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.