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  3. .Canadian Mathematical Society Société mathématique du Canada Editors-in-Chief Rédacteurs-en-chef K. Dilcher K. Taylor Advisory Board Comité consultatif G. Bluman P. Borwein R. Kane For other titles published in this series, go to http://www.springer.c

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  1. Canadian Mathematical Society Société mathématique du Canada Editors-in-Chief Rédacteurs-en-chef K. Dilcher K. Taylor Advisory Board Comité consultatif G. Bluman P. Borwein R. Kane For other titles published in this series, go to http://www.springer.com/series/4318
  2. Marián Fabian · Petr Habala · Petr Hájek · Vicente Montesinos · Václav Zizler Banach Space Theory The Basis for Linear and Nonlinear Analysis 123
  3. Marián Fabian Vicente Montesinos Mathematical Institute of the Academy Universidad Politécnica de Valencia of Sciences of the Czech Republic Departamento de Matematica Aplicada Žitná 25, Praha 1 Camino de Vera s/n 11567 Prague, Czech Republic 46022 Valencia, Spain fabian@math.cas.cz vmontesinos@mat.upv.es Petr Habala Václav Zizler Czech Technical University in Prague University of Alberta Department of Mathematics Department of Mathematical Faculty of Electrical Engineering and Statistical Sciences Technická 2 Central Academic Building 16627 Prague, Czech Republic Edmonton T6G 2G1 habala@math.feld.cvut.cz Alberta, Canada zizler@math.cas.cz Petr Hájek Mathematical Institute of the Academy of Sciences of the Czech Republic Žitná 25, Praha 1 11567 Prague, Czech Republic hajek@math.cas.cz Editors-in-Chief Rédacteurs-en-chef Canada K. Dilcher K. Taylor Department of Mathematics and Statistics Dalhousie University Halifax, Nova Scotia B3H 3J5 cbs-editors@cms.math.ca ISSN 1613-5237 ISBN 978-1-4419-7514-0 e-ISBN 978-1-4419-7515-7 DOI 10.1007/978-1-4419-7515-7 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010938895 Mathematics Subject Classicication (2010): Primary: 46Bxx Secondary: 46A03, 46A20, 46A22, 46A25, 46A30, 46A32, 46A50, 46A55, 46B03, 46B04, 46B07, 46B10, 46B15, 46B20, 46B22, 46B25, 46B26, 46B28, 46B45, 46B50, 46B80, 46C05, 46C15, 46G05, 46G12, 47A10, 52A07, 52A21, 52A41, 58C20, 58C25 c Springer Science+Business Media, LLC 2011 All 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, 233 Spring Street, New York, NY 10013, USA), 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 of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
  4. Preface Many problems in modern linear and nonlinear analysis are of infinite-dimensional nature. The theory of Banach spaces provides a suitable framework for the study of these areas, as it blends classical analysis, geometry, topology, and linearity. This in turn makes Banach space theory a wonderful and active research area in Mathematics. In infinite dimensions, neighborhoods of points are not relatively compact, con- tinuous functions usually do not attain their extrema and linear operators are not automatically continuous. By introducing weak topologies, compactness can be obtained via Tychonoff’s theorem. Similarly, functions often need to be perturbed so that the problem of finding extrema is solvable. To deal with problems in linear and nonlinear analysis, a good working knowledge of Banach space theory techniques is needed. It is the purpose of this introductory text to help the reader grasp the basic principles of Banach space theory and nonlinear geometric analysis. The text presents the basic principles and techniques that form the core of the theory. It is organized to help the reader proceed from the elementary part of the subject to more recent developments. This task is not easy. Experience shows that working through a large number of exercises, provided with hints that direct the reader, is one of the most efficient ways to master the subject. Exercises are of several levels of difficulty, ranging from simple exercises to important results or examples. They