals" [40], published ten years after the Lindenstrauss-Pelczynsky
paper, describes the state of the art at that time and contains much of the
Grothendieck theory, phrased in operator ideal terms. The publication in
1993 of the encyclopaedic monograph "Tensor Norms and Operator Ideals",
by A. Defant and K. Floret [6] draws the two strands of tensor products and
operator ideals together and shows their interconnections clearly.
We now make some suggestions to supplement particular chapters.
Chapter 2. Schatten's monograph [46] is worth reading to see the beginnings
of the theory. The book by Jameson [28] on nuclear and summing norms is
highly recommended. For the Bochner integral, see Diestel-Uhl [10].
Chapter 3. Further details on the Pettis integral can be found in the books
by Diestel-Uhl [10] and Talagrand [48].
Chapter 4. The approximation property was introduced by Grothendieck.
See the Memoir [21] and the Resume [19, 22]. Enflo [13] provided the first
example of a Banach space without the approximation property.
Chapter 5. The outstanding reference on vector measures and the Radon-
Nikodym property is the book of Diestel and Uhl [10] and the supplementary
material in [11]. See also the books of Bourgin [2] and van Dulst [49]. We
have followed the treatment of Diestel and Uhl closely in this chapter. The
papers of Gil de la Madrid [15, 16] motivated much of our treatment of
tensor products with spaces of measures. Much of the treatment of operators
on C(K) comes from the wonderful book of Dunford and Schwartz [12],
which continues to inspire over forty years since its publication. The proof
of the Principle of Local Reflexivity given in the text is adapted from that
of Dean [5]. For reasons of space, we have not included ultraproducts of
Banach spaces, which are needed to make full use of this principle and of the
local methods employed in the text. The survey article of Heinrich [24] is an
excellent introduction to this invaluable tool.
Chapter 6. The paper [45] by Saphar is recommended. Full treatments of
the Chevet-Saphar tensor norms can be found in the Gilbert-Leih paper [17]
and the book of Defant and Floret [6]. The class of p-summing operators was

A Suggestions for Further Heading 203
introduced by Pietsch [38]. This paper is clearly written and is recommended
reading. For p-surnming operators and all related matters, there is no
better reference than the book of Diestel, Jarchow and Tonge [9]. The proof
of Grothendieck's Inequality presented in the text is that of Kaijser and is
adapted from the version given in this book.
Chapter 7. We have drawn heavily on the paper of Gilbert and Leih [17]
and the book of Defant and Floret [6]. This latter work contains virtually
everything that one might wish to learn about tensor norms and is highly
recommended to the reader. For some early papers on the p-integral
operators, see [36, 37, 39]. The proof of the Kwapien Domination Theorem is taken
from the very accessible original [29].
Chapter 8. The paper of Gordon, Lewis and Retherford [18] is very readable
and illustrates the principle that tensor product and operator ideal methods
are often both needed. The books of Pietsch [40] and Defant-Floret [6] contain
complete accounts of the theory ©f operator ideals. Defant and Floret give an
exhaustive development of the interconnections between operator ideals and
tensor norms. •
Appendix B. For a highly readable account of sequences and series in
Banach spaces, see the book by Diestel [8].
Appendix C. For this material, we have drawn on the books of Diestel [8]
and Dunford-Schwartz [12].
i

B. Summability in Banach Spaces
A sequence (xn) in a Banach space is said to be summable if the series
J2n xn converges, or, in other words, the partial sums $3?=i xi ft>rm a Cauchy
sequence. The sequence (a;n) is absolutely summable if the series J^nH3""!!
converges. It is a straightforward consequence of the completeness of X that
every absolutely summable sequence in X is summable. We now consider
some other forms of summability.
We say that (a;n) is unconditionally summable if for every permutation a
of the set of natural numbers, the permuted sequence (x„(n)) is summable.
The sequence (a;n) is subseries summable if, for every strictly increasing
sequence (n*) of natural numbers, the sequence (xnk) is summable.
The sequence (a;n) is sign summable if the sequence (enxn) is summable
for every choice of signs en = ±1.
Finally, we have a condition involving convergence of the net of finite
partial sums. Let 7 be the set of finite subsets of N. Then fF is directed by
inclusion and we may define a net {Sa)a€5 by Sa = X.ne.4 Xn~ ^e sav *na*
the sequence (a;n) is unordered summable if this net converges. Thus, (a;n)
is unordered summable if and only if there exists x € X with the following
property: for every e > 0, there is a finite subset A of N such that if B is any
finite subset of N that contains A, then
-E
Xr,
neB
<e.
Equivalently, the net (Sa) satisfies the Cauchy condition: for every e > 0,
there exists a finite subset A of N such that if B is any finite subset of N that
is disjoint from A, then ||X3neBa:n|| < £-
Proposition B.l. The following statements are equivalent for a sequence
(xn) in a Banach space:
(i) (a:n) is unconditionally summable.
(ii) (a:n) is subseries summable.
(iii) (a:n) is sign summable.
(iv) (a:n) is unordered summable.
Proof, (i) implies (ii): we prove the contrapositive. Suppose that (a:n) is
not subseries summable, so that some subsequence (arm,) is not summable.

