An Introduction to Banach Space Theory

1 Basic Concepts.- 1.1 Preliminaries.- 1.2 Norms.- 1.3 First Properties of Normed Spaces.- 1.4 Linear Operators Between Normed Spaces.- 1.5 Baire Category.- 1.6 Three Fundamental Theorems.- 1.7 Quotient Spaces.- 1.8 Direct Sums.- 1.9 The Hahn-Banach Extension Theorems.- 1.10 Dual Spaces.- 1.11 The Second Dual and Reflexivity.- 1.12 Separability.- 1.13 Characterizations of Reflexivity.- 2 The Weak and Weak Topologies.- 2.1 Topology and Nets.- 2.2 Vector Topologies.- 2.3 Metrizable Vector Topologies.- 2.4 Topologies Induced by Families of Functions.- 2.5 The Weak Topology.- 2.6 The Weak Topology.- 2.7 The Bounded Weak Topology.- 2.8 Weak Compactness.- 2.9 James's Weak Compactness Theorem.- 2.10 Extreme Points.- 2.11 Support Points and Subreflexivity.- 3 Linear Operators.- 3.1 Adjoint Operators.- 3.2 Projections and Complemented Subspaces.- 3.3 Banach Algebras and Spectra.- 3.4 Compact Operators.- 3.5 Weakly Compact Operators.- 4 Schauder Bases.- 4.1 First Properties of Schauder Bases.- 4.2 Unconditional Bases.- 4.3 Equivalent Bases.- 4.4 Bases and Duality.- 4.5 James's Space J.- 5 Rotundity and Smoothness.- 5.1 Rotundity.- 5.2 Uniform Rotundity.- 5.3 Generalizations of Uniform Rotundity.- 5.4 Smoothness.- 5.5 Uniform Smoothness.- 5.6 Generalizations of Uniform Smoothness.- A Prerequisites.- B Metric Spaces.- D Ultranets.- References.- List of Symbols.

The purpose of this book is to bridge this gap and provide an introduction to the general theory of Banach spaces and functional analysis. It prepares students for further study of both the classical works and current resarch. It is accessible to students who have had a course in real and complex analysis and understand the basic properties of L_p spaces. The book is sprinkled liberally with examples, historical notes, citations, and original sources. Over 450 exercises provide students with practice in the use of the results developed in the text through supplementary examples and couterexamples.