Functional analysis and infinite-dimensional geometry. (English) Zbl 0981.46001

Fabian, Marián; Habala, Petr; Hájek, Petr; Montesinos Santalucía, Vicente; Pelant, Jan; Zizler, Václav

CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. 8. New York, NY: Springer. ix, 451 p. (2001).

This book is based on graduate courses taught at the University of Alberta in Edmonton. It is intended as an introduction to linear functional analysis and to some parts of infinite-dimensional Banach space theory. It is full of facts, theorems, corollaries; along with a large number of exercises with detailed hints for their solution.
Chapter 1 – Basic Concepts in Banach Spaces; Chapter 2 – Hahn-Banach and Banach Open Mapping Theorems; Chapter 3 – Weak Topologies; and Chapter 7 – Compact Operators on Banach Spaces, is basic material for functional analysis. Not so basic are: Chapter 4 – Locally Convex Spaces; Chapter 5 – Structure of Banach Spaces; and Chapter 6 – Schauder Bases. Most of the material in Chapter 8 – Differentiability of Norms; Chapter 9 – Uniform Convexity; Chapter 10 – Smoothness and Structure; Chapter 11 – Weakly Compactly Generated Spaces; and Chapter 12 – Topics in Weak Topology, has been developed in the past fourty years.
The authors have accomplished a text which is easily readable and as self-contained as possible. A very excellent book for the topics covered!

Reviewer: Joe Howard (Portales/New Mexico)

MSC:

46-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
46B20	Geometry and structure of normed linear spaces
46Bxx	Normed linear spaces and Banach spaces; Banach lattices
47B07	Linear operators defined by compactness properties

Keywords:

open mapping theorems; weak topologies; compact operators; locally convex spaces; structure of Banach spaces; Schauder bases; differentiability of norms; uniform convexity; Hahn-Banach