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CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 25 EDITORIAL BOARD D. J. H. GARLING , D. GORENSTEIN, T. TOM DIECK , P. WALTER...

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CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 25 EDITORIAL BOARD D. J. H. GARLING , D. GORENSTEIN, T. TOM DIECK , P. WALTERS

Banach spaces for analysts

Banach Spaces For Analysts

P. Wojtaszczyk Institute of Mathematics, Polish Academy of Sciences

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University of Cambridge

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CAMBRIDGE UNIVERSITY PRESS Cambridge

New York Port Chester Melbourne Sydney

Puhlished by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 40 West 20th Street, New York, NY 100 1 1 , USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia ©Cambridge University Press 1991 First published 1991 Printed in Great Britain at the University Press, Cambridge British Library cataloguing in publication data

Wojtaszczyk, P. Banach spaces for analysts. 1 Banach spaces I. Title 515.732 Library of Congress cataloguing in publication data available

ISBN 0 521 35618 0 hardback

Contents ix

Preface

Part I. Introduction I.A. LB.

Functional analysis Examples of spaces and operators

3 9

Part II. Basic concepts of Banach space theory II. A.

II.B.

II. C.

II.D.

II. E.

Weak topologies Weak and weak* topologies, Mazur's, Alaoglu's and Goldstine's theorems, reflexive spaces Isomorphisms, bases, projections Banach-Mazur theorem, block-bases and selection of block basic subsequences, examples of bases: Haar, FaberSchauder and trigonometric systems, small perturbations of basic sequences, decomposition method Weak compactness Eberlein- S mulian theorem, weakly compact operators; their factorization and basic properties Convergence of series Definitions and characterizations of unconditionally and weakly unconditionally convergent series, Orlicz theorem and Orlicz property, unconditional bases; spaces without unconditional basis, the Haar system is an unconditional basis in Lp for 1 .-spaces, BanachMazur distance and distances between £; -spaces and spaces of trigonometric polynomials, good c-nets in Auerbach lemma, principle of local reflexivity

27

35

49

57

69

Bx,

Part III. Selected topics liLA.

Lp-spaces; type and cotype Sobolev spaces ( 1 < p < oo ) and Bergman spaces (1 � p < oo ) are isomorphic to Lp-spaces, complemented subspaces of lp, stable laws and embeddings of lp into

83

vi

Contents

Lq , 0 < q � p � 2, type and cotype, Kahane's inequality, types and cotypes of Lp-spaces, Carleman theorem, Banach-Saks theorem. III.B Projection constants Leo is injective, extension and projection constants, Lewis's theorem on projections in Lp, Kadec-Snobar theorem, projection constants of various spaces of polynomials, existence of an inner function on the ball in ([d , polynomial bases in C( 'l') , projective tensor products and dual treatment of extensions of operators III.C. L1 { p,)-spaces Semi-embeddings and the non-existence of semiembeddings of L1 [0, 1) into Co, Menchoff theorem, Lyons theorem characterizing zero sets for measures with Fourier transforms tending to zero, Schur theorem that weakly compact sets in £1 are compact, uniform integrability and characterization of weakly compact sets in L1 ( p,) , weak sequential completeness of L1 (p,) , finite representability of £1 in a Banach space; connections with type, reflexive subspaces of L1 ( p,) , Nevanlinna theorem on unimodular functions in cosets of Leo/Heo III .D. C(K)-spaces M-ideals and best approximation, Heo +Cis a closed algebra, {Heo + C) /Heo is an M-ideal in Leo/Heo, Michael-PelczyD.ski linear extension theorem, Milutin theorem, Franklin system - construction and basic properties, Lipa ( 11') is isomorphic to leo, Dunford-Pettis property for rich subspaces of C(K), in particular for ball algebras and Sobolev spaces III. E. The disc algebra Rudin-Carleson theorem, interpolating sequences, interpolation of continuous functions by functions with finite Dirichlet integral, lacunary Fourier coefficients of functions in the disc algebra, projections in and infinite divisibility of existence of basis III.F. Absolutely summing and related operators Absolutely summing operators, Grothendieck's and Pietsch's theorems, unconditional bases in L1-spaces, local units in L1 (G) , Grothendieck 's inequality, polynomially bounded but not power-bounded operators, p-nuclear and p-integral operators, trace duality, characterization

A,

111

131

151

181

A

199

Contents

III.G.

III.H.

III.I.

vii

of Sidon sets, extrapolation inequalities for spaces of all p-absolutely summing operators, Grothendieck-Maurey theorem, p-absolutely summing operators and cotype 2 Schatten-von Neumann classes Approximation numbers of operators between Banach spaces, Weyl's inequality, Schatten-von Neumann classes and eigenvalues, Hilbert-Schmidt operators, Fourier coefficients of Holder functions, summability of eigenvalues of p-absolutely summing operators between Banach spaces with applications Factorization theorems Nikishin's theorem and some applications, Maurey's factorization of operators into Lp, reflexive subspaces of L 1 , connections between factorization and p-absolutely summing operators, structure of series unconditionally convergent in measure, dilation theorem and connections with orthogonal series, Menchoff-Rademacher theorem on almost everywhere convergence, more about Fourier coefficients of Holder functions, some multipliers into Bergman spaces Absolutely summing operators on the disc algebra Proper uniform algebras are uncomplemented, Bourgain's analytic projections onto Hp(ll.d>.), extrapolation inequalities between p-absolutely summing and qintegral norms for operators on has cotype 2 and the Grothendieck theorem holds for extensions of @ finite rank operators on is closed in C (Y) @C (Y) , reflexive subspaces of Ll/ H1 . extrapolation inequalities for the Riesz projection, extension of operators from reflexive subspaces of Ll / H1 into Hoc, interpolation for certain hi-analytic functions

237

257

291

A, A* A*, A, A A

Hints for the exercises List of symbols References Index

323 341 345 377

P reface Banach space theory became a recognised part of the mathematical scene with the appearance of Banach [1932). From its birth it maintained close ties with the rest of analysis. It turned out that Banach space theory offered powerful tools to other branches of analysis. The most useful ones are duality theory for spaces and operators, results about infinite dimensional convexity and results connected with the Baire category theorem, most notably the closed graph theorem. These powerful gen eral concepts are now well known among analysts and appear in almost every textbook on functional analysis or real variable theory. They were already well understood in the forties and fifties, and at that time it seemed to many that Banach space theory was dead or at least rele gated to an obscure corner of science. However, this was not the case. The sixties, and especially the seventies and eighties, saw an enormous eruption of activity in the field. Old problems were solved and, more importantly, new problems and new ties with the rest of mathematics emerged. Also, new and powerful methods and directions of research appeared. For those in the field progress seemed to be constantly accelerating, but to those outside it may have looked as if the theory was dying again. Probably one of the reasons for this perception was that this research activity (maybe because it was so dynamic ) unfortunately did not pro duce many books on the subject, and those that did appear were usually devoted to a special topic. There are two notable exceptions to this state ment, Beauzamy [1982) and Lindenstrauss-Tzafriri [1977, 1979). There are old favourites, still beautiful and in good shape like Banach [1932), Dunford-Schwartz [1958) and Day [1958 and newer versions) , but nat urally they do not present the more recent results. So the newcomer to the field, after learning what was generally known thirty years ago, had a choice of either starting on more specialized books or turning to Beauzamy or Lindenstrauss-Tzafriri. Now I like both these books and the present one is not intended to replace either of them. Beauzamy is a nice, easily readable introduction and Lindenstrauss-Tzafriri, although difficult in places for the novice, has every mark of a classic, especially if the long-promised volumes III and IV are added to the first two. But both these books have one feature in common: they study Banach spaces for their own sake and their own beauty. In this way they are great for

X

Preface

the future specialist who is already under the spell of Banach spaces. However, there are some mathematicians (even a majority) who are not convinced that Banach space theory is the most enchanting branch of mathematics, and although I am not one of them, I understand them. In fact, Banach spaces are not only beautiful and interesting, but also useful. This point is not, in my opinion, made clear in Lindenstrauss-Tzafriri or Beauzamy. The methods created in Banach space theory since the late 60's can be applied in other areas of analysis. Hence somebody interested primarily in harmonic analysis, functions of a complex variable, orthonormal series, approximation theory or proba bility theory can find it useful to know some Banach space theory. This book is directed towards such a person. Ideally I think of it as a text book for a graduate course for students who have already learned some functional analysis and are interested in analysis or in some area of it. I also hope that a mature analyst may read it, or some of it, as part of the ongoing education process which is an important part of the life of any active mathematician. I hope too that Banach space specialists will find some portions of the book interesting because they present some applications of the subject I was not aware of. Let me digress a bit and comment on the possible profit a classical analyst might derive from Banach space theory. I do not claim that Banach space theory can solve all his important problems. But it may help him to see the problem in a new light which makes it easier to isolate essential features. He can also use general theorems and techniques in the solution. The general framework can also suggest interesting new problems. To mention one example, the power of duality methods makes it a standard procedure to try to find the dual of any Banach space considered, a problem that could not even arise without this more general framework. Another example is the corona theorem. It is now a major part of the theory of analytic functions, but its origin lies in the attempt to get some understanding of the maximal ideal space H00 , a question which is unthinkable without the general theory of Banach algebras. Mathematics and each of its parts lives and grows on the exchange of ideas, between mathematicians, between various branches of math ematics and between mathematics and other areas of human activity. Banach space theory is no exception to this rule. It utilises ideas and techniques from other fields and in turn provides other branches of math ematics with some of its own. In this respect this book is one-sided. It concentrates heavily on ideas and results that have a clear potential to be useful in other branches of analysis.

Preface

xi

Formally the book is divided into three parts, numbered by roman numerals. Each of these parts is divided into chapters, distinguished by capital letters. Each chapter is divided into sections with arabic num bers. Each such section contains at most one Theorem, Proposition or Lemma. The Theorem appearing in section II. B. 7 is later referred to as Theorem II.B.7, or within the same chapter just as Theorem 7 (or Lemma or Proposition as the case may be) . Part I has an introduc tory chapter. It contains well known and some less well known results (without proofs) that will be used later. Chapter LA contains basic results from general functional analysis and Chapter I.B discusses the examples of Banach spaces that are considered later and quotes some analytical results about these spaces. The main function of Part I is to establish notation and conventions and to serve as a refresher and refer ence for the background material needed later on. Part II is essentially an introduction to the language and basic techniques of Banach space theory. The real heart of the matter is Part III, where a selection of topics is studied in depth. The reader can get an idea of the contents of each chapter from the Table of Contents, so I will not attempt a more detailed description here. Also, each chapter of Parts II and III starts with a short description of its contents, so the reader can find additional details there. Each chapter concludes with Notes and Remarks containing biblio graphical data and comments about generalizations, ext ensions or appli cations of the results presented in the chapter. I have tried really hard to find the correct reference and credits. On the other hand, I have not conducted a full scale historical investigation into the origin and devel opment of each idea and result. I would like to offer sincere apologies for any omissions and inaccuracies in this respect. In the main text theorems are only given names if it is common usage. The absence of a name in the text or of a credit in the Notes and Remarks does not mean that the result is due to the author. It means either that the result is so well known that I judged it to be folklore or simply an omission on my part. Each chapter of Parts I and II contains exercises. These exercises range from routine to very hard and I have not given any indication of their di fficulty. There is a hint for each exercise located in the special chapter at the end of the book. These hints range from almost complete solutions to the reference only. I have tried to point out, if possible, where the solution of an exercise can be found in the literature and to give proper credit. There are also some repetitions in the exercises. I have simply put the same or similar problems into different chapters if

xii

Preface

they fit well into the material. This should be useful for those read ers (the majority?) who read only selected parts of the book. It also indicates different approaches to the same question. The bibliography contains only the works actually referred to some where in the book. I have made no attempt to make it complete. Also, it does not include any data about translations (this is particularly impor tant with respect to publications in Russian) or reprints and republishing (it happens sometimes that an East European book is published origi nally in English and later republished without any changes by a West European or American company) . If the work appeared in Russian, this is indicated in the references and the author 's English translation of the title is given. This translation should be close to the one used in Mathematical Reviews, but need not be identical. As with most mathematics books, it will be unusual for a reader to read this book from beginning to end. It is also unnecessary. The reader interested in a particular theorem or chapter should start right there. The choice of material in the book reflects my philosophy, taste and, last but not least, knowledge and ability. Here I would like to mention some subjects which really should have been included but which were not because of limitations of space and time and (probably most impor tant) my poor understanding of them. The first is the deep connections between Banach spaces, descriptive set theory and the classical theory of sets of uniqueness for trigonometric series (see Kechris-Louveau [1987] and Lyons [1989] ) . The second is the applications of the study of finite dimensional spaces to problems on convex bodies. This in turn has ap plications to harmonic analysis, number theory and other subjects. This whole area is currently one of frantic activity and is undergoing constant and fascinating changes. Probably anything I could write about it now would be outdated by the time this book reaches the reader. As an in troduction to this area I suggest Pisier [P] , Milman-Schechtman [1986] or Milman [1986] . The next subject which I regrettably had to omit is the connection between Banach space theory and probability theory. Actually probabilistic methods underly much of the current research in Banach spaces. This shows even in this book despite my unfortunate lack of knowledge of probability theory. The study of probability in Ba nach spaces is developing too: see Linde [1983] . The last subject I would like to have included is the general area of vector valued functions and operators acting on such functions. There is considerable activity in this area as well. As an introduction to it I suggest Burkholder [1986] and Pisier [P 1] .

Preface

xiii

While writing the book I received helpful advice from many math ematicians. I am grateful to all of them for their time, advice and help. First of all I would like to express my profound gratitude to my teacher and colleague, Prof. Aleksander PelczyD.ski. He convinced me that I should write the book in the first place and offered plenty of advice on what it should contain. Much of the time I did not follow his advice, but the effort needed to refute his arguments helped greatly to clarify my own ideas. I would like to thank Prof. Keith Ball and Prof. Ben Garling for reading large parts of the manuscript and providing numerous and invaluable pieces of advice on language and presentation. The following other mathematicians helped me greatly by generously offering their ad vice, knowledge and insight: Dan Amir, Sheldon Axler, Don Burkholder, Joe Cima, John Fournier, Ben Garling, Nassif Ghossoub, Yehoram Gordon, Pawel Hitczenko, Bill Johnson, Serguey Kislyakov, Boris Kashin, Stanislaw Kwapien, Elton Lacey, Joram Lindenstrauss, Niels Nielsen, Gilles Pisier, Richard Rochberg and Walter Schachermayer. Some work on the book was done during visits to St. John's College Cambridge; Monash University, Clayton; University of British Columbia, Vancouver; Odense University and Texas A &M University, College Station. I would like to thank Ben Garling, Ala Sterna-Karwat, Nassif Ghossoub, Niels Nielsen and Elton Lacey for arranging these vis its and for being great hosts, each in his or her unique way. Clearly most of the actual work was done in Warsaw where my own Mathemat ical Institute of the Polish Academy of Sciences gave me the freedom to do the job and an excellent library to help me. The final version was written, and all typing done, at Texas A&M University. I would like sincerely to thank Robin Campbell for her unique ability to transform my scratchy and unreadable (sometimes even to myself) handwriting into the beautifully typeset text. Finally I would like to express my deep gratitude to my wife Anna and my children Ola and Kuba for all the love, support and distractions they generously provided over the years. Without their presence (and at suitable times their absence!) writing this book would have been much more di fficult.

Part I Introduction This introductory part contains background material. It is not intended to be a course in any subject. It is simply a collection of definitions and facts given without proof (with one exception) . We provide references to works which contain detailed exposition, full proofs, examples and motivation. In a sense this whole part is a quick reference guide to re sults which will be used in the later parts. The introduction is divided into two chapters, Chapter LA describing what we will need from gen eral functional analysis and Chapter I.B which contains results about concrete spaces and operators. Since I.A is really a review of a standard course in functional analysis, references are given only at the end of the chapter. In I.B we give references after each paragraph. The references given in this part are usually to standard textbooks and monographs, not to original works. If a particular result or subject cannot be easily located using the table of contents or index we try to provide more detailed information (sections or pages) . Sometimes we formulate a result in a form which is more convenient to us but different from the one given in the reference. Usually in such a case it is easy to derive our formulation from the one given in the reference.

I.A. Functional analysis 1. A linear topological space X is a linear space over the real or complex numbers endowed with a topology T such that the map (x, y) �--+ x + y is continuous from (X, r ) x (X, r ) into {X, r ) and the map (t, x) �--+ tx is continuous from R x X (or CC x X) into X. Such a topology is fully described by a basis of neighbourhoods of 0. A subset V c X is called convex if whenever x �, x2 E V then the whole interval ax1 + { 1 - a)x2 for 0 ::; a ::; 1 is in V. A linear topological space is called locally convex if it has a basis of convex neighbourhoods of 0. A functional on X is a continuous linear map from X into scalars. The set of all functionals on X will be denoted X* , and called the dual space. A linear operator (or just operator) T: X --+ Y {where X and Y are linear topological spaces) is a continuous linear map. A subspace of X will always (unless explicitly stated otherwise) denote a closed linear subspace. Given a set V C X by span V we denote the closure of the set of all linear combinations of elements from V (i.e. the subspace of X spanned by V).

2.

A linear topological space X i s called an F-space i f its topology is given by a metric p such that p(x, y) = p(x-y, 0) and X is complete with respect to this metric. A quasi-norm on a linear space X is a function q from X into the nonnegative reals satisfying (a)

q(x) = 0 if and only if x = 0,

q(Ax) = I Ai q(x) for all scalars A and all x E X, (c) there exists a constant Cx such that q(x + y) ::; Cx ( q(x) + q(y)) for all x, y E X.

{b)

One easily checks that each quasi-norm defines a linear topology on X. The basis of neighbourhoods of the point x consists of 'balls' around x , i.e. sets {y E X: q(x-y) < e }, e > 0. A linear space X with a quasi-norm will be called a quasi-normed space. A very important special case of quasi-norm is a norm. This is a quasi-norm on X for which the constant in (c) above equals 1 (one gets from {b) that � 1 ) . The usual notation for the norm of x is ll x ll . Every norm 11 · 11 on X defines a metric p (x, Y ) = ll x - Yll · A Banach space X is a linear space equipped with a norm II · II and such that X is complete with respect to the metric p. Clearly every Banach space is an F-space. The symbol will denote

Cx

Cx

Bx

4

l.A.

Functional analysis §3.

the closed unit ball of X, i.e. Bx = {x E X : l l x l l $ 1 } . Its interior is an open convex set, so every Banach space is locally convex. This need not be true for general quasi-norms. Note also that a subspace of a Banach space {resp. quasi-normed space) is a Banach space (resp. quasi-normed space) . We simply have to restrict the norm {resp. quasi-norm) . Let X and Y be two Banach spaces and let T: X -+ Y be a linear map. Then T is continuous if and only if II TII = sup{ II Tx l l : l l x l l $ 1 , x E X} < oo. The quantity II Til is a norm on the linear space L(X, Y) of all operators from X into Y. The space L(X, Y) with the above defined norm is a Banach space. Unless otherwise indicated the convergence of operators will be understood in this norm.

3.

In particular, for a Banach space X the space X* is also a Banach space with the norm of a functional x* E X* defined as llx* ll = sup{ lx* (x) l : x E X, l l x l l $ 1 } .

4.

Let X and Y b e Banach spaces and let T: X -+ Y be a linear operator such that T(Bx ) contains some open ball in Y. Then T(X) = Y and there exists a positive number r such that T(Bx ) :::> r · y = {y E Y: II YII < r} .

5 . Open Mapping Theorem.

B

6. Closed Graph Theorem. Suppose that T: X -+ Y is a linear map (not assumed to be continuous, but defined everywhere on X) from an F-space X into an F-space Y. Assume that { (x, Tx) : x E X} C X x Y is closed in the product topology. Then T is continuous. 7. Banach-Steinhaus Theorem. Suppose (T-y ) -yer is a family of linear operators from a Banach space X into a Banach space Y. As sume that for every x E X we have sup{II T-yxii='Y E r} < oo. Then sup{ II T'YII ='Y E r} < oo . In particular we get that the pointwise limit of a sequence of linear operators {if it exists everywhere) is a linear operator. Let X be a linear space over the real numbers (without any topology) and let Y C X be a linear subspace. Assume also that we have a function p: X -+ R such that p( x + y) $ p(x) + p(y) for all x, y E X and p(tx) = tp(x) for all x E X and t E R, t � 0. Assume moreover that we have a linear map f: Y -+ R such that J(y) $ p(y) for all y E Y. Then there exists a linear map F: X -+ R such that F I Y = f and -p( -x) $ F (x) :::; p (x) for all x E X .

8. Hahn-Banach Theorem.

l.A. Functional analysis §9.

5

This general algebraic theorem has many very important special cases. Some of them are stated below in 9-11. Note that despite the fact that 8 is true only for real spaces, the consequences listed below are true also for spaces over the complex scalars. Taking in 8 p(y) = IIYII we obtain: If X is a Banach space and Y C X is a subspace and if y* E Y* , then there exists x* E X* such that ll x* II = IIY * I and x* I Y = y* . In particular we get 9.

ll x ll = sup{ l x * (x) l : x * E X* , ll x* ll � 1 } . A judicious choice of p yields also the following: If X is a locally convex space and B C X are disjoint closed convex sets with being compact, then there exists a continuous linear functional I on X and a real number a such that Re I < a and 10.

Re

I(B)

> a.

A

A,

(A)

In particular we see that X* separates the points of X.

A, I(A)

We also have the following version of 10 . If X is a locally convex space and B c X are disjoint convex sets with open, then there exists a continuous linear functional I on X and a real number a such that Re < a and Re I(B) 2: a. 11.

A

12. If T: X --+ Y is a linear operator then it induces an operator T* : Y* --+ X* , called the adjoint (or dual) operator and defined as T* (y* ) (x) = y* (Tx). One easily checks that II T II = li T* II · 13. If T: X --+ Y is onto then T* is 1-1 and there exists a constant c > 0 such that II T*y* ll 2: cii Y * II for all y* E Y*.

If T: X --+ Y is such that there exists c > 0 such that II Tx ll for all x E X then T* maps Y* onto X*. 14.

2:

cllx ll

15. A linear map T: X --+ Y (X, Y Banach spaces) is compact if the set T( Bx) is compact in the norm topology of Y. One easily checks that each compact map is automatically continuous. The set of all compact operators from X into Y, denoted K(X, Y), is a subspace (i.e. closed and linear) of L(X, Y) . Also an operator T: X --+ Y is compact if and only if T* : Y* --+ X* is compact. It is easy to see that if T E K(X, Y) and 81 E L(ZI. X) and 82 E L(Y, Z2 ) then S2 TS1 E K(Z1 . Z2 ) .

I.A. Functional analysis § 1 6.

6

An operator T: X -+ X is called power-compact if Tn is compact for some n E N. 16.

An operator T: X -+ Y is called invertible if there exists an 17. operator S: Y -+ X {usually we write r - 1 instead of S) such that ST = idx and TS = idy. By idx (or idy) we mean the identity operator on X (or on Y), i.e. idx (x) = x for all x E X. It is important that we need both conditions ST = idx and TS = idy; one of them is not enough. If X is a complex Banach space (i.e. it is a linear space over the complex numbers) and T: X -+ X is a linear operator, then the spectrum of T, denoted u(T) , is the set of ali A E cr such that ( >-.idx - T) is not an invertible operator. The set u(T) is a non-empty, compact subset of cr. A number ).. E cr is called an eigenvalue of T if there exists a vector x E X, x -=F 0, called an eigenvector associated with the eigenvalue ).. , such that Tx = >-.x . Clearly each eigenvalue of T belongs to u(T) . With each eigenvalue ).. we associate its spectral manifold E>.. = E>.. ( T) = Un>l ker( >-.i dx - T) n . Clearly E>.. is an increasing union of subspaces of X. The number dim E>.. (possibly oo ) is called the multiplicity of the eigenvalue .>-.. 18.

19. If an operator T: X -+ Y is power-compact then u(T) is finite or consists of a sequence of points tending to zero together with zero itself. Every point ).. E a(T) , except possibly zero, is an eigenvalue of T of finite multiplicity. 20. An operator P: X -+ X is called a projection if P2 = P. Then P ( X ) is a closed subspace of X and Px = x for x E P(X ) . A subspace

Y C X which equals P(X) for some projection P: X -+ X is called complemented.

21. Let V be a convex set in a locally convex topological vector space X. A point v E V is called an extreme point if it is not in the interior of any closed interval contained in V, i.e. if v 1 , v2 E V and v = av1 + (1 - a)v2 with 0 < a < 1 then v = v 1 = v2 . If is any subset of X then the convex hull of denoted conv equals

A A, A, n n n x E X: x f;ajaj with f;aj = f; ia31 1 { and a3 E A for 1, 2, . .. EN · =

=

j

=

, n, n

}

I.A. Functional analysis §22.

7

22. Krein-Milman Theorem. If V is a compact, convex subset of a locally convex, topological vector space, then V equals the closure of the convex hull of its extreme points. In particular this theorem implies that each convex, compact subset of a locally convex, topological vector space has an extreme point. References. Everything said above can be found in most textbooks on

functional analysis. In particular everything can be found in Dunford Schwartz [1958] or Edwards [1965] and everything except power-compact operators can be found in Rudin [1973] .

I.B. Examples of spaces and operators 1. Whenever we consider a measure space {0, E, J.L) we assume that the measure J.L is complete and that there are no atoms of infinite measure. We say that the measure space (0, E, J.L) is separable if there exists a countable family of sets C E such that the smallest complete is E. The following characterizes separable u-field containing measure spaces. Suppose that {0, E, J.L) is a separable non-atomic measure space, with J.L a positive measure and J.L (O) = 1 (i.e. J.L is a probability measure) . Then (0, E, J.L ) is isomorphic to the unit interval (0, 1] with the Lebesgue measure. This can be found in Halmos (1950] §41 . A general classification theorem for arbitrary measure spaces is due to Maharam (1942] and can also be found in Lacey (1974] § 14.

(Aj)� (Ai)�1 1

J

For any measure space {0, E, J.L) with J.L positive we define Lp(O, E, J.L) , 0 < p :5 oo, to be the space of E-measurable functions {more precisely of classes of functions where we identify functions which are equal J.L-a.e.) such that Jn l f(w ) I P dJ.L (w ) < oo. For p = oo we mean supess l f(w ) l < oo. Usually the u-field E is clear so we will use the notation Lp (O, J.L) and when either the measure or the set are clear from the context we will suppress them also. We will use the notation I f li P =
Thus Lp{O, J.L) , 0 < p :5 oo, are linear spaces. For 1 :5 p :5 oo they are Banach spaces. For 0 < p < 1 we can introduce the natural met ric on Lp (O, J.L) by p(f, g ) = II! - gi l� · Lp (O, J.L) is complete with re spect to this metric and the topology induced by this metric makes Lp(O, J.L) an F-space. It is easy to see that Lp {O, J.L) is not locally con vex if 0 < p < 1 . By Lo{O, J.L) we mean the space of all {classes of ) measurable, almost everywhere finite functions on 0 with the topology of convergence in measure. If J.L (O) is finite, this topology can be given by the metric p( f, g ) = J0( (l !!;f��l) )dJ.L. When (0, J.L) is u-finite we

10

I. B. Examples of spaces and operators §3.

U�1 Ei Ei L�12-iJ.L(Ej)-1 1

J.L(Ei)

write n = with the 's disjoint and < oo and define p( f, g ) = fE/ ( �J!�� I l )dJ.L. Clearly Lo {f2, J.L) (at least for a a-finite measure space) is an F-space. All this can be found in Dunford-Schwartz (1958] , Edwards (1965] , Banach (1932] and in most textbooks on functional analysis. For a number p, 1 � p � oo we will denote by p' the conjugate exponent, i.e. p' satisfies � + '? = 1 {so clearly p' = (p� l ) ) with the con vention 1 ' = oo and oo' = 1 . The following inequality, called HOlder's inequality, is of fundamental importance: For any p, 1 � p � oo and any two functions J, 9 we have 3.

1

In l f(w)9(w) l dJ.L(w) � ( In l f (w) IPdJ.L(w) ) (In J;

l u (w) I P' dJ.L (w )

)

1

pr

or in short II ! · ull 1 � llfllv iiY IIv ' · If this inequality is actually an equality then g (w ) = c · sgn{f{w)) l f(w) I P- l J.L-a.e. where c is a constant and sgn(a) is the complex signum of a, i.e. sgn a = a f l a l . The proof can be found in almost every textbook of functional anal ysis or measure theory, e.g. Dunford-Schwartz (1958] , Edwards (1965] . The duals of Lv (n, J.L) spaces for 1 � p < oo are described as fol lows {this is really a restatement of HOlder's inequality) : Lp{n, J.L) * = Lp' (O, J.L) and the duality is given as g { f ) = f0 g (w ) f (w )dJ.L (w ) for 9 E Lp' {O, J.L) and f E Lp {f2, J.L) . The space L00(0, J.L) * is {unless {O, J.L) consists of a finite number of atoms) much bigger than L1 {f2, J.L) and is impossible to describe explicitly. For p < 1 the dual depends on the measure space. If {0, J.L) has no atoms then 0 is the only continuous linear functional on Lp (O, J.L) . If (0, J.L) is purely atomic and each atom has measure 1 then Lv (n, J.L) * = £00 (0, J.L) for 0 < p < 1 . All this can be found i n Dunford-Schwartz (1958] or Edwards (1965] or in most textbooks on functional analysis. 4.

5. The following notational conventions will be used throughout this book. When the measure on the set n and the a-field are clear from the context, we will use the notation Lp(O) , in particular Lp(R.) and Lp [O, 1] will mean that we consider the usual Lebesgue measure and Lp (T) will be with respect to Lebesgue measure on 11', but normalized to make the measure of the whole circle 1 . The Lebesgue measure of a set A is denoted by IAI. If the measure space consists of a certain number

I. B. Examples of spaces and operators §6.

11

of atoms each having measure 1 , then Lp(O) will be denoted lp (O). If those atoms are naturally identified with the integers 7l or the natural numbers N we will use the notation lp . If the measure space n consists of n atoms each of measure 1 , then lp ( 0) will be denoted by t;. All this applies to 0 < p � oo . The spaces Lp (O, J..L ) for various values of p are clearly related. One of the expressions of this relationship are interpolation theorems. In this book we will use only the simplest cases of the best known results.

6.

Let (O, J..L ) be a measure space and let B C Lp 1 ( 0, J..L) nLp2 ( 0, J..L ) be a linear set which is dense in both Lp; (n, J..L ) , i = 1 , 2, with 1 � Pl < P2 � oo. Assume that we have a linear map T

Riesz-Thorin Theorem.

defined on B with values in the measurable functions on a measure space (Ot . J..L t ) · Assume also that for every I E B we have li Tl llv ; � C ll l llv. for i = 1 , 2. Then for every p , p1 � p � P2 and I E B we have

so T extends to a continuous linear operator from Lp(O, J..L ) to Lp (Ot . J..L t ). The proof of this theorem can be found in Katznelson [1968] , Zyg mund [1968] , Dunford-Schwartz [1958] , or in books devoted to interpola tion of operators like Bennet-Sharpley [1988] or Krein-Petunin-Semenow [1978] .

7. We say that a function I on a measure space ( 0, J..L ) is of weak type p, 0 -; p < oo and write I E Lp,oo (n, J..L ) if there exists a constant K such" that for every .X E IR+ we have

J..L { W E 0: l l(w) l

� .\} �

( �r.

We denote the space of all functions of weak type p by Lp,oo and the in£ of all constants K which can appear in the above inequality for a given I by ll l llv ,oo · This quantity is not a norm but it is a convenient notation. If T is a linear map defined on a dense, linear subset B c Lp (O, J..L ) with values in the measurable functions on (Ot . J..L t ), then we say that T is of weak type (p-p) if there exists a constant K such that

12

I.B. Examples of spaces and operators §8.

for every A E IR+ and f E B . If p mean II Tf ll oo :5 K ll f ll oo ·

= oo

we interpret this condition to

Marcinkiewicz Theorem. Suppose that B is a linear set dense in both Lp , (n, JL) and Lp2 (n, JL) , 1 :::; P1 < P2 :::; oo and suppose that T is a linear map defined on B with values in the measurable functions on a measure space ( n 1 , JL 1 ) . If T is of weak type (P1 - P 1 ) and of weak type (P2 - P2 ) then for every p, p 1 < p < P2 there exists a constant Cp such that li T /li P :5 Cp ii ! II P for all f E B . So we get that T extends to a continuous operator from Lp ( fl, JL) into Lp (fl1 , JL 1 ) for P 1 < p < P2· For the constant Cp we have the es timate Cp :5 K max ( (p_!p,) , ( 2�p) ) . The proof can be found in Stein P [1970] , Zygmund [1968] or in any of the books on interpolation of oper ators mentioned in 6.

