Classical Banach Spaces I: Sequence Spaces (Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge)

The appearance of Banach's book [8] in 1932 signified the beginning of a syste­ matic study of normed linear spaces, which have been the subject of continuous research ever since. In the sixties, and especially in the last decade, the research activity in this area grew considerably. As a result, Ban:ach space theory gained very much in depth as well as in scope: Most of its well known classical problems were solved, many interesting new directions were developed, and deep connections between Banach space theory and other areas of mathematics were established. The purpose of this book is to present the main results and current research directions in the geometry of Banach spaces, with an emphasis on the study of the structure of the classical Banach spaces, that is C(K) and Lip.) and related spaces. We did not attempt to write a comprehensive survey of Banach space theory, or even only of the theory of classical Banach spaces, since the amount of interesting results on the subject makes such a survey practically impossible. 1. Schauder Bases a. Existence of Bases and Examples b. Schauder Bases and Duality c. Unconditional Bases d. Examples of Spaces Without an Unconditional Basis e. The Approximation Property f. Biorthogonal Systems g. Schauder Decompositions 2. The Spaces c0 and lp a. Projections in c0 and lp and Characterizations of these Spaces b. Absolutely Summing Operators and Uniqueness of Unconditional Bases c. Fredholm Operators, Strictly Singular Operators and Complemented Subspaces of lp? lr d. Subspaces of c0 and lp and the Approximation Property, Complement-ably Universal Spaces e. Banach Spaces Containing lp or c0. f. Extension and Lifting Properties, Automorphisms of l?c0 and l1 3. Symmetric Bases a. Properties of Symmetric Bases, Examples and Special Block Bases b. Subspaces of Spaces with a Symmetric Basis 4. Orlicz Sequence Spaces a. Subspaces of Orlicz Sequence Spaces which have a Symmetric Basis b. Duality and Complemented Subspaces c. Examples of Orlicz Sequence Spaces d. Modular Sequence Spaces and Subspaces of lp? lr e. Lorentz Sequence Spaces References. Front Matter....Pages i-xiii Schauder Bases....Pages 1-52 The Spaces c 0 and l p ....Pages 53-112 Symmetric Bases....Pages 113-136 Orlicz Sequence Spaces....Pages 137-179 Back Matter....Pages 180-190