206 B Summability in Banach Spaces
Therefore there exists e > 0 such that, for every natural number k, there
exist natural numbers / and m with the property that k < I < m and
3=1
>e.
Hence there exist mutually disjoint finite subsets Im of N such that
E
>e
for every m. """""airing a permutation of N that maps the sets Im into blocks
of consecutive natural numbers yields a rearrangement of the sequence (a;n)
whose partial sums do not satisfy the Cauchy condition. Therefore (a;n) is
not unconditionally summable.
(ii) implies (iii): suppose that (a;n) is subseries summable and let en = ±1
for every n. Let A = {n £ N : en = 1} and B = {n G N : en = -1}.
Then the series J^neA xn and X3n<=B(-a"n) both converge. Since the series
J2n£n.xn is obtained by interlacing these series, it follows that this series is
also convergent. Therefore (a;n) is sign summable.
(iii) implies (iv): suppose that (a;n) is not unordered summable. Then
there exists e > 0 and a sequence of finite subsets Ak of N such that max -A* <
min -Afc+i for every jfc and
^2, xn >e
neAk
for every k. Let
f 1 ifn<EUfc^fc,
1—1 otherwise.
Then the series J2n(^ + £n)xn does not satisfy the Cauchy condition. But
this series is the sum of the series J2n xn and J2n £nxn an^ so a* least one of
these series diverges. Therefore (a;n) is not sign summable.
(iv) implies (i): suppose that (a;n) is unordered summable. Let a be a
permutation of N and let e > 0. There exists a finite subset A of N such that
||5ZneB ^nll < e f°r every finite set B that is disjoint from B. Choose k G N
such that A is contained in the set {a(n) : 1 < n < k). Then ||£^nx}\\ <e
if k < n < m. Therefore the series X.na"<7(„) satisfies the Cauchy condition.
The strong summability condition expressed in this result is stronger than
summability; indeed, there are scalar sequences that are summable but not
absolutely summable and it follows easily that the same is true in every
Banach space.
In finite dimensions, unconditional and absolute summability are the
same. However, the Dvoretzky-Rogers Theorem states that every infinite

B Summability in Banach Spaces 207
dimensional Banach space contains an unconditionally summable sequence
that is not absolutely summable (see Corollary 6.21 for a weak form of this
theorem).
Apart from absolute summability, all the modes of summability defined
above have weak counterparts, where convergence with respect to the norm
topology is replaced by weak convergence. The weak version of the proposition
is false. For example, consider the sequence (a;n) in cq given by a-i = e| and
xn = en — en_i for all n > 1. It is easy to see that this sequence is weakly
unordered summable, with sum zero. However, (a;n) is clearly not weakly
subseries summable.
Proposition B.2. Let (xn) be a sequence in a Banach space.
(a) (a;n) is weakly unconditionally summable if and only if it is weakly
unordered summable.
(b) (a;n) is weakly subseries summable if and only if it is weakly sign
summable.
(c) if (a;n) is weakly subseries' summable then %t is weakly unconditionally
summable.
Proof, (a) Suppose that (a;n) is weakly unconditionally summable. Let x
be the sum of the weakly convergent series "£n xn. Then <p(x) is the sum
of the unconditionally convergent scalar series ]T)n <p(xn) for every tp £ X*
and it follows that the value of the weak sum ]T")n xn is independent of the
ordering of the terms. Now, weak unconditional summability of (a;n) means
that the series "£"]„ <p(xn) is unconditionally convergent to ip(x) for every
tp £ X*. On the other hand, weak unordered summability of (a;n) means
that there exists z £ X* such that the net of finite partial sums of the series
^n<p(xn) converges to <p{z) for every <p £ X*. The result now follows from
the equivalence of these definitions of convergence for scalar series.
(b) The proof of the fact that subseries summability implies sign
summability in the previous proposition also applies to the corresponding weak
notions. Conversely, suppose that (a;n) is weakly sign summable and let (xnk)
be a subsequence. Let en be 1 if n lies in the range of the sequence (n*);
otherwise, let en = — 1. Then the series J2n Xn an(^ 5Zn £nxn are both weakly
convergent. Adding, we see that the subseries J2k x"-k 's a^so weakly
convergent.
(c) Suppose that (a;n) is weakly subseries summable. Then (a;n) is weakly
sign summable, by (b). Let x be the sum of the weakly convergent series
Y^nxn and let ip £ X*. Then Yln'P(x"-) converges to ip(x). But the sign
summability of (a;n) implies that the series X)ne™V,(a;n) converges for every
choice of signs en. Therefore the scalar series ]T)n <p(xn) is unconditionally
convergent to f(x). Since this holds for every tp £ X*, it follows that (a;n) is
weakly unconditionally summable. O