8. The Rademacher functions (rn (t) )�=1 are defined on [0, 1] as rn (t) = sgn sin 2n t1r. The alternative descriptio!!- is that r1 ( t) = 1 if 0 :::; t < � and r1 (t) = - 1 if � < t :5 1 and Tn +1 (t) takes value 1 on the left hand half of each interval where rn ( t) is constant and takes value - 1 on the right hand half of each such interval. Using probabilistic language we can say that (rn (t) )�=1 is a sequence of independent random variables, each taking value 1 with probability � and value - 1 with probability � . Whatever the description, we easily see that (rn (t) )�=1 is an or thonormal system in £2 [0, 1] . Clearly this system is not complete. The main fact about Rademacher functions, which will be used repeatedly in this book is Khintchine's inequality. There exist constants Ap, Bp , 0 < p < oo such that for all (finite ) sequences of scalars (an ) �=1 and every p, 0 < p < oo we have

There is also a similar inequality for lacunary sequences of exponents. More precisely we have the following: If (nk ) k:,1 is a sequence of natural numbers such that infk (nk + l /nk ) = A > 1 then there exist constants Ap = A(p, A) and Bp = B (p, A) for 0 < p < oo such that for every sequence of scalars (a k ) k:,1 and every p, 0 < p < oo we have

I. B. Examples of spaces and operators §9.

13

This can be found in many places, e.g. Zygmund [1968] or Stein [1970] . If K is a topological space, then by C(K) we mean the space 9. of all scalar valued (i.e. real or complex valued) , bounded, continu ous functions on K. This is a Banach space with the norm 11/ llao = sup{ l/(k) l: k E K}. Clearly if K is a compact space then C(K) con sists of all continuous, scalar valued functions. If the space K is locally compact then by C0 (K) we denote the subspace of C(K) of functions 'vanishing at infinity' , i.e. such that for every c > 0 there exists a com pact set Ve C K such that sup{ l/(k) l: k E K\Ve} < c. If K has the discrete topology then C(K) is denoted by lao(K) and if K = N then lao(N) is abbreviated to lao. The symbol co denotes Co (N) , i.e. the space of all sequences converging to zero. If the set K happens to be of finite cardinality n then C( K ) = lao(K) is also denoted l�. The symbol c denotes the subspace of lao consisting of convergent sequences.

10. If on a Banach space X we have also defined a multiplication (not necessary commutative) x · y irr such a way that x · ( z + y) = X · Z + X · y, ( z + y) · X = Z · X + y · X and ( .\x) · y = X · ( .\y) = A(X y) (.X is a scalar) and llx · Y ll � llxiiii Y II then we have a Banach algebra. A linear multiplicative functional on such an algebra is a non-zero linear functional which also preserves the multiplication. The set of all linear multiplicative functionals on X is denoted M(X) . If X is a commutative algebra with unit then M(X) C X* is an w*-compact set (see II.A.9) . There is a natural, norm-1 map i: X -+ C(M(X) ) defined as i(x) (
Gelfand-Naimark Theorem. Suppose that on a commutative, com plex Banach algebra X with unit we have an involution * such that (x + y ) * = x* + y * , ( x · y ) * = y * · x* , ( .Xx ) * = Xx * , x ** = x and llxx* II = l x l l2 • Then the map i is a linear multiplicative isometry from X onto C(M(X ) ) . This theorem implies in particular that Lao(n, J..L ) i s isometric t o a C(K) for some compact space K. Also any space C(K) as defined in 9 is isometric to a space C( K) for some compact space K. This can be found in any book on Banach algebras, e.g. Zelazko [1973] , and in many general texts on functional analysis, e.g. Rudin [1973] .

14

I.B. Examples of spaces and operators § 1 1 .

1 1 . Riesz Theorem. The dual of the space C(K), K compact, equals the space of all regular, Borel measures (scalar valued, but obviously not necessarily positive) on K. The duality is defined as J.L { f ) JK fdJ.L and the norm of the measure as a functional on C(K) equals its total variation. The space of all Borel, regular measures with finite total variation on a space K will be denoted M(K). So for compact K we have C(K)* = M(K). This is a classical theorem and can be found in most texts on measure theory and functional analysis, e.g. Dunford-Schwartz [1958] or Edwards [1965] . =

Suppose that A c C(K) , K compact, is an algebra (i.e. sums, products and scalar multiples of functions from A are in A). Suppose also that if f E A then f E A (this condition is empty if we consider real scalars) and that 1 E A. If A separates points of K (i.e. for every k1 , k2 E K, k 1 f:. k2 there is an f E A such that f(kl ) f:. j(k2 )), then A is dense in C(K). If 1 f/. A but all the other assumptions are satisfied then A nfE ,- 1 (0) = {a} for some a E K and the closure of A equals {! E C(K): f(a) = 0}. If A does not separate points, then we can pass to the quotient space K/ ,...., where ,...., is the equivalence relation on K defined by k 1 ,...., k2 if and only if f(kl ) = j(k2 ) for all f E A. The space K/,...., is compact and the algebra in C(KJ "') corresponding to A separates points of K/ "' · This can be found in many texts on functional analysis, topology or approximation theory. A detailed discussion can be found in Edwards [1965] .

12. Stone-Weierstrass Theorem.

A locally compact abelian group G is an abelian group which at the same time is a locally compact space and such that all group operations are jointly continuous. On each such group there exists a unique (up to multiplication by a scalar) invariant, Borel measure, i.e. a Borel measure m such that m(A) = m(A + g ) for every Borel set A and every g E G. This measure is called the Haar measure. Note that A+g = {h E G: h = a+g with a E A}. If J.L is any measure in M ( G) and f is a Borel function on G, then the convolution f* J.L is defined to be the function f * J.L (s) = J0 f(s - t)d�-t(t) , provided that this integral exists. If f E Lp(G, m), 1 $; p $; oo then f * J.L E Lp(G, m) and ll f *J.LIIP $; 11�-tllllfllp· Also if f E C(G) then f * J.L E C(G). We can define the convolution of two functions f and g by the formula f * g(s) = f0 f (s - t)g(t)dm(t) if this integral exists.

13.

l.B. Examples of spaces and operators § 14.

15

If g E £1 ( G, m ) this definition clearly coincides with the previous one after we identify 9 with the measure 9dm. If I E L1 ( G, m ) and 9 E L00 ( G) then I * g E C ( G) . We can also define the convolution of two measures J..L, v E M ( G) as the measure defined for A C G as f..L * v(A) = fa J..L(A- t) dv(t) . One checks that (in all the above cases) the convolution is commutative and distributive, i.e. I* ( A1 91 + A292) = Ad * 91 + A2 I * 92 where A1, A2 are scalars. This means that every measure f..L defines an operator Tp.: Lp ( G) -+ Lp ( G) , 1 � p � oo and Tp.: C(G) -+ C(G) defined by Tp.( f ) = I * f..L · We have II Tp. ll � II J..L I I in all cases. When we consider Tp.: L1 ( G) -+ L1 ( G) or Tp.: C(G) -+ C(G) then we actually have II TP. II = II J..L I I · All this is basic harmonic analysis and can be found (in various degrees of generality) in any textbook on harmonic analysis, e.g. Rudin [1962a] . If G is a compact abelian group, then a character of G is a con tinuous group homeomorphism from G into '1'. The set of all characters of G is usually denoted by r. If we treat characters as complex valued functions on G then they form a complete orthonormal set in £ 2 ( G, m ) where m is normalized so that m ( G) = 1 (this is possible because the group is assumed to be compact) . The characters are also linearly dense in C ( G) . Given a measure f..L E M ( G) (or a function I E L1 ( G) ) we define its Fourier coefficients

14.

for The map ·: M ( G) -+ ioo(r) defined above is called the Fourier trans form. Since characters are linearly dense in C ( G) this map is one-to-one, i.e. the Fourier coe fficients determine the measure. One easily checks that (J..L * v)" ( 'y) = [l,('y) v('y) . If f..L E M ( G) has Fourier coefficients [l,{'Y) then we will write f..L = L"Y E r [1,('Y) · 'Y and call this series the Fourier series of f..L even when we know nothing about the convergence. On the circle group 'I' the characters can be identified with the functions (ei nB) n E Z· We write [l,(n) for J"D' e -i n8dJ..L (0) . Let us also recall here that a subset H C r is called a A(p) set, p > 1 , if there exists a constant K such that

for every sequence of scalars

(a"YhEH ·

I.B. Examples of spaces and operators § 1 5.

16

Let us also recall that a subset exists a constant K > 0 such that

H c r is called a Sidon set if there

for every sequence of scalars ( a-y ) -yEH· The details can be found e.g. in Rudin [1962a] .

15.

The Dirichlet kernel is defined as

n

'Dn(O) =

L

k= - n

e ik9 ,

n = 0, 1 , 2, . . .

Jt E M(11') and Jt = L::�: P,(k)eik!J then 'Dn * Jl = k L;":_nP,(k)ei !J. This means that the convolution with the Dirichlet ker

If we have

nel realizes the partial sum projection with respect to the Fourier series. We have r I'Dn(O) Idm(O) = 42 log(n + 1) + o( 1 ) .

I�r

1r

Note that this equals the norm of the operator f 1-t f * 'Dn in the spaces £1 ( 11') and C(11') . This operator will sometimes be called the Dirichlet projection. For details see Zygmund [1968] , Katznelson [1968] p. 50, Kashin Saakian [1984] or any book on trigonometric series.

16.

The Fejer kernel is defined as

Fn(O)

1 = - n+ 1

) ik!J. t vk(O) = tn ( 1 - J!1._ n+ 1 e

k=O

k=-

One checks that Fn(O) ?: 0 for n = 0, 1 , 2, . . . and so J1r IFn(O) Idm(O) = JyFn(O)dm(O) = 1 . If f E L p( 11') , 1 � p < oo, or f E C(11') in the case p = oo, then II/- f * Fnllp ---+ 0 as n ---+ oo. The norm of the Fejer operator, f 1-t f * Fn, is 1 in spaces Lp ( 11') , 1 � p � oo . The details can be found in Zygmund [1968] , Katznelson [1968] p. 1 2, Kashin-Saakian [1984] or in any book on trigonometric series.

17. The de la VallOO-Poussin kernel is defined as Vn = 2F2n-l -Fn-l· Clearly IIVnll1 � 3 and II/-Vn* fliP---+ 0 as n---+ oo for f E L p( 11') , 1 � p < oo, or f E C(11'). This follows immediately from 16. The nice

17

I.B. Examples of spaces and operators § 1 8.

property of the de la VallOO-Poussin kernel is that for f = L�n ajeii 9 we have Vn * f = f. All this follows easily from properties of the Fejer kernels. The details are in Zygmund [1968] and Katznelson [1968] .

18.

The Poisson kernel is defined as

+oo - oo

( 1 - r2 ) Pr(8) = :�:::> l i l eijfJ = (1 2r - cos 8 + r2 )

for

0 < r < 1.

Clearly Pr (8) � 0 and J'lf Pr(8)dm(8) = 1 for 0 < r < 1. As for the Fejer kernel we have II/ - f * Pr ll p - 0 as r - 1 for f E Lp(1l'), 1 � p < oo , or f E C(1l'). The important feature of the Poisson kernel is that for J.L E M(1l'), Pr*J.L can be treated as a function on D by h(rei 9 ) = Pr*J.L( 8) . Such a function is always harmonic i n D. Basic properties of the Poisson kernel can be found in Zygmund [1968] , Katznelson [1968] , Hoffmann [1962] , Duren [1970] , Koosis [1980] and in many other places.

19.

The Hardy spaces Hp(D),

{

Hp(D) = f(z): f(z)

11/IIH, =

0 < p � oo , are defined as

is analytic in

D and

)

�

1 2� f(rei9 ) P d8 " 1 l �� 271"

( 1

<

oo

}·

For 0 < p < 1 the Hardy space Hp( D) is a complete, linear metric space and for 1 � p � oo it is a Banach space. The limit limr-1 f(rei 9 ) = f(ei 9 ), called the boundary value of f, exists for almost all 8 E 1l', and 11/II H, = 11/IIL , ('If) · This shows that we can identify Hp(D) with span{ e i nfJ} n?: O C Lp(1l'). This space of functions on 1l' will be denoted by Hp(1l'). If f E Hp(1l'),p � 1 then we can recover the original analytic function as f(re i 9 ) = f * Pr (8) where Pr denotes the Poisson kernel. This correspondence is also natural in terms of Fourier series. If f = E�= O anzn E Hp(D), p � 1 then the boundary value f E Lp(1l') has Fourier series E�= O anei nfJ. We will use the notation H�(D) or H�('l') for the subspace of Hp consisting of functions vanishing at 0 E D, or equivalently such that

Jy / = 0.

The details can be found in Hoffmann (1962] , Zygmund (1968], Katznel son (1968] , Duren (1970] or Koosis (1980] and in many other places.

18

I. B. Examples of spaces and operators §20.

20. There is an orthogonal projection from L2 ( T) onto H2 ( T) . In terms of the Fourier coefficients this projection can be written as 1?.(2:�: anein9 ) = L�= O anein 9 • This projection is called the Riesz pro jection. It is important to know how the Riesz projection acts on other Lp (1l') spaces. The basic fact is that 'R. is of weak type (1-1) (this is called Kolmogorov's theorem) , so it is a continuous projection from Lp ( T ) onto Hp (1l') for 1 < p < oo (see 7). We have II'R. II L.,-L., � Cp2 (p - 1) - 1 . The Riesz projection 'R. is actually not continuous in Lt(T) and in L00(T) . This projection is clearly closely related to the Cauchy formula. If J.L E M (1l') then the Cauchy formula

2 _ 1 _ { .. dJ.L (eit ) f ( z ) = 211'i lo eit - z D. One easily checks that f (z) = 00 2: jL(n )z n Because 'R. is of weak type (1-1) we obtain that f (z) E n=O Hp (D) for p < 1. In a sense the Riesz projection is the Cauchy formula acting on T. As in 19 the details can be found in Hoffmann [1962] , Zygmund [1968] , Katznelson [1968] , Duren [1970] , Koosis [1980] and in many other places, e.g. a nice presentation of a proof that 'R. is bounded on Lp(1l') for 1 < p < oo can be found in Lindenstrauss-Tzafriri [1979] . defines an analytic function in .

21. Most of what has been said in 19 and 20 can be extended to polydiscs Dn and balls 1Bn in ern . For 0 < p � oo we define

{

Hp (Dn ) = f : f is analytic in D and II I IIH., = sup r< l

where

(

m is the probability Haar measure on yn,

{

)

Hp(1Bn ) = f : f is analytic in D and IIJIIH., = sup r< l

.!

{ IJ (rz) I P dm(z) " }yn

( Jsnr lf (rz) I Pdu(z) )

<

1

p

<

oo

oo

}

}

where u is the normalized rotation invariant measure on Sn. The boundary values limr 1 f(rz) = f(z) exist almost everywhere on yn or Sn , and we have IIIII H., = llfllp· Thus we can identity Hp(Dn ) -+

19

l.B. Examples of spaces and operators §22.

with a subspace of Lp(P) denoted by Hp(lf'n) and we can identify Hp (ffin) with a subspace of Lp(§n) denoted by Hp(§n)· The natural orthogonal projection from L2 (P) onto H2 (P) extends to a bounded projection from Lp(P) onto Hp(P) for 1 < p < oo (apply the Riesz projection R in each variable separately) but is not of weak type (11). On the ball this natural orthogonal projection from L2 (Sn) onto H2 (§n), called the Cauchy projection and denoted by C , is of weak type (1-1), and so it is a continuous projection from Lp(§n) onto Hp(§n)· Using the Cauchy kernel for the ball mn we can express this projection as

C (f)( z ) = Jsr n (1 -J(
for

and f E Lv (Sn)· This is simply the Cauchy formula for the ball ffin . The standard references for all this are Rudin [1969] and Rudin

[1980] .

L1 (1f') then we know (see

*

f(rei6 ) = ( ! Pr) (O) is harmonic in ID, so there exists a unique harmonic function j(rei 6 ) in ][) such that j(O) = 0 and f(rei 6 ) + if(rei6 ) is an analytic function. We know that f(rei 6 ) = 2:: � : r l n l ] (n)e i n(} so 22.

If f E

18) that

+oo

J(rei6 ) = - i L sgn n r l n l ] (n)ei n(} 00 -

with the convention that sgn 0 = 0. One can show that limr-+l j(rei 6 ) = f(ei6 ) exists almost everywhere on lf'. There is also a very useful integral representation of j(ei 6 ), namely

1 j(ei6 ) = P.V. 2 11'

1211" f(t 0

T

) cot T

2

T

d

.

The function j on 1f' is called the trigonometric (or harmonic) conjugate of f. There is a close connection between the trigonometric conjugation and the Riesz projection. Looking at Fourier coefficients we see that Rf = � J( 0) + ( ! + i j) . This shows that the map f �----+ j is continuous on Lv (lf') for 1 < p < oo and of weak type (1-1). This is called Kolmogorov's theorem. This is a standard result in the theory of Fourier series. The details can be found in Zygmund [1968] , Katznelson [1968] , Koosis [1980] and many other books.

20

I.B. Examples of spaces and operators § 23.

23. One of the basic tools in the study of spaces Hp(DJ. 0 < p � oo is the so-called canonical factorization. For f E Hp(Dl let (zn )�= 1 denote the zeros of f each counted according to its multiplicity. The sequence (zn )�= 1 satisfies the Blaschke condition 2:: ::'= 1 (1 - lzn l ) < oo. Conversely, if we have a sequence of points (with repetitions) (zn )�= 1 C such that 2:: ::'= 1 (1 - l zn l ) < oo, then the following prcduct (called the Blaschke product)

D

B(z) =

00

IJ l zZnn l 1Zn- -ZZZn -

n= 1

converges almost uniformly in D and the function B(z) (also called the Blaschke product) has zeros exactly at the points (zn )�= J · Moreover From this we see that each II B IIoo = 1 and I B(ei 6 ) 1 = 1 a.e. on f E Hp(D) can be written as f = B · ft where ft is zero-free and 11/llv = ll !t llv and l f(ei 6 ) 1 = l ft (ei6 ) 1 a.e. on In order to factor ft we need to present the construction of an outer h � 0 be such that J'lf log h(t)dt > -oo. function. Let h E L 1 It is known that for f E Hp ( ) , 0 < p � oo, 1!1 satisfies these assumptions. We take the harmonic conjugate of log h(t) and form the analytic function H(z) = exp(log h + ilog h) .

1I'.

(1I'),

1I'.

1I'

One checks that limr-+1 H(rei 6 ) exists almost everywhere and has mod ulus h on Applying this to l f(ei 6 ) 1 , for f E Hp (ID) we get the outer function F(z). This function is in Hp (D) and en Y we have l f(ei 6 ) 1 = I F(ei6 ) 1 a.e. Thus we can write every f E Hv (D) aB f = B · F · l where B ( z) is the Blaschke product formed from the zeros of f (counting multiplicity) and F is an outer function with I F(ei 6 ) 1 = l f (e i6 ) 1 a.e. on The remaining factor I can be defined simply as f · (B ·F)- 1 . It is an analytic function with ll l lloo = 1 and I I(ei 6 ) 1 = 1 a.e. on T. This factor is called a singular inner function (if it is not a constant) . In general an inner function is a function g in H00 ( ) such that l g (ei 6 i = 1 a.e. on Thus the Blaschke product is also an inner function, and a singular inner function is a zero-free inner function (non-constant). The singular inner functions have the representation

1I'.

1I'.

1I'

1I'.

1 I(z) = exp - 271'

!11' -.e•i-6 -d + p, (O) - 1r

z

e6-z

where p, is a positive, singular measure on [ - 11' , 11' ] . The a)ove described factorization f = B · F · I is called the canonical factcrization. The

21

I. B. Examples of spaces and operators §24.

outer functions provide the tool to build analytic functions in Hp(D) with given modulus on 1l'. This will be used extensively in III.I. Detailed proofs of the canonical factorization can be found in Hoff mann [1962] , Duren [19 70] , Koosis [1980] and Garnett [1981] .

24. The following inequality due to R.E.A.C. Paley will be very useful. Suppose that f = E:=o an z n E H1 ( D) . Then

Since this inequality will be used later to prove Grothendieck's The orem III.F.7. which is the basis for many results presented in chapters III.F -I, I have decided to present sketches of the proofs here.

Proof A. (Paley [1933] )

It follows from 23 that one can write

1 1 f(z) = (E:=o bn zn ) (E:=o en zn ) where fE:=o lbn l 2 ) 1 2 (E:=o l en l 2 ) 1 2 = ll f ll 1 · From this we infer that a2 k = L s ,r: s+ r= 2 k c8 br, so

00

00

� 4 L lcs l 2 L lbs l 2 = 4 11 ! 11 � s=O s=O Proof B. (Smith [1983] ) define inductively

Let

o:k

=

10- 1 a2 k (E� 1 l a 2• l 2 ) - 1 / 2

91 = o: 1e2i9 k k 9k = 21 o:k e2 i9 + (1 - lo:k l 2 )9k - 1 - 21 o:k e - 2 i9 9k2 - 1 · -

and

22

-

I. B. Examples of spaces and operators §25.

D we have I� + (1 - I W)a1

I

a;b � 1 we see that IIYk I � 1. One checks that for l � k we have Yk (2 ) = (1 - la1 l 2 )(1 - l a2 l 2 ) · · · (1 ln1 - 1 l 2 )a 1 and Yk (s) = 0 for s � 0 and s =F 21 . Since L� 1 1ak l 2 � 100- 1 this product is bounded away from 0, so 2 .. ll f ll 1 � 2 Yk f k 2 = L (1 - l a1 1 ) · · · (1 - l a 1 1 l 2 ) a1 a 2 t 1=1

Since for a, b

E

� �1 l

�c Since this holds for all

(t, I a,. 1 ')"'

k we have the claim.

We will also use the following inequality due to Hardy. If f = then I: :'= o � 1r ll f l h · The proof can be found in Duren [1970) p . 48, Hoffmann [1962) p. 70, Katznelson [1968] p. 91. The inequalities of Paley and Hardy fail for £ 1 ('1'). This is one possible way to show that the Riesz projection is not continuous on

25.

l: :'=o anzn E

H1(D)

(�a+!)

£ 1 ('1').

A(D)

D

The disc algebra is the space of all functions analytic in which admit a continuous extension to D. We can identify with the subspace span{ e i n9 }n�o C C('l'). This is the space of all functions f E C( ) such that j(n) = 0 for n = - 1, -2, . . . . It will be denoted by The symbols Ao(D) and Ao('l') will denote the subspace of A(D) consisting of functions vanishing at 0 E D. The annihilator of C C(T) is described as follows.

26.

T A(T). A('I')

F.-M. Riesz Theorem. If J.L E M('l') and JT fdJ..L = 0 for all f E A('l') then J..L is absolutely continuous with respect to the Lebesgue measure. Consequently df..L = hdm with h E

Hf('I').

The proof can be found in Hoffmann and Katznelson [1968) p. 89.

[1980]

D, Hp (D) Hp (D),

[1962] ,

Duren

[1970) ,

Koosis

and z E then the map f �--+ f(z) defines a If f E linear functional on 0 < p � oo. The norm of this functional is estimated in the following simple inequality

27.

I.B. Examples of spaces and operators §28.

The proof is in Duren

[1970]

23

p. 36.

V E ([n is an open set, then we can define the Bergman space Bv (V), 0 < p as the space of all holomorphic functions in Lp (V ) , where on V we have 2n-dimensional Lebesgue measure. We will be mostly interested in the case V = D or sometimes V = IBn and V = Dn . In all those cases Bv (V) is complete. Also the point evaluations are continuous and we have inequalities of the form: for every compact set K V there exists a constant C = CK such that l f (z) l C l f l v for z E K. We will also discuss the ball algebra A ( IBd ) which is the space If

28.

:::;

oo

:S:

C

of all functions analytic in IBd C ([d which have a continuous extension to IBd. The polydisc algebra A(Dd ) is the space of functions analytic in Dd which admit a continuous extension to Dd . The space A(Dd ) can d) . be naturally identified with a subspace of A good reference is Rudin [1980] where the case IBn (so in particular D) is discussed in detail. Also Axler [1988] discusses in detail. For the polydisc the standard reference is Rudin [1969] .

C(1I'

f

Bp (D) (K

f

, p) , we say 29. If is a scalar valued function on a metric space that satisfies the Holder condition of order a, 0 < a :::; 1 (for a = 1 the term Lipschitz condition is also used) if

k K

l f (k) l 'Po:U) Lipo:(K,

Clearly for every E the quantity is a norm and + the corresponding Banach space of all functions satisfying the Holder condition of order a is denoted by p) . We will consider only being or [0,1] with the natural metric. One can also replace the function in the definition by a more general function.

1I',R. o: t-

K

Now we want to discuss the Sobolev norms which measure the smoothness of functions of several variables. The theory is usually done in a much more general setting, but we restrict our attention to functions on s = 1, 2, . . . . Let = 0, 1, 2, . . . be the set of all multiindices A= such that the j ' s are nonnegative integers and p A) = z::; = l :S: Each A E defines a partial derivative denoted OA . We define the Sobolev space for k � 0, s � 1, to be the space of all functions on such that oA is continuous for all A E The norm which makes into a Banach space can be defined as

30.

1I'8 , (i1. i2 , . . . , is ) ij k.

D(k), k i D(k) k 8 C (1I' ) ys k 8 f C (1I' )

11 ! 11 � = sup { i i &Aflloo : A E D(k) } .

(

D(k).

24

I.B. Examples of spaces and operators §31 .

If we want to extend this definition for p < oo we encounter some techni cal problems, so to avoid them, we first define the norm on functions as

11 / 1 1 ;

= ( AED(k) L I 8A f l � )

c=

1

p.

By the Sobolev space w; (1fB ) we understand the completion of the functions under this norm. Quite often we will treat as a subspace of where V is the disjoint union of copies of11' 8 • (Here the modulus signs represent cardinality.) The natural isometric injection is defined by the rule that on the torus 1rs which is a subset of V with the index E is simply We can also treat as a subspace of E) , the space of continuous, E-valued functions on 11'8 where E is the Banach space The natural injection maps � The same operations work also for w; ( 11'8 ) and we can treat w; (11'8 ) as a subspace of Lv (V) (V the same as above) or a subspace of Lp ( 11's ; The standard reference for the theory of Sobolev spaces is Adams [1975] . It is also presented in Stein [1970] and Stein-Weiss [1971] .

c=

Ck (1fB)

C(V) I D (k) l j: Ck (11'8 ) ---+ C(V) j(f) A D(k) 8A f· kC (1fB) 8 C(11' ; t'�(k) l . f (8A f(t)) AED(k)· f1D(k) l ) . c=

31. We have defined w; ( 11'8 ) as the completion of the functions under a certain norm. Nevertheless it is interesting to know up to what degree the elements of w; ( 11'8 ) are really functions. This is answered by the so-called embedding theorems. We will use the simplest one.

c=

functions ex The identity on tends to a continuous, linear operator from w; ( 11'8 ) into w; - m ( 11'8 ) where 0 :=:; m :=:; and mp < s and p :=:; q :=:; (s �:T.v ) . In particular w; ( 11'8 ) embeds into Lq ( 11'8 ) if p :S q :S s �lv .

Sobolev Embedding Theorem.

k

( )

The proof can be found in Adams [1975] or Stein-Weiss [1971] .

ein1 91 ein2 92 eins 9s

32. We will also use the general multiplier theorem on 11'8 • Let us recall that the characters of the group 11'8 are when ( n t , . . . , n 8 ) E 7L8 • A bounded function m(�) , for � E 7L8 , defines an operator on £2 ( 11'8 ) , called a multiplier and defined by •

.

• .

The following theorem describes sufficient conditions for a multiplier to act on Lp (1I'8 ) , 1 

� we have

Suppose that r.p is a function on

R8 such that

A E D(k) (the description of D(k) is in 30)

nr sup R2 A 1 8Am(x) l dx < oo O
for

zs

)-

00 .

c

1 < p < oo and of weak type (1-1).

R8

This theorem is a well known Hormander-Mihlin type multiplier theorem. It is usually formulated on and in even greater general ity, so I was unable to locate the exact reference. It follows e.g. from Theorem 4.4 of Chapter XIV of Torchinsky [1986] together with an easy observation that a map bounded on some < 1 and some q > 1 is of weak type (1-1) (see Folland-Stein [1982] Th. 3.37). Also the proof in Garcia-Cuerva-Rubio de Francia [1985] pp. 210-14 gives it.

Hp ,P

Lq ,

Part II Basic concepts of Banach space Theory II.A. Weak Topologies In this chapter we discuss two topologies which are weaker than the norm topology. They play an important role, both in the general theory and in applications. These topologies allow easy use of compactness arguments, and so they are helpful in existence proofs. This is particularly true of the w* -topology which can be defined on any dual space. We also introduce and study the class of reflexive spaces.

(

1. On each Banach space X there exists a weak topology {or a X, X * ) topology or w-topology) . For each point xo E X its basis of neighbour hoods is defined as

U(xo; c, xi , . . . , x�) = { x e X: l xj (x) - xj (xo) l < c

for

j = 1,

. . . ,n

}

where xi , . . . , x� is an arbitrary finite set in X* and c is an arbitrary positive number. Obviously this defines a locally convex topology on X. A sequence (xn )�=l C X such that for some x E X, Xn --+ x in the a X, X* )-topology is said to be weakly convergent. We write this convergence as Xn � X. This is clearly equivalent to the condition that for every x* E X* the sequence x* (xn) --+ x* (x) as n --+ oo. A sequence (xn )�=l C X such that for every x* E X* the scalar sequence x* (xn) is convergent is said to be weakly Cauchy.

(

Xn = {1 , . . . , 1, 0, 0, . . . ) . Then Xn ..___..., n is weakly Cauchy because for every x* = (�j )� 1 E c0 = £ 1 we have x* (xn) = E;=l �j --+ E�1 �j as n --+ oo . On the other hand {xn )�=l is not weakly convergent, because {1, 1, 1, . . . ) fj CQ . 2 Examples.

{a)

Let X

= eo

and

times

Usually it is quite difficult to describe the weak topology o n the whole space. Quite often it is easier to describe this topology when restricted to the unit ball of X.

28

II.A. Weak Topologies §3.

B

(b) On the unit ball of eo the a(co , t'l )-topology coincides with the topology of pointwise (i.e. coordinatewise) convergence (i.e. the Ty chonoff topology on n:=l [-1, 1] restricted to the unit ball of eo ) . Clearly every open set in the topology of pointwise convergence is open in the a(eo , t'1 )-topology. Conversely, given a neighbourhood n U(xo; c, xi , . . . , x�) with ll xo ll � 1 we fix N such that L �N l xi (k) i < 1e0 for j = 1, . . . , n and put M = m i = l , ... , n ll xj ll . One checks that {x E eo : ll x ll � 1 and l x(k) - xo(k) l < 1 C:M for k = 1, . . . , N } C n U ( x o ; c, xi , . . . , x �). On the other hand the a( eo , t'1 )-topology and the topology of pointwise convergence do not coincide on the whole space Co · To see this let us consider Xn = 2 n en for n = 1, 2, . . . . Clearly E t'1 we have Xn converges pointwise to zero, but for x* = n- 2

B

a.x

B

( ) �= l

X n ¢ U(O; 1, x*).

For more examples see Exercises

1,2,3.

3 Lemma. A weakly Cauchy sequence is norm-bounded. Proof: For a given weakly Cauchy sequence (x n )�=l C X we define an operator T: X* ---+ c by T(x* ) = (x* ( xn ))�=l · By the closed graph theorem I.A.6, the operator T is continuous so IIT II = supn llxn ll < oo.a 4 Theorem. (Mazur)

If A is a convex set in X then the norm closure r

A of A equals its a(X, X*)

closure-

.

Clearly A c r . On the other hand if X E r\A then by the Hahn-Banach theorem I.A.lO there exists x* E X such that x* (x) > sup{x* (a): a E A } = sup{x* (a): a E A } . This implies that • x ¢ r.

Proof:

Suppose Xn � x. Then, by Theorem 4 for each j we have conv{xn } �=j · Thus we have the following useful

5.

xE

Corollary. If Xn� x then there exists a sequence of convex combinations

Yi = E��] AkXk such

we

that II Yi

- xll

----. 0.

a

6. On a dual space X* can introduce the w* -topology or a(X* , X) topology. The basis of neighbourhoods of a point x 0 E X* is given by

U(xo; c, X b

, , .

.

j = 1, . . . , n} an arbitrary

{x* E X* : I x * (xi ) - x(;(xi ) l < c for Xn is an arbitrary finite subset of X and c is

•

, Xn) =

where x 1 positive number. Clearly this defines a locally convex topology on .

•

•

X* .

29

II. A. Weak Topologies § 7.

7 Examples. (a) On £00 = li the w*-topology restricted to the unit ball coincides with the topology of pointwise convergence. The argument is the same as in Example 2b) .

(b) On lp and Lp [O, 1] for 1 < p < topology coincide.

oo

the w-topology and the w*

8 Remark. Let T: X --+ Y be a continuous linear operator. Then T is continuous from (X, a( X, X* ) ) into (Y, a(Y, Y* ) ) and T* is continuous II from (Y* , a(Y* , Y)) into (X* , a(X* , X)). 9. The great usefulness of the w* -topology stems from the fact that it allows us to use compactness. We have Theorem. (AJaoglu)

The closed unit ball Bx · of X* is a(X * , X)

compact.

