Köthe-Bochner Function Spaces

This monograph is devoted to the study of Köthe–Bochner function spaces, an active area of research at the intersection of Banach space theory, harmonic analysis, probability, and operator theory. A number of significant results---many scattered throughout the literature---are distilled and presented here, giving readers a comprehensive view of the subject from its origins in functional analysis to its connections to other disciplines. Considerable background material is provided, and the theory of Köthe–Bochner spaces is rigorously developed, with a particular focus on open problems. Extensive historical information, references, and questions for further study are included; instructive examples and many exercises are incorporated throughout. Both expansive and precise, this book’s unique approach and systematic organization will appeal to advanced graduate students and researchers in functional analysis, probability, operator theory, and related fields. Dedication......Page 2 Title Page......Page 3 Copyright Information......Page 4 Contents......Page 5 Preface......Page 7 Notation......Page 11 1.1 Preliminaries......Page 14 1.2 Basic Sequences......Page 21 1.3 Banach Spaces Containing l1 or c0......Page 42 1.4 James's Theorem......Page 55 1.5 Continuous Function Spaces......Page 61 1.6 The Dunford-Pettis Property......Page 70 1.7 The Pełczynski Property (V*)......Page 82 1.8 Tensor Products of Banach Spaces......Page 87 1.9 Conditional Expectation and Martingales......Page 94 1.10 Notes and Remarks......Page 107 1.11 References......Page 109 2.1 Strict Convexity and Uniform Convexity......Page 114 2.2 Smoothness......Page 137 2 3 Banach-Saks Property......Page 143 2.4 Notes and Remarks......Page 150 2.5 References......Page 152 3.1 Köthe Function Spaces......Page 156 3.2 Strongly and Scalarly Measurable Functions......Page 175 3.3 Vector Measure......Page 180 3.4 Some Basic Results......Page 190 3.5 Dunford-Pettis Operators......Page 201 3 6 The Radon-Nikodým Property......Page 208 3.7 Notes and Remarks......Page 224 3.8 References......Page 229 4.1 Extreme Points and Smooth Points......Page 232 4.2 Strongly Extreme and Denting Points......Page 239 4.3 Strongly and w*-Strongly Exposed Points......Page 246 4.4 Notes and Remarks......Page 255 4.5 References......Page 258 5 Stability Properties II......Page 260 5.1 Copies of c0 in E(X)......Page 261 5.2 The Díaz-Kalton Theorem......Page 270 5.3 Talagrand's L1(X)-Theorem......Page 274 5.4 Property (V*)......Page 291 5.5 The Talagrand Spaces......Page 303 5.6 The Banach-Saks Property......Page 308 5.7 Notes and Remarks......Page 320 5.8 References......Page 322 6.1 Vector-Valued Continuous Functions......Page 326 6.2 The Dieudonné Property in C(K,X)......Page 339 6.3 The Hereditary Dunford-Pettis Property......Page 344 6.4 Projective Tensor Products......Page 361 6.5 Notes and Remarks......Page 368 6.6 References......Page 376 Index......Page 380 This monograph isdevoted to a special area ofBanach space theory-the Kothe­ Bochner function space. Two typical questions in this area are: Question 1. Let E be a Kothe function space and X a Banach space. Does the Kothe-Bochner function space E(X) have the Dunford-Pettis property if both E and X have the same property? If the answer is negative, can we find some extra conditions 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. Does either E or X contain an lp-sequence ifthe Kothe-Bochner function space E(X) has an lp-sequence? To solve the above two questions will not only give us a better understanding 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 the first two chapters we provide some some basic results forthose students 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. This monograph is devoted to the study of Köthe-Bochner function spaces, an area of research at the intersection of Banach space theory, harmonic analysis, probability, and operator theory. A number of significant results--many scattered throughout the literature--are distilled and presented here, giving readers a comprehensive view of Köthe-Bochner function spaces from the subject's origins in functional analysis to its connections to other disciplines. Key features and topics:* Considerable background material provided, including a compilation of important theorems and concepts in classical functional analysis, as well as a discussion of the Dunford-Pettis Property, tensor products of Banach spaces, relevant geometry, and the basic theory of conditional expectations and martingales * Rigorous treatment of Köthe-Bochner spaces, encompassing convexity, measurability, stability properties, Dunford-Pettis operators, and Talagrand spaces, with a particular emphasis on open problems * Detailed examination of Talagrand's Theorem, Bourgain's Theorem, and the Diaz-Kalton Theorem, the latter extended to arbitrary measure spaces.