Synopses & Reviews
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. It presents a comprehensive view of Köthe-Bochner function spaces, from the subject's origins in functional analysis to its connections to other disciplines. Key topics and features: • 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 • "Notes and remarks" after each chapter, with extensive historical information, references, and questions for further study • Instructive examples and many exercises 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.
Review
From the reviews: "This book is a nice and useful reference for researchers in functional analysis who wish to have a quite comprehensive survey of geometric properties of Banach spaces of vector-valued functions." ---Mathematical Reviews "This book ... gives in fact an exhaustive and very up-to-date account of several aspects of the general theory (isomorphic) and geometry of Banach spaces. This book is self-contained with an exhaustive list of references at the end of each chapter. Apart from well thought-out exercises at the end of each section, the `Notes and Remarks' section at the end of each chapter contains several open questions with additional comments and references. This book is worth having on the shelves of anyone interested in Banach space theory. I thoroughly enjoyed going through it."(ZENTRALBLATT MATH) "This book, though somewahte restrictively entitled, gives in fact an exhaustive and very up-to-date account of several aspects of the general theory (isomorphic) and geometry of Banach spaces. . . This book is self-contained with an exhaustive list of references at the end of each chapter. Apart from well thoght-out exervises at the end of each section,t he 'Notes and Remarks' section at the end of each chapter contains several open questions with additional somments and references. This book is worth having on the shelves of anyone interested in Banach space theory. I thoroughly enjoyed going through it." ---Zenteralblatt MATH
Synopsis
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.
Table of Contents
Preface * Notation * Classical Theorems * Convexity and Smoothness * Köthe-Bochner Function Spaces * Stability Properties I * Stability Properties II * Continuous Function Spaces * References * Index