Proof: Let Kt for t � 0 denote the set of scalars of absolute value � t. We define a map cp: Bx · --+ P = H�: e x Kll x ll by cp(x* ) = {x* (x) }x eX · By the Tychonoff theorem P is a compact space when equipped with the product topology. The very definitions of w* -topology and product topology give that cp is a homeomorphic embedding. The proof is finished once we show that cp(Bx · ) is closed in P. If Po = Po (x) E P\cp(Bx · ) then either

(a) for some Xt , X 2 such that Xt

=

>.x 2 we have Po (xt ) =f. APo (x 2 )

or (b) for some X1 , X 2 , x3 such that Xt + x2 Po (x3) ·

=

X3 we have po (xt ) +Po (X 2 ) =f.

In case (a) we define U = {p E P: Jp(x t ) - Po (xt ) l < 8 and Jp(x 2 ) - Po (x 2 ) 1 < 8} where 8 = ! I Po (Xt ) - >.po (x 2 ) J . Case (b) is treated analogously. In both cases we get an open set U C P with Po E U and U n cp(Bx · ) = 0. Thus cp(Bx · ) is closed. II Let X be a Banach space. There exists a canonical embedding i: X --+ X** given by i(x) (x* ) = x* (x) . When X = eo one easily sees that i: co --+ £00 is the identity map. For X = f.p, 1 < p < oo we have X** = X and i is the identity. Thus the following is not very surprising. 10.

Proposition.

The map

i: X --+ X**

is a linear isometry.

II.A. Weak Topologies § 1 1 .

30

Proof:

We have i(ax1 + ,8x 2 ) (x * )

x * ( ax1 + .8x 2 ) = ax * (xi ) + ,8x * (x 2 ) = [ai(x1 ) + ,8i(x 2 )] (x* ) . =

Since this holds for arbitrary scalars a, ,8 and arbitrary x1 , x 2 E X and x* E X* we see that i is a linear map. From the Hahn-Banach theorem I.A.9 we get, for x E X, , ll i(x) ll

=

l xsup• l $ 1 l i(x) (x * ) l l xsup• l $ 1 l x* (x) l =

thus i is an isometry.

=

ll x ll ; •

Quite often we identify X with i(X) and treat X simply as a sub space of X** . Note that this identification identifies the a(X, X*) topology with the a( X** , X* )-topology restricted to i(X). In other words i is a homeomorphism of (X, I!( X, X*)) onto i(X) with the a( X** , X* )-topology. 11. One can view the a( X, X*)-topology on X as the weakest topology such that all norm-continuous linear functionals are still continuous. Analogously the a( X* , X)-topology is the weakest topology which makes all functionals in i(X) C X** continuous. It is a useful fact that these are all the functionals which are continuous in those topologies. We have the following

(a) The space of all continuous linear Eunctionals on (X, a( X, X*)) equals X* .

Proposition.

(b) The space of all continuous linear Eunctionals on (X* , a( X* , X)) equals X.

Proof: Part (a) is easy since the a(X, X* )-topology is weaker than the norm topology. The proof of (b) is as follows: Let r.p be any linear funct ional on X* continuous in a(X* , X) . Then {x* E X* : l r.p(x* ) l < 1 } :::> {x* E X * : l xj (x* ) l < c j = 1 , 2, . . . , n} for some c > 0 and some x1 , . . . , Xj E X*. The application of the following algebraic lemma com pletes the proof. ,
12 Lemma.

•

•

•

II.A. Weak Topologies § 1 3.

(b) ker <po

:::>

31

n

ker <{Jj . =l jn

Proof: (a)=9- (b) is obvious. To prove (b)=9-(a) let K denote the scalar field and consider a map 1r: X - Kn defined by 1r(x) = (r.pj (x)) j=l · Since ker 1r = nj=1 ker r.pi the condition (b) implies that r.po induces a linear form A on Kn , i.e. r.po (x) = A1r(x) . Thus there exists a sequence a of scalars (ai ) J=l such that r.po (x) = Ej=1 ai r.pi (x) . 13. We know that eo is pointwise dense in c0* = £00 • This suggests the following Theorem. (Goldstine) The dosed unit ball of X is a(X** , X* )-dense in the closed unit ball of X** .

Note that we identify X with its canonical embedding into X** ( cf. 10) . Proof: Let V denote the a(X** , X* )-closure of Bx and assume that there exists x** E X** with ll x** ll � 1 and x** f/. V. From Theo rem 9 we infer that V is a( X** , X* )-compact. Thus (use the Hahn Banach theorem I.A. 10) there exists a continuous linear functional r.p on (X** , a(X** , X* ) ) such that r.p(x**) > sup{r.p(v): v E V}. Proposition 1 1 shows that r.p(x**) = x** (x0) for some x0 E X* . We have

ll x0 11 = sup { l x0 (x) l : x E X, ll x ll � 1 } � sup{ l v(x0 ) 1 : v E V} = sup{ l r.p(v) l : v E V} < r.p(x ** ) = x ** (x 0 ) � ll x ** ll · ll xo ll � ll xo ll · This contradiction completes the proof.

a

14. We now turn to the study of a class of spaces for which the duality theory is particularly simple. The space X is said to be reflexive if i(X) = X** where i is the canonical embedding. The spaces lp and Lp, 1 < p < oo, are reflexive while £1 . £1 . co, C(K) , L00 are not, unless they are finite dimensional. Theorem. The following conditions on the Banach space X are equiv alent

(a) X is reflexive;

II.A. Weak Topologies § 1 5.

32 ( b ) X* is reflexive;

( c ) Bx is a ( X, X* ) compact;

( d ) every subspace of X is reflexive;

( e ) every quotient space of X is reflexive.

( a) => ( c ) . Bx with the a ( X, X* ) -topology is homeomorphic to Bx·· with the a ( X** , X* ) -topology which is compact by Theorem 9. ( c ) => ( a) . We get that i ( Bx ) is a compact subset of X** with the a ( X** , X* ) -topology. Thus by the above Theorem 13 we have i ( Bx ) = Bx•• , so also i ( X ) = X** . ( d ) => ( a) . Obvious. ( a) => ( d ) . If Y is a norm-closed subspace of X then the Hahn Banach theorem yields that Y is a ( X, X* ) -closed. Thus By is a ( X, X* ) compact. But on Y the a ( X, X* ) -topology coincides with a ( Y, Y* ) so the implication (( c ) => ( a)) gives that Y is reflexive. ( a) => ( b ) . If X is reflexive then the a ( X*, X ) and a ( X* , X** ) topologies coincide on X* , so Theorem 9 gives that Bx· is a ( X* , X** ) compact. Thus ( c ) => ( a) gives that X* is reflexive. ( b ) => ( a) . If X* is reflexive then using the implication ( a) => ( b ) we get that X** is reflexive so by ( a) => ( d ) X is reflexive. ( e ) {:} ( a) . We have to note that the dual of a quotient space of X is a a subspace of X* and use a previous equivalences. Proof:

We would like to conclude this chapter with the following useful observation. 15.

Proposition. able.

( a)

If X is separable then (Bx· , a (X* , X)) is metriz

If X* is separable then (Bx , a (X, X* )) is metrizable. Proof: ( a) Let (xn)�=l be a dense set in Bx . We define a metric on Bx· by p(xi , x;) = nL=l T n l xi (xn) - x;(xn)i. Clearly p is a a ( X*, X ) -continuous metric. Let U = {x* E Bx· : i x *(x) < c-} for a fixed x E X and x0 E X*. Changing c we can assume x0(x)i x E Bx . If we take such that Xn is close enough to x then U ::J {x* E Bx· : ix* (xn) - x0 (xn) i < �} ::J {x* E Bx· : p(x0 x* ) < � } . ( b)

00

n

,

33

II.A. Weak Topologies §Notes.

Taking finite intersections we infer that every a(X * , X)-open set in Bx· contains a ball in metric p, so p defines the a( X * , X)-topology on Bx· . (b) is an immediate consequence of (a) and the remarks made after Proposition 10. a Notes and

remarks.

Weak convergence of sequences in some special instances was used at the beginning of the century by D. Hilbert and M. Riesz. The general theory is presented in Banach [1932] . Banach, however, did not use the notion of weak topology, he relied exclusively on sequential arguments. This resulted in unnecessary separability assumptions in many theorems. The weak topology defined by weak neighbourhoods appears (for algebras of operators on Hilbert spaces) already in von Neumann [1930] . In the thirties it became clear that general topological notions are needed in order to establish non-separable theorems. Theorem 9 for separable X is in Banach [1932] and the general form was announced by several authors with the proof first published by Alaoglu [1940] . Theorem 4 and Corollary 5 are from Mazur [1933] and Theorem 13 was first proved in Goldstine [1938] . The notion of reflexive space is implicit in Banach [1932] . Reflexive spaces, under the name 'regular spaces', were an object of intensive study in the late thirties and early forties. Theorem 1 4 summarizes some of this work. The implication ( c) =? ( a) for separable X is in Banach [1932] chapter XI Th. 13. It is interesting to note that he did not realize the converse, despite having a sequential version of Theorem 9. The equivalence ( a ) <=? (b) was known to Plessner before 1936 (see Lusternik [1936] ) . The equivalence ( a ) <=? ( c) was proved in Smulian[1939] , Bourbaki [1938] and Kakutani [1939] . As is clear from the above the content of this section is classical. The reader will find much more in Dunford-Schwartz [1958] . This material is also covered in most textbooks on functional analysis.

Exercises 1. 2. 3.

Show that, if X is an infinite dimensional Banach space then the a( X, X* )-topology on X is not metrizable.

Let r be an uncountable set. Show that the weak topology on the unit ball of is not metrizable.

£2 (f) Let fn E C(K), ll fn ll ::5 1 for

n = 1 , 2, . . . . Show that fn �O if and only if fn (k) ---> 0 for every k E

K.

34 4. 5.

6.

7.

8.

II.A. Weak Topologies §Exercises

(x�)�=l

Find a Banach space X and a sequence c X* , such that tends w* to 0 but every convex 1 for n 1 , 2, . . . and combination of 's has norm 1 .

l ! x� l =

= x�

x�

Show that on the unit ball o f H00 (D) the topology of uniform conver gence on compact subsets of D coincides with the C!(H00 , (Ld H1 )) topology.

=

Let T: H00 (D) ---+ A(D) be defined by Tl ( z) 1( � ) . Show that T maps w* -convergent sequences onto norm-convergent sequences but is not continuous from the C!(H00 , (Ll / H1 ) )-topology into the norm topology.

=

l an l (cpn)�=l

(an)�= l (qn)�=l qn+ l 3qn · = + an qnt + �Pn))

(Riesz products) Let be a sequence of real numbers such that ::; 1 for n 1 , 2, . . . . Let be a sequence of nat ural numbers such that Show that for every sequence � the products n : 1 (1 cos( converge in the C!(M(Y) , C(Y))-topology as N ---+ oo to a positive measure J.L. Show also that

fi,(qn) = ei'l'n an. Suppose that ( an ) �� oo is a sequence of numbers such that lim infN-+oo II L �N aneinOI I < Show for some p, 1 p < P that if 1 < p < then there exists a function I E Lp(Y) such that }(n) = an for n = 0, ±1, ±2, . . . , and if p = 1 then there exists a measure E M(Y) such that fl(n) = an for n = 0, ±1, ±2, . . . . Let G be a compact abelian group with dual group r. For g E G let 19 ( / ) ( h ) = l( h g ) . Suppose that T: L1 (G) ---+ L 1 (G) is a linear operator such that I9T = T 9 for all g E G. Show that there exists a measure on G such that TI = I * -

::;

oo

oo,

oo .

J.L

9.

-

J.L

/

J.L

10. Show that every positive, harmonic function in D is a Poisson inte gral of a positive measure on Y. This means (show this) that every analytic function in D with values in a half plane is in Hp(D) for all p < l.

II.B. Isomorphisms , Bases , P rojections

In this chapter we discuss a diverse set of topics all centred on the concept of isomorphism, the basic equivalence relation between Banach spaces. We discuss projections and closely related direct sum decompositions of a Banach space. We present the 'decomposition method' which allows easy proofs of the existence of isomorphisms between Banach spaces. The concept of Schauder basis is introduced and some fundamental results about block-basic sequences are proved. 1. We will say that two Banach spaces X and Y are isomorphic (in symbols X Y) if there exists an invertible operator I (called an isomorphism) from X onto Y. An operator X ----. Y such that � for some > 0 and all E X is called an isomorphic embedding (or simply embedding) . Clearly then T(X) is a closed subspace of Y and X is isomorphic to T(X). An isometry is an operator I: X Y such that for all E X. Two Banach spaces X and Y are = isometric (in symbols X £':! Y) if there exists an isometry from X onto Y.

T:

,....,

cJ J x l

c

x

I I(x)J I l x l

co ----. c c0 c.

I J Tx l ----.

x

+

c

2 Examples. (a) The identity embeds eo into and into l00 • The operator I: defined by I((�i )� 1 ) = (6 �i+ 1 )� 1 is an isomor phism between and Obviously is not isomorphic to l00 since eo is separable while l00 is not. (b) If (S1, p,) is a measure space which contains an infinite se quence of disjoint sets (Aj )� 1 with 0 < p,(Aj ) < oo, then lp em beds into Lp(n, p,), 1 :5 p :5 oo. The embedding is given by T(�i ) =

co

Lj �j P, (Aj ) - ; XA;

(c) The space Lp (1R) is isometric to Lp [O, 1] for every p, 1 :5 p :5 oo. The isometry I: Lp [O, 1] ----. Lp (1R) can be defined as

I(f)(x) =

1

( tan 1r (x - 21 )) (1r ( 1 + tan2 (1r (x - 21 ))))

.!

P .

More generally if (S1, p,) is a a-finite measure space then Lp(S1, p,), 0

:5

p :5 oo, is isometric to Lp on a probability measure space. In order to see this let us take a sequence of disjoint subsets (Ai)� 1 of S1 with 0 <

36

II.B. Isomorphisms, Bases, Projections §3.

J.L (Aj ) < oo for j = 1, 2, . . . and 0 = U�1 Aj . Let us define a measure J.L l on 0 by J.Ll (A) = L:�1 2- j J.L (Aj ) - 1 J.L (A n Aj ) · One checks that J.L1 is a probability measure on 0 and that the map I: Lp (O, J.L ) ---+ Lp (O, J.L l ) defined by 00

I( f) =

Ll.

j=l

XA; .

2j fp J.L (Aj ) t

is an isometry onto.

Let X1 and X be two closed subspaces of a Banach X both of codimension 1 . 2Then X1 X2 . Proof: If X1 = X2 there is nothing to prove. Otherwise, the space Xo = xl n x2 has codimension 2 in X so there are X l E xl and X2 E x2 such that l x 1 l = l x2 l = 1 and X1 = Xo + ..\x 1 and X2 = Xo + ..\x 2 . We define T: xl x2 by the following rule: T I Xo = id and T(x l) = X 2 . a

3 Proposition.

space

,....,

---+

Now we will give examples of rather non-obvious isomorphic (even isometric) embeddings. 4 Theorem. (Banach-Mazur)

Every separable Banach space isometric to a subspace of C ( Ll.) , where Ll. is the Cantor set.

X is

Proof: Let B* be the closed unit ball of X* with w* -topology. By II.A.9 and II.A. 15 it is a compact metric space so by the well known Alexandroff-Hausdorff theorem (see Kuratowski [1968] 4§41.VI or Lacey [1974] §6.3) there exists a continuous map cp: Ll. � B*. We define C(Ll.) by

T: X ---+

T (x ) ( ) = t5

cp

x

( t5 ) ( ) for t5 E Ll.. a

It is easy to check that it is an isometric embedding.

We can replace C(Ll.) by C[O, 1] in Theorem 4. To see this, it is enough to find an isometric embedding of C(Ll.) into C[O, 1] . Such an embedding can be realized as an extension operator; to each I E C(Ll.) we assign F E C[O, 1] such that F I Ll. = I and F is linear on each interval of [O , 1] \ Ll..

(xn)�=l

x X,

X

of a Banach space is called 5. The sequence of elements a Schauder basis (or simply a basis) if, for every E there exists

37

Il.B. Isomorphisms, Bases, Projections §6.

)�

E:

a unique sequence of scalars ( an = l such that x = = l anXn· Let us emphasize that the sign '=' means here that the series = l anXn converges to x in the norm of X. Sometimes we will consider bases indexed by countable sets other than the natural numbers, but then the order of summation has to be specified. Let us recall (I.A.20) that an idempotent operator P: X -+ X, i.e. an operator such that P2 = P, is called a projection. Obviously P acts as the identity on P(X), so P(X) is a closed subspace of X. A subspace Y C X such that there exists a projection P: X -+ X with P(X) = Y is called complemented. Bases and projections are connected as follows:

)�

E:

6 Proposition. Let ( xn = l be a basis in X. For each N we define a partial sum projection PN : X -+ X by PN = l anXn = = l anXn . Then sup N II PN II < oo.

( E:

) E�

)�

The number sup N I l PN II is called the basis constant of the basis

( xn = l and is denoted by bc( x n) ·

The linearity of PN and the identity P'J. = PN are obvious. The real difficulty is that we do not know that the PN 's are continuous. We define a new norm on X by l l lxl l l = supN II PNxll · We want to show that (X, 1 1 1 · 1 1 1 ) is complete. Once this is established the closed graph theorem I.A.6 will give a C such that l l lxl l l :::; C llxll for x E X, i.e. sup N II PN II :::; C. Let us take (y k ) C X which is Cauchy in I l l · I l l · This implies that there are ZN E X such that II PN ( Y k ) - ZN II -+ 0 as k -+ oo, uniformly in N. The sequence ( zN = l is Cauchy in II · II because, given c > 0, if k is fixed so that II PN (Y k ) - ZN II < � for all N we have Proof:

)�

II ZN - zM II :::; 3c

2

+ II PN (Yk ) - PM (Yk ) II < c

for N and M big enough. Let z E X be such that li z - ZN II -+ 0 as N -+ oo. Since every linear operator on a finite dimensional space is continuous and dim PM (X) = M we have PN ( ZM ) = PN (lim PM (y k )) = lim PN PM ( Y k ) k

k

= lipt Pmin(N,M) (Y k ) = Zmin(N,M) ·

38

II. B. Isomorphisms, Bases, Projections § 7.

ZN r::= 1 anXn,

(an)�= 1 such that PN (z) ZN so

z r::= 1 anXn ·

This shows that there exists a sequence of scalars = so that = This gives

=

a

If (xn)�= 1 is a basis in the space X and for x E:= 1 anXn we define x�(x)· an,2 then (x�)�= 1 are continuous lin ear functionals and l xn l l x� l ::::; bc(xn) · The functionals x�, called biorthogonal or coefficient functionals, uniquely determined by the conditions x�(xm) 6n, m · Thus if (xn)�= 1 is a basis in X , then for each x E X we have x E:= 1 x�(x)xn, where (x�)�= 1 are biorthogonal functionals and the series converges in norm. The partial sum projection PN admits the representation PN (x) r::= 1 x�(x)xn. 7 Corollary.

=

=

are

=

a

=

=

The following routine but useful proposition gives an equivalence between bases and some systems of projections. 8.

If(xn)�= 1 is a basis in X then its partial sum projections for all x E X we have PN (x) -+ x as N -+ dim PN ( X ) N,

PropoSition. satisfy

(a) (b)

oo ,

=

)�= 1 2 , (xn)�= 1

Conversely, if we are given a sequence of projections (PN satisfying (a) , (b) , (c) above then any sequence of non-zero vectors such that E (X) and for n = 3, . . . is a basis E n ker a in X .

X1 p1

Xn Pn(X)

Pn- 1

9. Any complete orthogonal system in a separable Hilbert space is a basis. Also the standard unit vectors in the spaces lp, 1 ::::; p < oo and in eo form a basis. Now we will discuss more interesting examples of bases. We start with the Haar system, which is one of the most important orthonormal systems. Each number n = 1, 2, 3, . . . we represent as n =

39

Il.B. Isomorphisms, Bases, Projections § 1 0.

2i + k with j = 0, 1, 2, . . . and k = 0, 1, . . . , 2i - 1. We define the Haar functions on [0, 1] , {hn)�=O as follows: ho ( t ) =

{;

1;

for k2- i :::; t < { 2k + 1)2 - i - 1 , - 2� hn (t) = 2 for { 2k + 1)2 -i - 1 t < (k + 1)2 -i ,

:::;

otherwise.

It is imperative that the reader draws the picture and understands how the supports of consecutive Haar functions are located. 10 Proposition.

(a) The Haar system

is

an orthonormal system in

£2 [0, 1] . { b) The Haar system is a basis in Lp [O , 1] , 1 :::; p < oo .

(a) This is obvious once one realizes that J; hn (t) dt = 0 for m > n , hn (t) · hm(t) equals either zero or a constant times hm . { b) Let FN , N = 2i + k, j = 0, 1, 2, . . . , k = 0, 1, . . . , 2i - 1, be the space of functions which are constant on intervals of the family Proof.

n

= 1, 2, . . and that for .

SN = {(s2 - i - I , {s + 1)2 - i - 1 ): s = 0, 1 . . . , 2k + 1 , and (s2 -i , (s + 1)2 - i ) s = k + 1, . . . , 2i - 1}. Since h n E FN for n = 0, 1, . . . , N and dim FN = N + 1 we infer that span {hn ) �=O = FN . This description implies that span {hn ) �=O is dense in Lp [O , 1] , 1 :::; p < oo. One also easily sees that the projection PN ( f)

=

L I I I - 1 I f(t) dtxl

IESN

I

is a partial sum projection. Applying the Holder inequality we have

p 1

I PN !I I � = L i I f (t) dt i I XI :::; L I I I 1 -P I J I P/q I I J I P = L I I J I P = 1 ! 1 � So I PN I = 1 for N = 0, 1, 2, . . . and Proposition 8 completes the proof. P IESN

IESN

a

I I I -p

0

I

I

IESN I

40

II.B. Isomorphisms, Bases, Projections § 1 1 .

Let us consider the trigonometric system (e i n9 )��- oo · We know (see I.B.20) that the Riesz projection 'R( f) = E n > o j(n)ei nB is bounded in Lp (T), 1 < p < oo. Since the operator of �ultiplication by e i nB , In (f)(O) = einB f(O), is an isometry from Lp (T) into Lp ('I') we infer that for -oo < N < M < oo, the map SN,M = IM (I - 'R)IN - M'RL N is a bounded operator on Lp (T), 1 < p < oo. One checks that SN ,M(f) = 11.

E nM= N f(n)em9 . A

•

Trigonometric polynomials are dense in Lp (T) so Proposition 8 gives that the characters (ei n B )��- oo ordered 0, 1, - 1, 2, - 2, . . . form a basis in Lp (T), 1 < p < oo . The Faber-Schauder basis in C[O, 1] . Actually we will describe a whole class of bases in terms of partial sum projections. Let (tj )� 1 be a sequence of distinct points of [0,1] with t 1 = 0, t 2 = 1 and such that {tj }� 1 = [0, 1] . We define projections in C[O, 1] as 12.

Pt (f)(t) =

/ (0),

PN (f) is a piecewise linear function with nodes

at tj , j = 1, 2, . . . , N and PN (f)(ti ) = f(tj), j = 1, 2, . . . , N. Clearly the projections PN , N = 1, 2, . . . satisfy conditions (a) , (b) , (c) of Proposition 8. The classical Faber-Schauder system is defined as r.p 1 ( t) = 1, r.pn (t) = ll hn - 2 11 1 1 J; hn - 2 (s)ds for n = 2, 3, . . . , where (hj )�0 are the Haar functions. The Faber-Schauder system corresponds to the sequence (tj)� 1 being naturally ordered dyadic points, i.e. 0 , 1, 1/2, 1/4, 3/4,

1/8, 3/8, . . . .

13. A system (x n )�=l C X is called a basic sequence if it is a basis in its closed linear span. Obviously every subsequence of a basis is a basic sequence. Also if (x n ) is a basic sequence in X and i: X --+ Y is an isomorphic embedding then (i( xn )) is a basic sequence in Y. We can also note that the Haar system is a basic sequence in £00 [0, 1] . Two basic sequences (x n )�=l in X and ( Yn )�=l in Y are called equivalent if the series E:= l an Xn converges if and only if E:=l an Yn converges. This is equivalent to saying that there exists an isomorphism be tween span( xn)�=l and span(yn)�= l which sends onto Yn for n = 1, 2, 3, . . . .

Xn

Now we will present two useful ways to find basic sequences. 14. One is a small-perturbation argument and the other involves the so called block-basic sequences. They are of fundamental importance in

41

II.B. Isomorphisms, Ba.ses, Projections § 1 5.

Banach space theory and in many applications. Let us start with the perturbation arguments. They are ba.sed on the following well known

U: X -+ X and l idx - U l < 1. Then U is invertible. The series L::'=0 (idx -U) n is absolutely convergent and defines

Lemma. Let Proof:

u- 1 •

a The following proposition sums up the most useful perturbation arguments.

(xn)�= 1

(x�)�= 1

x�(xm) 8n,m · (Yn )�= 1

15 Proposition. Let be a ba.sic sequence in X and let be its biorthogonal functionals, i.e. = (a) C X is a sequence of vectors such that is a ba.sic sequence equiv = < 1 , then alent to is complemented by a projection with (b) If, moreover, span < is complemented in X. then span y

If (Yn)�=1 2::::'= 1 l xn - Ynl l l x� l 8 (xn)�= 1 . (xn)�= 1 1 ( n)�= 1 I PI 8

T:

(xn)�= 1

P

-+

T(x)

T(xn) Yn· 2::::'= 1 x�(x) Yn· (1) l x - T(x) l :S nL= 1 l x�(x)l l Yn - Xn l :S 8 l x l we infer that (1 - 8) l x l ::; I T (x) l ::; (1 + 8) l x l so T is an isomorphic embedding. This shows that ( Yn)�= 1 is a ba.sic sequence equivalent to (xn)�= 1 · (b) Let us define A: X -+ X by A idx - P + TP where T is defined above. By (1) I I - A l = l i P - TP I ::; I P I 8 < 1, so Lemma 14 yields that A is invertible. One ea.sily checks that APA - 1 is a projection Proof:

(a) Let us define Obviously

=

span Since

X by

00

=

onto span(yn)�= 1 ·

(xn)�= 1

a

Un (an)

anXn (nk ) (uk )/:'= 1

16. If is a ba.sic sequence in X then a sequence of non zero vectors of the form = I:�:�\ for a strictly increasing a sequence of scalars is called a block-ba.sic sequence of integers and is a ba.sic sequence and sequence (block ba.sis for short) . Clearly

be(uk ) :S bc(xn) ·

17 Proposition. Let biorthogonal functionals

(xn)�= 1 be a ba.sis in X with l xn l 1 and (x�)�= 1 . Let (zk )� 1 be a sequence in X such =

42

Il.B. Isomorphisms, Bases, Projections § 1 9.

that

(2) l zkl l :2: 8 > 0 for k = 1, 2 ,3, . . . (3) k-+limoo x�(zk ) = 0 for n = 1, 2 , 3, . . . Then there exists a subsequence (zn . ) which is equivalent to a block basic sequence of (xn)· Proof: Let us put K = bc(xn) and let Pn denote the partial sum projections of the basis (xn) · Fix < 3 /4 8(K + 1) - 1 . Inductively we choose two increasing sequences of integers n8 and N8 in such a way that (4) I PN. - 1 (zn.) l < 'f/4- s , 8 (5) l zn. - PN.(Zn.) l < 'f/4- • We start with n1 = 1 and N1 chosen to satisfy (5). This is possible since Pn(x) - x as n for x E X. Having n and N8 for = 1, 2 , 3, . . . we pick nr+l so big that (3) ensures that8 (4) holds. Next we find Nr+l so big that (5) holds for Zn r+ 1 • The sequence U 8 = PN.Zn. PN _ zn. is a block-basic sequence. We can find biorthogonal functionals u: .such1 that l 2u: l - :5s 2K I us l - 1 :5 2K(8 - 'f//4) - 1 (Corollary 7). Since l zn. - us I :5 .,.,4 we infer from Proposition 15 (a) that (zn. ) is the "'

oo

, r

s

a

desired subsequence.

Corollary. Every weakly null sequence has a basic subsequence.

(xn)�= 1 in X with l xn l = 1

(xn) 4

1]

Proof: Since span is separable we can assume that X is separable as well. Using Theorem we can treat X as a subspace of C[O, which has a basis ( 12) . Now the corollary follows from Proposition 17. a

1, . . . Xn

Ej= 1 x :L:j= 1 1, . . . , Xn (xj)j= 1

Xn

be subspaces of a Banach space X. We say that X 19. Let X is a direct sum of Xj 's and write X = Xi or X = X1 E9 · · · E9 if every E X has a unique representation = Xj with Xj E Xj . If we are given Banach spaces X we can form their direct sum X= Xi as follows: X is the set of n-tuples of elements with Xj E Xi , with coordinatewise algebraic operations and product topology. We can introduce a norm in X in many equivalent ways. The most = ( i Y. common are lp-sums, 1 :5 p :5 oo , where I

x :L:j= 1

Ej=1 I x l i P 1 The direct sum with such a norm will be denoted ( Ej= 1 Xj ) . P (xi ) l i P

43

II.B. Isomorphisms, Bases, Projections §20.

20. The following proposition states the easy but fundamental prop erties of these constructions:

(a) The space X is a direct sum of its subspaces X1 and X2 if and only if there exists a projection P: X -+ X such that X1 = P(X) and X2 = ker P. {b) IfT; : X; -+ }j for j = 1 , . . . , n, then we can define an operator

Proposition.

by the formula T((x; ) j= 1 ) = (T; (x; )) j=l · (c) If in (b) every T; is an isomorphism then T is.

X2

{b) and (c) are obvious. For (a) note that for x = x1 + x2 with X1 E X1 and E X2 we define P(x) = X 1 . P is continuous by the closed graph theorem. Conversely given a projection P we write x = Px + (x - Px), and Px E P{X) and (x - Px) E ker P. The a decomposition is unique.

Proof: Parts

Quite often we will need infinite direct sums. Since a basis provides an infinite direct sum decomposition of a space into one-dimensional subspaces, we see that, unlike the finite case, the form of the norm on the whole sum is crucial. In practice we will consider only lp-sums. They are formed as follows: Given a sequence of Banach spaces {X; )� 1 we put for 1 � p < oo 21.

00

00

l.

( J�=l xi) p = { (x;)� l : x; E X; and ll (x;) II P = ( J�=l l x; II P) " < oo } . We also define

( jtx i) =l

and

00

=

{{x; )� 1 : x; E X; and

l (x; ) l oo = s�J p l x;l l < oo}

= { (x;)� l E ( fx;) ( Jfx;) l x;l l - o } . =l J =l 0

00

:

In all the above sums the algebraic operations are performed coordi natewise. It is routine to check that all the above sums are Banach spaces.

44

Il.B. Isomorphisms, Bases, Projections §22.

Very similarly to the case of lp spaces one checks that

( L:xi 00 ) ( I:00 x; ) , ( jL:xi 00=1 ) 0 ( jI:00=1 x; ) 1 if *

*

p

j= 1

=

=

1 � p < oo

q

j= 1

1

1

and - + - = 1 . p

q

In particular if the X; 's are all reflexive and 1 < p < oo then E�1 X is reflexive.

i)

(

P

p�

(

(

(a) Lp [O, 1] is isometric to E:= 1 Lp [O, 1]

22 Examples. 00

(

(b) lp is isometric to E:= 1 lp

)

)

P

for 1 � p �

oo

)

P

for 1 �

and co is isometric

to E:= 1 co . 0 (c) C(�) is isomorphic to E:= 1 C(�) where � is the Cantor 0 set. To see (a) is relatively easy. We put An = [(n + 1) - 1 , n - 1 ] and identifying f E Lp [O, 1] with the sequence ( !IAn)�= 1 we see Lp [O, 1] = E:= 1 Lp(An )) p. Since Lp(An ) � Lp [O, 1] we have the claim. (b) is analogous. In order to prove (c) we start with the same idea. Let us fix an increasing sequence tn /' 1 such that t1 < 0, t 2 > 0 and � n (tn , tn+ l ) = � n is homeomorphic to �. Identifying the n-th summand of E:= 1 C(�) with C( �n ) we can identify E:= 1 C(�) with 0 0 the subspace Co (�) = { ! E C(�) : / (1) = 0}. So we have to prove that Co (�) ....., C(�). This follows from the following

(

)

(

)

(

23 Lemma. morphic to

Co .

(

)

The space C(�) contains a complemented subspace iso

With �n as above we put en = sup �n· The map P f = E:= 1 [/ (en) - / (1)]x.6., is a projection from C(�) onto a subspace of

Proof:

Co (�) of functions constant on each �n· One easily checks that this a subspace is isometric to CO · Now we can write C(�) ....., Z $ co . Since C0 (�) has codimension 1 in C(�) Proposition 3 gives that Co (�) is isomorphic to Z $ V where

45

Il.B. Isomorpliisms, Bases, Projections §24.