208 B Summability in Banach Spaces
This result is not the whole story. The Orlicz-Pettis Theorem
(Proposition 3.12) asserts that weak subseries summability is equivalent to
unconditional summability.
We now present a useful characterization of unconditional summability.
Proposition B.3 (Bounded Multiplier Test). A sequence {xn) in a
Banach space is unconditionally summable if and only if the sequence (anxn) is
summable for every bounded sequence of scalars (<xn)-
Proof. Suppose that (anxn) is summable for every bounded scalar
sequence (a„). Then (a;n) is sign summable and hence is absolutely summable.
Conversely, suppose that (a;n) is unconditionally summable. Let (<*„) be
a bounded sequence of scalars. We shall prove that the partial sums of the
series J2n anxn form a Cauchy sequence. First, we have
^2<*kXk
k=n
sup
<peBx.
Yiak<p(xk)
k=n
<ll(a»)lloo sup £>(a-fc)|
ipeBx.
k=n
for n < m. In the complex case, we use the inequality \<p(x)\ < |Rey>(a;)| +
|Imp(a:)| and apply the argument below to the real linear functionals Rep
and Im <p.
Proceeding with the real case, let A consist of the natural numbers k
between n and m for which <p(xk) > 0 and let B consist of those k between
n and m for which <p(xjc) < 0. Then
fc=n
k£A keB
= <p[Y,xk) -<p[Y,Xk) ^ lC
Xk
KkeA
HeB
keA
+
Y,xk
keB
for every <p £ Bjv- Since (a;n) is unordered summable, it follows that the
series J2n anxn satisfies the Cauchy condition. O
We conclude with a weak version of the Bounded Multiplier Test. A series
]T)n xn is said to be weakly unconditionally Cauchy if the partial sums of the
series ]T)n a"<7(„) form a weak Cauchy sequence for every permutation a of N.
Equivalently, the scalar series ^,nf{xn) is unconditionally convergent for
every <p € X*.
Proposition B.4. Let {xn) be a sequence in a Banach space X. The scalar
sequence (<p{xn)) is unconditionally summable for every ip £ X* if and only
if the series ]T)n anxn converges for every (an) £ Co-
Proof. Suppose that "jTJn |p(a:n)| < oo for every ip £ X*. Then we may
define a linear mapping T: X* -*■ (t by T(ip) = ((p(xn)). An application of
the Closed Graph Theorem shows that T is bounded. Therefore

B Summability in Banach Spaces 209
supfc \>p(xn)\ •• f e Bx. I < ||r|| < oo.
Let (an) £ cq. Arguing as in the proof of the previous proposition, we have
^2<*kXk
k=n
<sup\ak\\\T\\,
k>n
which tends to zero with n. It follows that the S6ri6s 2_.„ ocn
x„ satisfies the
Cauchy condition.
Conversely, suppose that Ylnanxn converges for every (an) £ cq. Then
the series J2n an'P(xn) converges for every (an) £ Co and every ip £ X*.
Hence the sequence (<p(xn)) is unconditionally summable for every ip £ X*.
a

C. Spaces of Measures
In this appendix, we give an account of some of the structural properties of
Banach spaces of scalar measures that we need. We are particularly interested
in questions relating to the weak topology. We shall assume that the reader
has a basic knowledge of measure theory up to and including the Radon-
Nikodym Theorem.
We begin by recalling some basic results. Let E be a cr-algebra of subsets
of a set Q. We shall refer to the elements of Q simply as measurable sets. We
denote by M(E) the vector space of scalar valued measures on E, that is,
functions from Q into K = R or C that are countably additive and send 0*to
zero. We emphasize that our measures take only finite values. Nevertheless,
in order to avoid any confusion, we shall refer to a measure that takes its
values in (0, oo) as a finite, positive measure.
The variation of a measure \i is defined on E by
|/t|(E) = sup I J2 \fi(Ai)\ :{Au...,An}& partition of E I,
where a partition of E is a finite collection of mutually disjoint measurable
sets whose union is E. The variation of p is a finite, positive measure on E;
it can be characterized as the smallest finite, positive measure that satisfies
\/i(E)\ < \ij,\(E) for every E £ E. Furthermore, we have^.
\fi(E)\ < \(j,\(E) < 4 sup{|j»(F)| :FCE, Ag £}. (C.l)
In particular, every scalar measure is bounded. The variation norm is defined
on the vector space M(E) by
IMI = W(ft)-
We will see shortly that M(E) is complete in the variation norm.
Let A be a finite, positive measure on E. The measure (i is absolutely
continuous with respect to A if, for every e > 0 there exists S > 0 such that
if A(E) < 5, then \fi(E)\ < e. Equivalently, if A(E) = 0, then fi(E) = 0. We
also say that A is a control measure for y,. The Radon-Nikodym Theorem
states that fi is absolutely continuous with respect to A if and only if there
exists / G £i(A) such that fi(E) = fEfd\ for every measurable set E.

212 C Spaces of Measures
Furthermore, we have \fi\(E) = fE \f\ dX and it follows in particular that the
variation norm of fi coincides with the L\ (A)-norm of /.
For each finite, positive measure A, we denote by M,x(E) the subspace of
M(E) consisting of all the measures that are absolutely continuous with
respect to A. The remarks above show that the correspondence between a
measure and its Radon-Nikodym derivative establishes an isometric isomorphism
between M,\(E) and L|(A). This observation is invaluable when
investigating properties of M(E) relating to the behaviour of sequences of measures.
Indeed, if (fin) is any sequence of measures on E, we may define a finite,
positive measure by
A(E) = f;2-"(i-i-iKii)-i|/Ltn|(E).
It is easy to see that if A(E) = 0, then fin(E) = 0 for every n. Thus A is a
control measure for all the measures fin. To summarize, we have shown that
every sequence in M(E) lies in a subspace that is isometrically isomorphic to
some L\ (A). We now give some applications of this idea.
Proposition C.l. The space M(E) is complete in the variation norm.
Proof. Let (fin) be a Cauchy sequence in M(E). This sequence lies in a
subspace of M(E) that is isomorphic to the Banach space Li(A) for some A.
Therefore (fin) converges. O
Next, we have a characterization of weak convergence:
Proposition C.2. A bounded sequence (fin) in ^Vt(E) converges weakly to fi
if and only if (fin(E)) converges to fi(E) for every E £ E.
Proof. The necessity of the stated contition follows from the fact that the
mapping E ■-* /i(E) defines a bounded linear functional on M(E) for every
EGE.
To prove sufficiency, it suffices to deal with the case \i = 0. Arguing as in
the proof of the preceding proposition, the sequence (fin) lies in a subspace
Ma(E) that is isometrically isomorphic to Li(A), so that fhi(E) = fE fnd\
for every E e E and every n, where fn € Li(A). Then (/„) is a bounded
sequence in Lx (A) and the condition fin(E) -»■ 0 is equivalent to fE fnd\-t0
for every E.
We wish to show that fE fgd\-*0 for every g S L^, (A). By the bounded-
ness of the sequence (/n) and the density of the measurable simple functions
in Loo (A), it is enough to show that fngfnd\ -»■ 0 for every measurable
simple function g. This is easily seen to be equivalent to the condition that
fE fn d\ -t 0 for every measurable set E. Therefore (/„) converges weakly
to zero, in Lx (A) and hence (fin) converges weakly to zero. Q