illustrate delicate points in the theory and introduce the reader to additional lines of research. In this respect, they should be considered an integral part of the text. A list of remarks and open problems ends each chapter, presenting further developments and suggesting research paths. An effort has been made to ensure that the book can serve experts in related fields such as Optimization, Partial Differential Equations, Fixed Point Theory, Real Analysis, Topology, and Applied Mathematics, among others. As prerequisites, basic undergraduate courses in calculus, linear algebra, and general topology, should suffice. The text is divided into 17 chapters. In Chapter 1 we present basic notions in Banach space theory and introduce the classical Banach spaces, in particular sequence and function spaces. v
  5. vi Preface In Chapter 2 we discuss two fundamental principles of Banach space theory, namely the Hahn–Banach Theorem on extension of bounded linear functionals and the Banach Open mapping Theorem, together with some of their applications. In Chapter 3 we discuss weak topologies and their properties related to compact- ness. Then we prove the third fundamental principle, namely the Banach–Steinhaus Uniform Boundedness principle. Special attention is devoted to weak compactness, in particular to the theorems of Eberlein, Šmulyan, Grothendieck and James, and the theory of reflexive Banach spaces. In Chapter 4 we introduce Schauder bases in Banach spaces. The possibility to represent each element of the space as the sequence of its coefficients in a given Schauder basis transfers the purely geometric techniques of the elementary Banach space theory to the analytic computations of the classical analysis. Although not every separable Banach space admits a Schauder basis, the use of basic sequences and Schauder bases with additional properties is one of the main tools in the inves- tigation of the structural properties of Banach spaces. In Chapter 5 we continue the study of the structure of Banach spaces by adding results on extensions of operators, injectivity, and weak injectivity. The core of the chapter is the theory of separable Banach spaces not containing isomorphic copies of 1 . Chapter 6 is an introduction to some basic results in the geometry of finite- dimensional Banach spaces and their connection to the structure of infinite- dimensional spaces. We do not discuss the deeper parts of the theory, which essen- tially depend on measure theoretical techniques. We introduce the notion of finite representability, and prove the principle of local reflexivity. We use the John ellip- soid to prove the Kadec–Snobar theorem and give a proof of Tzafriri’s theorem. We indicate the connection of this result with Dvoretzky’s theorem. Last part of the chapter is devoted to the Grothendieck inequality. In Chapter 7 we present an introduction to nonlinear analysis, namely to varia- tional principles and differentiability. In Chapter 8 we study the interplay between differentiability of norms and the structure of separable Asplund spaces. Chapter 9 introduces the subject of superreflexive spaces, whose structure is nicely described by the behavior of its finite-dimensional subspaces. Chapter 10 studies the impact of the existence of higher order smooth norms on the structure of the underlying space. Special effort is devoted to countable compact spaces and p spaces. Chapter 11 deals with the property of dentability and results on differentiation of vector measures. We prove some basic results on Banach spaces with the Radon– Nikodým property. Chapter 12 introduces the reader to the nonlinear geometric analysis of Banach spaces. Results on uniform and nonuniform homeomorphisms are presented, includ- ing Keller’s theorem and basic fixed points theorems (Brouwer, Schauder, etc). We discuss a proof of the homeomorphisms of Banach spaces and results on uniform, in particular Lipschitz, homeomorphisms.
