To myadvisors W.J. Davis andD.R. Lewis Pei-Kee Lin Function Spaces Kăthe-Bochner Springer-Science+Business Media, LLC Pei-Kee Lin Department of Mathematics University of Memphis Memphis, TN 38152 U.S.A. Library of Congress Cataloging-in-Publication Data Lin, Pei-Kee, 1952- KlIthe-8ochner function spaces I Pei-Kee Lin p. cm. Includes bibliographical references and index. ISBN 978-1-4612-6482-8 ISBN 978-0-8176-8188-3 (eBook) DOI 10.1007/978-0-8176-8188-3 1. Normed linear spaces. 2. 8anach spaces. 3. Vector-valued functions. 4. Operator-valued functions. 1. Title. QA322.2.L52 2003 515'.732-dc22 2003063007 CIP AMS Subject Classifications: Primary: 46820, 46E40; Secondary: 46822, 46828, 46842, 46E30, 28805 ISBN 978-1-4612-6482-8 Printed on acid-free paper. ©2004 Springer Science+Business Media New York Originally published by Birkhauser Boston in 2004 Softcover reprint ofthe hardcover Ist edition 2004 Ali rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer-Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication oftrade names, trademarks, service marks and similar terms, even ifthey are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to property rights. 987654321 SPIN 10953037 www.birkhasuer-science.com Contents Preface vii Notation xi 1 Classical Theorems 1 1.1 Preliminaries . . . . . . . . . . . . 1 1.2 Basic Sequences . 8 1.3 Banach Spaces Containing £1 or CO 29 1.4 James's Theorem . 42 1.5 Continuous Function Spaces . . 48 1.6 The Dunford-Pettis Property . 57 1.7 The Pelczynski Property (V") . 69 1.8 Tensor Products ofBanach Spaces 74 1.9 Conditional Expectation and Martingales 81 1.10 Notes and Remarks. 94 1.11 References . . . . . . . . . 96 2 Convexity and Smoothness 101 2.1 Strict Convexity and Uniform Convexity . 101 2.2 Smoothness ... .. . . 124 2.3 Banach-Saks Property 130 2.4 Notes and Remarks. 137 2.5 References .. . ... . 139 3 Kothe-Bochner Function Spaces 143 3.1 Kothe Function Spaces . 143 3.2 Strongly and Scalarly Measurable Functions . 162 3.3 Vector Measure . . . . . . 167 3.4 Some Basic Results . . . . . . . 177 3.5 Dunford-Pettis Operators . . . 188 3.6 The Radon-Nikodym Property 195 3.7 Notes and Remarks. 211 3.8 References . 216 vi CONTENTS 4 Stability Properties I 219 4.1 Extreme Points and Smooth Points. 219 4.2 Strongly Extreme and Denting Points .. 226 4.3 Strongly and w*-Strongly Exposed Points 233 4.4 Notes and Remarks. 242 4.5 References . .. ..... . .... . . . . 245 5 Stability Properties II 247 5.1 Copies ofCO in E(X) . 248 5.2 The Diaz-Kalton Theorem . 257 5.3 Talagrand's L . 261 1(X)-Theorem 5.4 Property (V*) . 278 5.5 The Talagrand Spaces . . . 290 5.6 The Banach-Saks Property 295 5.7 Notes and Remarks . 307 5.8 References .... . .... . 309 6 Continuous Function Spaces 313 6.1 Vector-Valued Continuous Functions 313 6.2 The Dieudonne Property in C(K,X) 326 6.3 The Hereditary Dunford-Pettis Property. 331 6.4 Projective Tensor Products 348 6.5 Notes and Remarks. 355 6.6 References . 363 Index 367 Preface Thismonographisdevotedto aspecial areaofBanachspace theory-theKothe Bochner function space. Two typical questions in this area are: Question 1. Let E be a Kothe function space and X a Banachspace. Does the Kothe-Bochner function space E(X) have the Dunford-Pettis property if both E and X have the same property? Ifthe answer is negative, can we find some extraconditions on E and (or) X such that E(X) has the Dunford-Pettis property? Question 2. Let 1 ~ p ~ 00, E a Kothe function space, and X a Banach space. DoeseitherE orX containan lp-sequence iftheKothe-Bochnerfunction space E(X) has an lp-sequence? Tosolvetheabove two questionswillnot only giveusa betterunderstanding of the structure of the Kothe-Bochner function spaces but it will also develop some useful techniques that