C Spaces of Measures 213
Now fix a finite, positive measure A. For each measurable set, E, the
characteristic function \E is integrable with respect to A. Two characteristic
functions, xe and xf, belong to the same equivalence class in L\ (A) if they are
equal almost everywhere with respect to A or, equivalently, if the symmetric
difference E A F has A-measure zero. This defines an equivalence relation
on E. We denote the quotient set by Ea and for each E £ E, we denote its
equivalence class by E. We may now consider S^asa subset of L\ (A) via the
injective mapping E •-* xe- This embedding induces a metric on Ea'
dx(E,F) = f\XE- Xf\ d\ = X(E A F).
We claim that the metric space (Ea, d\) is complete. To see this, let (En)
be a Cauchy sequence. Then the sequence (xe„) is Cauchy in I>i(A) and so
converges. Let the function / be a representative of the limit of this sequence.
Since every Li-convergent sequence has a subsequence that converges almost
everywhere, it follows that / takes the values 0, 1 almost everywhere with
respect to A. Therefore / lies in the same equivalence class as the
characteristic function of some measurable set E. It follows that the sequence (fEn)
converges to E.
The fact that (Ea,^a) is a complete metric space opens the door to the
Baire Category Theorem. All we need to add to the picture is the fact that
absolute continuity is the same as continuity in the metric sense. Suppose that
fi is absolutely continuous with respect to A. Then ^ is a well-defined function
on Ea, since A(E A F) = 0 implies fi(E A F) = 0 and hence fi(E) = |t(F).
Then the e — S definition of absolute continuity shows that y, is a continuous
function on the metric space (Ea^a)-
Proposition C.3 (Vitali-Hahn—Saks Theorem). Let (fin) be a sequence
of measures on the a-algebra E such that (pn(E)) converges for every BeE,
and suppose that each fin is absolutely continuous with respect to the same
finite, positive measure A. Then fin are uniformly absolutely continuous with
respect to A and the formula y,{E) = limn fin(E) defina a measure on E that
is also absolutely continuous with respect to A.
Proof. By the remarks preceding the proposition, we may consider the
measures fin as continuous functions on the metric space (Ea, ^a)- Fix e > 0 and
consider the sets
Ffc = {2 g EA : \^(A) - fim(A)\ <e Vm,n > fc} .
The continuity of the fx„ ensures that these sets are closed and the feet that
the sequence (fin(A)) converges for every A S E implies that the union of the
Ffc is all of EA. Hence, by the Baire Category Theorem, there exists k such
that Ffc has non-empty interior. Thus there exists Ao € E and 5 > 0 such
that

214 C Spaces of Measures
X(E AA0)<5 => \Hn(E) - /Xm(£)| < e Vm,n>k.
Let E be a measurable set such that X(E) < S. We can write E as E\\ Ei,
where Ei = E U A0 and Eh = Ao\E. Then A(E, A Aq) < S for t = 1,2.
Therefore, if n > fc, we have
ME)|<ME)| + M^)-^(^)I
= ME)| + ME,) - pn(Eh) - pfc(E,) + pfc(E2)|
< |^fc(£?)| + |p„(E,) - fik(Ei)\ + |pn(E2) - ^(£"2)1
< |/ifc(E)|+ 2e.
By the absolute continuity of pi,..., pt, we may assume that X(E) < 5
implies |pt(£)| < e for 1 < t < fc. Hence we have shown that if X(E) < 5,
then |/i„(£)| 5: 3e for every n. This proves the uniform absolute continuity
of the sequence (p„).
Now consider the function fi(E) = limnp„(E). It is easy to see that fi
is finitely additive and so, in order to establish the countable additivity of
p, it suffices to show that p(Efc) -»■ 0 for every decreasing sequence (E*) of
measurable sets with empty intersection. But this follows immediately from
the uniform absolute continuity of the pn, as does the absolute continuity
of p. O
The presence of the measure A here is something of a red herring, as we
have seen that every sequence of measures has a control measure. We may
rephrase the result without explicit mention of A:
Proposition C.4 (Nikodym Convergence Theorem). Let (fin) be a
sequence of measures on E such that the sequence (fin(E)) converges for every
EeS. Then the (in are uniformly countably additive and (i(E) = limn p„(E)
defines a measure on E.
Combining the Nikodym Convergence Theorem with the characterization
of weak convergence given in Proposition C.2, we obtain:
Corollary C.5. The Banach space M(E) is weakly sequentially complete.
As a byproduct of this result, we see that the space I>i(A) is weakly
sequentially complete, since this space is a subspace of M(E).
We can also deduce a striking property of the space t\. If fi is the set of
natural numbers and E the power set, then M(E) is l\, where the measure
(i is given by its values p* = p({fc}) at the singletons and the variation of fi
on a set E is given by %2keE Ia**I- A. Banach space is said to have the Schur
property if every weakly convergent sequence converges in norm.
Corollary C.6. The Banach space t\ has the Schur property.