  6. Preface vii Chapter 13 contains a basic study of an important class of non-separable Banach spaces, the weakly compactly generated spaces. In particular, we discuss their decompositions and renormings. We also study weakly compact operators, abso- lutely summing operators, and the Dunford–Pettis property. Chapter 14 deals with results on weak topologies, focusing on special types of compacta (scattered, Eberlein, Corson, etc.). Chapter 15 presents basic results in the spectral theory of operators. We study compact and self-adjoint operators. Chapter 16 deals with the basic theory of tensor products. We follow the Banach space approach, focusing on the Grothendieck duality theory of tensor products, Schauder bases, applications to spaces of compact operators, etc. We include Enflo’s example of a Banach space without the approximation property. A short appendix (Chapter 17) has been included collecting some very basic definitions and results that are used in the text, for the reader’s immediate access. In writing the text we strived to avoid excessive technicalities, keeping each sub- ject as elementary as reasonably possible. Each chapter ends with a brief section of Remarks and Open Questions, containing further known results and some problems in the area that are—to our best knowledge—open. Several more specialized books and survey articles appeared recently in Banach space theory, as [AlKa], [BeLi], [BoVa], [CasGon], [DJT], [HMVZ], [JoLi3], [Kalt4], [KaKuLP], [LPT], [MOTV2], [Wojt], among others. We hope that the present text can help both the student and the professional mathematician to get acquainted with the techniques needed in these directions. We also made an effort to make this text closer to a reference book in order to help researchers in Banach space theory. We are grateful to many of our colleagues for suggestions, advice, and dis- cussions on the subject of the book. We thank our Institutions: the Institute of Mathematics of the Czech Academy of Sciences, the Czech Technical University in Prague, the Department of Mathematical and Statistical Sciences at the Univer- sity of Alberta, Edmonton, Canada, the Universidad Politécnica de Valencia, Spain, and its Instituto Universitario de Matemática Pura y Aplicada. This work has been supported by several Grant Agencies: The Czech National Grant Agency and the Institutional Research Plan of the Academy of Sciences (Czech Republic), NSERC Canada, the Ministerio de Educación (Spain) and the Generalitat Valenciana (Valen- cia, Spain). The grants involved are IAA 100 190 610, IAA 100 190 901, GACR ˇ 201/07/0394, No. AVOZ 101 905 03 (Czech Republic), Proyecto MTM2008-03211 (Spain), BEST/2009/096 (Generalitat Valenciana) and PR2009-0267 (Ministerio de Educación), NSERC-7926 (Canada). We would like to thank the Springer Team for their interest in this project. In particular, we are thankful to Keith F. Taylor, Karl Dilcher, Mark Spencer, Vaishali Damle, and Charlene C. Cerdas. We thank also Eulalia Noguera for her help with the tex file, and to Integra Software Services Pvt Ltd, in particular Sankara Narayanan, for their assistance in editing the final version of this book. Above all, we are indebted to our families for their moral support and encour- agement.
  7. viii Preface We would be glad if this book inspired some young mathematicians to choose Banach Space Theory and/or Nonlinear Geometric Analysis as their field of interest. We wish the reader a pleasant time spent over this book. Prague, Czech Republic Marián Fabian Prague, Czech Republic Petr Habala Prague, Czech Republic Petr Hájek Valencia, Spain Vicente Montesinos Edmonton, AB, Canada Václav Zizler Spring, 2010
  8. Contents 1 Basic Concepts in Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Hölder and Minkowski Inequalities, Classical Spaces C[0, 1], p , c0 , L p [0, 1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Operators, Quotients, Finite-Dimensional Spaces . . . . . . . . . . . . . . 13 1.4 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.5 Remarks and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Exercises for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2 Hahn–Banach and Banach Open Mapping Theorems . . . . . . . . . . . . . 53 2.1 Hahn–Banach Extension and Separation Theorems . . . . . . . . . . . . 54 2.2 Duals of Classical Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.3 Banach Open Mapping Theorem, Closed Graph Theorem, Dual Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.4 Remarks and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Exercises for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3 Weak Topologies and Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.1 Dual Pairs, Weak Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.2 Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.3 Locally Convex Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.4 Polarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.5 Topologies Compatible with a Dual Pair . . . . . . . . . . . . . . . . . . . . . 100 3.6 Topologies of Subspaces and Quotients . . . . . . . . . . . . . . . . . . . . . . 103 3.7 Weak Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.8 Extreme Points, Krein–Milman Theorem . . . . . . . . . . . . . . . . . . . . . 109 3.9 Representation and Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 3.10 The Space of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 3.11 Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.11.1 Banach–Steinhaus Theorem . . . . . . . . . . . . . . . . . . . . . . . 119 3.11.2 Banach–Dieudonné Theorem . . . . . . . . . . . . . . . . . . . . . . 122 ix