V C eo , V we have

=

{(ai )� 1 E eo: a1 Co (a)

=

0}. Since V is clearly isometric to

"' z a1 "' z a1 "' v

eo

C(a).

Co

a

24. Arguments like that used in 22 . ( c ) are used quite often in proving that two Banach spaces are isomorphic. In the above case it is possible to write an explicit formula for the isomorphism, but in more complicated cases it is practically impossible. The following theorem is the most common variant of the 'decomposition method' which is the elaboration of such arguments.

X Y

X Y Y X. X X) P X "' Y. Let us write X "' Z $ Y and Y "' V $ X. Then

Theorem. Let and be two Banach spaces such that is iso morphic to a complemented subspace of and is isomorphic to a complemented subspace of Assume moreover that is isomorphic to E:'= 1 for some p with 1 � p � oo or 0. Then

(

Proof:

and also

X "' ( � x)P "' ( � za1Y) P "' ( � z) P $ ( � y)P "' ( � z)P $ ( � y) P $Y "'X$Y. So X "' Y. One easily checks that each step is justisfied.

Notes

and Remarks.

a

Much of the material in this chapter is folklore of Banach space theory. The notion of isomorphic embedding appeared in the early 30's, and var ious cases of embedding and impossibility of embedding were studied by Banach and his collaborators. Those results are summarized in Banach [1932] where in particular Theorem 4 is proved and in Banach-Mazur [1933] . The comparison between Banach [1931] and Banach [1932] shows the crystallization of the notion. The general notion of a Schauder basis was introduced in Schauder [1927] . By now the subject has grown enormously. An encyclopaedic

46

ll.B. Isomorphisms, Bases, Projections §Exercises

treatment is in Singer [1970] and [1981] and a very good, concise pre sentation in Lindenstrauss-Tzafriri [1977] . The Haar system, introduced in Haar [1910] , is one of the most important orthonormal systems (see Kashin-Saakian [1984] ). It is also a prototype of the general notion of martingale difference sequence. Haar studied his system in order to show that the expansion with respect to this system of every f E C[O, 1] is uniformly convergent {Exercise 7{a) ) . Proposition l O. (b) was proved by Schauder [1928] . The classical Faber-Schauder system was introduced and shown to be a basis in C[O, 1] in Faber [1910] . The general case was independently treated in Schauder [1927] . The notion of a basic sequence and a block basis emerged in the late 50's in the work of Pelczyiiski and Bessaga. The main ideas of Proposition 1 5 are in Krein-Milman-Rutman [1940] . They were rediscovered and applied in Bessaga-PelczyD.ski [1958] . This paper contains also Proposition 1 7 and Corollary 1 8. Direct sums appear already in Banach [1932] . Theorem 24 is taken from PelczyD.ski [1960] but germs of such ideas are due to Borsuk (see Borsuk [1933] and Banach [1932] chap XI. §7) .

Exercises 1.

Show that, if X1 and X2 are closed subspaces of a Banach space X , both of codimension n , then xl x2 . A norm 1 1 1 · 1 1 1 on a Banach space {X, 11 · 11) is called an equivalent norm if for some C > c > 0 we have cllxll ::;; l l l x l l l ::;; C ll x ll for all .....,

2.

x E X.

3.

(a) Suppose Y c X and d ( Y, Z) < c {here d ( Y, Z) is the Banach Mazur distance defined in II.E.6) for some Banach space Z. Construct an equivalent norm 1 1 1 · 1 1 1 on X such that c- 1 ll x ll ::;; l l l x l l l ::;; ll x ll for all x E X and {Y, 1 1 1 · 1 1 1) is isometric to Z. (b) For f E C[O, 1] put I I Ifi l l = ll f ll oo + ll f ll2 · Show that I l l · I l l is an equivalent norm on C[O, 1] and £ 1 is not isometric to any subspace of (C[O, 1] , 1 1 1 · 1 1 1 ) . (c) A norm is strictly convex if for all x, y, with x f. y and ll x ll = II Y II = 1 we have ll x + Yll < 2. Show that every separable Banach space has an equivalent strictly convex norm. {d) If r is a set of continuum cardinality then ioo (r) does not have any equivalent strictly convex norm. Suppose that (xn)�= l is a basis in X. Show that the biorthogonal functionals (x;'.)�= l form a basic sequence in X* . This need not be a basis (even if X* is separable) .

Il.B. Isomorphisms, Bases, Projections §Exercises

47

C X be a bounded sequence which is not norm rela tively compact. Show that there are sequences nk and mk such that (xn,. - Xm ,. ) � 1 is a basic sequence.

4. Let (xn)�= l 5.

Suppose X and Y are subspaces of a Banach space Z, such that X n Y = {0}. Show that, if the algebraic sum X + Y is not closed in Z, then there exist a basic sequence (xn)�= l C X and a basic sequence ( Yn )�= l C Y such that ll xn ll = llYn I = 1 for n = 1, 2, . . . and ll xn - Yn ll < 4 - n for n = 1 , 2, . . . . In particular X and Y have isomorphic subspaces.

6.

Prove the following. (a) The space C[O, 1] has a basis { !n )�= l consisting of polynomials. {b) There exists a sequence of polynomials which is a basis in all Lv [O, 1] , 1 � p < oo.

7.

(a) Show that for I E C[O, 1] the Haar-Fourier expansion converges uniformly. {b) Construct a basis in C( �) .

8.

Let (cpn(t))�= l be the Faber-Schauder system. Show that I = L:�= l an'Pn E LipQ [O, 1] , 0 < a < 1 if and only if l an l = O{n - Q ), so LipQ [O, 1] is isomorphic to £00 •

9.

Show the following isomorphisms: (a) Hp (T) "' Lp(T) for 1 < p < oo;

{b) Lipl [O, 1] "' Loo [0, 1] ; (c) C k [O, 1] "' C[O, 1] for k = 1, 2, 3, . . . ; {d) W;' [O, 1] "' Lv [O, 1] for k = 1, 2, 3, . . . and 1 � p < oo.

10. Suppose S C � is a closed subset. Prove the following.

(a) If S has continuum cardinality then C(S) "' C( � ) . (b) If S is countable then C(S) is not isomorphic to C{�). 11. Show the following isomorphisms. (a) C[O, 1] "' {EC [O , 1] )o ;

{b) C [O , 1 ] EB eo "' C[O , 1] ;

(c) Lv [O, 1] EB lv "' Lv [O, 1] .

48

II.B. Isomorphisms, Bases, Projections §Exercises

12. Prove the following. (a) Every isometric embedding of fp into Lp (J.L) (J.L-arbitrary mea sure) , 1 � p < oo , p =F 2, maps unit vectors onto disjointly supported, norm-one functions. (b) Every subspace of Lp (J.L) isometric to ip , 1 � plemented.

p

<

oo

is com

13. Suppose X n �0 in X. Show that for every e > 0 and every N E N there exists a finite sequence such that = ei)..i xi ll < e for every choice of ei 1 >-.i l = 1 and II with l ei l = 1.

"L-f=j.M

(>-.j )f=+ff + "L-f=NM

"L.f=+,_M Aj

14. Let E be a a-algebra of subsets of n and let J.L be a probability measure on E. Let E1 be a sub-a-algebra of E. Prove the following. (a) For every I E £1 (E, J.L) there exists a unique function PI which is E 1 -measurable, such that for every B E E 1 we have J8 l dJ.L = f8 P ldJ.L. (b) P is a linear map. (c) If I � 0 then PI > 0.

(d) P is a .norm-one projection in Lp(E, J.L) for 1 � p � (e) If g E L oo (Et . J.L) and I E Lt (E, J.L) then P(gf)

=

oo.

gPI .

15. Prove the following. (a) Every separable Banach space X is a quotient of £1 .

(b) If r is a set of continuum cardinality then i 1 (r) is isometric to a subspace of £00 •

(c) If r is a set of continuum cardinality then lp (r), 1 < and eo(r) are not isomorphic to any subspace of £00 •

p

<

oo

16. (The weak basis theorem) . Suppose that (x n )�= l C X is such that for every x E X there exists a unique sequence of scalars (tn )�= l such that the series tnXn converges weakly to x. Then (xn)�= l is a Schauder basis in X.

"L-:'= 1

II. C . Weak Compactness

The sets compact in the u (X, X* ) -topology are important in many ap plications. We study such sets in this section. The main result is the Eberlein- S mulian theorem which says that weak compactness of a set is determined by properties of sequences, even when the u ( X, X* ) -topology on this set is not metrizable. We apply this to study weakly compact operators, i.e. operators such that the image of any ball is contained in a weakly compact set. We show that each weakly compact operator fac torizes through a reflexive space, and use this to investigate properties of such operators. 1. This section is devoted to the study of weakly compact sets in Banach spaces, i.e. subsets A C X which are compact in the u (X, X* ) topology. We say that the set A C X is relatively weakly compact if its u ( X, X* ) -closure in X is weakly compact. From Theorem II.A.14 we infer that every bounded subset of a reflexive space is relatively weakly compact. Also by Theorem II.A.4 and II.A.14 we get that every convex, bounded, norm-closed subset of a reflexive space is weakly compact. Also if X is a reflexive space and if T: X --+ Y is a continuous linear operator, then T(Bx) is a weakly compact set. 2.

We have

A C X is relatively weakly compact if and only if it is bounded and the u ( X** , X* ) -closure of i(A) in X** is contained in

Lemma. A subset

i( X )

.

Proof: Immediate from Alaoglu's theorem II.A.9 and the remarks after Proposition II.A. lO.

3. We have seen ( Exercises II.A. l and 2) that the weak topology ( even on a weakly compact set ) need not be metrizable. Thus it seems that we are not permitted to use sequential arguments. Actually it is not so. The following important and quite surprising theorem says essentially that sequential arguments about weakly compact sets are permissible.

Theorem. (Eberlein-Smulian).

compact if and only if every sequence subsequence.

A set

AA hasX aisweakly relatively weakly convergent

(an) c

C

50

II. C. Weak Compactness §3.

Assume A is relatively weakly compact in X and fix (an)�= l C A. Denote V = span{an}�= l · Clearly V is a separable subspace of X .

Proof:

Let us fix a sequence (x;)�=l C X * such that i f x E V and x; (x) = 0 for n = 1, 2, . . . then x = 0. (There are many ways to construct such a sequence, e.g. take (vn)�= l dense in V and use the Hahn-Banach theorem to get x; E X* such that ll x; ll = 1 and x; (vn) = ll vn ll for n = 1, 2, 3, . . . . ) Using a standard diagonal argument we find a subsequence (ank )�1 such that limk -+oo x; (ank ) exists for every n = 1, 2, . . . . Let jj E X be any weak cluster point of the set {ank }� 1 C A; thus x; (y) = limk -+oo x; (ank ) for n = 1, 2, . . . . Since V is a closed subspace, it is weakly closed, so jj E V. Thus jj is the unique cluster point of the sequence ( ank )� 1 • We have to show that ank �jj as k -+ oo, i.e. that for every x* E X* we have x* (ank ) -+ x* (y) as k -+ oo. But if for some x0 E X* we have x* (ank . ) --+ a =f. x0 (jj) , then there exists a cluster point of {ank . }� 1 , thus a cluster point of {ank }� 1 , different from jj. This contradicts the fact that jj was the unique cluster point of {ank }�1 . So in fact ank �jj. To prove the converse let us assume that A C X is not relatively weakly compact. We have to produce a sequence (an)�= l C A without a weakly convergent subsequence. Using Lemma 2 we find F E X** \i(X) such that F is in the a (X** , X*) closure of A. Let 9 = dist(F, i(X)) > 0. We will inductively construct (an , Yn)�= l C A x X* such that (1) Yn E X* and IIYn ll = 1, (2) an E A, (3) Re F(gn) > £9 for n = 1, 2, 3 . . . , (4) i Yn (a; ) i < a 9 for j < n, (5) Re Yn (a; ) > £9 for j � n. Such (an)�= l has no weakly convergent subsequence, because if (ank ) �a then by Corollary II.A.5 there is a convex combination such that From (4) we see that for n > nM we have IYn CL: ! N a k ank ) i < a 9 . Thus we infer that for n > nM we have iYn (a) i < �9 . On the other hand (5) implies Re Yn (a) � £9 for n = 1, 2, . . . , a contradiction. Digression. Let us consider the following simple example: A = Bco c Co · This is clearly not a weakly compact set. Put an = E;= l e; . Then F = (1, 1, . . . ) E £00 = cO* , 9 = 1 and Yn = e;. Conditions (1) (5) are clearly satisfied. Our construction is an attempt to imitate this example in every set not weakly compact. End of digression.

51

II. C. Weak Compactness §4. Inductive construction. Since II F II 2::: �< x · · ' x· )

() there exists 91 satisfying there exists a 1 E A such that

(1) and (3) . Since F E i(A) I F(91 ) - 91 (at ) l is small enough so that (5) holds. Suppose we have (ai , 9i) 'J= 1 satisfying (1)-(5) . Use the Hahn Banach theorem to obtain

�() and ll �.p ll < 1. The Goldstine theorem II.A. 13 gives 9n + 1 E X* such that (1), (3) and (4) hold. Next we choose an + l E A which approximates F on 91 ! . . . , 9n + l so well that (3) gives w.

a

Now we would like to introduce and study the concept of a weakly compact operator. An operator T: X -+ Y is said to be weakly compact if the set T (Bx ) is relatively weakly compact. We infer from Theorem 3 that this is equivalent to the condition that for every bounded se quence (x n )�= 1 C X the sequence (T(x n ))�= 1 has a weakly convergent subsequence. From Theorem II.A. 14 we get easily that T: X -+ Y is weakly compact if X or Y is reflexive. On the other hand the operator T: Lt [O, 1] -+ C[O, 1 ] defined as Tf(x) = J; f(t)dt is not weakly compact since T(2n(X [ .!2 - .!.n ' .!2 ] - X [ .!2 ' .!2 + .!.n ] )) has no weakly convergent subsequence. We should note that the requirement that T (Bx ) be weakly compact is far too strong. For example the identity id: C[O, 1] -+ L 2 [0, 1 ] is weakly compact but id(BC[o, 1 J ) is not norm-closed in L 2 [0, 1] . We can also find 1 1 a functional on C[O, 1] , e.g.
5.

R.

Our investigation of weakly compact operators is based on the following:

Theorem. An operator T: X -+ Y is weakly compact if and only if T factors through a reflexive space, i.e. there exist a reflexive space R and operators a: X -+ R and /3 : R -+ Y such that T = /3 · a. Moreover such a factorization can be chosen such that ll a ll · ll/311 :::; 4 · II T II .

The 'if' part is obvious from 4, so we will discuss the 'only if' part. Proof: We can assume that T is 1-1. (If not consider T: X/ker T --+ Y

defined by T([x] ) = Tx.) Let us put T (Bx ) = W. The set W is convex and weakly compact. We define

Un = 2 n w + 2 - n By = {yo E Y: Yo = 2n w + 2 - n y with w E Wand

I I Y II :::; 1 } .

52 Each Un is convex and 2 - n By C Un induces a norm II · lin on Y such that

II. C. Weak Compactness § 6. C

(2n ii T II + 2 - n )By . Thus Un for all

y E Y.

We define R = { y E Y: I I IYI I I = CE�= l' IIYII �) ! < oo } . Obviously ( R, 1 1 1 · 1 1 1 ) is a closed subspace of (E� 1 (Y, II · lln )h so it is a Banach space. One checks ( see II.B.21 ) that R** can be canonically identified with {y** E Y** : (L:� 1 II y** ll �) ! < oo } . Since W is weakly compact the unit ball in ( Y** , I · lin ) equals 2 n w + 2 - n By.. . Thus for y ** E Y** \ i ( Y ) we have ll y** lln � 2 n inf { ll y** y ll : y E Y } . This shows that for y ** E Y** \ i ( Y ) the series E�= 1 ll y** ll � diverges, in other words R** C (E�= l ( Y, I · ll n )h so R is reflexive. Now we define a : X - R as a(x) = T(x). Since Tx E ll x ll · W we have II Tx ll n :::; 2 - n so Tx E R and a ll a ll :::; 2. We define f3(y ) = y for y E R. Clearly 11/311 :::; 2 II T II . -

We will now use Theorem 5 to deduce the fundamental properties of weakly compact operators from corresponding properties of reflexive spaces.

- Y is weakly compact and U: X1 - X and V: Y - Y1 are continuous. Then VTU is weakly compact. (b) An operator T: X - Y is weakly compact if and only if T* : Y* - X* is weakly compact. (c) An operator T: X - Y is weakly compact if and only if T** (X**) c i( Y ) . (d) The set of all weakly compact operators from X into Y is a norm closed linear subspace of L(X, Y ) . 6 Theorem. (a) Suppose T: X

( a) follows immediately from the fact that norm-continuous linear operators are also weakly continuous and the well known fact that a continuous image of a compact set is compact. (b) Follows from Theorem 5 and Theorem II.A.14. (c ) If T is weakly compact then from Theorem 5 we get that T** (X**) C f3** ( R** ) = f3(R) C Y. Conversely, suppose T** (Bx·· ) C i(Y). Since T** (Bx·· ) is u ( Y** , Y* ) -compact (see Theorem II.A.9 ) we infer ( cf. remarks after Proposition II.A. lO ) that i - 1 (T** (Bx··) is u ( Y, Y* ) -compact. Obviously T(Bx) C i - 1 (T** (Bx··)) so T(Bx) is relatively weakly compact, so T is weakly compact. {d ) The fact that the set of weakly compact operators is linear can be seen easily from ( c ) . To show that it is complete it is enough to show

Proof:

II. C. Weak Compactness § 7.

53

that if T = L:: := l Tn with Tn weakly compact and II Tn ll :::; 4 - n then T is weakly compact. For each Tn let us take a factorization as in Theorem 5 with Rn reflexive and llan ll :::; 2 · 2- n and II.Bn ll :::; 2 2 - n . We define a: X - R by a (x) = (an(x))�= l and ,B (R) - Y R = (L:: := l Rn) 2 and 00 by ,B ((rn )�= 1 ) = L .Bn (rn ) · The space R is reflexive (II.B.21) and ·

n= l ,Ba (x) = T(x) so T is weakly compact.

a

7. A more elementary proof of the fact that the norm limit of weakly compact operators is weakly compact follows from the following

Lemma. A subset A C X is weakly compact provided for every e there exists a weakly compact set

>0 Ac- c X such that A c Ac- + eBx .

Since the ( X** , X*)-closure of Ac- + eBx in X** equals Ac- + eBx·· (use Lemma 2) we infer that the a(X** , X*)-closure of A is contained in n c->o Ac- + eBx·· c X. Lemma 2 gives that A is weakly a

Proof:

a

compact.

The reader undoubtedly observed that the above arguments have already been used in the proof of Theorem 5. 8 Corollary.

If A

C

X is a relatively weakly compact set then conv A

is weakly compact.

Assume first that X is separable. Replacing A by its weak clo sure we can assume that A is actually weakly compact. Let f 1 (A) denote the space of functions x(a) defined on A such that L aE A l x(a) l < oo . This is clearly a Banach space. Define an operator T: £1 (A) - X by T((x(a)) aE A ) = L aE A x(a) · a. Since A is bounded (Lemma 2) the operator T is bounded. The adjoint operator T* : X* - foo (A) acts as T* (x* ) (a) = x* (a) ; thus actually T* (X*) C C(A, a ( X, X* )). Let denote the same map as T* but acting from X* into C(A). Let (x�) C Bx• . Since X is separable, passing to a subsequence we can as sume that x� - in the a ( X* , X)-topology, so � point wise in C(A). Thus for every E C ( A ) * , the Riesz representation Theo rem I.B.ll and the Lebesgue theorem give fA � fA so (x�) converges weakly to Theorem 3 gives that S(Bx· ) is relatively weakly compact, so T* is weakly compact so by Theorem 6 the operator T is weakly compact. Since conv c T(Bt1 ( A ) ) we see that conv is weakly compact. Proof:

S

x0

S

A

J.L

S(x0).

S(x�) S(x0) S(x�)dJ.L S(x0)dJ.L, A

II. C. Weak Compactness §Notes.

54

For non-separable X we argue as follows. If conv A is not weakly compact then by Theorem 3 there is a sequence in conv A that is not relatively weakly compact. Each element of such a sequence is the norm limit of some convex combinations of a countable subset of A. Thus we get a separable subspace Y C X such that conv{Y n A) is not weakly compact. This is impossible in view of the first part of our argument.a

Notes and remarks.

The fundamental Theorem 3 is, as usual, the result of much work by many mathematicians. The standard references to papers where various pieces appeared in full generality are Eberlein [1947] and Smulian [1940] . Our proof of sufficiency is taken from James [1981] . Other proofs can be found in Dunford-Schwartz [1958] , PelczyD.ski [1964] , Whitley [1967] . Basic facts about weakly compact operators ( Theorem 6 (a, b,c)) have been proved in Gantmacher [1940] . Theorem 5 which is our basis for dis cussing weakly compact operators was proved by Davis-Figiel-Johnson PelczyD.ski [1974] . This paper contains numerous applications of the technique of the proof to problems of the geometry of Banach spaces. Theorem 5 is the first instance of an application in this book of a factor ization idea. The reader will encounter many more examples later on. It should also be noted that the proof is a very elementary application of an interpolation argument. Corollary 7 was proved by M. Krein-V. Smulian [1940] .

Exercises 1.

Let X be a Banach space. Show that i(X*) is complemented in

X*** .

( ) IYI

()

:5 !} is 0. Show that {g E Lt IL : weakly compact, and if IL i suppf is not purely atomic then it is not norm-compact.

2. Let f E Lt IL and f � 3.

4.

£1 a((en)�= l ) = 0:::::=1 en );:': 1 .

Let X and Y be Banach spaces and let T: X - Y be an operator that is not weakly compact. Show that there exist S: - X and where U: Y - £00 such that UTS

=

=a

Let IL be a measure on T and let T,.. : Lt (T) - Lt (11') be given by *

T,.. { ! ) f IL· Show that the following conditions are equivalent: (a) T,.. is weakly compact; {b) T,.. is compact; (c)

1L

E Lt (11') .

II. C. Weak Compactness §Exercises

55

5.

Suppose that TK f(x) = J� K(x, y)f(y)dy where K(x, y ) is a measurable 1 function on [0, 1] x [0, 1] . Show that if supessx J0 jK ( x, y) j 2 dy < oo, then TK maps £00 [0, 1] into £00 [0, 1] and is weakly compact. Show that for K ( x, y) = x - 1 X [O , xj (Y ) the operator TK0 maps C[O, 1] into itself but is not weakly compact.

o

6.

Let SN (IL) = l: �N [L(n)e i nB . Show that if IL is a measure on T such that sup N II SN (IL) Ih < oo then [l,(n) - 0 as l n l - oo. Let X be the space of all measures IL such that l l l�tl l l = sup N II SN (IL) Ih < oo. Show that the Fourier transform X - co is a weakly compact operator. •

7.

Let IL be a measure on T such that [L(n) = 0 or 1 for n = 0, ±1, ±2, . . . . Show that {n E N: jJ,(n) = 1 } = (F1 U V)\F2 with F1 , F2 finite sets and V a periodic set, i.e. a finite sum of arithmetic progressions.

8.

:

=l

A basis ( x n) � in a Banach space X is called boundedly complete if

an = l =1 a a = l =o Show that if ( x n) �= l is a shrinking basis in X then ( x� ) �= l is a boundedly complete basis in X* . Let ( x n) �= l be a boundedly complete basis in X and let Y = span( x� ) �= l X* . Show that X is naturally isometric to Y* .

for every sequence of scalars ( ) � such that sup N 11 2:: n x n ll < oo the series l::' n X n converges. A basis ( x n) � is called shrinking if for every x* E X*, limN-oo ll x* I span(xn)n� N II = 0. (a) (b)

C

(c) Show that every basis in a reflexive space is both shrinking and boundedly complete.

xn = l

(d) Show that if ( ) � is a shrinking basis in X then X** can be naturally identified with the set { ( ) � sup N 1 1 2:: ll < oo } . 9.

(James space) . Let

<ej ) � � =

ll < ei )II J = sup

us

an = l :

= l an xn

define the following norm on sequences

{ ( }; l ei·+' - ei. l 2 )

�:

}

1 � i1 < h < · · · < in ·

The James space J = { (ei )�1 e eo : ll (ei )IIJ < oo } . (a) Show that J is a Banach space.

Il. C. Weak Compactness §Exercises

56

(b) Show directly that the unit ball in J is not relatively weakly compact. (c) Show that the unit vectors form a basis in J and that this basis is shrinking. (d) Show that J** can be naturally identified with { (�i )� 1 E li (�i ) II J < oo}. Thus dim J** fi (J) = 1 .

c:

(e) Show that J is isomorphic to J** . (f) Show that J is not isomorphic to J EB J.

10. Apply the construction of Theorem 5 to the operator E: i 1 -+ c defined by E ( (�)� 1 ) = E;= 1 �i = . Show that the resulting 1 space X have the property that dim X** fi(X) = 1 .

(

)�

II. D . Convergence Of Series

We start this section with various characterizations of unconditionally and weakly unconditionally convergent series in Banach spaces. We prove the classical Orlicz theorem that every unconditionally convergent series in Lp , 1 � p � 2, has norms square summable. We also introduce the notion of an unconditional Schauder basis and show that C[O, 1] and £1 [0, 1] are not subspaces of any space with an unconditional basis. We conclude with the proof that the Haar system is an unconditional basis in Lp [O, 1] for 1 < p < oo . 1. This section deals with various types of convergence of series of elements of a Banach space X. The series E:=l X n is said to converge absolutely if E:=l ll xn I < oo. It is an obvious consequence of the triangle inequality that absolutely convergent series converge. The series E:=l Xn is said to converge unconditionally if the series E:=l c n Xn converges for all en with en = ±1 for n = 1, 2, 3, . . . . When X is finite dimensional then the classical Riemann theorem asserts that absolute and unconditional convergence coincide. This as sertion actually characterizes finite dimensional Banach spaces {Exercise III.F.8 ) . For the time being let us note that E:=l enen converges un conditionally in '-v • 1 < p < oo, whenever (en )�=l E lp but converges absolutely only when {en )�=l E '-1 · The following characterize unconditional convergence. 2 Theorem. For a series E:= l X n in a Banach space X the following conditions are equivalent:

( a) the series

00 E=l Xn is unconditionally convergent; n 00 ( b ) the series E an Xn is convergent for every (an )�=l E i00; n =l ( c ) there exists a compact operator T: c0 --+ X such that T(en ) = X n for n = 1, 2, 3, . . . ; ( d ) for every permutation of the integers the series E:=l X u(n) con a

verges;

( e ) for every increasing sequence of integers

E:=l Xn k

converges.

(n k )�=l

the series

II.D. Convergence Of Series §3.

58

Proof:

We will prove the following implications: (c)

--

(b) - (e) -- (a)

� (d) /

---

(c)

co.

(c)::::} ( b) . For every N we put VN = 2::= 1 anen E Since VN is weakly Cauchy T(vN ) is norm-Cauchy, so 2:::"= 1 anXn converges. (b)::::} ( e) . Take appropriate sequence of zeros and ones. (e)::::} ( a) . Given a sequence en = ±1 we define n k in such a way that €nk = 1 for k = 1 , 2, 3, . . . and en = - 1 if n =1- n k for k = 1 , 2, . . . . Then 2:::"= 1 €nXn = 2 2:�= 1 Xnk - 2:::"= 1 Xn · Since both series on the right hand side converge the left hand side series also converges. (a)::::} ( c) . We will give the proof for real spaces. The changes for complex spaces are straightforward. Every vector in eo with a finite number of non-zero coefficients is a convex combination of vectors tak ing values 1, - 1, 0. Thus for every N we have li T I span(en)n �N II :::; sup., n = ±1 ll l: ::"= N €nXn ll · Note that limN-+oo sup., n = ±1 ll l: ::"= N €nXn ll = 0 since otherwise we can inductively produce a sequence of signs (cn)�= 1 such that 2:::"= 1 €nXn is not Cauchy. This shows that T is compact. (c)::::} ( d) . The permutation a induces an isometry of c0 defined as Iu (�n) = (�u (n) ) · The operator T o Iu: co -+ X is compact and sat isfies (T o Iu ) (en) = Xu(n) so the already proven implication (c)::::} (a) gives that 2:::"= 1 Xu(n) converges. (d)::::} ( e) . If (e) does not hold then we can find c > 0 and increasing se quences (n k )k'= 1 and (Ns )':: o with No = 0 such that 11 2: :,;,iJ. +1 Xnk I � c for k = 1, 2, 3, . . . . Let m8 be an increasing enumeration of N\{n k }f: 1 . One checks that for the permutation a defined by a

the series 2:::"= 1 Xu (n) diverges.

3. There is also a related notion of weakly unconditionally convergent series. The series 2:::"= 1 Xn is said to be weakly unconditionally conver gent if for every functional x* E X* the scalar series 2:� 1 x* (xn) is unconditionally convergent. Actually the name 'weakly unconditionally convergent' is a bit misleading, because such series need not converge (even weakly) . As an example take 2:::"= 1 en in More generally, the series l: ::"= o fn in the space C(K) , K compact, is weakly unconditionally convergent if and only if there is a such that

co.

c

Il.D. Convergence Of Series §4.

59

c < oo for every k E K. This can be easily checked using the Riesz representation theorem (I.B. l l ) .

E::'= o l/n (k) l :5

4 Proposition. For a series E;:'= 1 Xn in a Banach space X the follow ing conditions

are

equivalent:

(a) the series E ;:'= 1 X n is weakly unconditionally convergent; (b) there is a constant C such that

eo

L l x* (xn ) l :5 C ll x* ll for every x* E X*; n= l (c) there exists a constant C such that for every ( tn ) �= l E leo

(d) there exists an operator T: eo -+ X such that T(e n ) = X n · (a)::::} ( b) . We define S: X* -+ l 1 by S(x*) = (x* (x n )). The closed graph theorem implies that S is continuous, thus (b) holds. (b)::::} ( c) . For every N and every ( tn)�= l E leo we have Proof:

I nt= l tnXn I =

l ( nt= l tnxn ) I N I nL= l tnx* (xn ) l :5 ll (tn) lleo

S�p x •

ll x 11 9

= sup

ll x* ll 9

:5 C ll (t n ) lleo ·

sup

ll x * ll 9

eo

L l x* (xn) l

n= l

(c)::::} ( d) . Obvious. (d)::::} ( a) . For x• E X* we have 00

eo eo L l x * (xn) l = L lx * (Ten ) l = L I T* (x * ) (en ) l . n= l n =l n= l Since T* (x*) E l1 = c0 this is finite.

a

Il.D. Convergence Of Series §5.

60

5. Now we will investigate weakly unconditionally convergent series which are not convergent. It turns out that the example we gave in 3 is a canonical one. More precisely we have

I::'= t Xn P t r t P2 r2

Suppose that is a non-convergent weakly un conditionally convergent series. Then there exist increasing sequences of integers with < < < < · · · such that the vectors

Proposition.

Pk , Tk

(1) form a basic sequence equivalent to the unit vector basis in CQ .

I::'= t Xn does not converge we can find sequences (Pk )k';. (rk ) k= l as above such that the vectors (uk ) k= l given by (1) satisfy1 l uk l � c > 0 for some c. From Proposition 4(d) we infer that uk �O. Corollary II.B. 18 yields a subsequence (which we will still denote u k ) which is basic. Since (u k ) k= l is basic, there exists a constant C such that for all finite sequences ( t k ) we have

Proof: Since

and

On the other hand Proposition 4(d) yields a constant

I for all

(tk ) E

Co ·

�::> k k ukl l :::; C l (tk ) l oo

This completes the proof.

C1 such that a

We would like now to investigate the unconditionally convergent series in Lp , 1 :::; p :::; 2. We have the following 6.

I::'= t Xn

converges uncondition Theorem. (Orlicz) . If the series ally in Lv (n, J..L ) with 1 :::; p :::; 2 and J..L is a probability measure then

I::'= t l xn l 2 < oo.

Let (rn(t))�=l denote the Rademacher functions. It fol lows from Propositkn 4( e) that there exists a constant such that

Proof:

II. D. Convergence Of Series § 7.

sup N

61

SUPte[o,1J I z:::= 1 rn (t)xn l v C. Thus we have 1 N 1 N p CP � I I n�- 1 rn(t)xn l pdt = I I I n�- 1 Tn(t)xn(w) I PdJL(w )dt 1 N = I I I nL= 1 rn (t)xn(w) I PdtdJL(w). �

0

0

n

n

{2)

0

The Khintchine inequality I.B.8 and the fact that p � 2 yield

CP � K: I ( nt= 1 l xn(w )1 2 ) P/2 dJL(w) N K = : I ( nL= 1 (l xn(w) IP)� ) 2 dJL(w ) n

n

E

{3)

� K: ( n�- 1 ( I l xn(w) IPdJL(w) ) ) 2 N = K: ( � l xn l ; ) 2 �

N

P

n

E

E

•

Since N was arbitrary the theorem follows.

a

The important technical feature of the above proof is the use of the Rademacher functions to represent all possible choices of signs, each occurring with the same probability. This will be used extensively in the sequel.