C Spaces of Measures 215
Proof. Suppose the sequence (/j,n) converges weakly to zero in t\, where
fin = (fink)k for each n. By Proposition C.3, there exists a positive element,
A, of t\ with respect to which the fin are uniformly absolutely continuous. Let
e > 0. There exists 5 > 0 such that, if 52keE^k < ^> tnen l-CfceE/^n/cl < £
for every n. It follows from (CI) that, if J^ei"* ^k **"* ^> then J^ei"* 1^**1 ^
4e for every n. Choose a positive integer JsT so that J2k>K ^k < $• Then
Ylk>K lA*«fcl < ^e ror every n. Since (//„) is weakly convergent to zero, there
is a positive integer N such that J2k=i l/*n*l < £ if ^ ■> -W- Therefore
llMnll = ]C lA*»*l + X) lA*»*l < 5C
7c=l fc>K
when n> N. Hence (fi„) converges to zero in norm. Q
We conclude with a characterization of the weakly compact sets in M(E).
We say that the elements of a subset, K, of M(E) are uniformly countable
additive if, for every sequence" (En) of mutually disjoint measurable sets, the
convergence of the series J2n M-^n) is uniform in ft s K. Equivalently, if (An)
is a decreasing sequence of measurable sets with empty intersection, then the
sequence (fi(An)) converges to zero uniformly in ft £ K.
Proposition C.7. The following statements are equivalent for a bounded
subset K o/M(E):
(i) K is relatively weakly compact.
(ii) There exists a finite, positive measure X on E such that the elements of
K are uniformly absolutely continuous with respect to X.
(iii) The elements of K are uniformly countably additive.
Proof, (i) implies (ii): suppose that K is relatively weakly compact. We
claim that for every e > 0 there exists 5 > 0 and a finite subset {fii,..., fim}
oiK such that if \fi,\(E) < S for 1 < j < m, then \fi(E}\ < e for every fi£ K.
Suppose for the moment this is true. Taking e = %~n, we get <S„ > 0 and
elements finj of K, for 1 < j < mn, such that for every n,
max \fin]\(E)<S => \fi(E)\ < 2~n for every fi e K. (*)
Let
Then \\vn\\ < 1 for every n and so
oo
X=Y,2~nvn
n=l

216 C Spaces of Measures
defines a finite, positive measure on E. Now suppose that
where M = sup^jf \\n\\. Then
un(E) < 5n/mn(l + M)
and it follows that
"*><^n(E)<^
1 + W V ' 1 + M
and so \(iin] \{E) < Sn for each j = 1,..., mn. Hence, by (*), we have \fi(E)\ <
2_n for every fi € K. This establishes the uniform absolute continuity of the
elements of K with respect to A.
We prove our claim by contradiction. Suppose it is false for some e > 0.
Taking S = 2_1 and fixing any m £ K, there exists Ei G E and p-i^K such
that ||ti|(Ei) < 2_1 and b/2(Ei)| > e. Similarly, taking S = 2-2, there exists
E2 S E and fi3 £ K such that
\lii\(E2), M(E2)<2-2 and \n3(E2)\ >e.
Proceeding in this way, we get a sequence (fin) in K and a sequence (En) of
measurable sets such that
\fi, \(En) < 2~n for 1 < j < n and |^n+, (En)| > e
for every n. By the relative weak compactness of K, we may assume that the
sequence (/*„) is weakly convergent. Consider the control measure for this
sequence given by
{)~h i+iwi "
By the Vitali-Hahn-Saks Theorem, the fin are uniformly absolutely
continuous with respect to v. However, we have
HEn)<^ y + ^ 2
fc=l T'l/**ll fc=„+l
n oo
< ]T2-fc2~n + ^ 2~* < 2_n+1
fc=l fc=n+l
for every n, while |/tn+i(En)| > e, a contradiction. Thus our claim is proved.
(ii) implies (iii) is trivial.
(iii) implies (i): suppose that the elements of K are uniformly count-
ably additive. Let (fin) be a sequence in K. Choose a control measure A for