  9. x Contents 3.11.3 The Bidual Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.11.4 The Completion of a Normed Space . . . . . . . . . . . . . . . . 126 3.11.5 Separability and Metrizability . . . . . . . . . . . . . . . . . . . . . 127 3.11.6 Weak Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 3.11.7 Reflexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 3.11.8 Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 3.12 Remarks and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Exercises for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 4 Schauder Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 4.1 Projections and Complementability, Auerbach Bases . . . . . . . . . . . 179 4.2 Basics on Schauder Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 4.3 Shrinking and Boundedly Complete Bases, Perturbation . . . . . . . . 187 4.4 Block Bases, Bessaga–Pełczy´ ski Selection Principle . . . . . . . . . . 194 n 4.5 Unconditional Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 4.6 Bases in Classical Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 4.7 Subspaces of L p Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 4.8 Markushevich Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 4.9 Remarks and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 Exercises for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 5 Structure of Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 5.1 Extension of Operators and Lifting . . . . . . . . . . . . . . . . . . . . . . . . . . 237 5.2 Weak Injectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 5.2.1 Schur Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 5.3 Rosenthal’s 1 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 5.4 Remarks and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 Exercises for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 6 Finite-Dimensional Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 6.1 Finite Representability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 6.2 Spreading Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 6.3 Complemented Subspaces in Spaces with an Unconditional Schauder Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 6.4 The Complemented-Subspace Result . . . . . . . . . . . . . . . . . . . . . . . . 309 6.5 The John Ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 6.6 Kadec–Snobar Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 6.7 Grothendieck’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 6.8 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Exercises for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 7 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 7.2 Subdifferentials: Šmulyan’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 336
  10. Contents xi 7.3 Ekeland Principle and Bishop–Phelps Theorem . . . . . . . . . . . . . . . 351 7.4 Smooth Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 7.5 Norm-Attaining Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 7.6 Michael’s Selection Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 7.7 Remarks and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 Exercises for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 8 C 1 -Smoothness in Separable Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 8.1 Smoothness and Renormings in Separable Spaces . . . . . . . . . . . . . 383 8.2 Equivalence of Separable Asplund Spaces . . . . . . . . . . . . . . . . . . . . 385 8.3 Applications in Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 8.4 Smooth Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 8.5 Ranges of Smooth Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 8.6 Remarks and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 Exercises for Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 9 Superreflexive Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 9.1 Uniform Convexity and Uniform Smoothness, p and L p Spaces . 429 9.2 Finite Representability, Superreflexivity . . . . . . . . . . . . . . . . . . . . . . 435 9.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 9.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 Exercises for Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 10 Higher Order Smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 10.2 Smoothness in p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 10.3 Countable James Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 10.4 Remarks and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 Exercises for Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 11 Dentability and Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 11.1 Dentability in X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 11.2 Dentability in X ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 11.3 The Radon–Nikodým Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 11.4 Extension of Rademacher’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 504 11.5 Remarks and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 Exercises for Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 12 Basics in Nonlinear Geometric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 521 12.1 Contractions and Nonexpansive Mappings . . . . . . . . . . . . . . . . . . . . 521 12.2 Brouwer and Schauder Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 526
