Elements of Functional Analysis

Table of Contents

• Preface to the second edition
• Preface to the first edition
• Part I. Basic Set Theory and Analysis:
• 1. Sets and functions
• 2. Real and complex numbers
• 3. Sequences of functions, continuity, differentiability
• 4. Inequalities
• Part II. Metric and Topological Spaces:
• 1. Metric and semimetric spaces
• 2. Complete metric spaces
• 3. Some metric and topological concepts
• 4. Continuous functions on metric and topological spaces
• 5. Compact sets
• 6. Category and uniform boundedness
• Part III. Linear and Linear Metric Spaces:
• 1. Linear spaces
• 2. Subspaces, dimensionality, factorspaces, convex sets
• 3. Metric linear spaces, topological linear spaces
• 4. Basis
• Part IV. Normed Linear Spaces:
• 1. Convergence and completeness
• 2. Linear operators and functionals
• 3. The Banach–Steinhaus theorem
• 4. The open mapping and closed graph theorems
• 5. The Hahn–Banach extension
• 6. Weak topology and weak convergence
• Part V. 1. Algebras and Banach algebras
• 2. Homomorphisms and isomorphisms
• 3. The spectrum and the Gelfand–Mazaur theorem
• 4. The Weiner algebra
• Part VI. Hilbert Space:
• 1. Inner product and Hilbert spaces
• 2. Orthonormal sets
• 3. The dual space of a Hilbert space
• 4. Symmetric and compact operators
• Part VII. Applications:
• 1. Differential and integral problems
• 2. The Sturm–Liouville problem
• 3. Matrix transformations in sequence spaces
• Appendix
• Bibliography
• Index.
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