Remark.

We will discuss this type of question in more detail in Let us note that what we actually proved is the following inequal ity: There exists a constant such that 7

liLA.

C

{4)

(xn)�= 1

C Lv (f!, J.L) , 1 � p � 2, and J.L a probability measure. for all finite A general Banach space X in which (4) holds is said to have the Orlicz property.

Il.D. Convergence Of Series §8.

62

(x�)�= l

(xn)�=l I:::'= t x�(x)xn

with biorthogonal functionals 8. A Schauder basis in a Banach space X (see II.B.5) is called an unconditional basis if for every E X the series converges unconditionally. A basis which is not unconditional is called conditional. Clearly the unit vectors in lp , 1 :5 p < oo, or in co form an unconditional basis. One easily checks that if is an unconditional basis in X then is an unconditional basic sequence in

x

(xn)�= l

9

Proposition.

tional basis in

Lp

(x�)�= l

x·.

()

The trigonometric system T only when p = 2.

(ein9 )��- oo is an uncondi

2( )

Proof: The trigonometric system, being complete and orthonormal is clearly an unconditional basis in L 1l' . By duality it is enough to consider only the case 1 :5 p < 2. Suppose that the series represents a function I E Lp T and converges unconditionally. Then Theorem 6 gives < oo, thus I E L 2 (1l') . •

() 2 I:�: l an l

I:�: aneinB

In view of Proposition 9 the question arises whether Lp , p =/:. 2, has an unconditional basis at all. The answer is positive for 1 < p < oo and negative for p = 1. We have the following. The space Lt [O, 1] does not embed into any space with an unconditional basis.

10

Theorem.

The proof of this theorem relies on the following general 11 Proposition. A block basic sequence of an unconditional basis is a an unconditional basic sequence.

Proof of Theorem 10.

(rn(t))�= l be the sequence of Radx E Lt [O, 1]

Let emacher functions. Note that for every

x · Tn�O

as n -+ oo

(5)

(6) l x + xrn l t l x l t as n -+ Assume that £1 [0, 1] embeds into a Banach space Y with an uncondi tional basis ( Yn)�= t· Starting with Xt = 1 we define

and

-+

oo .

Il.D. Convergence Of Series § 1 2.

63

where kn increases so fast that

the sequence (x n )�= l considered in Y is equivalent to a block-basic sequence of the basis ( Yn )�= l ·

{8)

Condition {8) can be ensured using {5) and arguments like those in II.B. 17. Condition {7) follows from (6) because x1 + · · · + Xn - 1 = (x1 + · · · + Xn - 2 ) + (x1 + · · · + Xn - 2 )rkn - l . Conditions (7) , (8) and Proposition 1 1 show that for some constant C (depending on the norm of the embedding of £ 1 (0, 1] into Y) and for every choice of e n = ± 1 and for every N II

But this and (7) clearly contradict the inequality {4) .

12 Corollary. The space C(O , 1] does not embed into any space with an unconditional basis.

Proof: It is a special case of II.B.4 that £ 1 (0, 1] embeds into C(O , 1] , II so the proof follows from Theorem 10.

The existence of an unconditional basis in the spaces Lp[O, 1] , 1 < p < oo, is contained in the following theorem, which actually gives more precise information. 13 Theorem. numbers with

Let 1 < p < oo. If ( a k )k:,0 and (bk )k:, 0 l bk l ::5 lak l for k = 0, 1 , 2, then for all n � 0

are

complex

(9) where (h k ) k= O is the Haar system and p* =

ma:x.(p, pf(p - 1)).

Let us recall that the Haar system was defined in II.B.9. Proof: One easily checks using duality that it is enough to consider only the case 2 < p < oo (when p* = p) . Let v: ([ x ([ --+ 1R. be defined as

v(x, y) = I Y I P - (p - 1)P ix i P.

( 10)

Il.D. Convergence Of Series § 1 4.

64

The crucial part of the proof is the following 14 Lemma.

There exists a function then

a, b, x, y E CC and l b l � I a I

u: CC

x

CC

-+

R.

such that if

v(x , y ) � u (x, y ), u(x, y ) u( -x, -y ), u (O, 0) 0, u(x + a , y + b) + u(x - a, y - b) � 2u(x, y ).

= =

= E �= O

{11) {12) {13) {14)

==E�=O =

Assuming this lemma for a moment we complete the proof as fol lows. Put fn bk h k . Assume also that ak h k and Yn (h k )k:, 0 is normalized so that ll hk ll oo 1 for k 0,1, . . . . This clearly does not influence the theorem but some formulas are a bit shorter. Then by (10) and {11)

= I v(fn (t), gn (t))dt � I u(fn (t), gn(t))dt. {15) Since both fn - 1 and Yn - 1 are constant on the In = supp hn we get from {14) 11 Yn ll � - {p - 1) P II fn ll � "

1

I0 u(fn(t), gn(t))dt = (0,1I)\1.. uUn- 1 (t), gn- 1 (t))dt + I u{!n - 1 + an , 9n - 1 + bn)dt { h,. >O } + I uUn - 1 - an , Yn - 1 - bn)dt { h,. < O} � I u Un - 9n - d dt [0 , 1 ] \1.. + � I [u{ !n - 1 + an , Yn - 1 + bn ) b

+ uUn - 1 - an , Yn - 1 - bn)]dt

::::::; J uUn- 1 , 9n-1 ) dt 1

0

(16)

II.D. Convergence Of Series § 14.

and thus inductively

llYn II � - (p - 1) P II fn ll � �

65

J u(fo , 9o)dt = u(ao, bo)

= 21 (u(ao, bo) + u(-ao, -bo)) � u(O, O) = o. a

Proof of Lemma 14.

The desired function u (x, y ) is given by

( )

p 1 u(x, y ) = av( l x l + lyi ) P - 1 ( IYI - (p- 1) l x l ) where ap = p 1 - p1 - . (17) Conditions (12) and (13) are clearly satisfied. To prove (11), by homo geneity it is enough to assume l x l + I YI = 1. Letting l x l = s, (11) reduces then to the inequality

F(s) = ap( l -ps) - (1 - s) P + (p - 1) P sP � 0 for 0 � s � 1 and p � 2. (18)

One checks by a direct computation that

F(O) > 0, F(1) > 0, F

G) = F' G) = 0, F" ( � ) > 0

and F" has only one zero in [0, 1] . This is enough to see that (18) holds. To prove (14) it is enough to assume that x and a are linearly indepen dent over IR and the same is true for y and b. Under this assumption the function G(t) = u(x + ta, y + tb) is infinitely differentiable. A routine but rather tedious computation gives

G"(O) = av { - p(p - 1)( 1al 2 - I W ) ( I x l + lyi ) P - 2 (19) - p(p - 2) [ l b l 2 - Re < 1 1 ' b > 2 ] 1YI - 1 ( 1xl + lyi ) P- 1 - p(p - 1)(p - 2) [Re < 1 1 ' a > +Re ] 2 1 x l ( l x l + lyi ) P- 3 }

� :

�

Since l b l � l a l , the Cauchy-Schwarz inequality gives G"(O) � 0. Since this holds for all x, y , a and b with l b l � Ia I we see that in general G"(t) � 0. This clearly implies (14). a Notes and remarks.

The interplay between conditional and unconditional, i.e. absolute con vergence for scalar series, was already the subject of research in the

II.D. Convergence Of Series §Exercises

66

nineteenth century, and probably even earlier. Thus, with the emer gence of a theory of Banach spaces, investigation of series in Banach spaces became an important topic. In this section we only scratch the surface of this vast area. We will present further development of these ideas in some of the subsequent chapters, most notably in liLA and III.F. The fundamental early paper on the subject is Orlicz [1933] . It contains our Theorem 2 (actually it contains the proof of (d) <* ( e ) and condition is used without comment in the proofs) and Theorem 6 with basically the same proofs. The notion of weakly unconditionally convergent series also appeared quite early. Already in Orlicz [1929] the following theorem is proved.

(a)

Theorem.

Every weakly unconditionally convergent series in a weakly sequentially complete space is unconditionally convergent.

Let us point out that a space X is weakly sequentially complete if every weakly Cauchy sequence is weakly convergent. This theorem is a forerunner of our Proposition 5 which is due to Bessaga-Pelczynski [1958] . Unconditional bases (under the name of absolute bases) were already investigated in Karlin [1948] where among other results Propo sition 9 and a weak version of Corollary 12 were proved. Theorem 1 0 was shown in Pelczynski [1961] and the proof we present was given in Milman [1971] . Theorem 13 except for the constant p - 1 is a classi cal result of Marcinkiewicz [1937] but the main work was done by Pa ley [1932] . This theorem was the starting point of much of the later work on orthogonal series and martingale inequalities. The proof we present is due to Burkholder [1988] . It has the advantage of being rather elementary and it gives the best constant in the inequality (9); see Exercise 9.

Exercises 1. (a) Show that if f E C[O, 1] then the series

"'£';:=0 ( , hn)hn, where (hn)�=O is the Haar system, converges uniformly! to f. (b) Show that if w (8) is the modulus of continuity of f E C[O, 1] then I L:= l (! , hn)hn - /lloo Cw( -f;r ) . Show that "'£';:= 1 an'P n ( t ), where ('Pn)�= l is the Faber-Schauder sys tem, converges unconditionally in C[O, 1] if L k sup2 k < n 9 k+1 ! ani < Show that this is not a necessary condition. :::;

2.

oo.

Il.D. Convergence Of Series §Exercises

67

3. {Orlicz-Pettis theorem) . Suppose that for every subsequence (n s ) of the integers the series 2::, 1 xn. converges weakly to an element in X. Show that the series 2::"= 1 converges unconditionally. where { )�= O 4. For I E Lp [O, 1] , 1 2, does not embed isomorphically into any space Lq (/-L) , 1 � q � 2. 7. Let X be a Banach space and let 2::"= 1 be a conditionally con vergent series in X with 2::"= 1 0. Let

=

--+

Xn

hn)hn)hn,! - hn. hn

--+

Xn = Xn

U(xn) = { x E X: there exists a permutation of such that X = � Xu(n) } (a) (Steinitz) . Let X be an n-dimensional Banach space and let (xi )J'= 1 X be such that Ej= 1 Xj = 0 and ll xi ll � 1 for Show that there exists a permutation of j = 1, 2, u

N

C

u

. . . , m.

numbers {1, 2, . . . , m} such that

and the constant C(n) depends only on the dimension n.

Un

{b) {Steinitz). Show that if X is finite dimensional then ( x ) is a linear subspace.

Xn Un {d) Show that in L 2 [0, 1] there exists a conditionally convergent series 2::"= 1 Xn such that U ( x n ) consists of exactly two points.

{c) Show that in L 2 [0, 1] there exists a conditionally convergent series 2::"= 1 such that ( x ) is not a linear subspace.

8.

Suppose that X is a Banach space with an unconditional basis (xn)�= 1 {where N is finite or infinite) . We define the unconditional

II.D. Convergence Of Series §Exercises

68

basis constant of this basis ubc(xn) = sup

{I I t, cnanxn l = l en I = 1 for

n = 1, 2, . . . , N and

We define the unconditional basis constant of the space X as

ubc(X) = inf{ u bc { x n ) : ( xn ) �= l is an unconditional basis in X } .

9.

(a) Suppose that ( Yn) is a block basis of the basis ( xn ) �= l · Show that ubc ( yn ) $ ubc ( x n )· {b) Suppose that gn, k , n = 0, 1 , 2, . . . , k = 1, 2, . . . , 2 n , is a system of functions on [0,1] such that {1) supp gn, k = An , k and for every n, the sets An, k • k = 1, 2, . . . , 2 n , are disjoint and each has measure 2- n , {2) An+ 1 ,2 k - 1 = {t: gn, k = 2 i } and An+I,2 k = {t: gn, k = - 2 -i } . Show that {gn, k } is an orthonormal system. Show that in Lp [O, 1] , 1 $ p $ oo it is a basic sequence equivalent to the Haar system in Lp [O, 1] . (c) Show that every basis in Lp [O, 1] , 1 $ p < oo, has a block-basic sequence equivalent to the Haar system. {d) Show that for every complete orthonormal system ( 0. Let w > p satisfy xP +pwP - 1 - wP = 0. Let us set (J = 1 - � = ;, and f3k = 1 - (x�tc5) for k = 1 , 2, . . . , where 0 < 6 < ! . Let us define d l = X . X(O,l] • d2 = 6X(O,/hl + [fJ ( x + 6 ) - x] X[.Bt o l] • d3 = 6X[o , .B1 ·.B2 J + [fJ ( x + 26 ) - ( x + 6 )] X(.81,82 , .Bt ] , d4 = 6X[o ,,B1,82,83 ] + [fJ ( x + 36 ) - ( x + 26 )] X[,81,82,83 ,,81,82] , etc. {a) Show that limx-+o limc5-+ 0 limn-+oo ii E�= l {- 1) k dk ii = 1 and P limx-+ 0 limc5-+ 0 limn-+oo II L �= l dk ll p = P - 1 . (b) Show that the constant (p* - 1 ) in {9) is the smallest possible. (c) Show that ubc(Lp [O, 1] ) = (p* - 1 ) .

II.E. Local Properties

In this chapter we present some results and notions concerning finite dimensional Banach spaces and the relation between an infinite dimen sional Banach space and its finite dimensional subspaces. We start with a discussion of the bounded approximation property and the 1r�-spaces. We also prove the local reflexivity principle which connects the local properties of X and X** . We prove the Auerbach lemma which allows a good identification of an n-dimensional Banach space with n or ccn . We also study the concept of Banach-Mazur distance.

R

1. By local properties of a Banach space we mean the properties which depend on the structure of finite dimensional subspaces of the space. Some examples of such properties will be pointed out in this chapter and many more will be encountered in the sequel. The basic aim of this chapter is to provide an elementary under standing of local phenomena. Even at this early stage it is apparent that one needs a clarification of two points:

(a) how the general Banach space is built up from finite dimensional subspaces; (b) what are the relevant properties of finite dimensional spaces. Let us start with some definitions and examples which explain point (a) a little . What we are really thinking about in (a) is the approxima tion problem: how well can we approximate the identity operator on the space X by finite dimensional operators? 2. A Banach space X is said to have the bounded approximation prop erty (b.a. p.) if there exists a constant C such that for every finite subset { . , Xn } C X and for every c > 0 there exists a finite dimensional operator T: X ---+ X such that li T Xj I $ c, for j = 1, . . . , n and

x 1,

.

xi

.

II T II $ C.

-

A Banach space X is said to be a 1r�-space if there exists a family (X-y )-rer of finite dimensional subspaces of X such that

3.

/'I ./'2 f there exists

E for every and X-y2 c X-ra •

1'3

E

f such that X-y1 C X-y3

(1)

70

II.E. Local Properties §4.

U "Y E r X-y is dense in X,

(2)

for every 'Y E r there is a projection P"Y : X � X-y with II P"Y I :5 A.

(3)

Many similar properties can be thought of and have been investi gated in the literature but we will discuss only these and the existence of a basis. We have 4

Proposition. A Banach space with a Schauder basis is a 71). -space A. A 11>.-space has the bounded approximation property.

for some

Proof: Let (xn)�= l be a Schauder basis for X. We let Xn span(x k )k= l and Pn be the partial sum projection. One obviously has (1)-(3) . Now let X be a 7r>. space and x� , . . . , Xn be a fixed finite sub set of X. Given b > 0 we use (1) and (2) to find 'Y E r such that dist(x3 , X-y) < b for j = 1, . . . , n. Fix x3 E X-y such that l txj - x3 11 < b for j = 1 , . . . , n. Then we have ll x3 - P-yxj I :5 ll x3 - Xj II + ll x3 - P-yxj II :5 b + II Poy (Xj - Xj ) ll :5 (A + 1) b. This shows that X has the b.a. p.

5

a

Examples. (a) Let (!1, t-L) be a probability measure space. Then Lp ( n, 1-L), 1 :5 p :5 oo is a 1r1-space. We define r as the set of all finite partitions of n into sets of positive measure. For a parti tion 'Y = (!1 � , . . . , !1n ) we put X-y = span{xn }j= 1 and P-y (f) = j L:7= 1 t-L(nj ) - 1 J fXni dt-L · Xni . The properties (1)-(3) are clearly sat isfied (see the proof of II.B. lO(b) or Exercise II.B. 14) . Observe that Lp(n, t-L), 1 :5 p :5 oo, may not be separable and then it does not have a Schauder basis. Note also that for each 'Y we have ImP-y � e; . (b) The disc algebra A(JD) has the bounded approximation prop erty. To see this, it is enough to consider operators Fn (f) = f * Fn where Fn is the n-th Fejer kernel (I.B. 16) . It also has a Schauder basis (see III.E.17) but is not a 1r1-space (Exercise III.E.9) . (c) Every C(K), K compact, is a 7r i +e;-space for every c > 0. (Ac tually it is a 1r1-space, but this requires more care.) This can be proved directly. The alternative argument is to note that C(K)** = L00 (t-L) for some (usually non-a-finite) measure 1-L · From this and the principle of local reflexivity (Theorem 15) we infer that there exists an increasing net (Xoy) of finite dimensional subspaces of C(K) such that U X-y is dense

71

II.E. Local Properties §6.

x

in C{K) and d(X..,. , ta:,m ., ) � 1 + c. {The notion d(X, Y) is defined in 6. ) It follows from the Hahn-Banach theorem that there is a projection P..,.: C(K) �X..,. with li P..,. II � 1 + c. 6. Since all n-dimensional spaces are isomorphic, in order to investigate finite dimensional spaces more precisely we will use a more quantitative notion. Let X and Y be two isomorphic Banach spaces. The Banach-Mazur distance between X and Y denoted as d( X, Y) is defined as

d(X, Y) = inf{ II T II · II T - 1 1 1 : T: X�Y is an isomorphism} .

{4)

If the spaces X and Y are not isomorphic we set d(X, Y) = oo . The Banach-Mazur distance has the following, almost obvious, properties: (5) d(X, Y) � d(X, Z) · d(Y, Z); if X and Y are isometric then d(X, Y)

=

1.

{6)

Thus the Banach-Mazur distance is not a metric in the geometrical sense, but its logarithm is. We follow, however, the long established custom, and use the definition {4) and the name 'Banach-Mazur distance'. We have the following converse to {6). 7

Proposition. If X is finite dimensional and d(X, Y) = 1

isometric to Y.

then

X is

Proof: Let us take Tn : X ---+ Y such that II Tn ii · II T; 1 II -+ 1. Multiplying Tn by an appropriate scalar we can assume II Tn ll = 1 for n = 1, 2, . . . . Since L(X, Y) and L(Y, X) are finite dimensional Banach spaces, passing to a subsequence we can assume II Tn - T il 0 as n ---+ oo and II T; 1 S ll 0 as n ---+ oo for some T E L(X, Y) and S E L(Y, X). Obviously T is invertible and r- 1 = s. Moreover II T II = II S II = 1 so T is the --+

--+

desired isometry.

II

8. As an easy example of a computation of the Banach-Mazur distance we offer the following

Proposition. For every n = 1, 2, 3, . . . and p such that 1 � p � oo have d(l;, .ey) = nl t - ! 1 .

we

Proof. Note that for reflexive spaces X and Y, d(X, Y) = d( X* , Y* ) ; thus it i s enough to consider 2 � p � oo . T he upper estimate i s obvious

72

ll.E. Local Properties §9.

T:id: l�l�. T I 1. n; = I t, ±ei 1 2 � I t, ±Tei 1 2 � j I t, rj (t)Tei 1 2dt = Jt l t, ri (t)Tei (k) l 2dt = t t, l rei (k) l 2 = l rej t, r Tei 2 n ;-! I T-111 � n !-; . f: T n 1 p if we take £;: -+ operator £;: -+ II.D. { 4) we get

In order to check the lower estimate take any with l i Analogously as in the proof of �

s�p

s�p

Thus i�f ii 3

•

and

ll �

of trigonometric polynomials of order Let us consider the space with Lp-norm. More precisely the elements of Tf:, � � oo, are trigonometric polynomials of the form 9.

We have

Theorem.

There is a constant C such that

d(Tn•P lp2n+l ) - p-p--2 1 · = e'· <2n+ l) k = 0, 1, n 1 2 (2n +-1 L ) 2 )1 .! p 2� 1 ( 2n 1+ 1 Ln 1
.

21tk

Let fh for . . . , 2n. The theorem follows from the following two estimates valid for every h E Tf:.

Proof:

.!.

l h( lh W p � 3 l lhllp.

� �

oo ,

p ,

<

(7)

k=O

llh ii P

�C

p

lh(Ok ) I P

k=O

p

<

oo .

(8 )

73

II.E. Local Properties §9.

Proof of (7) : Let Vn be the de la Valle-Poussin kernel (cf. I.B. 17.). Clearly h Vn = h for h E Tf: . Thus *

so from the properties of the Fejer kernel i.B.16 we get

Summing these inequalities we get

2n

E�m ak eikB with

Let us observe that for every trigonometric polynomial cp( 0) m � we have

2 '��"

2n + 1 kL=O cp(Ok ) = a.o = 21!' I cp(O) dO. 1

2n

1

0

{10)

Interchanging the order of summation and integration in {9) and apply ing {10) for .1'2n - 1 and Fn - 1 we get {7) .

Proof of (8) . The condition {8) follows by duality from (7) . Let Pn be the natural projection from Lp(T) onto Tf: . We know {see II.B.ll) that II Pn ll � (;�tl) . Take g E Lq(T), � + � = 1 , IIYIIq = 1 such that

74

Il.E. Local Properties § 1 0.

(27r) - 1 J�'lr g(9)h(9)d9 = ll h ll p · ll h llp = = =

Then, using

(10) we get

21<

� 1 g (9)h(9) d9

2

0

21<

� I Pn (g)(9)h(9)d9

2

0

2n

LO Pn (g)(Bk )h(Bk) · 2 n + 1 k= 1

Applying Holder's inequality and

(7) we get

(8) .

•

One of the most powerful tools available for investigating finite dimensional spaces which does not exist in the infinite dimensional sit uation is volume. Since we can treat an n-dimensional space as Rn (or cr:n ) with some norm it is clear what the volume is. The normalization is arbitrary but in most cases it does not matter. As an illustration of it let us prove

10.

Proposition.

-

Let X be an n-dimensional Banach space. If {xj }f=, 1 is an c-net in Bx , then N � g - n for real X and N � c 2 n for complex X. There exists an c-net in Bx , {xj }f=, 1 with N � ( �t for real X and

N�

{� ) 2n

for complex X.

Proof:

Let us consider real spaces first. If {xj }f= 1 is an c-net in Bx then Uf= 1 B(xj , c) ::> Bx so vol Bx � vol

�

N

N

U B(xi , c)

j=l

vol B(xj , c) = N · vol cBx = N · en · vol Bx . L j= l

Thus N � e - n . In order to find an c-net of a small cardinality let us fix a maximal 2c-separated set, i.e. a maximal set {xj }f= 1 C Bx such that ll xi - xi I > 2c for i =f. j . Such a set is obviously an c-net in Bx . Since B(xi , c) n B(xi, c) = 0 for i =/:- j and B(xi , c) c (1 + c)Bx for j = 1, . . . , N we have N ·c n · vol Bx

=

vol

(ldN B(xi ,

c)

) � vol((1 +c)Bx ) = (1 +c) n vol Bx

75

II.E. Local Properties § 1 1 . n

so N :5 ( �) . Since an n-dimensional complex Banach space is a 2n dimensional real space, the claim for the complex spaces also follows.a Another application of volume considerations is the following. 11 Lemma. {Auerbach) . Let X be an n-dimensional Banach space. There exists a biorthogonal system (xj , xj ) j= 1 in X x X* with ll xi ll = 1 and ll xj II = 1 for j = 1 , . . . , n.

Let (yj , yj ) j= 1 be any biorthogonal system in X. The function V(zt , . . . , Zn ) = det(yj (zi )) i,j = l is a continuous function on the n-fold product of Bx . Let V (x t , . . . , X n ) = max{V{zt , . . . , Zn ) : ll zi I :5 1 for j = 1 , . . . , n} . Clearly ll xi ll = 1 for j = 1 , . . . , n. We define Proof:

A moment's reflection gives that (xi , xj) j= 1 is the desired biorthogonal system. a As an application of Auerbach's lemma we show certain general perturbation results. 12 Proposition. (a) Let X be a Banach space and let (X,), e r be a fantily of subspaces satisfying (1) and (2). Let Xo be a finite dimensional subspace of X. For every c > 0 there exists a 'Y E r and XfJ C X, such that d(Xo, XfJ) < 1 + c. {b) Let X have the b.a.p. Then there exists a constant C such that for every finite dimensional subspace Xo C X there exists an operator T: X -+ X with II T II :5 C and T I Xo = idx0 • Proof: Let ( xj , xj ) j= 1 c X0 x X0 be given by Auerbach's lemma. In order to show (a) fix 'Y E r such that there are Xj E X, with I xi - Xj I :5 � - Define T: Xo -+ X, by T(xi ) = Xj and put XfJ = T(Xo). For x E Xo we have

I f: l I f: l t It ll x ll =

J=l

xj (x)xj =

J=l

xj (x) xj +

Since

xj (x)(xi - Xj ) :5

we get

lxj (x) l ll xi - Xj ll :5

:5 8 11 x ll

l

f: xj (x)(xi - Xj ) ·

J=l

t ll xj ll · ll x ll · ll xi - xi I

76

II.E. Local Properties § 1 3.

II Tx ll - 6ll x ll :5 ll x ll :5 II Tx ll + 6ll x ll ·

If 6 is suitably chosen this gives ( a) . To show ( b ) we use the defini tion of b.a.p. and we fix an operator T1 : X ---+ X such that II T1 (xi ) - xi ll :5 � and II T II :5 cl . We define T(x) = Tl (x) + L:j= l xj (x)(xj - Txj) where xj is a Hahn-Banach extension of xj to a functional on X, j = 1, . . . , n. Clearly T(xj ) = Xj for j = 1, . . . , n so T I Xo = idx0 • Also

II Tx ll :5 II T1x ll +

n xj (x) x - Tx l l ll i i ll L j=l

Thus actually ( b ) holds with any constant C bigger than the constant given by (2) . a For every n-dimensional Banach space X and for > 0 there exists an embedding i: X ---+ t� with (1 - c) ll x ll :5 ll i(x) ll :5 ll x ll where N :5 1 �e if X is a real space and N :5 1 �e if X is a complex space.

13 Proposition.

every c

( r

( fn

Let (xj )f= 1 be an c-net in Bx· given by Proposition 10. We define i: X ---+ t� by i(x) = (xj (x))f= l · Given x E X let x* E Bx· be such that ll x ll = x* (x), and let j be such that ll xj - x* ll < c. Then

Proof:

l xj (x) l = l x* (x) + xj (x) - x* (x) l � l x* (x) l - l (xj - x*)(x) l � (1 - c) ll x ll . a Thus ll i(x) ll � (1 - c) ll x ll . Obviously ll i ll :5 1.

Remark: This result is a finite dimensional or 'local' version of Theo rem II.B.4. Actually without an estimate for N this result easily follows from Theorem II.B.4. 14. Now we will return to the interplay between local and global properties of Banach spaces. We start with the important Theorem. (Principle of local reflexivity.) Let X be a Banach space and let E c X** and F C X* be finite dimensional subspaces. Given c > 0 there exists an operator T: E ---+ X such that

I I T II · II T - l I T(E) II :5 1 + c, T J E n X = id, f(Te) = e(f) for all f E F and e E E.

(11)

(12) ( 13)

77

Il.E. Local Properties § 1 5.

This theorem asserts that finite dimensional subspaces of X** are basically the same as finite dimensional subspaces of X. The proof relies on the following

and let X** .

A;

(A; )f= 1

be bounded, norm-open convex subsets of X denote the norm interior of the a( X** , X * )- clos ure of A; in

15 Lemma. Let

N N If n;=1 A; =f. 0 then n;=1 A; =f. 0. (b) If we have a map T: X Y with Y space then T** C n_f= 1 A; ) = T(nf= 1 A;). -

(a)

--+

Proof:

a finite dimensional Banach

(a) Let XN be the direct sum of N copies of X. The set

is a bounded, norm-open, convex subset of XN . If n.f=, l A; = 0 then A n V = 0 where V = {(x;)f= 1 E XN : x; = x1 for j = 1 , 2, . . . , N} . Let

A- -- {( X;**) jN= l

and

{( X;** )jN= l

E X** N ·. X;** E

3,

A- ·

X;**

J. -

1 , ... , N}

= x 1** £or J = 1 , . . . , N} . If A n V = 0 then there exists 0 for all a E A. Since A is a(X** , X* )-dense in A (use Goldstine's theorem II.A. 13) we get 0 for all a** E A, N so A n V** = 0, i.e. n A; = 0 . j= l N N (b) Clearly T** C n;=1 A; ) and T(n;=l A;) are open, convex sets and N N T(ni = l A; ) c T** (n;=l A;). If they are not equal, then there exists a point p E T** C nf= 1 A; ) and a functional

(3 V**

such that

=

E X** N

· •

.

N

a > (3 >
x* E X* equal T* ( a}. Since ij = A; n {x** E X** : x** (X* ) > a} see that nf= 1 ij =f. 0 but nf= 1 Aj is empty. This contradicts (a) and Let

so proves (b) .

we

a

II.E. Local Properties § 1 5.

78

Let dim E = n and dim E n X = n - k. Fix a biorthogonal system (xj* , ej ) j=1 in E x E* such that span(xj* ) j= k+ l = E n X, and ll xj* ll = 1. The identity id: E � X** can be written as id ( e) = Ej=1 ej ( e )xj* . We want to find X t , . . . , Xk in X such that the map T: E � X defined as T (e) = E7=l ej (x)xi + E�+ l ej (e)xj* will have the desired properties. Property ( 12) is satisfied with this definition. Let Z be the direct sum of k copies of X and let 6 > 0 be a small number. Fix the following finite sets: { /i }� 1 is a basis in F, {xj }� 1 c Bx· is such that for every e E E ll e ll $; (1 + 6) sup{ l xj (e) l : j = 1, . . . , R } , { ei }_f= 1 i s a 6-net i n BE. We have n "'""' A, jr X r** e3. - L..J

Proof of Theorem 14.

r= l

•

Now we form the following subsets of Z: Cj

= { (x s )==l : and ll x s ll

I t, A

� Xs +

< 1 + 6,

s

t

l

,\�x : • < I l ei II 1 8 = 1, . . . k}, j = 1, . . , N. .

Those subsets of Z are norm-open, bounded and convex. Since (x: • )==l E n_f=, 1 Ci C Z** (where - has the same meaning as in Lemma 15) Lemma 15 gives (x s )== l E nf=1 Ci =f. 0 . Let us consider an operator s : z � RM · k $ R R · k (or into ccM · k $ (CR · k ) , defined as

From Lemma 15 we infer that there exists (x 8 )==l such that (x s ) != t E and

N

n=l Ci

j

S((x s)== l ) = S ** ((x : • )==l ) ·

With this choice of (x8 )== l we clearly get (13). Also we get ixj ( Te) i II Te ll ;::: j =sup l , ,R . . .

=

sup ixj (e) i ;::: ( 1 + 6 ) - 1 ll e ll . j = l , ,R ...

( 14)

79

II.E. Local Properties § 1 5.

Given e E BE let us fix have II Tei ll :5 ll e; ll · So

II Te ll

:5

ei

with

II Tei ll + II T(e - e;) ll

A very crude estimate for

:5

li e - ei ll

:5

6.

Since

(x s )!=1

E

C;

li e; II + II T II · 6 :5 ll e ll + 6 + 6II T II .

li T II is ( 1 + 6 ) :E7=1 ll ej II 6 is small enough ( 1 1 ) follows from ( 14 ) and (15 ) .

:5

2 :E 7= l

we

(15 )

ll ej II , so if a

15. We say that a Banach space X is finitely representable in a Banach space Y if there exists a constant C such that for every finite dimensional subspace X1 c X there exists Y1 c Y with d(X1 , YI ) :5 C. In other words X is finitely representable in Y if finite dimensional subspaces of X are subspaces of Y. Note that Proposition 12 yields that every Banach space X is finitely representable in c0 • From Proposition 12(a ) and Example 5(a) we get that Lp (O, ft ) is finitely representable in lp for 1 :5 p :5 oo. On the other hand from II.D. (4) we easily get that lp for p > 2 is not finitely representable in any Lq(O, ft ) for 1 :5 q :5 2. Theorem 14 shows that X * * is finitely representable in X . Notes and remarks. The general idea of approximation of a separable Banach space by finite dimensional ones is quite old and was around in Lw6w in the thirties. Banach ( 1932] asked the question if every separable Banach space has a basis. A notion of approximation property (a. p. ) , the concept even weaker than b.a.p. , was also invented then. We say that a Banach space X has the approximation property if for every norm-compact set K c X and for every e > 0 there exists a finite dimensional operator T: X � X such that sup { II Tk - k ll : k E K} :5 e. The first deep study of a.p. and b.a.p. is contained in Grothendieck ( 1955] . The notion of 71'A-space emerged in the sixties (see Lindenstrauss ( 1964] and Michael-Pelczynski ( 1967] ) . The real breakthrough in the study of approximation proper ties come with Enflo's ( 1973] example of a Banach space without a.p. Many examples differentiating various approximation properties have been produced later. We refer the interested reader to Lindenstrauss Tzafriri ( 1977] and ( 1979] , Pisier ( 1986] and Szarek ( 1987] . All this is a very fascinating subject but beyond the scope of our book. As a first result of the local theory of Banach spaces one can consider the following well known fact proved in Jordan-von Neumann ( 1935] . Ac tually the analogous three dimensional characterisation was given earlier by Frechet [1935] .