  11. xii Contents 12.3 The Homeomorphisms of Convex Compact Sets: Keller’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 12.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 12.3.2 Elliptically Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 535 12.3.3 The Space T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 12.3.4 Compact Elliptically Convex Subsets of 2 . . . . . . . . . . . 538 12.3.5 Keller Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 12.3.6 Applications to Fixed Points . . . . . . . . . . . . . . . . . . . . . . . 541 12.4 Homeomorphisms: Kadec’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 542 12.5 Lipschitz Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 12.6 Remarks and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 Exercises for Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 13 Weakly Compactly Generated Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 13.2 Projectional Resolutions of the Identity . . . . . . . . . . . . . . . . . . . . . . 577 13.3 Consequences of the Existence of a Projectional Resolution . . . . . 581 13.4 Renormings of Weakly Compactly Generated Banach Spaces . . . . 586 13.5 Weakly Compact Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 13.6 Absolutely Summing Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 13.7 The Dunford–Pettis Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596 13.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 13.9 Remarks and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 Exercises for Chapter 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 14 Topics in Weak Topologies on Banach Spaces . . . . . . . . . . . . . . . . . . . . . 617 14.1 Eberlein Compact Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 14.2 Uniform Eberlein Compact Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 622 14.3 Scattered Compact Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 14.4 Weakly Lindelöf Spaces, Property C . . . . . . . . . . . . . . . . . . . . . . . . . 629 14.5 Weak∗ Topology of the Dual Unit Ball . . . . . . . . . . . . . . . . . . . . . . . 634 14.6 Remarks and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 Exercises for Chapter 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 15 Compact Operators on Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 15.1 Compact Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 15.2 Spectral Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 15.3 Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668 15.4 Remarks and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678 Exercises for Chapter 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678 16 Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687 16.1 Tensor Products and Their Topologies . . . . . . . . . . . . . . . . . . . . . . . 687 16.2 Duality of Injective Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . 696
  12. Contents xiii 16.3 Approximation Property and Duality of Spaces of Operators . . . . 700 16.4 The Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708 16.5 Banach Spaces Without the Approximation Property . . . . . . . . . . . 711 16.6 The Bounded Approximation Property . . . . . . . . . . . . . . . . . . . . . . . 717 16.7 Schauder Bases in Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . 721 16.8 Remarks and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726 Exercises for Chapter 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727 17 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 17.1 Basics in Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 17.2 Nets and Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735 17.3 Nets and Filters in Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . 736 17.4 Ultraproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737 17.5 The Order Topology on the Ordinals . . . . . . . . . . . . . . . . . . . . . . . . . 737 17.6 Continuity of Set-Valued Mappings . . . . . . . . . . . . . . . . . . . . . . . . . 738 17.7 The Cantor Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739 17.8 Baire’s Great Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741 17.9 Polish Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741 17.10 Uniform Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741 17.11 Nets and Filters in Uniform Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 742 17.12 Partitions of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743 17.13 Measure and Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744 17.13.1 Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744 17.13.2 Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 17.14 Continued Fractions and the Representation of the Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751 Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781 Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807
