Köthe-Bochner Function Spaces (Progress in Mathematics) - Hardcover

Lin, Pei-Kee

Hardcover

ISBN 10: 0817635211 ISBN 13: 9780817635213

Publisher: Birkhäuser, 2003

Synopsis

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.

About this title

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 Kothe-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 Kothe-Bochner function spaces from the subject's origins in functional analysis to its connections to other disciplines. Key features and topics include a 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.It also covers rigorous treatment of Kothe-Bochner spaces, encompassing convexity, measurability, stability properties, Dunford-Pettis operators, and Talagrand spaces, with a particular emphasis on open problems. It presents detailed examination of Talagrand's Theorem, Bourgain's Theorem, and the Diaz-Kalton Theorem, the latter extended to arbitrary measure spaces.

'Notes and remarks' are provided after each chapter, with extensive historical information, references, and questions for further study. The book contains 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.

Edition details

• Publisher: Birkhäuser
• Publication date: 2003
• Language: English
• ISBN 10: 0817635211
• ISBN 13: 9780817635213
• Binding: Hardcover
• Number of pages: 383