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Introduction to Banach spaces and their geometry. 2nd rev. ed. (English) Zbl 0585.46009

North-Holland Mathematics Studies, 68. Notas de Matemática (86). Amsterdam, New York, Oxford: North-Holland. XV, 338 p. hbk: $ 46.25; Dfl. 125.00; pbk: $ 22.25; Dfl. 60.00 (1985).
The present book gives an opportunity for students to become familiar with an elementary theory of Banach spaces, and to reach a level which will allow them to understand more specialized topics and read research papers.
The content of the book. Part 1. Functional analysis (Ch. 1 Baire’s property and its consequences; Ch. 2. Infinite-dimensional Banach spaces; Ch. 3. Reflexive Banach spaces, separable Banach spaces). Part 2. The structure of some common Banach spaces (Ch. 1. Hilbert spaces; Ch. 2. Schauder bases in Banach spaces; Ch. 3. Complemented subspaces in Banach spaces; Ch. 4. The Banach spaces \(\ell_ p\) (1\(\leq p\leq \infty)\) and \(c_ 0\); Ch. 5. Extreme points of compact convex sets and the Banach space C(K); Ch. 6. The Banach spaces \(L_ p(\Omega,{\mathcal A},\mu)\) \((1\leq p<\infty))\). Part 3. Some metric properties in Banach spaces (Ch. 1. Strict convexity and smoothness; Ch. 2. Uniform convexity and uniform smoothness). Part 4. The geometry of super-reflexive Banach spaces (Ch. 1. Finite representability and super-properties of Banach spaces; Ch. 2. Basic sequences in super-reflexive Banach spaces; Ch. 3. Uniformly non- square and J-convex Banach spaces; Ch. 4. Renorming super-reflexive Banach spaces). Part 5. Type and cotype of Banach spaces (Ch. 1. Type and cotype; Ch. 2. Introduction to the local theory of Banach spaces). Appendix. The unconditional constant of the Haar system.
The part 5 and the appendix are added in the present second edition of the book. [The first edition (1982) has been reviewed in Zbl 0491.46014.]
Reviewer: M.Kadets

MSC:

46B20 Geometry and structure of normed linear spaces
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
46B25 Classical Banach spaces in the general theory
46B10 Duality and reflexivity in normed linear and Banach spaces

Citations:

Zbl 0491.46014