  13. Chapter 1 Basic Concepts in Banach Spaces In this chapter we introduce basic notions and concepts in Banach space theory. As a rule we will work with real scalars, only in a few instances, e.g., in spec- tral theory, we will use complex scalars. K denotes simultaneously the real (R) or complex (C) scalar field. We use N for the set {1, 2, . . . }. All topologies are assumed to be Hausdorff, unless stated otherwise. In particular, by a compact space we mean a compact Hausdorff space. By a neighborhood of a point x in a topological space T we mean any subset of T that contains an open subset O of T such that x ∈ O. If (T, T ) is a topological space, and S is a nonempty subset, we shall write T S for the restriction of the topology T to S (and so (S, T ) becomes a topological space). If there is no possibility of misunderstanding, the restricted topology will be called again T . For a brief review on basic topological notions see, e.g., the Appendix. 1.1 Basic Definitions Definition 1.1 A non-negative function · on a vector (i.e., linear) space X is called a norm on X if (i) x ≥ 0 for every x ∈ X , (ii) x = 0 if and only if x = 0, (iii) λx = |λ| x for every x ∈ X and every scalar λ, (iv) x + y ≤ x + y for every x, y ∈ X (the “triangle inequality”). A vector space X with a norm · is denoted by (X, · ), and is called a normed linear space (or just a normed space). Note that the function ρ(x, y) := x − y , where x, y ∈ X , is indeed a metric on X . To check the triangle inequality we write ρ(x, z) = x − z = x − y + y − z ≤ x − y + y − z = ρ(x, y) + ρ(y, z). n n By induction, i=1 x i ≤ i=1 xi for a finite number of vectors x1 , . . . , x n in X . M. Fabian et al., Banach Space Theory, CMS Books in Mathematics, 1 DOI 10.1007/978-1-4419-7515-7_1, C Springer Science+Business Media, LLC 2011
  14. 2 1 Basic Concepts in Banach Spaces All topological and uniform notions in normed spaces refer to the canonical met- ric given by the norm, unless stated otherwise. In situations when more than one normed space is considered, we will sometimes use · X to denote the norm of X . Definition 1.2 A Banach space is a normed linear space (X, · ) that is complete in the canonical metric defined by ρ(x, y) = x − y for x, y ∈ X , i.e., every Cauchy sequence in X for the metric ρ converges to some point in X . Let (X, · ) be a normed space. The set B X := {x ∈ X : x ≤ 1} is said to be the closed unit ball of X , and S X := {x ∈ X : x = 1} the unit sphere of (X, · ). Given x0 ∈ X and r > 0, the set B(x0 , r ) := {x ∈ X : x − x0 ≤ r } is said to be the closed ball centered at x0 with radius r . If M ⊂ X , then span(M) stands for the linear hull—or span—of M, that is, the intersection of all linear subspaces of X containing M. Equivalently, span(M) is the smallest (in the sense of inclusion) linear subspace of X containing M, or the set of all finite linear combinations of elements in M. Similarly, span(M) stands for the closed linear hull of M, i.e., the smallest closed linear subspace of X containing M. If no misunderstanding can arise, by a “subspace” of a vector space we will mean a linear subspace and, in case of normed spaces, a closed linear subspace. Definition 1.3 Let E be a vector space. Given x, y ∈ E, the set [x, y] := {λx + (1 − λ)y : 0 ≤ λ ≤ 1} is called the closed segment defined by x and y. If x = y, the set (x, y) := {λx + (1 − λ)y : 0 < λ < 1} is called the open segment defined by x and y. A subset C of a vector space E is called convex if [x, y] ⊂ C whenever x, y ∈ C. If M ⊂ X , the convex hull of M is the smallest convex subset of X containing M, and will be denoted by conv(M); conv(M) denotes the closed convex hull of M, i.e., the smallest closed convex subset of X containing M. Definition 1.4 Let U be a convex subset of a vector space V . We say that a function f : U → R is convex if f λx+(1−λ)y ≤ λ f (x)+(1−λ) f (y) for all x, y ∈ U and λ ∈ [0, 1]. We say that f is strictly convex if f λx +(1−λ)y < λ f (x)+(1−λ) f (y) for all x, y ∈ U , x = y, and λ ∈ (0, 1). For instance, every norm of a normed space X is a convex function on X . Observe that a function f : U → R is convex if and only if the epigraph of f , i.e., the set epi f := {(x, r ) ∈ U × R : f (x) ≤ r } ⊂ X × R, is convex (the linear structure of X × R is defined coordinatewise). For subsets A, B of a vector space X and a scalar α we also write A + B := {a + b : a ∈ A, b ∈ B} and α A := {αa : a ∈ A}. A set M ⊂ X is called symmetric if (−1)M ⊂ M, and balanced if α M ⊂ M for all α ∈ K, |α| ≤ 1. Let Y be a subspace of a normed space (X, · ). By (Y, · ) we denote Y endowed with the restriction of · to Y if there is no risk of misunderstanding. Fact 1.5 Let Y be a subspace of a Banach space X . Then Y is a Banach space if and only if Y is closed in X .
