Banach Spaces

Introduction.

Some Notation and Terminology.

1. Banach Spaces.

1.1 Normed Spaces.

1.1.1 General Results.

1.1.2 Some Standard Examples.

1.1.3 Minkowski's Theorem.

1.1.4 Locally Compact Normed Spaces.

1.1.5 Products of Normed Spaces.

1.1.6 Summable Families.

Exercises.

1.2 Operators.

1.2.1 General Results.

1.2.2 Standard Examples.

1.2.3 Infinite Matrices.

1.2.4 Quotient Spaces.

1.2.5 Complemented Subspaces.

1.2.6 The Topology of Pointwise Convergence.

1.2.7 Convex Sets.

1.2.8 The Alaoglu-Bourbaki Theorem.

1.2.9 Bilinear Maps.

Exercises.

1.3 The Hahn-Banach Theorem.

1.3.1 The Banach Theorem.

1.3.2 Examples in Measure Theory.

1.3.3 The Hahn-Banach Theorem.

1.3.4 The Transpose of an Operator.

1.3.5 Polar Sets.

1.3.6 The Bidual.

1.3.7 The Krein-Šmulian Theorem.

1.3.8 Reflexive Spaces.

1.3.9 Completion of Normed Spaces.

1.3.10 Analytic Functions.

Exercises.

1.4 Applications of Baire's Theorem.

1.4.1 The Banach-Steinhaus Theorem.

1.4.2 Open Mapping Principle.

Exercises.

1.5 Banach Categories.

1.5.1 Definitions.

1.5.2 Functors.

1.6 Nuclear Maps.

1.6.1 General Results.

1.6.2 Examples.

1.7 Ordered Banach Spaces.

1.7.1 Ordered Normed Spaces.

1.7.2 Order Continuity.

Name Index. Subject Index. Symbol Index.