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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