  15. 1.2 Hölder and Minkowski Inequalities, Classical Spaces 3 Proof: Assume that Y is closed. Consider a Cauchy sequence {yn }∞ in Y . Since n=1 the norm on Y is the restriction of the norm of X , the sequence is Cauchy in X and therefore converges to some y ∈ X . As Y is closed, y ∈ Y and yn → y in Y . The other direction is proved by a similar argument. Definition 1.6 A subset M of a normed space (X, · ) is called bounded if there exists r > 0 such that M ⊂ r B X . M is called totally bounded if for every ε > 0 the set M can be covered by a finite number of translates of ε B X . A sequence {xn } in X is called bounded (totally bounded) if the set {xn : n ∈ N} is bounded (respectively, totally bounded). Note that every totally bounded set is already bounded. See also Exercises 1.47 and 1.48 for a description of total boundedness by using ε-nets, Definition 3.11, and Section 17.10. 1.2 Hölder and Minkowski Inequalities, Classical Spaces C[0, 1], p , c0 , L p [0, 1] We will now turn to some examples of Banach spaces. Definition 1.7 The symbol C[0, 1] denotes the vector space of all scalar valued continuous functions on the interval [0, 1] (the vector addition and the scalar mul- tiplication being defined pointwise), endowed with the norm f ∞ := sup{| f (t)| : t ∈ [0, 1]} (= max{| f (t)| : t ∈ [0, 1]}). Proposition 1.8 The function · ∞ introduced in Definition 1.7 is indeed a norm, and (C[0, 1], · ∞ ) is a Banach space. Proof: We easily check that C[0, 1] is a normed space. Consider a Cauchy sequence { f n }∞ in C[0, 1]. As | f k (t) − fl (t)| ≤ f k − fl ∞ , the sequence { f n (t)}∞ is a n=1 n=1 Cauchy sequence for every t ∈ [0, 1]. Set f (t) := lim f n (t). This defines a scalar n→∞ valued function f on [0, 1]. It remains to show that f is continuous and f n → f uniformly (i.e., in · ∞ ). Given ε > 0, there is n 0 such that | f n (t) − f m (t)| ≤ ε for every t ∈ [0, 1] and every n, m ≥ n 0 . By fixing n ≥ n 0 and letting m → ∞ we get | f n (t) − f (t)| ≤ ε for every n ≥ n 0 and every t ∈ [0, 1]. Let t0 ∈ [0, 1] and ε > 0 be fixed. Choose δ > 0 so that | f n 0 (t) − f n 0 (t0 )| < ε whenever |t − t0 | < δ. Then, whenever |t − t0 | < δ, | f (t) − f (t0 )| ≤ | f (t) − f n 0 (t)| + | f n 0 (t) − f n 0 (t0 )| + | f n 0 (t0 ) − f (t0 )| < 3ε. Therefore f ∈ C[0, 1]. It has been shown above that, for every n ≥ n 0 , fn − f ∞ ≤ ε. This proves that f n − f ∞ → 0, so C[0, 1] is complete. Analogously, the space C(K ) of continuous scalar functions on a compact space K , endowed with the supremum norm, is a Banach space.