80

II.E. Local Properties §Exercises

A Banach space X is isometric to a Hilbert space if and only if for all x, y E X . Note that this result implies that X is isometric to a Hilbert space if and only if every two-dimensional subspace of X is isometric to a Hilbert space. For an isomorphic version of this see Exercise 9(b) . The local theory of Banach spaces gathered momentum in the sixties with the study of L1-preduals (see Lacey [1974] ) and p-absolutely summing operators (III. F) . Today it is a vast subject having connections with operator theory, harmonic analysis, geometry of convex bodies etc. Some of it will be presented later. For a more detailed presentation of different aspects of the theory, the reader should consult Milman-Schechtman [1986] or Tomczak-Jaegermann [1989] or Beauzamy [1985] . The notion of Banach-Mazur distance is in Banach [1932] . Theorem 9 is a classical result of Marcinkiewicz [1937a] (see also Zygmund [1968] chapter X §7) . Proposition 8 is an easy special case of a result of Gurarii -Kadec-Macaev [1965] . The Auerbach lemma is mentioned without proof in Banach [1932] . The principle of local reflexivity ( Theorem 14) was proved in Lindenstrauss-Rosenthal [1969] . Exercises

1.

A reai Banach space X is called uniformly convex if there exists a function c,o (e) > 0 for e > 0 (called the modulus of convexity) such that if x, y E X, l l x l l = IIYII = 1 and l l x - Y ll > 2e then II (xty) I $ 1 - c,o(e) (draw the picture) . (a) Let X be uniformly convex and let x• E X* , llx* ll that diam{x E X: l l x l l $ 1 and x* (x)

=

1. Show

> 1 - e:} --+ 0 as e --+ 0.

(b) Show that uniformly convex spaces are reflexive. (c) Show that Lp [O, 1 ] , 1

< p < oo,

is uniformly convex.

(d) Show that if L:;::"=1 Xn converges unconditionally in X then n= l (e) Suppose that X is a uniformly convex space and that Y C X is a closed subspace. Show that for every x E X, there exists a unique y E Y such that ll x - Y l = dist(x, Y) .

i

81

Il.E. Local Properties §Exercises

2.

Let X be a complex Banach space. We say that X is complexly uniformly convex if there exists a function cp( e) > 0 for e > 0 (called the complex modulus of convexity) such that if y E X with IIYII � f: and llx + ei 6 y li :5 1 for all () then ll xll :5 1 - cp(e).

x,

(a) Show that

L 1 [0, 1]

is complexly uniformly convex.

(b) Show that if cp is a complex modulus of convexity of X and the L: ::'= 1 Xn converges unconditionally in X then

L: ::'= 1 cp( ll xn ll ) <

00 .

,

=

3.

Find two Banach spaces X and Y such that d (X Y) Y are not isometric.

4.

Let T;:" be the space of trigonometric polynomials of the form L � n dk ei k(l with the sup-norm. (a) Show that d (T;:"

, £�+ 1 ) :5 C log(n + 2) for n

=

1 , 2, 3, . . . .

(b) Show that T;:" contains a subspace isometric to

5.

1 but X and

£'�+ 1

Let T� 2 be the space of trigonometric polynomials of degree at most N in two variables (i.e. f(O , t) E T� 2 if and only if ' f(O, t) = L:: m = N an, m ei n(leimt ), equipped with the norm 11/llv =

; ( (411" 2 ) - 1 f02., fg.,. 1/(0, t) I P dOdt) t . cp2 � for 1 0 for every open set U and such that J.L(UI ) < oo for some open set U1 . Translation invariant means that J.L(A + x ) = J.L(A) for every x e X.

7.

Suppose that T: X - Y. Let (Ya)aer be a net of finite dimensional subspaces of Y, ordered by inclusion and such that Uaer Ya = Y. Assume that there is a C such that for each a E r there is an operator Sa : Ya ---+ X such that l i Sa I :5 and TSa = idy"' . Show that T * (Y * ) is complemented in X* .

onto

c

8.

(a) Show that £2 is isomorphic to a complemented subspace of (b)

( L: ::'= 1 £�) 00 . Let Mn denote the set of all n-dimensional Banach spaces (up

to isometry, i.e. we identify isometric spaces). Show that for every n E N the set Mn with the Banach-Mazur distance (or rather, its logarithm) is a compact space.

82

Il.E. Local Properties §Exercises

(c) Let (Bk )� 1 be a sequence of finite dimensional spaces such that {Bk } � 1 n Mn is dense in Mn for all n E lN. Show that for every separable Banach space X the space (:�:::.: ;:'= 1 Bk) 00 contains a complemented copy of X* . 9.

(a) Show that if X is a Banach space finitely representable in a uniformly convex Banach space Y (Exercise 1 ) , then X has an equivalent uniformly convex norm. (b) Show that if a Banach space X is finitely representable in £2 then X is isomorphic to a Hilbert space.

10. A Banach space X has the uniform approximation property ( u.a.p.) if there exist a constant C and a function cp( e, n) , n E lN, e > 0, such that for all X 1 , . . . , Xn E X there exists an operator T: X --+ X such that IITx; II :::; e l l x; II for j = 1 , 2, . . . , n and liT II :::; C and dim T ( X ) :::; cp ( e , n) . Show that Lp [O, 1] , 1 :::; p :::; oo, have the

x2 , -

x;

u.a.p.

1 1 . Show that

£� is isometric to a subspace of £00 but is not isometric to any subspace of co .

Part III Selected Topics III . A Lp- Spaces; Type And Cotype. In this chapter we investigate the Lp (JL)-spaces, 1 < p < oo . We start by proving the isomorphisms of some natural spaces to spaces Lp (JL) . We show that the Sobolev space W� ("11'2 ) is isomorphic to Lp ("11'2 ) for 1 < p < oo and that the Bergman space Bp (D) is isomorphic to lp for 1 $ p < oo . Along the way we prove a useful criterion for the boundedness of integral operators on Lp(f!, JL) (Proposition 9). Later we borrow from probability theory and show the existence and basic properties of stable laws. These provide isometric embeddings of lp into Lq , q $ p $ 2. We continue the line of thought started in II.D.6 and introduce the general notion of type and cotype of a Banach space. In order to study these notions efficiently we prove the vector valued generalization of Khintchine inequality (Kahane's inequality) . A gener alization of a classical result of Carleman from the theory of orthonormal series is also presented. We conclude this chapter with the Banach-Saks theorem and its generalizations to almost everywhere convergence. 1.

We start this chapter with some general observations.

A separable space Lp (f!, JL) , 1

to one of the following spaces: e; , n

Proposition.

t;)p, n = 1 , 2, . . . , (Lp (O, 1]

E9

lp)p o

=

$

p

<

oo,

is isometric

1, 2, . . . , lp , Lp [O, 1] , (Lp (O, 1]

E9

The proof is an immediate consequence of the characterization of non-atomic, separable measure spaces given in I.B. l. Let us also note that the above list contains at most two non-isomorphic infinite dimen sional spaces, namely ip and Lp (O, 1] (see II.B Exercise. 1 1 ) . That for P =I= 2, 1 $ p < oo these spaces are really non-isomorphic follows from Propositions 5 and 7. Thus there are rather few separable Lp-spaces. Some questions about non-separable Lp-spaces can be reduced to the separable case using

84

III.A Lp-Spaces; Type And Cotype §2.

2 Proposition. Every separable subspace X c Lv(f2, f..L ) , 1 � p < oo, is contained in a separable Y c Lv(n, f..L ) isometric to some Lv(n, f..L I ) .

)� 1 A;,a ,b ·

Proof: Let us fix a countable dense subset ( x; in X, and consider sets = { w E n: a < x; ( w ) < b } where a, b are rational numbers. Let E be the a-algebra generated by all sets The space Lp(f2, E, f..L ) a of all E-measurable, p-integrable functions is the desired Y.

A;,a ,b

3. One of the reasons why Lp-spaces are important is that many other spaces common in analysis are isomorphic to Lp-spaces (see II.B.Exercise 9). We want to present one more example of this type. Proposition. The space WJ (']['2) is isomorphic to Lp (1['2 ) , 1

< p < oo .

(an,m )�.:= - oo

For a doubly indexed sequence of numbers we say n i i that ( a ) is a multiplier on Lp(']['2) if the map L. } (n, m)e 6 l e m 62 1---+ E a f ( n, m)e i n61 eim()2 extends to a continuous operator from Lp(Y2) into Lp(Y2 ) .

n,mn

,m

4 Lemma.

The doubly indexed sequences

( ( 1 + l nl+ l m l ) ) :.m= - oo

Lp(Y2) for 1 < p <

oo .

and

( ( l + l ni+ l ml ) ) :.m= - oo

( ( l +l ni+ l ml ) ) :.m= - oo '

are multipliers on

This Lemma is a special case of the multidimensional multiplier Theorem I.B.32. Proof of Proposition 3. Let T: Lv(']['2) ---+ WJ (Y2) be defined by It follows easily from Lemma 4 the multiplier that T is continuous. We define E: WJ (Y2) ---+ Lp(Y2) by E( f ) = f + 8d + 82 / where

( ( l + l ni+ l m l ) ) :,m= - oo .

1 ( n,Lm }(n, m)ein61 eim62 ) n,Lm l n i }(n, m)ein61 eim62

8 and

n,Lm

fh is defined analogously. } n (n, m)ein61 eim62 E Lp(']['2)

=

Since f E WJ (']['2) the function and using the Riesz projection (see

Theorem I.B.20 ) in the variable fh we get that 81 : WJ (Y2) ---+ Lp(Y2 ) . Analogously for [h , so E is continuous. A routine calculation shows that E Tf = f for f E Lp(Y2). Since E is clearly 1-1 we get the desired isomorphism. a

85

Ill.A Lp-Spaces; Type And Cotype § 5.

Now we will investigate the spaces lp .

5.

Proposition. Let X be an infinite dimensional subspace of lp , 1

:5 p <

or of Co . Then X contains a subspace y such that y "' ep (or Co ) and is complemented in lp (or eo). oo,

(zn)�=1 =

be a block-basic sequence of the unit vector basis Proof: Let in ep (or co , so Zn E Z:�11 with kn increasing to 00 and = 1. One checks that span{zn }�=1 is isometric to Let z� b e a functional and z� = E Z:�11 (or co such that z� (zn) on 1 by P n 1, 2, 3, . . . . We define P: = L:::'= 1 z� (x)zn . P is algebraically a projection onto span{zn}�=1 and

)

i =v

)

a ; e;

l zn l

iv . = = iz iv ---+ iv l �(x)I

/3; ej ,

v )e x( I P(x) l v = ( n�- 1 i z� (x) I P) ; :5 ( n�- 1 � z�( k,.+ ; E1 j ) l ) ; :5 ( � I E x(j )e; I P ) t n - 1 k,. + 1 :5 ( � l x (j ) I P ) p = l xl v so I P I = 1 . From II.B. 17 we infer that X contains a sequence (xn)�=1 very close to such a (zn)�=1 and II.B.15 gives that span(xn)�=1 is com 00

!.

plemented in lp (or Co) and isomorphic to span(zn)�=1 so to ep (or eo).a Using the above Proposition 5 and Theorem II.B.24 we get

ip , 1 :5

p

a

Every infinite dimensional complemented subspace of

6 Theorem.

< oo , or of Co is isomorphic to the whole space.

iv

7. This simple structure of complemented subspaces of is rather exceptional. For example Lp [O, 1] , 1 < p < oo, contains complemented subspaces isomorphic to Hilbert space as well as those isomorphic to or Lp [O, 1] itself (see II.B.2(b)) .

iv

Proposition. Let (rn)�= 1 b e Rademacher functions. Then the space span(rn)�= 1 Lp [O, 1] is isomorphic to £2 for 1 :5 < and is comple c

mented for 1

< p < oo .

p

oo

86

Ill.A Lv-Spaces; Type And Cotype §8.

Proof: The first claim is just the Khintchine inequality I.B.8. For the second let P: L2 [0, 1] --+ L2 [0, 1] be an orthonormal projection onto span(rn)�=l · For oo > p � 2 the Khintchine inequality gives

for By duality we get P: Lv [O, 1] --+ Lv [O, 1] for 1

f E Lv [O, 1] .

 1 . The proof i s exactly the same, only uses the analogue of the Khintchine inequality for lacunary sequences of characters (see I.B.8) .

lzl

{z

v

Let D = E ([: < 1} and be Lebesgue measure on D. Bv c Lv (D, denotes the Bergman space Bv (D) (see I.B.28 for definitions) . 8.

dv)

Theorem. For every 8

Ps f(z)

=

>0 8

the operator

+1 7r

f l wzl w)2t2f(+sw) dv(w)

D

(1 (1 -

is a continuous projection from Lv (D,

dv) onto Bp , 1 :5

(1) p

< oo .

Note that this theorem is false for p = oo . Obviously Boo = H00 and there is no continuous linear projection from L00 onto H00 ; see re marks after Proposition III.E.15. In the proof we will need the following criterion for the boundedness of integral operators on Lv ( n , J.L ). 9 Proposition. Let (n, J.L ) b e a measure space an d let K(w t , w2 ) b e a measurable function on

n

X

n. Let us define

Then

(a) if I supessw 1 E n n

I K(wt , w2 ) 1 dJ.L (w2 ) < oo then

(b) if supessw2E n I I K(wt , W2 ) idJ.L(Wt ) n

< oo

then

87

III.A Lp-Spaces; Type And Ootype § 1 0

( c ) if 1

< p < oo and there exists a measurable positive function 0 and constants a, b such that for � + /J = 1 we have

g on

J I K(wt , w2 ) ig(wi )P' dJ.L(wi ) :::; [ag(w2 )]P' J.L - a.e. J I K(wt , w2 ) lg (w2 )PdJ.L(w2 ) :::; [bg(w1 )]P J.L - a.e.

!1

and

!1

Proof: The argument for ( a) and ( b ) is obvious, so we will prove ( c ) . We have

I T J (w 2 ) l I K(wl , w2 ) i l f(wl ) l dJ.L(w i )

:::; J J [I K(w1 , w2 ) l -? g(wi )] [I K(wl , w2 ) l � l f(wl ) l g(wl ) - 1 ]dJ.L(wi ) < ag(w2 ) { J I K(wl , w2 ) i ( l f l /g) P (w l )dJ.L (w i ) } ;; . !1

·

!1

1

·

Hence

!1

[! ( '�1 r(wl ) J gP (w2 ) I K(wt , W2 ) idJ.L(w2 )dJ.L(WI )] :::; ab [ j ( l f l fg ) P (wi ) gP (wi ) dJ.L (WI ) ] ab [ J if (wi ) I P dJ.L(wl ) ] ;; .

II T/IIv :::; a

!1

!1

=

10.

!1

1

p

1

v

1

!1

a

The other fact we will use is the following For every a and s such that - 1 < a < 1 and cons tant Co:,s such that for l w l < 1 we have

Lemma. exists a

s

>

0, there

(2)

III.A Lp-Spaces; Type And Cotype § 1 0.

88

> 0 such that for all p with 0 p 1 1 1 'Y- pei8 1 'Y (1 - p + 1 0 1 ) . This gives � 2� d� 2 s < J 1 1 - pe•8 1 2+s - 'Y / (1 - p + 1 0 1 ) - 2- sdO

Proof: There exists a constant and all with < 1r we have

0

101

::£

�

0

5

::£

-�

Cs J o - 2 - s dO (3) 1-p 1 Cs (1 - p) - -s . Since the integral in (2) depends only on p l w l , passing to polar coordinates and using (3) we get -- l z� (1,..:--1-'- 22)"+:-s dv(z) -1 2� 1 r (1 - r•2. ) a2+s drdO (4) 211" J J 1 1 - pre 8 1 J 1 1 - zw l 1 /1 (1 - r2 ) a (!2� dO + ) dr 211" 11 - prei8 1 2 s 1 C (1 - r)"(1 - pr) - 1 - 8 dr. ::£

::£

=

=

D

0

::£

0

0

0

J

::£

0

Integrating the last integral by parts we see that it equctl.s

1

c1 + c2 J (1 - r) 1 +"(1 - pr) - 2 - 8 dr 1 c1 + c2 J (1 - pr) 1 +" (1 - pr) -2 - 8 dr Ca,s (1 - p) a - s , 0

::£

0

::£

so we get (2) .

( 1) 1 , v). (z) P*s (f) ( z ) s + 1 {1 - l z l 2 ) s J {1g(z)d - zw)� 2+s

a

Proof of Theorem 8. First we show that defines a bounded operator on L ( D The adjoint operator is given by the formula =

7r

D

(5)

89

III.A Lp-Spaces; Type And Cotype § 1 1 .

()

P; is bounded on ( ) g (z) = (1 - l z l 2 ) - p\

so we infer from Proposition 9 b and Lemma 10 that L00 (D, so is bounded on L1 (D, For 1 < p < oo we apply Proposition 9 c for where � + � = 1 . Lemma 10 yields

v) P8

v).

$

[Cg(zW

and analogously

P8

so is bounded. Clearly for f E L1 (D, 11 ) , Moreover for n = 0, 1 , 2, . . . we have

P8 (!) is analytic in D.

Ps (zn )(z) = s +1r 1 J ( 1( 1--l wzwl 2))2s+wsn dv(w) 2 + ;) r s 1 (zw) k dv(w) = ; [ ( 1 - l w l 2 ) s w n � ��72 +s s + 1 r(� 7 2 + ; ) zn / ( 1 - w 2 ) s w 2n dv(w). = ll ll 1r n. r 2 + s ][)

][)

P8 (zn ) zn

Evaluating the last integral in polar coordinates and using the well for known properties of Euler's beta function we get that = n = 0, 1 , 2, . . . . This shows that a is a projection onto Bv (D) .

P8

As an application of Theorem 8 we show 11 Theorem.

The space Bp is isomorphic to fp , 1 $ p <

oo .

The proof of this theorem will be based upon the following 12 Lermna. Every compact operator T: X

X

T

Lp

Q� f � p

--+

Lp admits a factorization

90

III.A Lv-Spaces; Type And Cotype § 1 2.

with

ll a ll · ll fi ll

:::; 8 II T II ·

Proof: Since T is compact and Lp is a 1r1-space (II.E.5(a)) there exists a sequence of norm-one projections Pn : Lp --+ Lp such that II T - PnT II :::; 4- n and d(ImPn , f�im ( ImPn ) ) = 1, for n = 1, 2, . . . . We identify fp with o = �=l ImPn )p and we define

and ,B ( fn ) = 2:: �=1 2- n + 1 fn · One checks the desired properties.

a

Proof of Theorem 11. Let us fix an increasing sequence of numbers (rn ) ;:,o= l tending to 1 with r1 > 0. Let us put

IDo =

{z E ID: l z l

::=;

r1 } and IDn =

{z E ID:

rn <

lzl

::=;

Tn + l } ·

Let In : Bp --+ Lp (IDn , dv) be the natural restriction operator. From I.B.28 and a standard normal family argument we infer that each In is compact. Let (an , fin ) be a factorization of In given by Lemma 12, with II an II = 1, II fin II :::; 8. Thus we have the commutative diagram

where a( f) = (an (f i1Dn ) ) ;:,o= 1 , I denotes the identity embedding, ,B ((xn ) ;:,o= 1 ) = ( ,Bn (xn ) ) ;:,o= l and � ((fn) ;:,o= l ) = L�=l fn and P is any projection onto Bp (see Theorem 8) . Since PI = idBv we get that a is an isomorphic embedding of Bp into (�fp)p � fp and P�,B is a projection a onto a( Bv) . Theorem 6 gives the claim. Remark: It is easy to see from (5) that Im(P; ) C L 00 (1D, v) is exactly

{f(z): f(z) = (1 - l z 2 l ) s · g(z) with g(z) analytic} so we infer from Theorem 1 1 that the space Xs of all analytic functions f( z ) such that suplzl < l ( 1 - l z l ) s lf (z) l < oo is isomorphic to foo , for s > 0.

III.A Lp-Spaces; Type And Cotype § 1 3.

91

13. Our aim now is to introduce the so-called stable laws. These are well known probability distributions, but because of their importance in Banach space theory we will discuss them here in some detail. Let us recall some general notions. To each real valued random variable f on a probability space ( 0, P) there corresponds a probability measure I-ll on JR, called the distribution of J, determined by the relations P { w : f (w) < .>.} = I-ll (( - oo , .>.)), >. E JR. Conversely for each probability measure p, on JR there exist random variables f such that p, 1 = p, . It is clear and well known that the integrability properties of f are reflected in properties of p, 1 . More precisely we have the following formula:

I

n

F (J(w) ) dP(w)

=

+co

I

-

oo

F ( x ) dp, 1 ( x ) ,

(6)

valid for every bounded continuous or positive continuous function F : JR -+ JR.

14 Theorem. For every 0 < p � 2 there exists a distribution P,p such that

I-co eio:xdp,p(a) = e- lxiP . 00

(7)

Every function ( random variable ) whose distribution equals P,p is called p-stable. Those variables do not exhaust the class of all p-stable variables considered in probability theory. This is a simple special case, but sufficient for our purposes. Proof: Note that (7) defines what in probability theory is called the characteristic function and in harmonic analysis the Fourier transform of the measure p, . Its basic properties are well known and can be found in many books, e.g. Katznelson [1968] Chapter VI. Let B denote the class of functions on JR which are Fourier transforms of positive measures on JR. This class satisfies the following properties:

if JI , h E B, a1 , a2 ;::: 0 then ad1 + a2 !2 E B; (8) if ft , h E B then fi · h E B; (9) if Un )�=l C B and fn converges almost uniformly on JR to f then f E B. ( 10)

92

III.A Lv-Spaces; Type And Cotype § 1 4.

Our aim is to show that e- l x l �' E B, 0 < p $ 2, because then we get from (7) that JLp(1R) 1 , so JLp is a probability measure on 1R . The case p 2 is the classical Gaussian ( normal ) distribution so dJL 2 (x) ., 2 (211") 21 e- T dx. From now on we assume 0 < p < 2 . From the formula

=

=

=

(11)

which is easy to check using the substitution �

=

u

we get

Approximating this integral we see that e - lx i P is an almost uniform limit of the functions

So it is enough to check ( see (9) and (10)) that exp

( - 1 +{! ) 2 ) E

B.

But

Since the convergence is almost uniform, from (8) , (9) and ( 10) we see that it is enough to check that (x 2 + b2 ) - 1 E B. This follows from the formula

-00co = � cos ax e- ba da

J 0

( 1 2)

III.A Lp-Spaces; Type And Cotype § 1 5.

93

which is easily verified using integration by parts twice.

a

15 Proposition. Let I be a p-stable function on a probability measure space

(f!, JL).

( a) if p

Then q

= 2 then I E Lq (f!, JL) for 0 <

<

oo,

( b) if p < 2 then I E Lq (f!, JL) for 0 < q < p. The case ( a) easily follows from (6) , since we know that .,2 dJL2 (x) = (27r) 21 e - T dx. Let p < 2 and I be a p-stable variable with the distribution JLp satisfying (7) . We have to estimate J�:; l x l q dJLp(x) ( see (6) ) . Since l x l q = Cq J000 (1 - cos xt)r 1 - q dt ( substitute xt = u here the condition q < 2 is important ) we get using (7) Proof:

oo

Joo

Joo 10 �lcz: xt dtdJLp(x) oo + oo Cq I t l�q I cos xt) dJLp(x)dt 0 oo oo + oo Cq I t l�q I Re eixt )dJLp (x)dt 0 oo 00 e - tP ] dt. Cq I t l +q 0

ixl q dJLp(x) Cq

=

-

=

=

=

oo

Substituting in the last integral

00

I0 which is finite for 0 <

1 - e - tP

1

(1 (1 -

1 [ 1-

tP = u we get 1

00

I

t l +q dt = p- 0

q

1 - e -u !±.i

up

du, a

< p.

16. Suppose now that (/n)':'= 1 is a sequence of independent p-stable functions on a probability measure space ( f!, JL). Let ( an )':'= 1 be a finite sequence of real numbers with E:= l ian i P 1 and put E:=l an ln ·

=

I=

94

If

III.A Lp-Spaces; Type And Cotype § 1 7.

J.Lf is the distribution of f then we have + oo Ioo eiaxdJ.LJ (a.) nI eixf(w) dJ.L(w) =

-

=

=

In n:fi= l eixanfn (w) dJ.L(w ) :fi I eiXanfn (w) dJ.L(w) n= l n II e - l an xi P 00

=

n= l =

Thus

f is also p-stable.

exp -

00

(L:: l an i P ) I x i P

=

n= l

e- lxiP .

In particular ( see Proposition 15) we have

( )

J.L)

Let Un)':= 1 be a sequence of independent p-stable func tions, 0 < p � 2. The span fn ':= 1 in real Lq (S1, is isometric to f.p if q < p and p < 2. For p 2 it is isometric to £2 for 0 < q < oo. D

Corollary.

=

17. We now wish to return to the circle of ideas connected with un conditional convergence of series in Banach spaces which were discussed in II.D.7. Motivated by Orlicz's theorem and in particular by II.D. ( 4 ) we introduce the following definitions. Definition. A Banach space X is said to have cotype p, 2 � p � there exists a constant C such that for all finite sets

(xi ) J= l

oo,

if

CX

( 13)

A Banach space is said to have type p, 1 � p � 2, if there exists a constant C such that for all finite sets ( xi ) j=1 C X ( 14)

III.A Lp-Spaces; Type And Cotype § 1 8.

95

Recall that (rj)� 1 are the Rademacher functions. A few comments about these definitions are in order. ( a) Since scalars satisfy neither ( 13 ) for p < 2 nor (14) for p > 2 ( see Khintchine's inequality I.B.8 ) we see that the above restrictions for p are essential if we hope to get non-trivial concepts. ( b ) If a Banach space X has type p and cotype q and Y is finitely representable in X then Y also has type p and cotype q. In particular X and X** have the same type and cotype; cf. principle of local reflexivity II.E. 14. ( c ) Every Banach space X has type 1 and cotype oo . Also if X has cotype p it has also cotype q for q > p and if X has type p it has also type q for q < p. (d) The smallest constant for which ( 13 ) holds for a given space X is called the cotype p constant of X and is denoted Cp(X). Similarly we define the type p constant of X, denoted by Tp(X). The following vector valued generalization of the Khintchine in equality is a fundamental tool for investigating types and cotypes. 18 Theorem. (Kahane's inequality) There exist constants Cp , 1 � p < oo such that for every Banach space X the inequality

holds for every finite sequence

(xi ) J= l

C

X.

The proof follows from the following distributional inequality. Let V(t) = II E.i=o ri (t)xi ll · {t: V(t) > 2a} l l � 4 l {t: V(t) > aW .

19 Proposition. we have

Proof: For k � n let us put the following sets:

Then for every

Vk (t) = II E7=o ri (t)xi ll

n

0

and let us define

Am = {t: Vk (t) � a , k = 0, . . . , m - 1 , Vm (t) > a}, n A = U Am = {t: sup Vk (t) > a}, k m= l B = {t: V(t) > a}, C = {t: V(t) > 2a }, Cm = { t : I I L rj (t) xj l l > a } . J =m

a>

III.A Lp-Spaces; Type And Cotype § 1 9.

96

Let us write Ej=o Tj (t)xi = E;: o Tj (t)xj + Ej= m + l rj (t)xj = a(t) + b(t). It follows from properties of the Rademacher functions that b(t) is symmetric on every set where a(t) is constant. Since ll x ll � max( ll x + Yll , ll x - Yll ) for every x, y E X, we see that at least on half of the set Am we have ll a ll � ll a + b ll · Thus

(16) Analogously

(17) We put Am = (A m n{t: rm (t) = 1})U (A m n{t: rm (t) = - 1}) = A;\:. uA;;;, and Cm = (Cm n {t: Tm (t) = 1}) u (Cm n {t: rm (t) = - 1}) = c� u c;;;, . The independence of the Rademacher functions gives

and Since

l A� n C� I = 2 I A� I · I C� I l A;;;, n C;;;, l = 2 I A;;;, I · I C;;;, I . l A;\:. I = l A;;;, I = ! I Am l and I C� I = I C;;;, I = ! I Cm l we have (18)

Since obviously

C C B C A,

from

(18), (17)

and

(16) we get

I C I � L I Am n C I � L I Am n Cm l = L I Am i iCm l � sup i Cm l · L I Am l � 2 I B I · I A I � 4 I B I 2 m

•

Proof of Theorem 18. The left hand side inequality is obvious, while the right hand side inequality is a standard passage from a distribu 1 tional inequality to an integral one. We can assume J0 V(t)dt = 1, so l {t: V(t) > 8} 1 � �- Applying Proposition 19 inductively for k = 1, 2, 3, . . we obtain .

This gives

1

k . 8) P · l {t: 2 k - l . 8 < V(t) � 2 k . 8} 1 J0 V(t)Pdt � 8 + kL(2 =l k � 8 + L (2 k . 8) P . T 2 = c:. lc = l 00

00

a

97

III.A Lp-Spaces; Type And Cotype §20.

20 Remarks. ( a) An obvious and immediate consequence of Theorem 18 is that in (13) and ( 14) instead of J II Erj Xj I we can use (J II Erj Xj ll q ) � for every q, 1 � q < oo. (b) For some applications the magnitude of the constant Cp is im portant. Our proof, as can be easily verified, gives Cp � C p for p ;::::: 2. The correct order of magnitude is Cp � C.,jP (see e.g. Milman Schechtman [1986] ) .

·

The following repeats arguments from II.D.6.

21.

Proposition.

If X has cotype p and E:: 1 X n is an unconditionally

convergent series in X, then

Proof.

E:: 1 ll xn li P < oo.

We have for every N

(� ) 1 I I n�- 1 rn (t)xn l dt N

.!

ll xn ll p

P

N

� Cp (X)

0

� Cp (X) s�p

I n�- 1 rn (t)xn l N

� C. a

22. A part of the relation between type and cotype is explained by the following. Proposition. If X has type p then X* has cotype q , where � + �

x1 , . . . , X n E X and xi , . . . , x� E X* 1 n n n xi (x i ) = ri (t)xi ri (t)x i dt •= 1 •= 1 •= 1 0

Proof:

For arbitrary

1.

we have

I ( ?:

?:

�

�

Since

=

) ( ?: ) ] I t•-1 ri (t)xi 1 · 1 t•-1 ri (t)xi l dt 2 ! 2 ! r (t)x r dt (t)x i l ) ( } I t•- 1 i ( } I t•- 1 i i l ) . 0

0

0

( 19)

98

III.A

Lp-Spaces; Type Cotype And

§23.

from (19) we get

1

!

( ?:t=1 l xill q) ; (Ji i ?=t=n1 ri(t)xi11 2) 2 { ( } I tt= 1 ri(t)xill 2) ! : tt=1 l xiiiP } n

�

0

· sup

� 1 ·

0

a

Using Theorem 18 we get the claim. 23.

The types and cotypes of Lp-spaces are as follows.

The space Lp O, J.L , ) ( cotype L00(0, J.L)

Theorem. max(2, p) .

1 �p�

oo,

is of type

min(2, p) and

(x;) Lp (O, J.L)

Proof: Clearly has type 1 and cotype exactly like in II.D.6 we obtain for every c

For 1 � p <

oo.

( } I � r;(t)x; I pPdt) ; 1 = ( / � � � r;( t )x;(w ) I P dtdJ.L (w ) ) ; ( / ( � l x;(w W ) dJ.L (w ) ) ; ( / [ � (l x;(w) IP) � ] 2 dJ.L(w )) 0

J

n

""

=

n

n

1

o

of oo

(20)

J

�

J

.!

1?.

P

J

(x;) .

where '""' indicate, that there are inequalities in both directions with constants independent of the set If 1 � p � 2 then Theorem 18 and (20) give

1

x; l pdt J II �r;(t) J o

$

c (j L.J l x;(w )I PdJ.L (w )) n

1

p

$

C(� J

l x;ll�);

99

Ill.A Lp-Spaces; Type And Cotype §24.

and

! II � Tj (t)xj ll pdt � c ( � (! l xj (w) IPdJL(w) ) ) 2 2

1

0

For 2

3

=c

3

(� ) 3

0

ll xi ll �

p

!

.

:::; p < oo Theorem 18 and (20) give 1 ri (t)xi dt :::; l xi (wW p 3 3 0 0

JI �

l c(J (� ) ) )) :::; c ( � (! 3

0

1

£ 2

dJL(w)

l xj (w ) I P dJL(w)

A

p

� !