can be applied to other fields, such as harmonic analysis, probability theory, and operator theory. Let us outline the contents of the book. In thefirst twochaptersweprovide somesomebasicresultsforthosestudents who do not have any background in Banach space theory. We present proofs of Rosenthal's l1-theorem, James's theorem (when X is separable), Kolmos's theorem, N. Randrianantoanina's theorem that property (V*) is a separably determined property, and Odell-Schlumprecht's theorem that every separable reflexive Banach space has an equivalent 2R norm. In Chapter 3weintroduce Kothe-Bochner function spaces and proveseveral basic results regarding those spaces: • Weprove that Vitali's lemma and Lebesgue's dominated convergence the orem arestill trueifone replaces £1 byanorder-continuousKothefunction space over a finite measure space. • For any order-continuous Kothe function space E and any Banach space X, the Kothe-Bochner function space E(X) is (weakly) uniformly convex if and only if both E and X are (weakly) uniformly convex. • For any order-continuous Kothe function space E and any Banach space X, the Kothe-Bochner function space E(X) has the Radon-Nikodym viii PREFACE property if and only if both E and X have the Radon-Nikodym prop erty. Chapter4isdevotedto thegeometric propertiesoftheKothe-Bochnerfunc tion spaces. One of the fundamental questions regarding the Kothe-Bochner function spaces is the following: Question 3. Let E be a Kothe function space and X a Banach space. Suppose that I is an extreme point of the unit ball of the Kothe-Bochner function space E(X). Is lI~mx an extreme point of the unit ball of X for I? almost all t in support of The answer to Question 3 is dependent on whether there is a measurable selection. It is known that there is a nonseparable Banach space X and a unit vectorI inLp(/-L,X) such thatforallt, lI~mlx isnotanextremepointoftheunit ball of X, but I is an extreme point of Lp(/-L,X). On the other hand, Zhibao Hu and Bor-Luh Lin showed that if X is separable, then such a measurable selection exists. Using this technique, we prove that for any 1 < p < 00 and any unit vector I in Lp(X), if for almost all t in support of I, I(t)/III(t)lIx is a strongly exposed point of X, then there is a linear functional F E (Lp(X))* which strongly exposes the unit ball of Lp(X) at f. In Chapter 5wepresent several deep results on the Kothe-Bochner function spaces: • Bourgain's co-average Theorem. J. Hoffmann-Jorgensen and S. Kwapieri first proved thatforany 1::;p < 00, the Lebesgue-Bochnerfunction space Lp(X) contains a co-sequence if and only if X contains a co-sequence. Later, J. Bourgain gave another proof. But there is an advantage to Bourgain's proof, i.e., it does not require In'sto be strongly measurable. Itisknown that ifE isaseparableorder-continuousKothefunction space, the dual of the Kothe-Bochner function space E(X) is isometrically iso morphic to the space of weak" measurable X*-valued function spaces. By Bourgain's proof and a classical theorem on Banach spaces, for any order-continuous Kothe function space E and any Banach space X, the Kothe-Bochner function space E(X) contains a complemented copy off 1 ifand only ifeither E or X contains a complemented copy of fl• • The Diaz-Kalton Theorem. Ultraproduct is an important tool for Ba nach space theory. Using the ultraproduct technique, C. Stegall showed that there isa Banach space withthe Dunford-Pettis propertywhose dual does not have thesame property. Modifying Stegall'sproof,B.W. Johnson showed that there is a separable Banach space X such that for any other separable Banach space Y, Y* is isomorphic to a complemented subspace of X*. In Section 5.2, we use this idea again to show that foo(X) con tains a complementedf ifand only ifX