  16. 4 1 Basic Concepts in Banach Spaces We note that C[0, 1] is an infinite-dimensional Banach space. To see this, it is enough to produce, for any n ∈ N, a linearly independent set of n elements in C[0, 1]. The set of functions {1, t, t 2 , . . . , t n−1 } has this property. More generally, the space C(K ), where K is a compact topological space, is infinite-dimensional as soon as K is infinite; indeed, given a finite set of distinct points S := {ki : i = 1, 2, . . . , n} in K , define the function δki on S for i = 1, 2, . . . , n, where δk is the Kronecker delta function at k, i.e., δk (k) = 1 and δk (k ) = 0 for all k = k. Extend each δki to a continuous function on K by using the Tietze–Urysohn theorem (see Corollary 7.55). The resulting set of extended functions {δki : i = 1, 2, . . . , n} is linearly independent in C(K ). Definition 1.9 The symbol n denotes the n-dimensional vector space of all n- ∞ tuples of scalars (that is, Rn or Cn ), the vector addition and the scalar multiplication being defined coordinatewise, endowed with the supremum norm · ∞ defined for x = (x1 , . . . , xn ) ∈ n by ∞ x ∞ = max{|xi | : i = 1, . . . , n}. Note that n is a special case of a C(K ) space, where K := {1, . . . , k}, endowed ∞ with the discrete topology. In order to introduce the class of p spaces for 1 < p < ∞ we need to prove the following classical inequalities. Theorem 1.10 (Hölder inequality) Let p, q > 1 be such that 1 p + 1 q = 1 and let n ∈ N. Then for all ak , bk ∈ K, k = 1, . . . , n, we have n n 1 n 1 p q |ak bk | ≤ |ak | p · |bk |q . (1.1) k=1 k=1 k=1 For p = 2, q = 2, the inequality (1.1) is known as the Cauchy–Schwarz inequality. In the proof of Theorem 1.10 we will use the following statement. ap bq Lemma 1.11 Let p, q > 1 be such that 1 p + 1 q = 1. Then ab ≤ p + q for all a, b ≥ 0. Proof: Consider the graph of the function y = x p−1 , x ≥ 0, and the areas A1 of the region bounded by the curves y = x p−1 , y = 0, x = a, and A2 of the region bounded by the curves y = x p−1 , x = 0, y = b (see Fig. 1.1). Clearly, a p b q A1 = 0 x p−1 dx = ap . As x = y 1/( p−1) = y q−1 , we get A2 = 0 y q−1 dy = b . q ap bq It follows that ab ≤ A1 + A2 = p + q . ap q An alternative proof is by checking extrema of the function ϕ(a) := p + bq −ab for a fixed b > 0. Proof of Theorem 1.10: We may assume that ai , bi ≥ 0 and neither all ai nor all bi are zero. For k = 1, . . . , n define
  17. 1.2 Hölder and Minkowski Inequalities, Classical Spaces 5 Y b A2 A1 0 a X Fig. 1.1 Two areas and a rectangle in the proof of Lemma 1.11 n n −1 p −q 1 A k = ak aj p and Bk = bk bjq . j=1 j=1 We note that n Ak p =k=1 n k=1 Bk = 1. By Lemma 1.11, we have for k = q 1, . . . , n that Ak Bk ≤ p Ak 1 p + 1 B q . Summing up this inequality for k = 1, . . . , n q k we get n n n 1 1 1 1 Ak Bk ≤ Ak p + Bk q = + = 1, p q p q k=1 k=1 k=1 which implies the desired inequality. Theorem 1.12 (Minkowski inequality) Let p ∈ [1, ∞) and n ∈ N. Then for all ak , bk ∈ K, k = 1, . . . , n, we have n 1 n 1 n 1 p p p |ak + bk | p ≤ |ak | p + |bk | p . (1.2) k=1 k=1 k=1 Proof: The statement is trivial for p = 1. If p ∈ (1, ∞), let q ∈ (1, ∞) be such that p + q = 1. We may assume that ai , bi ≥ 0. Using the Hölder inequality (1.1) and 1 1 the fact that ( p − 1)q = p we obtain (ak + bk ) p = (ak + bk ) p−1 (ak + bk ) = (ak + bk ) p−1 ak + (ak + bk ) p−1 bk 1 1 1 1 (ak + bk )( p−1)q ak p (ak + bk )( p−1)q bk p q p q p ≤ + 1 1 1 1 (ak + bk ) p ak p (ak + bk ) p bk p q p q p = + . 1 Dividing by (ak + bk ) p q we get
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