= C ( L ll xi ll � ) ! j

and

24. As we said earlier ( 17) our Definition 17 was motivated by the Orlicz property ( II.D. ( 4 ) ) . There is however a more formal connection between these two notions. Suppose the Banach space X has the Orlicz property and suppose that X "' ( E:: 1 X ) for some p, 1 :::; p :::; 2. Then X has P cotype 2.

Proposition.

Proof:

Let us take a finite set ( xj)j; 1 C X. Let rj (k), k = 'Rademacher type' functions on the

1, 2, . . . , 2 m , j 1 , . . . , m be the set {1, 2, . . . , 2 m }. Let us define

=

Xj

= 2- � kL2m= 1 fJ (k)xj

100

Ill.A Lp-Spaces; Type And Cotype §25.

where

0::: �

xj denotes the vector x; considered in the k-th summand of 1 X) P . Using the isomorphism between X and 0::::= 1 X) P and

the Orlicz property we get

! (f: l x; l 2 ) ! :::; c sup l f r; (t) x; l 2 (f: l x ) l ; t E[O,l ] j = l j =l j=l 2"' ( )xj l = C tEsup(O,l ] 1 2 - W" jL= l r; (t) L f; k k= l 2"' I L (t) ( )x l p ) = C tE[supO,l] 2 - W" (I.: r; f; k ; k= l j = l 1 = C tE[supO,l ] ( / l jf= l r; (t) r; (u)x; I Pdu) ; = O " c ( I t, r; (u)x; l du) • . =

m

=

m

.l P

J

�

We see from Remark 20 that this completes the proof.

•

25. As a simple application of these ideas we will present the gener alization of a classical result of Carleman.

(cpn)� L [0, 1] l , ct'nW 1 for 2every c

Theorem. For every complete orthonormal system there exists an f E 1] such that E�= l ( f

p<

C [O ,

2.

= oo

2

Proof: Suppose it is not so. A standard category argument or the closed graph theorem applied to the space yields p < SUCh that < oo for all f E 1] . Thus we have a commutative E�= l I (!, diagram

ct'n W

C [O , 1]

C [O ,

-�

"d

Up< 2 £p

L2 [0, 1]

�'\ fp �

where operators and � are defined by ( f = (! , and 1] is clearly a non 1] --t Since � ) = E�= l compact operator the following lemma gives the contradiction.

(�n

26 Lemma.

cp �nct'n ·

cp ) ( n )� id: C [O , L2 [0, C,O ) 1 Every operator from C ( K) into fp, 1 < < 2, is compact. p

III.A Lp-Spaces; Type And Cotype §27.

101

Let T: C(K) --+ fp Take T* : fq --+ M(K), � + * = 1 . If T is not compact, nor is T* , thus there exists a sequence (x n )�= 1 C fq , ll xn I � 1 such that II T*xn - T*xm ll � 8 for n =1- m. Since f.q is reflexive we can pass to a subsequence such that X n k �x00 and to another subsequence such that Yk = X n k - X00 is equivalent to the block basis of the unit vector basis in f.q (apply II.B . 1 7) and thus to the unit vector basis in f.q . Since II T* yk I � 8 and M(K) has cotype 2, for N = 1 , 2, . . . we get Proof:

-

vN · 8 �

( k�-1 ) 2 C J I �k-1 rn(t)T* yk l dt 1 C II T II J I � rn (t) yk l dt C II T II ( � IIYk ll q ) k- 1 k- 1 N

II T * yk il 2

!

1

�

N

0

N

� Since

q

!

N

9

�

0

�

CN Jq . •

> 2 this is a contradiction.

This lemma is also true for £ 1 (Exercise II.D.5) .

27. We know from II.A.5 that for every weakly convergent sequence there exists a sequence of convex combinations convergent in norm. For Lp spaces, 1 < p < oo , this can be improved. Theorem. (Banach-Saks) .

Every bounded sequence of functions such

(x n )�= 1 C Lp (O, J.L) , 1 < p < oo, contains a subsequence (xn k )� 1 that N - 1 E�= 1 X n k converges in norm. This theorem is an obvious consequence of the following.

Every bounded sequence (x n )�= 1 c Lp (O, J.L) , 1 < contains a subsequence (x n k ) f= 1 such that for some x E Lp (O, J.L)

28 Proposition. p<

oo,

(21) for every finite subset of integers

A and for

s

= min(2, p) .

Propositions 1 and 2 show that it suffices to consider Lp [O, 1] . First we use reflexivity to choose the subsequence (x n k ) such that Xn k � x . If for some further subsequence (still call it Xn k ) we have llxnk - xll --+ 0 we take once more a subsequence such that llxn k - xll � 2 - k so (21 ) holds. Otherwise llxn k - xll � 8 for k = 1 , 2, . . so (xnk - x)

Proof:

.

III.A Lp-Spaces; Type And Cotype §29.

102

has a further subsequence equivalent to the block basis of the Haar sys tem (II.B. 17), thus unconditional (see II.D . l l and II.D. 13) . Theorem 23 easily gives 21 in this case. a

( )

29. We also wish to present a similar result for almost everywhere convergence. Because of the application in III.C.8. we formulate it for countable family of sequences.

Z(x�)� 1

Theorem Suppose that for every m E is a bounded sequence in Lp ( O, J..L ) , 1 < p < oo. Then there exists an increasing sequence of integers such that for every m E Z there exists E Lp ( O, J..L ) such that 1 '"' J..L a.e. (22)

(nk ) f= 1

N m m N kLJ= 1 xu(k) � X

for every m E Z and every permutation

a

xm

of natural numbers.

(nk )� 1 ( ) (x�)� 1 ( ) (x�k )� 1 Xn,.< k l - x hk N - 1 2::= 1 hk HN . 2:�= 1 (HN+ 1 - HN ) (xn) I E i xn l l i P El l xHn l , l - H N+ N l: N ( N( + ) ) I HN I ( N(N�1 ) ) HN l:N ( N (N� 1 ) ) HN converges J..L-a.e. It remains to show that 00 h converges J..L - a.e. (23) L n= 2 n

A standard diagonal procedure and Proposition 28 shows that there exists an increasing sequence of integers such that for each m E Z the sequence satisfies 21 with some constant depending on m. We will show that any sequence satisfying 21 satisfies separately, (22). Clearly it suffices to consider each sequence so in the rest of the proof we will omit the superscript m. and = Denote = We have to show that converges J.L-a.e. we see that Since for C Lp ( O, J..L ) we have � absolutely convergent series converge J.L-a.e. We write = < oo so + h};;; > . From (21) we get � l

Proof:

�

Fix numbers 0<

a and (3 such that

a < 1, (3 > 1, a(3 < 1 , as > 1 and s(3 + 1 > s + (3.

K we have from (21) [( K + l ) /3 ] h < C ( [( K'"'+ l ) l3 ] - f' s < C ( f' - 1 ) • . K (K ) I n=LJ[Ki3] I Kf's n=LJ[Ki3] )

For each integer '"'

� n

�

l

103

III.A Lp-Spaces; Type And Cotype §Notes. Since (3

> 1 we infer that 1.

[ K I3] hn ex1sts . 1m ""' K-. oo nL..J n =l

-

J.l - a . e .

(24)

Let us write hn = h�+h� where h� = hn · X {w: l hni:Sn <> } · Since ll hn ll p � C we get J..L (supp h�) � n-ap so L n J..L (supp h�) < oo. This easily implies

h" !. L ---.! n= l n

00

converges

J.l - a . e .

(25)

From (24) and (25) we infer that

h' lim ""' ---.!!. exists K-. oo nL..J =l n [ K I3 ]

J.l

(26)

a.e.

But for [K.B ] � N � [(K + 1 ) .8 ] we have

I t �I

� ( (K

[K/3 ]

+ 1 ) .8 - K.B ) (K

;; )a.B

� CK a .B - 1 .

The choice of a and (3 with (26) and (27) gives that lim K J.L-a.e. and this together with (25) yields (23) .

(27)

� ( �) exists

n l

a

There is an alternative argument for (23) . From the proof of In(n�l )hn s � C ! n-s ln(n + Proposition 28 we infer that

Remark:

E:=2

li

I E�= N

E N

In(:+ l ) hn converges unconditionally in Lp (O.,

J.L) ,

1 )8 so the series so it unconditionally converges in measure. Corollary III.H.25. shows that (23) holds.

(J.L

Notes and remarks. The Lp )-spaces are among the most important and widely used spaces in analysis. It is probably useless to trace back their first appearance in the literature but already in Banach [1932] they are the prime examples of Banach spaces. Proposition 3 is, as very often in this book, only a sample result. The same holds for spaces w; (M) for s 2: 0, 1 < p < oo , and M a sufficiently regular set in JR. or a differentiable manifold. It also holds for spaces analogously defined by different sets of derivatives. The reader should consult Pelczynski-Senator [1986] for generalizations.

n

III.A

104

Lp-Spac

es ;

Type And Cotype §Notes.

Proposition 5 and Theorem 6 are taken from Pelczynski [1960] . Much work has been done on complemented subspaces of 1] , 1 < p < oo . Many local and global characterizations have been given. It was shown in Bourgain-Rosenthal-Schechtman [1981] that there are uncountably many non-isomorphic such subspaces. We will not discuss this subject in our book. The interested reader should consult the above mentioned paper and references quoted there. For p = 1 the situation is different. All known complemented subspaces of 1] are isomorphic either to £1 or to 1] . It is unknown if this is true for all complemented subspaces of 1] . Theorem 8 is a special case of results proved by Shields-Williams [1971] . Our proof of this result follows the presentation of Forelli-Rudin [1976] where these results are extended to the unit ball in ccn . The same approach is given in Axler [1988] . Proposition 9 is well known. Parts (a) and (b) are almost obvious and (c) is usually called Schur's lemma. Actually Schur proved only a very special case of it and the result evolved gradually. Theorem 1 1 is taken from Lindenstrauss-Pelczynski [1971] . It is also true that is isomorphic to for 0 < p < 1 (see Kalton- Trautman [1982] ) but the proof has to be different since Theorem 8 is clearly false for p < 1 . Those results lead naturally to the following problem: Find a system of functions which is a basis in equivalent to the unit vector basis in £P " Wojtaszczyk [1984] has shown that for p :S: 1 analytic versions of spline systems analogous to the Franklin system have this property. In particular the Bockariov basis for A constructed in III.E. 16 and 17 is also a basis in � 1. In the case p > 1 different bases have been constructed in Matelievic-Pavlovic [1984] . Proposition 7 and Corollary 1 6 address special cases of the follow ing question: for what p, q is the space 1] or isomorphic to a subspace of 1] ? Under the name of linear dimension this was stud ied already in Banach [1932] and Banach-Mazur [1933] . Today the full answer is known. It is summarized in the Table 1 . From the results given in this section the interested reader can easily deduce all these facts except the case 2 < p < q < oo, which is due to Kadec-PelczyD.ski [1962] (see Exercises 3 and 4) . The answers are the same if we replace by 1] . This follows from Proposition 2 and the following

Lp[O ,

£1 [0,

L1[0 , LI[O ,

Bp(D)

fp

Bp(D)

Bp(D),

fp-

Lp[O ,

Lq[O ,

lp

fp Lp[O ,

:S:

Then there exists a measure subspace of oo.

Lp (J.t) .

:S:

J.t such that X is isomorphicfp, to1 a

Proposition. Let X be a Banach space finitely representable in

p

105

III.A Lp-Spaces; Type And Cotype §Notes.

Thble 1.

fp\Lq

q=1

1
q=2

2 < q < oo

q = 00

p= 1

Yes

No

No

No

Yes

1
Yes

If p � q yes if p < q no

No

No

Yes

p=2

Yes

Yes

Yes

Yes

Yes

2 < p < oo

No

No

No

If p = q yes p =/: q no

Yes

p = 00

No

No

No

No

Yes

The proof can be found in Lindenstrauss-Pelczynski [1968] ; it is basically a compactness argument. Corollary 1 6 in the context of Banach spaces was observed by Kadec [1958] , but all the probabilistic background as given in Theorem 14 and Proposition 15 was known much earlier. It can be found in Levy [1925] and our proof of Theorem 14 follows Bochner [1937] . We presented the existence of stable laws in detail not only to show the very useful Corollary 1 6 but also because they are important in Banach space theory (e.g. the notion of p-stable type explained in Notes and remarks to III.H) . As remarked already the class of p-stable variables investigated in probability theory contains many more variables than we discuss in this book. Note that our p-stable variables are symmetric (this follows immediately from (7)). The general treatment can be found in many books on probability theory, e.g. Feller [1971] or Lukacs [1970] . The notions of type and cotype were in the air in the early 70's. Type 2 under the name of 'subquadratic Rademacher average' ap peared in Dubinsky-Pelczynski-Rosenthal [1972] and the general notion of type and cotype was introduced by Maurey in the Seminaire Maurey Schwartz 72/73 and Hoffmann-J!Ilrgensen in the Aarhus University 72/73 preprint 'Sums of independent Banach space valued random variables'. It was the French group around L. Schwartz, B. Maurey and G. Pisier who showed the importance of the concept in Banach space theory and

106

III.A

Lp -Spaces;

Type And Cotype §Exercises

in operator theory. We will discuss some important applications of the notions of type and cotype in III.F, III.H and III.I. Let us also note that it is not known in general if a space with the Orlicz property has cotype 2. Our Proposition 24 (which is due to Figiel-Pisier [1974] ) represents all that is known about this problem. We will use it in III.I. Theorem 18 (Kahane's inequality) , which is basic to the theory of type and cotype, can be found in Kahane [1985] . It was first published in the first edition of this book which appeared in 1968. Theorem 25 for the trigonometric system was shown in Carleman [1918] . Actually the following was shown in Kahane-Katznelson-de Leeuw [1977] . Theorem.

Given a sequence of numbers (a n)��-= ' an < oo there exists f E C(1I') such that lf(n) l

L:!:'- oo a; n = 0, ±1, ±2, . . . .

> 0 and > an for

As was shown by Kislyakov [1981] one can even get the above f with uniformly convergent Fourier series. Some generalizations of Theorem 25 are presented in Wojtaszczyk [1988] . A classical but more complicated proof of this theorem and more precise results can be found in Olevskii [1975] , p. 77 and Chapter III sec. 4. It is interesting that our proof of this Theorem is only a small modification of the arguments in Orlicz [1933] . Theorem 27 was proved in Banach-Saks [1930] . In this formulation it clearly fails for L1 and L00 • On the other hand if we consider only weakly convergent sequences the theorem still holds in L1 . This was shown by Szlenk [1965] . Theorem 29 (also for p = 1) was proved by Berkes [1986] . It extends earlier results (without permutations being allowed) of Komlos [1967] and Aldous [1977] . Our proof is a modification of the arguments given in Lyons [1985] . Exercises

1.

1 !, 1

Let { f, },H c Lp (p,) , 1 � p � oo , be a subset such that �g for some g E Lp (f.L ) and all 1 E r. Show that there exists f E Lp (p,) , denoted f = sup{!, such that � f for all 1 E r and if h is such that h =1- f and h � f then there exists 11 E r such that we do not have J,, � h . All the inequalities between functions are understood to be p,-a.e.

hEr

!,

Warning. Note that elements of Lp( J..t ) are really classes of functions equal p,- a.e. , so if r is uncountable we have trouble with pointwise supremum.

107

III.A Lp-Spaces; Type And Ootype §Exercises 2.

3. 4.

Suppose X c Lp(J.L) is a closed subspace such that for some q < p 2:: we have for all E X. Show that 2:: for all s , 0 < s < p and E X.

l xl q cl xl p x

Suppose X C Lp(J.L) , 1 < p < oo and X "' lp - Show that there exists Y c X such that Y "' lp and Y is complemented in Lp (J.L) . 2:: 2. Show that every infinite dimensional subspace Lp [O, 1] contains an infinite dimensional subspace Y such that Y is complemented in Lp [O, 1] and either Y "' lp or Y

Suppose p

X

5. 6. 7.

l x l s Cs l x i P

x

c

"' £2 .

Let P: Lp (J.L) --+ Lp (J.L) , 1 � p � oo , be a projection of norm 1 with dim lm P = n. Show that Im P � £; .

0::::� 1 i�) P "' lp , 1 0. Show that

oo .

oo

8.

A family of projections 'P = {Pi h EJ on a Banach space X is called a boolean algebra of projections if (a) for

j1,j2 E J we have Ph Ph = Ph Pil E

'P ,

(b) if ii > h E J and Ph Ph = 0 then Ph + Ph E 'P.

I lj

A boolean algebra of projections is bounded if sup { Pi : E J} < oo. Assume that X is a subspace of Lp (fl., J.L) , 1 � p < oo. Show that if we are given two commuting, bounded boolean algebras of projections 'P = {Pj };EJ and Q = {Q E s (i.e. we assume Pj Qs = Q8Pj for all s E E J ) , then the boolean algebra of projections generated by 'P U Q is bounded.

S,j

9.

s ls

(/n)::': 1 (z2n l z2n l _ 1 ):'= l (t2n l t2n l ; 1 ):'= l

C Bp (D) , 0 < = Show that the sequence p < oo , n = 1 , 2, . . . is a basic sequence equivalent to the unit vector C Lp [O, 1] , 0 < p < basis in lp . Show also that oo, n = 1 , 2, . . . is equivalent to the unit vector basis in lp -

10. For

f defined on D its Bergman projection is defined as 1 f(w) d (w ) . P( f ) ( z ) = - i (1 1r )2

Show that

10

- zw

v

(a) P is the orthogonal projection from L2 (D, v ) onto B (D) , (b) P is a bounded projection from Lp (D, v ) onto Bp (D) for 1 < p<

oo ,

2

108 11.

III.A Lp-Spaces; Type And Cotype §Exercises

(c) P is not bounded on

L1 (D, v) .

Show that if g is an analytic function on D such that the operator Tg(f) f · g maps Bp(D) into Bq (D) for some p, q, 0 p q then g 0. 12. Let i: W� ('1l'n ) -+ 0::::��/ Lp (r ) ) P be the natural 1embedding, and let P be the orthogonal projection from 0:::: ��1 L 2 (r)) onto i(WJ(r)). Show that P is continuous on ( 2::��11 Lp (r ) ) P2 , 1 p and of weak type (1-1). 13. Show that the operator Ta f(x) J; (x - y)-a f(y)dy , 0 1, is bounded on Lp[O , 1] for 1 :5 p :5 14. Show that the operator Tf(x) x- 1 J; j(t)dt is a bounded opera tor on Lp (O , ) for 1 p 15. Show that every linear operator from fp into lq , 0 q p is compact and also every operator from co into fq , 0 q is compact and that the same is true for T: X -+ Y where X is any subspace of lp and Y any subspace of fq . 16. Let (cpn)� 1 be a complete orthonormal system in L2 [0, 1]. (a) Show that for every U C [0, 1] with l U I > 0 there exists f E £00[0 , 1], supp f C U such that E� 1 l(f, cpn} I P for all 2. (b) Show that there exists an f E C [O , 1] such that f is linear on every interval of [0, 1]\ � (where � is a Cantor set) and E� 1 I (/, IPnW for all p 2. (c) Show that there exists a set U C [0, 1] such that for all p 2. E� 1 I (xu , ct'n} I P (d) Suppose that ('l/ln )�= l is an orthonormal system in L 2 [ 0 , 1] (not necessarily complete) such that infn J I t/In I > 0. Show that for there exists an f E C [O , 1] such that 2:: � 1 I ( ! , 'l/ln } I P all p 2. (e) Find an example of a non-complete orthonormal system for all ('l/ln )� in L [0 , 1] such that E:'= 1 l(f,'¢'n }l f E C [O1, 1] . 2 <

=

=

< oo ,

<

<

< oo

< a <

=

oo .

oo

<

=

< oo .

<

<

<

< oo ,

< oo ,

= oo

p <

= oo

<

<

= oo

= oo

<

<

oo

17. Suppose that X is a Banach space of type p and that Y C X. Show

that the quotient space X/ Y has type p.

III.A Lp-Spaces; Type And Cotype §Exercises

109

18. Suppose that (Xn );;c'= 1 is a sequence of Banach spaces such that Tp(Xn) � C for n = 1 , 2, . . . and some p, 1 � p � 2. Show that 0:::: � 1 Xn) q has type min {p, If Cp (Xn ) � C for n = 1 , 2, . . . and some p, 2 � p � oo then 0:::: := 1 Xn) q has cotype max {p,

q).

q) .

Un )'::: 1

C

C[O, 1] such that ll fn ll = 1 , fn�O as n --+ oo but for every subsequence Un k ) �1 we have lim sup N-oo N - 1 11 E�=l fn k ll oo > 0.

19. { a) Find a sequence

{ b ) Show that a sequence like in ( a) can be found in a reflexive

space.

III.B. Projection Constants

We discuss projection and extension constants of Banach spaces. The Lewis estimate for the norm of a projection onto a finite dimensional subspace of Lp (JL) is given as well as the Kadec-Snobar estimate for the projection constant of n-dimensional Banach space. We compute the projection constant of the space of trigonometric polynomials (Lozinski Kharshiladze theorem) and apply it to show the divergence of the general interpolating processes in C(1l') and to estimates of degrees of polynomial bases for C(T) . We also show that spaces of n-homogenous polynomials on lBd have projection constants bounded uniformly in n. This is ap plied to some questions in function theory, in particular a construction of a non-constant inner function in lBd is presented. We conclude this chapter with the dual presentation of extension of operators; we intro duce the notion of the projective tensor product and present some basic observations about it.

an

1. In this paragraph we discuss quantitative notions connected with projections and extensions of linear operators. We say that a Banach space X is injective if for every Banach space Y and every subspace Z of Y and every operator T: Z --+ X there exists an extension T: Y --+ X. We define the extension constant e(X) by

e(X) = inf{c: for every Y :::> Z and T: Z --+ X there exists T : Y --+ X : T I Z T and II T II � c ii T II }.

=

Every injective space X has e(X) < oo , because if not then take Yn :::> Zn and Tn : Zn --+ X with II Tn ll = n - 2 such that every Tn : Yn --+ X such that Tn i Zn = Tn has II T il n. Put Y = ( L �1 Yn) 2 :::> Z = ( z= :=l Zn) 2 and define T: Z --+ X by T((zn )) = (Tn (zn )). One sees that T is a continuous operator from Z into X without a continuous extension.

;::::

2. The Hahn-Banach theorem applied coordinatewise gives e(loo ) = 1 . The following generalizes this observation. Theorem. For every measure p, we have e(L00 (p,) )

w = (wj)

= 1.

Proof. Let n be a space on which p, i s a measure. Let r{f!) denote the set of all partitions of n into sets of finite, positive measure.

112

Ill.B. Projection Constants §3.

Note that r(n) is a directed set, if we put w < w if and only if for every E w there exist wi E w such that w; c Wi · Given a pair Y :::> Z and T: Z -+ L00 (p,) we define Tw : Z -+ L00 (p,) by

w;

Tw (z)

=

L j

1 ) 1 T(z)dp,] [(w, w, Jl

·

x

wr

Since the mapping z t-+ p,(w; ) - 1 fwJ. T( z )dp, defines a linear functional on Z of norm not greater than II T II we extend each such functional to Y�; E Y* with IIY�; I � II T II . We define Tw: Y -+ L oo (JL) by Tw(Y) = L; Y�/ Y)Xwr Clearly, for every w E r(n) , Tw is an extension of Tw with II Tw ll � II T II . Let Bt be the closed ball in L00 (JL) with centre 0 and radius t, equipped with the a(L00 (p,) , Lt (JL) )-topology. By II.A.9 it is a compact set. Put B = n y E Y Burn II Y II · Then B is a compact set with the usual product topology. We consider each Tw as an element of B; Tw at the coordinate y has the value Tw (y) . Let T be the limit point of the set {Tw}w Er ( O ) C B. One easily checks that T can be interpreted as a linear operator from Y into L00 (p,) with II T II � II T II · Also, since for every z E Z, Tw (z) -+ T(z) along the directed set r(f!) a in a(L00 (p,) , Lt (JL) )-topology we get T I Z = T. Now we will define some other constants. For a pair (X, Y) con sisting of a Banach space Y and a subspace X of Y we define

3.

=

inf{ c : for every operator T: X -+ Z there exists an extension T: Y -+ Z : II T II � ciiTII }, A(X, Y) = inf{ II P II : P is a projection from Y onto X}. e(X, Y)

For a fixed space X we define e(X) A(X)

=

sup{e(Xt . Y) : X1 is a subspace of Y isometric to X}, = sup{A(Xt . Y) : X1 is a subspace of Y isometric to X}, 'Yoo (X) = inf{ ll aii · II.BII : X � L oo (JL) _!__. X with p, an arbitrary measure and ,Ba = idx }. The constant A(X, Y ) is called the relative projection constant while A(X) is called the projection constant. 4 Lermna.

A(X, Y) .

For every Y and its subspace X we have e(X, Y)

III.B. Projection Constants §5.

1 13

Proof: Since a projection from Y onto X is an extension of id: X

---+ X to an operator from Y into X we see that A( X, Y) :5 e(X, Y). On the other hand every projection P: Y � X gives rise to the extension • T = TP with II T II :5 II P II · II T II . This shows e(X, Y) :5 A(X, Y).

5 Theorem. Let X be a Banach space and let X1 be any subspace of

Leo (p,) isometric to X. Then

Moreover if X is finite dimensional and X1 is a subspace of some C(K) space isometric to X then A( X) = A(Xt , C(K) ) .

Proof: We infer from Lemma 4 that e(X) = A(X) and e(Xt , Leo (P,)) =

A(Xl , Leo (P,)) . Obviously 'Yeo (X) :5 A(Xt , Leo (P,)) :5 A( X). If we have X � Leo � X with f3a = idx and ll a. ll · ll /3 11 :5 'Yeo (X) + c and are given Y :J Z and T: Z ---+ X then Theorem 2 gives an operator 8: Y ---+ Leo with 8 I Z = aT and 11 8 11 = ll a.T II . Since for z E Z we have {38(x) = f3aT(z) = T(z) the operator {38 is the extension of T with ll /38 11 :5 ll a ll · 11/311 II T II , so e(X) :5 'Yeo (X) . If X is a subspace of Y we can extend id: X ---+ X to a projection P: Y ---+ X with II P II :5 e(X), so A(X) :5 ·e(X) . This gives (1). In order to see the 'moreover' part note that from the definition A(Xt , C(K)) :5 A(X). Conversely if P is a projection from C(K) onto X1 then P** : C(K) ** � Xl is a projection of the same norm, so A(Xt , C(K)) ;::: A(Xt , C(K))** ) . Since C(K)** � a Leo (!-L) for some measure p,, (1) gives the claim. 6 Corollary.

A(Y) · d(X, Y) .

For any two Banach spaces Xand Y we have A(X)

:5

From the definitions 'Yeo (X) :5 'Yeo (Y) · d(X, Y) so the claim follows from (1). a

Proof:

7. Before we proceed to more concrete questions, we want to address the general question how big A(X, Lp) , 1 :5 p :5 oo, can be for n dimensional spaces X. The answer is given in Theorem 10 below. We start with Proposition. Let X be an n-dimensional subspace of the space Lp(p, ) , 1 < p < oo . Then there exist a basis (hj )J=I c X and a system of functions (gj ) J=I C Lp• (p,) such that

III. B.

114 (a) (b)

� I ( L:7= 1 I hji 2 ) 1 P � n t , � I ( L:;= 1 193 1 2 ) 1 p, � n t ,

Projection Constants § 7.

1

H = ( L:7= 1 l hj l 2 ) 2 then g3 = hj · HP- 2 (we take § = 0) . Proof: The basis (h3 )' 'J= 1 is found as a solution of the following ex tremal problem: given ¢1 , . . . ,
1

1

1

X

1

:2::

Since both sides are polynomials in equal for 0 we have

t=

1

t of degree n and both sides are

III.B.

Projection Constants §8.

1 15

that is 1

so 1 1 1 '¢ 1 1 1 � nPT . Now we use the Hahn-Banach theorem and extend '¢ 1 to an element (g t , . . . , 9n) E ELp' with I l l (91 , . . . , 9n) I l i P ' � n P' . Since

n .P ( h, , · · · , h.. ) � �

t, I h; g; dp $ I ( t, l h; l 2 ) ( t, ) l

I Y; I ' l dp

� l l l (hi ) .i=t i i i P · l l l (gi ) .i= t l l l p' = n both inequalities are in fact equalities so the conditions for equality in a the Holder inequality (I.B.3) give (d) . 8 Lemma. With the notation of Proposition 7 we define a Hilbertian norm N(f ) N(f ) = (J l f l 2 wdJL) ! where w = HP- 2 The following inequalities hold: (a) if 1 < < 2, then I I ! I I P � n "P - 2 N(f) and for x E X we have N(x) � ll x ll p; (b) if 2 � < then N( f ) � n 2 - "P I I J II P and for x E X we have as

•

1

p

p

1

1

oo,

1

ll x ii P � N(x) .

Proof: It follows directly from Proposition 7(d) that (hj ) J= l is an orthonormal basis in the norm N(·). For 1 < p < and x = "'£ o:i hi E X we have

2

� ll x ll�

( � lo:i l 2 )

� 2 H2 -p · w l l oo = II

ll x ll� N(x) 2 - p

so N(x) � ll x ll w For 1 ) P.

w dJL 2

l f i P w ! w - ! dJL HPdJL

N (f ) P n ( 2 -2 p)

( 2- ) 2p

=

•

w < =f >< c 2 � p J ) dJL

(2- ) 2p

1 16

III.B. 1

1

Projection Constants §9

so II ! II P ::; n "P - 2 N(f). This proves part (a) . For p :2:: 2 and arbitrary f we have

N(f) 2 :S

(/

= ll f ll; = 11 ! 11 ;

ll x ll:

�

I lx l'

I t.

= N(x) P - 2 so ll x i i P

:S

o; h;

)

"P2 l f i P dJL

(/

( J HPdJL)

)

(p- )

P2 w � dJL -

(p - 2 )

P

n (p ; 2J

r-'d� I l x l ' ( L I a; I ') '" ;" w-'d� ,;

J lx l 2 wdJL = N(x)P

N(x). This gives (b) .

a

Let X be- an n-dimensional subspace of Lp(JL) , 1 ::; p ::; Then d(X,£�) ::; nl! tl. In particular for any n-dimensional space X we have d(X, £�) ::; y'n. 9 Corollary.

oo .

For 1 < p < oo the corollary follows directly from Lemma 8(a) and (b) . Let X be any n-dimensional Banach space. By the Banach-Mazur theorem II.B.4 X is isometric to a subspace X of C[O, 1] . Let Xp denote X with the norm from Lp [O, 1] . One easily sees that d(Xp, X) ---. 1 as p ---+ oo, so Proof:

The estimates given in this corollary are in general best possible; see II.B.8.

Let X be an n-dimensional subspace of Lp(JL) , 1 < p < Then there exists a projection P onto X with II P I I ::; nl!-tl. If Y is any Banach space and X is an n-dimensional subspace of Y, then there exists a projection P: Y �X with IIP II ::; y'n. In particular for any Banach space X we have A(X) ::; v'dim X. 10 Theorem. oo.

1 17

III.B. Projection Constants § 1 1 .

For 1 < p < oo we take as P the orthogonal projection with respect to the Hilbertian norm N(·) considered in Lemma 8. For 2 � p < oo Lemma 8 gives

Proof:

li P/lip � N(Pf) � N(f) � n ! - t 11/ll w Given any Banach space Y we can assume that Y is a subspace of some

L00(p,). Let Xs be the space X with the norm Ls (f..L ) , s � 2. Let Ps be a projection from Ls (f..L ) onto Xs with li Ps II � n ! - � . Let us fix any basis (xj )'J= 1 in X and write Ps( f ) = L:j= 1 cpj(f)xi . If cpj , j = 1, 2, . . . , n, is any cluster point of cpj as s --+ oo in L� , then P(f) = L:j= 1 cpi (f)xi is a projection from L00 (f..L) , and so from Y, onto X with II P II � ..fii . To show that the orthogonal projection P works also for 1 < p < 2

requires some additional duality arguments. The adjoint projection P* : Lp' --+ Lp' is given by P* (f) = L:j= 1 J fhi df..L Yi · This is a projec tion orthogonal in the scalar product (cp, t/J ) = J cp1jjw - 1 df..L , and (gj )'J= 1 is an orthonormal basis with respect to this product. Analogously as in Lemma 8(b) we get

Thus for any f E Lp' we have

li P* /li P' � 1

1

(/

1

)

l/l 2 w - 1 df..L 2 � II/Il P' n !- ; . a

So II P II = II P* II � n 2 - ; . 11 Corollary.

Let X

.A( X, Y) � ..fii + 1.

C

Y be a subspace of codimension n. Then

Let P be a projection from Y* onto X .L with li P II � ..fii . We write P( y* ) = L:j= 1 yj* (y* ) xf with xf E X .L and yj* (xt) = Oj , k · Using the principle of local reflexivity, II.E.14, for span ( yj* )'J= 1 we find ( yj ) 'J= 1 C Y such that xf (yk) = Oj k and such that P defined as P (y) = L:j=1 xf (y)yj is a projection with norm � ..fii + c. Obviously ker P = X so I - P is a projection onto X with II I - P ll � ..fii + 1 + c. Since c was arbitrary we have the claim. a Proof:

,

118

III.B. Projection Constants § 1 2.