containsfr's uniformly 1-sequence complemented. PREFACE ix • Talagrand'sL B. Maurey,G.Pisier and J.Bourgain proved 1(X)-theorem. that for any 1::;p< 00 and any uniformly bounded iI-sequence {ik}%"=1 inLp(X), 1<P< 00, there istEn, such that {fnk(t)}~1 containsan i1 subsequence. Later, M. Talagrand proved that for any bounded sequence {fn}~=I' thereisanessentially normalized iI-blocksequence {gn}~=1 such that for almost all tEn, either {gn(t)}~=1 is weakly Cauchy or there is k (depending on w) such that {gn(t)}~=k is an iI-sequence. Using this result, he proved that for any order continuous Kothe function space E, the Kothe-Bochner function space E(X) is weaklysequentially complete ifboth E and X are weakly sequentially complete. • The Bourgain-CembranosTheorem. The application ofthe Ramsey The orem in Banachspaces isan important topic in Banachspace theory. H.P. Rosenthal proved that every bounded sequence in a Banach space either contains a weakly Cauchy subsequence or an iI-subsequence. Recently, W.T. Gower discovered a block Ramsey theorem for Banach spaces. Us ing this technique, he also proved that every Banach space X either con tains a subspace Y with an unconditional basis or contains a hereditarily indecomposable subspace Z; i.e.,every infinite dimensional subspace of Z has no nontrivial complemented subspace. In Section 5.6, we use Ram sey's Theory and give a necessary and sufficient condition such that the Lebesgue-Bochner function space Lp(X), 1 < P < 00, has the Banach Saks property. Chapter 6 is devoted to C(K,X) and X0Y. Some interesting results are proved in this chapter. • The Cembranos-Diestel-Elton-Knaust-Odell Theorem. In Section 6.3, we apply Ramsey's Theorem again and weprove that a Banach space X hasthehereditaryDunford-Pettispropertyifforanyweaklynullsequence {xn}~=1 in X, either {Xn}~=1 is a null sequence or {xn}~=1 contains a co-subsequence. Using the result, we prove that C(K,X) has the hered itary Dunford-Pettis property if and only if both C(K) and X have the hereditary Dunford-Pettis property. • The Bu-Diestel Theorem. One question about Banach spaces is: Question 4. Let X and Y be two Banach spaces with the Radon Nikodym property. Does the projective tensor product X0Y have the Radon-Nikodym property? Ifthe answer is negative, can we find a non trivial condition on X and Y such that X0Y has the Radon-Nikodym property? J. Bourgain and G. Pisier showed that there is a Banach space X with the Radon-Nikodym property such that X0X contains a copy of CO (so X0X does not havethe Radon-Nikodym property). In the second part of Section 6.4, weprove that ifE is a reflexiveKothe function space and X x PREFACE is a Banach space with the Radon-Nikodym property,then the projective tensor product E0X has the Radon-Nikodym property. To help the reader keep track of what is known, we have adopted the no tation (Problem 1) or (Problem 3.2.5) for solved problems and (Question 1) or (Question 3.2.5) for open questions. The author thanks his friends and colleagues who pointed out and corrected some mistakes and typos in the preliminary versions. These include, in par ticular, H. Hudzik, J. Jamison, N.J. Kalton, A. Kaminska, E. Odell, N. Ran drianantoanina, and Huiying Sun. Particularly, he owesvery special thanks to J. Diestel and David Kramer, who read through our manuscript and offered suggestions and corrections (forour mistakes and typos) and M. Talagrandwho explained the idea ofthe extension ofhis £1(X)-theorem sothat a correct proof could be presented. We also like to thank to the referees and the Executive Editor of Birkhauser Boston, Mathematics and Physics, Ann Kostant, for their suggestions and comments. Of course, the author is responsible for any mistakes in this book. Pei-Kee Lin October 24, 2003