C Spaces of Measures 217
this sequence. Then this sequence lies in the subspace Ma(E) of M(E) that
is isometrically isomorphic to Li(A), each measure fin corresponding to its
Radon-Nikodym derivative /„. Taking a countable base {Om} for the
topology of K, let Enm = f~l{Om). Let Ei be the sub-er-algebra of E generated
by the countable family of sets {Enm} and let Ai be the restriction of the
measure A to Ei. Then the sequence (/n) lies in the closed subspace L\ (Ai) of
L\ (A). We shall show that the sequence (/n) has a subsequence that converges
weakly in Li(Ai).
Since the sequence (fin(E)) is bounded for every measurable set E, a
diagonal argument shows that (fin) contains a subsequence (fhik) with the
property that the sequence (fink(Enm)) converges for each member of the
countable family of sets {Enm}. Let T be the collection of all sets EeSj
for which the sequence (fink (E)) converges. Thus, every set in the algebra
generated by the sets Enm belongs to T. We claim that T is a monotone
class. To see this, let (Fj) be an increasing sequence in T and let F = \J- F3.
By the uniform countable additivity of the sequence (fin), we have fink (F3) —t-
fink(F) uniformly in k. Furthermore, the sequence (f*nk(F]))k converges for
each j. These facts together imply that the sequence (fink (F)) converges and
hence F belongs to E. A similar argument shows that T is closed under
countable decreasing intersections.
Therefore T is a monotone class that contains the algebra generated by the
sets Enm- By the Monotone Class Lemma, T contains the er-algebra generated
by the Enm and so the sequence (fink(E)) converges for every E £ H1. Since
the sequence (fink) is bounded, it follows that (fnk) converges weakly in
Li(Ai) and hence also in Li(A). Therefore (fink) converges weakly in M(E).
a
Since the spaces Li(A) are subspaces of M(E), we also obtain a description
of the weakly compact subsets of these spaces. A subset K of L\ (A) is said
to be uniformly integrable if, for every e > 0, there exists S > 0 such that
if A(E) < 5, then fE \f\ dX < e for every / e if. It is easy to see that this
notion is equivalent to uniform countable additivity. "Thus, we have:
Corollary C.8. Let X be a finite, positive measure. A bounded subset, K,
of Li(X) is relatively weakly compact if and only if the elements of K are
uniformly integrable.
Most of the results presented in this appendix have a "topological
version" , where the space of measures M(E) is replaced by the space of regular
Borel measures on a compact topological space. We shall need just one of
these topological results in the text. Let S be a compact topological space
and let "Bs be the a-algebra of Borel subsets. Consider the Banach space
C(S) of continuous scalar functions on S, endowed with the usual supremum
norm. The Riesz Representation Theorem states that the dual space C(S)m
is the space of regular Borel measures on "Bs, with the variation norm. We

218 C Spaces of Measures
note that C(S)* is a closed subspace of M(!Bs). We will need a
characterization of the weakly compact subsets of C(S)*. An examination of the proof of
Proposition C.7 shows that the control measure constructed in the proof of
the implication (i) => (ii) is regular if all the measures belonging to the set
K are regular. Thus, we have the following:
Proposition C.9. Let S be a compact topological space. The following are
equivalent for a bounded subset K ofC(S)*:
(i) K is relatively weakly compact.
(ii) There exists a finite, positive, regular measure A on 3$ such that the
elements of K are uniformly absolutely continuous with respect to A.
(iii) The elements of K are uniformly countably additive.

References
1. I. Amemiya and K. Shiga. On tensor products of Banach spaces. Kodai Math.
Sem. Rep., 9:161-178, 1957.
2. R. D. Bourgin. Geometric aspects of convex sets with the Radon-Nikodym
property. Springer-Verlag, Berlin, 1983.
3. S. Chevet. Sui certains produits tensoriels topologiques d'espaces de Banach.
Z. WahrschevnlichkeiUtheorie und Verw. Gebiete, 11:120-138, 1969.
4. Simone Chevet. Sur certains produits tensoriels topologiques d'espaces de
Banach. C. R. Acad. Sri. Paris Sir. A-B, 266:A413-A415, 1968.
5. D. W. Dean. The equation L{E, X") = L(E, X)" and the principle di local
reflexivity. Proc. Amer. Math. Soc., 40:146-148, 1973.
6. A. Defant and K. Floret. Tensor norms and operator ideals. North-Holland
Publishing Co., Amsterdam, 1993.
7. J. Diestel. The Radon-Nikodym property and spaces of operators. In
Measure theory (Proc. Con/., Oberwolfach, 1975), pages 211-227. Lecture Notes in
Math., Vol. 541. Springer, Berlin, 1976.
8. J. Diestel. Sequences and series in Banach spaces. Springer-Verlag, New York,
1984.
9. J. Diestel, H. Jarchow, and A. Tonge. Absolutely summing operators.
Cambridge University Press, Cambridge, 1995.
10. J. Diestel and J. J. Uhl, Jr. Vector measures. American Mathematical Society,
Providence, R.I., 1977. With a foreword by B. J. Pettis, Mathematical Surveys,
No. 15.
11. J. Diestel and J. Jerry Uhl, Jr. Progress in vector^ measures—1977-83. In
Measure theory and its applications (Sherbrooke, Que*, 1982), pages 144-192.
Springer, Berlin, 1983.
12. N. Dunford and J. T. Schwartz. Linear operators. Part I. John Wiley &
Sons Inc., New York, 1988. General theory, With the assistance of William G.
Bade and Robert G. Bartle, Reprint of the 1958 original, A Wiley-Interscience
Publication.
13. P. Enflo. A counterexample to the approximation problem in Banach spaces.
Acta Math., 130:309-317, 1973.
14. B. R. Gelbaum and J. Gil de Lamadrid. Bases of tensor products of Banach
spaces. Pacific J. Math., 11:1281-1286, 1961.
15. J. Gil de Lamadrid. Measures and tensors. Trans. Amer. Math. Soc., 114:98-
121, 1965.
16. J. Gil de Lamadrid. Measures and tensors. II. Canad. J. Math., 18:762-793,
1966.

