Köthe-Bochner Function Spaces

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.

Table of Contents:
1 Classical Theorems.- 1.1 Preliminaries.- 1.2 Basic Sequences.- 1.3 Banach Spaces Containing l1 or c0.- 1.4 James’s Theorem.- 1.5 Continuous Function Spaces.- 1.6 The Dunford-Pettis Property.- 1.7 The Pe?czynski Property (V*).- 1.8 Tensor Products of Banach Spaces.- 1.9 Conditional Expectation and Martingales.- 1.10 Notes and Remarks.- 1.11 References.- 2 Convexity and Smoothness.- 2.1 Strict Convexity and Uniform Convexity.- 2.2 Smoothness.- 2.3 Banach-Saks Property.- 2.4 Notes and Remarks.- 2.5 References.- 3 Köthe-Bochner Function Spaces.- 3.1 Köthe Function Spaces.- 3.2 Strongly and Scalarly Measurable Functions.- 3.3 Vector Measure.- 3.4 Some Basic Results.- 3.5 Dunford-Pettis Operators.- 3.6 The Radon-Nikodým Property.- 3.7 Notes and Remarks.- 3.8 References.- 4 Stability Properties I.- 4.1 Extreme Points and Smooth Points.- 4.2 Strongly Extreme and Denting Points.- 4.3 Strongly and w*-Strongly Exposed Points.- 4.4 Notes and Remarks.- 4.5 References.- 5 Stability Properties II.- 5.1 Copies of c0 in E(X).- 5.2 The Díaz-Kalton Theorem.- 5.3 Talagrand’s L1(X)-Theorem.- 5.4 Property (V*).- 5.5 The Talagrand Spaces.- 5.6 The Banach-Saks Property.- 5.7 Notes and Remarks.- 5.8 References.- 6 Continuous Function Spaces.- 6.1 Vector-Valued Continuous Functions.- 6.2 The Dieudonné Property in C(K, X).- 6.3 The Hereditary Dunford-Pettis Property.- 6.4 Projective Tensor Products.- 6.5 Notes and Remarks.- 6.6 References.