Functional Analysis: An Introduction to Banach Space Theory

Description

A powerful introduction to one of the most active areas of theoretical and applied mathematics This distinctive introduction to one of the most far-reaching and beautiful areas of mathematics focuses on Banach spaces as the milieu in which most of the fundamental concepts are presented. While occasionally using the more general topological vector space and locally convex space setting, it emphasizes the development of the reader's mathematical maturity and the ability to both understand and "do" mathematics. In so doing, Functional Analysis provides a strong springboard for further exploration on the wide range of topics the book presents, including: * Weak topologies and applications * Operators on Banach spaces * Bases in Banach spaces * Sequences, series, and geometry in Banach spaces Stressing the general techniques underlying the proofs, Functional Analysis also features many exercises for immediate clarification of points under discussion. This thoughtful, well-organized synthesis of the work of those mathematicians who created the discipline of functional analysis as we know it today also provides a rich source of research topics and reference material. Preface Introduction 1. Basic Definitions and Examples 1.1 Examples of Banach Spaces 1.2 Examples and Calculation of Dual Spaces 2. Basic Principles with Applications 2.1 The Hahn-Banach Theorem 2.2 The Banach-Steinhaus Theorem 2.3 The Open-Mapping and Closed-Graph Theorems 2.4 Applications of the Basic Principles 3. Weak Topologies and Applications 3.1 Convex Sets and Minkowski Functionals 3.2 Dual Systems and Weak Topologies 3.3 Convergence and Compactness in Weak Topologies 3.4 The Krein-Milman Theorem 4. Operators on Banach Spaces 4.1 Preliminary Facts and Linear Projections 4.2 Adjoint Operators 4.3 Weakly Compact Operators 4.4 Compact Operators 4.5 The Riesz-Schauder Theory 4.6 Strictly Singular and Strictly Cosingular Operators 4.7 Reflexivtty and Factoring Weakly Compact Operators 5. Bases in Banach Spaces 5.1 Introductory Concepts 5.2 Bases in Some Special Spaces 5.3 Equivalent Bases and Complemented Subspaces 5.4 Basic Selection Principles 6. Sequences, Series, and a Little Geometry in Banach Spaces 6.1 Phillips' Lemma 6.2 Special Bases and Reflexivity in Banach Spaces 6.3 Unconditionally Converging and Dunford-Pettis Operators 6.4 Support Functionals and Convex Sets 6.5 Convexity and the Differentiability of Norms Bibliography Diestel J. -EFGHJ Johnson W.B -KLMNP Pefczynski A. -RSTVWY Author Index ABCDEFGH IJKLMNOPRS TUVWYZ Subject Index a bc d efgh ijkl mno pqrs tuvw z Symbol Index