220 References
17. J. E. Gilbert and T. J. Leih. Factorization, tensor products, and bilinear forms
in Banach space theory. In Notes in Banach spaces, pages 182-305. Univ. Texas
Press, Austin, Tex., 1980.
18. Y. Gordon, D. R. Lewis, and J. R. Retherford. Banach ideals of operators with
applications. J. Functional Analysis, 14:85-129, 1973.
19. A. Grothendieck. Resume1 de la theorie metrique des produits tensoriels
topologiques. Bol. Soc. Mat. Sao Paulo, 8:1-79, 1953.
20. A. Grothendieck. Sui certaines classes de suites dans les espaces de Banach et
le theoreme de Dvoretzky-Rogers. Bol Soc. Mat Sao Paulo, 8:81-110 (1956),
1953.
21. A. Grothendieck. Produits tensoriels topologiques et espaces nucl&ures. Mem.
Amer. Math. Soc., 1955(16):140, 1955.
22. A. Grothendieck. Resume1 de la theorie metrique des produits tensoriels
topologiques. Resenhas, 2(4):401-480, 1996. Reprint of Bol. Soc Mat. Sao
Paulo 8 (1953), 1-79 [MR 20 #1194].
23. A. Grothendieck. Sur certaines classes de suites dans les espaces de Banach,
et le theoreme de Dvoretzky-Rogers. Resenhas, 3(4):447-477, 1998. With a
foreword by Paulo Cordaro, Reprint of the 1953 original.
24. S. Heinrich. Ultraproducts in Banach space theory. J. Reine Angew. Math.,
313:72-104, 1980.
25. J. R. Holub. Tensor product bases and tensor diagonals. Trans. Amer. Math.
Soc., 151:563-579, 1970.
26. J. R. Holub. Tensor product mappings. Math. Ann., 188:1-12, 1970.
27. J. R. Holub. Hilbertian operators and reflexive tensor products. Pacific J.
Math., 36:185-194, 1971.
28. G. J. O. Jameson. Summing and nuclear norms in Banach space theory.
Cambridge University Press, Cambridge, 1987.
29. S. Kwapien. On operators factorizable through Lp space. In Actes du Colloque
d'Analyse Fonctionnelle de Bordeaux (Univ. de Bordeaux, 1971), pages 215-
225. Bull. Soc. Math. France, M&n. No. 31-32. Soc. Math. France, Paris, 1972.
30. S. Kwapien and A. Peiczyiiski. The main triangle projection in matrix spaces
and its applications. Studia Math., 34:43-68, 1970.
31. J.-T. Laprest6. Sur un generalisation de la notion d'operateurs nucleaires et
sommants dans les espaces de banach. C. R. Acad. Sci. Paris., 275:45-48,1972.
32. J.-T. Laprest6. Operateurs sommants et factorisations. A travers les espaces
Lp. Studia Math., 57(l):47-83, 1976.
33. D. R. Lewis. Conditional weak compactness in certain inductive tensor
products. Math. Ann., 201:201-209, 1973.
34. J. Lindenstrauss and A. Peiczynski. Absolutely summing operators in £p-spaces
and their applications. Studia Math., 29:275-326, 1968.
35. J. Lindenstrauss and H. P. Rosenthal. The £p spaces. Israel J. Math., 7:325-
349, 1969.
36. A. Persson. On some properties of p-nuclear and p-integral operators. Studia
Math., 33:213-222, 1969.
37. A. Persson and A. Pietsch. p-nukleare une p-integrale Abbildungen in Ba-
nachraumen. Studia Math., 33:19-62, 1969.
38. A. Pietsch. Absolut p-summierende Abbildungen in normierten Raumen. Studia
Mathematica, 28:333-353, 1967.
39. A. Pietsch. Adjungierte normierte operatorenideale. Math. Nachr., 41:189-211,
1971.

References 221
40. A. Pietsch. Operator ideals. VEB Deutscher Verlag der Wissenschaften, Berlin,
1978.
41. G. Pisier. Counterexamples to a conjecture of Grothendieck. Acta Math.,
151(3-4):181-208, 1983.
42. G. Pisier. Factorization of linear operators and geometry of Banach spaces.
Published for the Conference Board of the Mathematical Sciences, Washington,
DC, 1986.
43. P. Saphar. Comparaison de normes sui des produits tensoriels d'espaces de
Banach. Applications. C. R. Acad. Sci. Pans Sir. A-B, 266.A809-A811,1968.
44. P. Saphar. Produits tensoriels topologiques et classes d'applications lineaires.
C. R. Acad. Set. Paris Sir. A-B, 266:A526-A528, 1968.
45. P. Saphar. Produits tensoriels d'espaces de Banach et classes d'applications
lin&iires. Studia Math., 38:71-100. (errata insert), 1970.
46. R. Schatten. A Theory of Cross-Spaces. Princeton University Press, Princeton,
N. J., 1950.
47. R. Schatten. Norm ideals of completely continuous operators. Springer-Verlag,
Berlin, 1960.
48. M. Talagrand. Pettis integral and measure theory. Mem. Amer. Math. Soc.,
51(307).ix+224, 1984.
49. D. van Dulst. The geometry of Banach spaces with the Radon-Nikodym
property. Rend. Circ. Mat. Palermo (2), (Suppl. 7):ix+81, 1985.