Suppose Z and V are finite dimensional subspaces of m, dim V = n. Then

12 Corollary.

X with Z

C

V, dim Z =

A(Z, X) � Jn - m + 1 , A(V, X) A(V, X) + 1 � Jn - m + 1. A(Z, X) + 1

(3) (4)

Clearly A(Z, X) � A(Z, V) · A(V, X) and Corollary 11 gives A(Z, V) � ..jn - m + 1. This yields (3) . Given a projection R from X onto Z and Q a projection from kerR onto kerR n V we put P = R + Q(I - R) . One checks that P is a projection from X onto V and II P II � II R II + II Q II ( II R II + 1). Thus

Proof:

A(V, X) + 1 � (A(Z, X) + 1)(A(kerR n V, kerR) + 1). Since dim(kerR n V)

=

n - m Theorem 10 gives (4) .

a

The following theorem is used quite often for computation of 13. projection constants of concrete spaces. Theorem. Let X be a Banach space and Y a complemented subspace. Let G be a compact group and let g 1-+ Tg be a representation of G into L(X) such that

(a) Tg (x) is a continuous function of g, for every x E X,

(b) Tg (Y) c Y for all g E G. Then for every c > 0 there exists a projection P: X ---+ Y such that TgP Tg-1 = P for all g E G and II P II � (A(Y, X) + c) supg E G 11 Tg ll 2 • Observe first that the Banach-Steinhaus theorem I.A.7 implies supg E G II Tg ll < oo. Let us fix a projection Q from X onto Y such that II Q II � A(Y, X) + c and for x E X define

Proof:

P(x)

=

i TgQTg-1 (x)dg

(5)

where dg denotes the normalized Haar measure on G. Condition (a) ensures that there is no problem with the definition of the integral. For every x E X we see that TgQTg-1 (x) E Y. Also for y E Y we see that TgQTg- 1 (y) = y so P is a projection onto Y. For h E G Th PTh - 1

=

L ThTg QTg- 1 Th - 1 dg L ThgQT( hg ) - 1 dg =

= P.

III.B. Projection Constants §14.

119

Also

II Px ll

:::; L II Tu ll II Q II II T9- 1 II ll x ll dg

a

:::; II Q II ll x ll sup 11 Tu ll 2 -

In many concrete application T9 are isometries and the invariant projec tion is unique. Then the theorem implies that the norm of the invariant projection equals .X(Y, X). 14.

Let us recall that

and that u is the normalized rotation invariant measure on Sd. By Wn (Sd ) we will denote the space of all homogenous polynomials of degree n on ([d . Equipped with the norm II P IIoo = sup < E Sd I P( ( ) I it will be denoted by W:' (Sd ) and equipped with the norm

it will be denoted by W�(Sd ) · Clearly every polynomial p(z1 , . . . , Zd ) can be uniquely written as p = L:Z =o Pk with Pk E Wk (Sd ) · The compact group of all unitary operators on ([d is denoted by U(d) . It acts on C(Sd ) by the formula U(d) : g 1--+ T9 where T9 ( ! ) = f o g. Obviously T9 is an isometry on C(Sd) · 15 Theorem. Proof:

defined by

+ n)r( l + j) .X(Wn00 (Sd)) = r(d r( l + n) r(d+ 2) "

Let us consider the operator Pd,n from C(Sd) into W:' (Sd ) Pd , n ( /) ( ( ) =

(d - 1 + n) ! { f(z) ( ( , z) n du(z). (d l ) !n! Jsd _

The binomial expansion of the Cauchy kernel 1

_

( 1 - (( , z))d -

'"""'

n f='o (d - 1) !n! (( , z) (d - 1 + n) !

'

120

III.B. Projection Constants § 1 6.

together with the Cauchy formula I.B.21 and the fact that homogenous polynomials of different degrees are orthogonal, show that Pd,n is actu ally a projection onto W:O (§d) · The operator Pd,n commutes with the action T9 of the unitary group U(d) and Theorem 12.3.8 of Rudin [1980] , i.e. the fact that U(d) acts irreducibly on Wn ( §d ), shows that it is the unique such projection. Theorems 5 and 13 give

In order to estimate this integral we use the formula (see Rudin [1980] 1.4.5)

which reduces it to

Passing to polar coordinates and substituting l z l 2 = r we get 1 (d - 1) J0 (1 - r) d - 2 r� dr. Known values of the Euler integrals of the first kind yield

�

r(d)f(1 + � ) l (z, () l n da( ( ) = f(d + ) sd 2

.

(7)

Substituting (7) into (6) we get the desired estimate.

a

!!

The integrals computed in (7) are clearly well known. An alternative argument can be found e.g. in Garling [1970] . Remark:

We want to note two special cases of Theorem 15. For n = 1 one easily checks that Wl' ( §d ) � cg. So using the properties of the r function we get 16 Corollary.

..fii � A(£�) = f(1 .5) n::!>>

An easy computation also yields

�

h/1r..fii .

121

III.B. Projection Constants §18.

18. Now we will present some applications to function theory on the unit ball of ([d . Our main goal is to present the construction of a non constant inner function in IBd . A function f(z) E H00 1Bd ) is called inner if 1/ (z) l = a-a.e. on Sd {by f(z) for z E Sd we mean the radial boundary value; see I.B.21). Clearly inner functions are characterized by = ll/ ll2 = 11/lloo so the following proposition can be considered as a 'first approximation'.

{

1

1

n 1, 2, . . . , such that Proposition. =

For every

d 1 there exist polynomials Pn E Wn (Sd ), �

{8)

Proof: Let I denote the identity map from W:' (Sd ) into w;(Sd ) · We have to show II l ii � - d .,;rr . From Corollary we get

2

6

{9)

A (W�(Sd)) :5 A (W:' (Sd )) · d(W:' (Sd ), W�(Sd )) :5 A (W:' (Sd )) · II I II · II I - 1 11 · The space w; (Sd) is obviously a Hilbert space. Since Pd,n is an orthog onal projection from L 2 (Sd , a) onto w;(Sd ) the dimension of w; (Sd ) equals the square of the Hilbert-Schmidt norm of Pd,n {III.G.12 and so { 2n d1. m Wn2 (Sd) = Jsd l (z, () l da( ( ) .

{7)

[ {d --1 +! n)! ! J2 {d 1) n

13)

From we infer that dim w;(sd ) = <�-=._11�,:r . The same conclusion can also be reached counting monomials. Thus Corollaries and applied to give

{9)

p p{1, 0, . . . , 0) U(d) {1, 0, . . . , 0). 0, . . . , 0) c

16

17

1

{10)

In order to compute II I - 1 11 take E Wn (Sd ) such that II P II 2 = and = II P IIoo · Let us aver II P IIoo = II I - 1 11 · We can assume age over the subgroup of fixing For the averaged 1 polynomial Po we have Po(1, = II I - 11 and II Po ll2 :5 II P II 2 , so Po(zb . . . , Zd ) = czf. The II Po ll = II P II 2 = But one easily sees that 1 2 normalization gives = ( fsd l z1 l da(z)r2 . Since = II I - 1 11 , using and substituting into we get the claim. a

p

1.

c

(10)

{7)

19. Our next result is a technical device to construct the inner function.

122

III.B. Projection Constants § 1 9.

Proposition. For every cp E C (Sd), cp > 0 there exists a natural num ber N = N ( cp) such that for every n > N there exists a polynomial R = R
(a) I R I < cp on §d, (b) fsd I R I 2 da > 4 - d fsd cp 2 da,

(c) R = '2:�!: Rk where Rk E Wk (§d) ·

It is enough to show the Proposition for cp polynomial in Z I , . . . , Zd and .Z1 . , Zd since such polynomials are dense in C(Sd) · For arbitrary f E C(Sd) we have

Proof:

•

•

.

Applying this for f = Pn , where the Pn satisfy (8) and using the fact that U(d) is connected we infer that there are polynomials Pn = Pn o 9n E Wn (Sd ), n = 1 , 2, . . . such that

II Pn ll oo = 1 , 11Pn ll 2 � 2 - d ..;:rr and l cp( ( )Pn ( ( ) l 2 da( ( ) � 4 - d 1r [sd cp2 da. [ s d l l

(11)

The desired polynomial R will be equal to �C(cp · Pn ) where C is the Cauchy projection (I.B.21) and n is big enough. Let a, (3, -y, 6 denote d-tuples of non-negative integers. Then

One sees that infer that if

fs4 z.8+6 zr+a da( z)

=

0 if (3 + 6 =f.

'Y

+

a.

From this we

k - K�I aai �k + K This shows that for n big enough �C(cp · pn ) satisfies (c) with N(cp) = 2K. That � C(cp Pn) for large n also satisfies (a) and (b) follows from ·

limoo II'PPn - C(cppn ) l ! c(S4) n-+

=

0.

(12)

123

III.B. Projection Constants §20.

This in turn is an immediate consequence of the fact that Pn tends weakly to zero in Hp (md), 1 :::; p < oo and the following lemma which will be also used later. 20 Lemma. Let cp E C(Sd ) satisfy the Lipschitz condition. Then Tcp ( / ) = cp · f - C(cpf) is a compact operator from Hq (md ) into C(Sd ) for q > 2d. Proof:

Using the Cauchy formula we can write

cp(z) - cp(() ( )da( ) = [ rz ( ) ( )da( ) . Tcp ( / ) (z) = J[sd (1 ( (/( ( - (z, () ) d / ( Jsd

Since l z - (I :::; C.J I 1 - (z, () I for ( E sd and z E md u sd, we have

so l rz ( ( ) l q is a uniformly integrable family of functions for q < ( 2J� 1 ) (see III.C. l l for definitions) . Since for (zn );:."= 1 c Sd u md such that Zn ----+ ZO We have rZn ----+ rzo pointwise 0"-a.e. , the Uniform integrability and the Egorov theorem imply that z 1---+ rz is a continuous map from Sd U md into Lq , for q < ( 2J� 1 ) . This yields the compactness of T"'. D 21.

Now we are ready to prove There exists a non-constant inner function in

Theorem.

1, 2, 3, . . .

Proof:

md, d

=

Inductively using Proposition 19 we construct a sequence

(Rn );:_"= 1 of polynomials such that (a) Rn (o) = 0, n = 1, 2, . . . , (b) fsd Rn R k da = 0 if n =f. k, (c) I Rn + l l < 1 - I E 7= 1 Ri l on sd, (d) fsd I Rn +I I 2 da > 4 - d fsJ 1 - I E 7= 1 Ri l ) 2 da. The series E :'= t Rn converges in H2 (md) since it is an orthogonal series (see (b) ) and its partial sums are uniformly bounded by 1 (see (c)). This implies that fsd 1 Rn l 2 da --+ 0 n --+ oo so by (d) I E7=l Ri l ----+ 1 a-a.e. , so E:'= t Rn is an inner function. Condition (a) gives that it is as

non-constant.

'

a

124

III.B. Projection Constants §22.

22. Now we will discuss the projection constant of the space T� of trigonometric polynomials of degree at most n equipped with the sup norm. When naturally embedded into C(1l') , T� is a rotation invariant subspace and the unique rotation invariant projection is the Dirichlet projection Vn . Its norm equals (I.B. 15) 47r- 2 log(n + 1) + o(1). Using Theorem 13 we obtain Theorem. (Lozinski-Kharshiladze) .

o(1).

.X(T�)

(�) log(n + 1)

a

+

23. This theorem implies some classical results about the divergence of trigonometric interpolation. For example we have Theorem. For every system of points x0, xf, . . . , x2n E 11', n = 1, 2, . . . there exists a function f E C(1l') such that the sequence of interpolat ing polynomials In ( / ) , i.e. polynomials in T� such that In ( / ) (xj ) = f(xj ), j = 0, . . . , 2n, is not uniformly bounded.

Observe that In is a projection from C(1l') onto T�, so II In il � .X(T�). Theorem 22 and the uniform boundedness principle (see I.A.7) give the claim. a Proof:

24.

We also have

Suppose ( /n )':= 1 is a Schauder basis for C(1l') such that with k(n) an increasing sequence of integers. Then k(n) � I + c log2 n for some c > 0.

Theorem.

fn E Tk(n)

Proof: Let Fn = span ( /k ) k= l · Since ( /n ) ';'= 1 is a basis for 0(11') , II.B.6 and Theorem 5 show that .X(Fn ) :5 C for some constant C, n = 1, 2, . . . . Using (4) we obtain �--:---:---

.X(Tk(n) ' C(1l') ) + 1 2k(n) + 1 - n + 1. .X(Fn , C(1l') ) + 1 :5 vf Thus Theorem 22 yields log n :5 log 2k(n) + 1 :5 C J2k(n) + 1 - n so k(n) �

i + c log2 n.

a

We would like to conclude this chapter with a more abstract 25. treatment of the extension of operators. This is connected with the

125

III.B. Projection Constants §26.

concept of tensor product. Actually we will consider only one tensor product. We assume that the reader is familiar with the algebraic notion of tensor product. Definition. If X and Y are Banach spaces then their projective tensor product X ® Y is a completion of the algebraic tensor product X ® Y with respect to the norm

l � xi ® Yi l N

J =l

II

= inf

{ I) xj ii iiYj ll : � xi ® Yi = � xj ® yj } . (13) N

3

3

J =l

26 Lemma. The dual space of X ® Y can be isometrically identified with L(X, Y* ) . Proof: The duality which we use is basically the trace duality ( compare with III.F. 16) . Explicitly it is given as

j

j

for T: X --+ Y* and I:j Xj ® Yi E X ® Y. One checks that this definition is independent of the representation of the tensor. Since

I (T, � xi ® Yi ) l � L I T(xi )(Yi ) l � II T II · � ll xi ii · IIYi ll 3

3

3

we infer from (13) that every T E L(X, Y* ) induces a linear functional on X ® Y of norm � II T II . Considering the elementary tensors we get

II T II = sup{ I (Tx)( y) l : IIYII = ll x ll = 1} = sup{ I (T, x ® y) l : ll x ii · IIYII = 1 } � sup (T, � xi ® Yj) : � Xj ® Yj

{I

3

l �

3

L � 1}.

This shows that each T E L(X, Y* ) gives a functional in (X ® Y)* of norm l iT II so L(X, Y* ) is isometrically a subspace of (X ® Y) * . Now suppose that we are given cp E (X ® Y)*. Note that for a fixed x E X, cpx (Y ) = cp (x ® y ) is a linear functional on Y with norm � ll cp ll - One easily checks that the map x ...... cpx is a linear operator from X into Y* . a This shows that L(X, Y*) = (X®Y ) * .

126

III.B. Projection Constants §27.

27 Corollary. Suppose Y conditions are equivalent:

C

X and Z* are given. The following

(a) every operator T: Y --+ Z* extends to an operator T: X --+ Z* ; (b) the map r: L(X, Z* ) --+ L(Y, Z*) given by r(T)

=

T I Y is onto;

(c) the identity map i: Y ® Z --+ x ® z is an isomorphic embedding. Proof: This is a routine duality (see I.A. 13 and 14) and the observation

that i*

=

a

r.

28. Now we will formulate the proposition which is the dual version of Theorem 2. Before we do so we will define the space £1 (S1, f.£, Y) of Bochner integrable Y-valued functions (Y is a Banach space) . First we consider the step functions of the form f(t) = Ej= 1 YiXA; (t) where Yi E Y, j = 1, . . . , N and (Aj ) f= 1 are disjoint subsets of S1 with f.£(Aj ) < oo for j = 1, 2, . . . , N. For such f we put

11/11

=

I: 11Yi ll f.£(Aj ) = In 11/(t) ll df.£(t) . i= l N

A Y-valued function f(t) is Bochner integrable if there exists a sequence (/n)':'= l of step functions, as above, such that Jn 1 1 /(t) - fn (t) ll df.£(t) --+ 0 as n --+ oo . The space of all Y-valued, Bochner integrable functions (when we iden tify functions equal f.£-a.e.) is denoted by £1 (S1, f.£, Y) . One easily checks that it is a Banach space. This description makes the following proposition easy to believe. Proposition. If (S1, f.£) is a u-finite measure space and Y is a Banach space then

We define a map cp: Y ® £1 (f.£) --+ £1 (S1, f.£, Y) by the formula cp( Ef= l Yi ® /i ) = L:f= 1 /j (t) · Yi · One notes that this map is well defined and

Proof:

127

III.B. Projection Constants §Notes.

so ll

( � Yi ® XA; ) N

=

f and II I II =

� N

This shows that

I� N

IIYi ii iiXA; I �

Yi ® X A;

t·

a

Notes and remarks.

The question when a linear operator can be extended from a subspace to the whole space is quite natural and important. It was asked in Banach [1932] , remark to Chapter IV, and the first example showing that it is not always possible was given by Banach and Mazur [1933] in the form of an uncomplemented subspace of C[O , 1] . Thus the connection between extensions of operators and projections was noted very early. Theorem 2 is due to Kantorovic [1935] with the proof that directly mimics the proof of the Hahn-Banach theorem. Some extensions in terms of Banach lattice theory were given by Akilov [1947] and the following theorem was proved by Kelley [1952] building on earlier work of Nachbin [1950] and Goodner [1950] . Theorem. e(X) = 1 if and only if X is isometric to C(K) for some extremaly disconnected compact space K.

The notion of the projection constant and its properties summa rized in Theorem 5 evolved from those works in the 50's (except for the constant 'Yoo which is the child of the theory of operator ideals and appeared later) . The beautiful estimate >.(X) :5 v'dim X was proved by Kadec Snobar [1971] . The original proof used the important result of John [1948] , a special case of which is our Corollary 9 for p = oo . Theorem 10 and Corollary 9 were proved by Lewis [1978] . Our presentation fol lows Lorentz-Tomczak-Jaegerman [1984] . Corollary 1 1 was observed by Garling-Gordon [1971] . The Kadec-Snobar theorem stimulated further research in the local theory of Banach spaces, which is summarized in Tomczak-Jaegerman [1989] . The Kadec-Snobar theorem is also quite useful in various questions of approximation theory. It may be of some interest that there exist finite dimensional spaces Yn , dim Yn = n such that >.(Yn) � .fii - Jn · Also an improvement in the Kadec-Snobar estimate is possible. There exists > 0 such that for any finite dimen sional Banach space X we have >. (X) :5 .fii - Jn · All this is beyond

c

128

III.B. Projection Constants §Notes.

the scope of this book (see Konig [1985] , Konig-Lewis [P] and Tomczak Jaegerman [1989] ) . It should be pointed out that methods used in those more sophisticated studies use extensively various Banach ideal norms, in particular those discussed in III.F. Corollary 12 and Theorem 24 are taken from Kadec [1974] . Theo rem 24 for k(n) = n and Theorem 23 are classical results of Faber [1914] . The estimate from below for k( n) in Theorem 24 was the best known for some time. Only recently Privalov [1987] has shown that k(n) � (1 + e)n (where e > 0 depends on the basis) . On the other hand Bockarev [1985] has constructed a basis Un )':'= ! for C(ll') such that fn E T4 n · Theorem 13 was proved by Rudin [1962] but its main idea, the averaging of projections, was already used in Faber [1914] in the case of the circle group and interpolating projections. Theorem 1 5, Corollary 1 7 and Proposition 1 8 are taken from Ryll-Wojtaszczyk [1983] ; Corollary 16 however is quite old. With basically the same proof the upper bound was given in Griinbaum [1960] and this bound was shown to be exact by Rutowitz [1965] . The non-constant inner function in 1Bd , d � 2 was constructed independently in Aleksandrov [1982] and L0w [1982] . This was an unexpected solution to the long standing problem. The proof presented here is due to Aleksandrov [1984] . Further results stemming from the solution of the inner function problem are discussed in detail in Rudin [1986] and Aleksandrov [1987] . Proposition 18 seems to be quite useful in function theory. Besides many applications presented in Rudin [1986] some are presented in Ryll-Wojtaszczyk [1983] and Wojtaszczyk [1982] . Alexander [1982] used Proposition 1 8 to construct a function f E A(1Bd ) such that / - 1 (0) has infinite (2d - 2)-dimensional volume. Also, the papers Ullrich [1988] and [P] should be consulted for further applications and improvements. Theorem 22 is due to S.M. Lozinski and F .I. Kharshiladze but the first published proof we are aware of is in Natanson [1949] . It is however one in the long line of averaging arguments starting from Faber [1914] or even earlier. The general theory of tensor products of Banach spaces was created in the 1940's by R. Schatten in a series of papers. The presentation of this work is contained in Schatten [1950] which even today seems to be the most complete exposition. Later Grothendieck [1955] and [1956] generalized the concept to more general linear topological spaces and applied it to Banach space theory in the most profound way. What we have presented here is mostly folklore and can be found in the works mentioned earlier.

III.B. Projection Constants §Exercises

129

The reader should be informed however that there are many impor tant norms on X ® Y besides the projective tensor norm. We have only scratched the surface.

Exercises

L00 [0, 1] . Find a one-complemented subspace in A (11'2 ) isometric to T::O . 3. Let Wn = span{1, t, t 2 , , t n } C C[0, 1] . Show that .X(Wn ) > c log n for some positive c. 4. Show the analogue of Theorem 24 for £1 (11') and A(11'). 1 1 5. (a) Show that if 2 $ p $ oo then cnii $ .X(t;) $ nii for some c > 0. (b) Show that lp, 1 < p $ 2, is not isomorphic to a complemented 1. 2.

Show that

£00

is isomorphic to

•

•

•

subspace of any £1 (p,).

.X(£1). 7. Show that, if X is separable and Y C X is isomorphic to co, then Y is complemented in X. 8. (a) Show that ioo /co contains a subspace isometric to co(r) , where r has continuum cardinality. (b) Show that co C £00 is not complemented. (c) Show that if X C £00 and X rv c0 then X is not complemented 6.

Estimate

in i00 •

9.

10.

Suppose that p (z) is a polynomial of degree N on CCd , d 2:: 1 such that < 1. Show that there exists an inner function cp on md such that if cp = E:=o 'Pn with 'Pn a polynomial homogeneous of degree n, then p = E �= l 'Pn· Show also that for every r < 1 and every e > 0 this inner function cp can be constructed in such a way that sup{ l cp(z) - p(z) l : l z l < r} < e. For d > 1 let v c md be the set { (zb . . . ' Xd): l zi l $ d - ! j = ' 1 , . . , d}. Show that there exists an inner function cp on md such that l cp(z) l < � for z E V.

II P IIHoo(IBd)

.

11.

If f(z) is holomorphic on md , then its radial derivative Rf (z) is defined as Rf ( ) = E�= l Show that if the d-th radial d derivative of J , R f , is in H1 (md) then f E A(md) .

z

(zi 8£t> ) .

130

III.B. Projection Constants §Exercises

£2 defined T( f ) = T: A ( JBd ) , where (Pk ) are the polynomials con =l structed in Proposition 18, is onto. 13. A function f(z) holomorphic in ID is called a Bloch function if sup 1/ '(z) l · (1 - l z l 2 ) < oo .

12 .

Show that the operator

( fsd f( ( )P2n ( ( )da( ( ) )

�

---+

as

zEID

( a) Show that L::= l anz 2 n is a Bloch function if and only if ( an ) ;::='= l E ioo . (b) Find a Bloch function which does not have radial limits a. e. on the circle 11'.

14.

For a function analytic in 1Bd and ( E Sd we define a slice function fc; (z) analytic in ID by fc; (z) = f(z · ( ) . We say that a function f analytic in 1Bd is Bloch if

Find a Bloch function in 1Bd which does not have radial limits a-a.e. on Sd.

III. C . L1 (JL)-Spaces This chapter discusses some topics connected with £1 ( JL ) -spaces. We start with the general notion of semi-embedding and investigate semi embeddings of £1 ( JL ) into various Banach spaces. This is applied to the class M0 (11' ) of all measures such that p,(n) --+ 0 as n --+ oo. We prove the classical Menchoff theorem that there are singular such measures and a theorem of Lyons characterising zero sets for the class Mo (ll' ) . Next we describe relatively weakly compact sets in £1 ( Schur's theorem ) and in general £1 ( JL ) -spaces for a probability measure JL ( the Dunford Pettis theorem ) . We also discuss the connection between type and finite representability of £ 1 . Some characterizations of reflexive subspaces of £1 are given. We conclude with some results connected with the classical result of Nevanlinna about cosets F + H00 (11') C £00 (11') . We have already seen that properties of £1 spaces differ from properties of Lp-spaces for 1 < p < oo. In particular, £1 being non reflexive, its unit ball is not weakly compact. Actually more is true.

1.

Proposition. If JL is a non-atomic measure and T: £1 ( JL ) --+ X is a 1-1, weakly compact linear operator then T(BL1 (p.) ) is not norm-closed.

Since BL1 (p.) has no extreme points and T is 1- 1 T(BL 1 (p.) ) also has no extreme points. The Krein-Milman theorem I.A.22 implies that T(BL1 (p.) ) is not weakly compact, thus it cannot be norm-closed.a

Proof:

2.

Let us introduce the following

Definition. A 1-1 linear operator T: embedding if T(Bx ) is closed in Y.

X

--+

Y is called a semi

Clearly every isomorphic embedding is a semi-embedding. Also if X is reflexive then every 1-1 map T: X --+ Y is a semi-embedding. Proposition 1 clearly says that there is no semi-embedding from £ 1 [0 , 1] into any reflexive space.

3. The following is a general, topological observation showing that r-1 I T(Bx ) has at least one point of continuity. We formulate it for Banach spaces only, because this is the way we will use it.

132

III. C.

L 1 (p,)-Spaces §4

Proposition. Let X and Y be separable Banach spaces and let be a semi-embedding. Then there exists an x E X, ll x ll = 1 such that if (x n )�= I C X, ll xn I ::; 1, n = 1, 2, . . . and Tx n ---. Tx then

T: X ---. Y Xn ___. X.

Since T is a semi-embedding, the image of every closed ball in X is closed in Y. Let us fix c: > 0 and a relatively open set V C T(Bx). Let us fix a sequence (vj ) � 1 C Bx such that we can write Bx = U;: 1 B(vj , c:) n Bx . Applying the Baire category theorem to the covering of V by closed sets T ( B ( vj , c:) n Bx) we get the following statement:

Proof:

> 0 and for every relatively open set V C T(Bx) there exists a non-empty relatively open set U C V such (1) 1 that diam T (U) < c:. for every c:

Applying (1) inductively we find a sequence of relatively open sets (Un )�= I in T(Bx) such that Un :J Un + I :J Un +I and diam T - 1 (Un ) < � and 0 � ul . Clearly n�=l T - 1 Un consists of exactly one point Xo . The a desired X equals .

I �� I

4.

Applying Proposition 3 for

X = L 1 [0, 1]

we get

Lemma. If T: L 1 [0, 1] ---. X is a semi-embedding, then there exists an f E L 1 [0, 1] with II f II = 1 and a number 8 > 0 such that for all 1 real-valued functions r.p with J0 r.p(t)dt = 0 and I'PI = 1, p,-a.e. we have

II T( rp f) ll

�

8.

From Proposition 3 we infer that there exists an f E L 1 [0 , 1] , 11 !11 = 1 and 8 > 0 such that II ! - gi l < � whenever IIYII ::; 1 and li T f - Tg ll < 8. Given r.p above we take '1/J = r.p or 'ljJ = -r.p so that we have J(1 + '1/J) I f l dp, ::; 1. For g = (1 + '1/J)f we have IIYII ::; 1 and IIY - ! II = 11 '1/Jf ll = 1, thus 8 < li T/ - Tg ll · But II T( rpf ) ll = li T/ - Tg ll

Proof:

as

so the claim follows.

5.

a

Now we are ready to prove

Theorem. Let p, be an atom-free, separable measure. There is no semi-embedding from L 1 (p,) into Co · We see from III.A. 1 that it is enough to consider LI [O, 1] only. Assume to the contrary that T: £ 1 [0, 1 J ---. eo is a semi-embedding, and

Proof:

III. C.

L 1 (p,)-Spaces §6.

133

II T II = 1. Take f E L 1 [0 , 1] and 8 > 0 as in Lemma 4. Fix an integer N > 2 /8 , and let A� . . . . , A N be disjoint sets such that fA - 1/1 = j;; . 1 For each n , n = 1 , 2, . . . , N let rj denote a sequence of Rademacher-like functions on An , i.e.

l r'J I = X An ' rj = 0,

J

= 1 , . . . , N, j = 1 , 2, . . . , n = 1, . . . , N, j = 1, 2, . . . , n

rj f �O as j ---+ oo ,

for every

n, n

= 1, 2, . . . , N.

(2)

(3) (4)

Using the standard 'gliding hump' argument (cf. proof of II.B. 17) con dition (4) yields numbers j (n) , n = 1 , . . . , N, such that

But (2) , (3) and Lemma 4 give

which is impossible. This contradiction proves the theorem.

a

6. The spirit of the above results is that it is rather difficult to put a weaker, linear topology on the unit ball of L 1 [0, 1] and make it compact or even complete. The intuition is that somehow we always have to add some singular measures in the completion. The following classical theorem of Menchoff is only an example of this. Theorem. (Menchoff) . Let G b e a compact, infinite, metrizable abelian group with dual group r. Let M0 (G) c M(G) denote the set of all measures v with C'(-y) E eo(r). Then Mo is a non-separable closed band in M(G) . Since the Fourier transform A: M(G) ---+ l00 (r) is continous and since Mo = (A - 1 )(c0(r)), we see that M0 is a closed linear subspace. Also for v E Mo and p = :L: ..,. eA a..,. 'Y with finite A c r one easily checks that p · v E Mo. Since Mo is closed this gives that Mo is a band. If M0 is separable, then there exists a positive measure p, E M (G) such that Mo = L1 (p,) . One easily checks that p, has no atoms. By Theorem 5 (B M0 )A is not closed in co ( r} , i.e. there exist o!(-y) E c0(r)\(BM0 )A

Proof:

134

III. C.

and fLn E such that a in of Since 1 11-LII � 1 and contradicts the choice of a.

BMo {!Ln} �=1·

P,n -+

P,

L1(IL)-Spaces § 7.

eo(r).

= a

Let fL be an w*-cluster point we infer that fL E This

BMo·

a

Note that the group structure plays almost no role in the above argument.

7. Given a class of measures it is natural to seek its zero sets, i.e. sets of measure zero with respect to every measure in the class. We will discuss zero sets for M0 (11'). This is a small part of the classical branch of the theory of Fourier series or harmonic analysis on more gen eral groups. In order to proceed we need some definitions. We say that a sequence C 11' has an asymptotic distribution if and only if N 8x n converges a(M(11') , C(T)) to some measure v E M(11'). Let

(xn )�=1 1 N- n=L:1 us recall that 8x denote the Dirac measure concentrated at the point x. 1 Since for every m E we have zl(m) lim N -+oo ( N- L::= 1 8x n ) " (m) 1 lim N -+oo ( N- L::=1 exp ( - im x n ) ) we see that the sequence (x n )�= 1 N 11' has an asymptotic distribution if and only if Nlim k L: exp( - im xn)) ( -+oo =1 n exists for every m E Z. A Borel subset E C 11' is called a Weyl set if there exists an increasing sequence of integers (nk)k:, 1 such that for ev ery x E E the sequence (nk · x)f= 1 has asymptotic distribution with 7l

=

the corresponding measure measure.

8.

Vx

=

C

an

different from the normalized Lebesgue

The following theorem characterizes

M0 (11').

The measure fL is in Mo(11') if an d only if fL(E) every Weyl set E C 11'.

Theorem.

=

0

for

Throughout the whole proof it is sufficient to consider only positive measures (see Theorem 6). =>. Let fL be a non-zero measure in Mo 11' and let E be a Weyl set with the corresponding sequence For m E Z, m =/= 0 we put

Proof:

c

m

(t) =

{

(nk)k:, 1 .

1

lim N-+ oo N

. 0,

()

t k =1 exp ( - imnk t)

,

t E E, t (/_ E .

(5)

III. C.

£1 (p,)-Spaces §9.

For a Borel subset

135

F c E we have (6)

Mo(ll') is a band (Theorem 6) p, I F E Mo(ll') so (6) gives JF Cm (t)dp,(t) = 0. Since F was an arbitrary Borel subset of E we infer that cm (t) = 0, p,-a.e. for m =/= 0. But E is a Weyl set, thus for every t E E there is an m =I= 0 such that em (t) =I= 0. This shows that p,(E) = 0. ¢=: . Let us take p, fJ. M0(11') and let us fix a sequence of integers n k � oo as k - oo (or n k � -oo as k - oo ) such that jL(n k ) � Since

a

=I= 0 as k -

oo.

Applying Theorem III.A.29 to the family of sequences

{exp(-imn k t) }f= 1 in L2 (11', dp,) we get a further subsequence (n�)� 1 such that JJV (t) = N - 1 L�= 1 exp( - imn�t) converges p,-a.e for each m E 7l. Let us put E = {t E 11' : N-+oo lim JJV(t) exists for each m E 7l and lim j].{t) is not zero}. N-+oo

This is a Weyl set. We have