Index
a, associated norm, 161
/a, at\, /at\, injective associates,
a/, \a, \a/, projective associates, 168
accessible tensor norm, 162
adjoint ideal, 197
adjoint tensor norm, 191
alternating tensor product, 13 -
approximable operator, 46
approximation property, 72
- bounded, 80
- metric, 80
Banach ideal, 194
Bartle integral, 112
bilinear form, 1
- 2-dominated, 183
- a-integral, 187
- a-nuclear, 193
- bounded, 22
- canonical extensions, 25, 132, 188
- Hilbertian, 180
- integral, 58
- nuclear, 39
- type a, 187
Bochner integral, 27
bounded approximation property, 80
canonical extensions of a bilinear form
25, 132, 188
Chevet-Saphar norms, 135
cofinitely generated, 161
conjugate ideal, 199
contragredient norm, 191
dp, 135
Da, &2, 183
dual tensor norm, 160
Dunford integral, 52
Dunford operator, 52
Dunford-Pettis property, 110
elementary tensor, 3
finitely generated Banach ideal, 198
function
- Bochner integrable, 27
- essentially separably valued, 26 ,
- ^-measurable, 26
- Pettis integrable, 52
- simple, 25
- weakly /n-measurable. 26
- weakly integrable, 52
gp, 135
Grothendieck constant, 153
Hilbertian tensor norm, 176
3, II • 11/. 62
3P, »p, 173
inequality K
- Grothendieck,'il52
- Khinchine, 36 >
injective norm, 45
injective space, 25
injective tensor product, 46
integral bilinear form, 58
integral norm
- of a bilinear form, 58
- of an operator, 62
integral operator, 62
Kg, Grothendieck constant, 153
X(£,X), 104
Khinchine Inequality, 36
jCw, II • Ik, 180

224 Index
.0]-space, 30
•Coo-space, 51
.Op-space, 155
Mi(E,Jf), 107
M(2,X), 95
MA(E), 103, 212
maximal Banach ideal, 197
metric approximation property, 80
minimal Banach ideal, 197
>fp. I! • K. 140
W. II • llw. 41
natural tensor norm, 183
norm
- 2-dominated, 183
- a-integral, 187, 189
- a-nuclear, 193
- Hilbert-Schmidt, 151
- Hilbertian, 180
- p-integral, 173
- p-nuclear, 140
- Pettis, 53
- Pietsch integral, 65
norming set, 46
nuclear bilinear form, 39
- on Co, 40
nuclear norm
- of a bilinear form, 39
- of an operator, 41
nuclear operator, 41
operator
- a-integral, 189
- a-nuclear, 193
- approximable, 46
- completely continuous, 64
- Dunford, 52
- Hilbert-Schmidt, 151
- Hilbertian, 180
- integral, 62
- nuclear, 41
- p-integral, 173
- p-nuclear, 139
- p-summing, 142
- Pietsch integral, 65
- representable, 115
7„ 7T,,, 142
W. I! • ||w, 65
Pettis integral, 52
Pettis norm, 53
Pietsch integral operator, 65
Principle of Local Reflexivity, 124
projective norm, 15
Rademacher functions, 33
Radon-Nikodym property
- and the approximation property, 122
- coincidence of integral and nuclear
operators, 113
- for vector measures, 106
- for/i, 102
- for Banach spaces, 102, 108
- for reflexive spaces, 120
- for separable duals, 119
- is separably determined, 121
rank of a tensor, 3
reasonable crossnorm, 127
Schatten dual, 160
semivariation, 94
separating set, 4
space
- Lebesgue-Bochner, 29
square ordering, 88
symmetric
- tensor product, 13
- tensor norm, 129
tensor, 2
tensor diagonal, 23
tensor duality, 9
tensor norm, 130
- accessible, 162
- adjoint, 191
- associated with a Banach ideal, 195
- Chevet-Saphar, 135
- cofinitely generated, 161
- dual norm, 160
- Hilbertian, 176
- injective, 129
- injective associate, 166
- left (right) accessible, 165
- left (right) injective, 151
- left (right) injective associate, 166
- left (right) projective, 137
- left (right) projective associate, 168

Index 225
- natural, 183
- projective, 129
- projective associate, 168
- Schatten dual norm, 160
- totally accessible, 162
tensor product
- injective, 46
- of linear mappings, 7
- of operators, 17
- of vector spaces, 1
- projective, 17
tensor product basis, 88
Theorem
- Bartle-Dunford-Schwartz, 97
- Bochner, 28
- Bounded Convergence, for vector
measures, 101
- Dvoretzky-Rogers, 149
- Egoroff, for vector measures, 101
- Grothendieck Factorization, 191
- Kwapien Domination, 181
- Lewis-Stegall, 116
- Local Reflexivity, 124
- Orlicz-Pettis, 56
- Pettis Measurability, 26
- Pietsch Domination, 146
- Pitt, 85
totally accessible tensor norm, 162
trace
- class, 43
- for nuclear operators, 77
- of a linear mapping, 8
trace duality, 10
transpose, 7, 129
uniform crossnorm, 128
variation
- of a vector measure, 93
variation norm, 94
vector measure, 93
- compact, 104
- of bounded variation, 94
- regular, 109
- semivariation, 94
- semivariation norm, 95 .
- variation, 93
- variation norm, 94
weakly subseries summable sequence,
56