The second edition of this successful textbook, first published in 1970, retains the aims of the first, namely to provide a truly introductory course in functional analysis, but the opportunity has been taken to add more detail and worked examples. The main changes are complete revisons of the work on convex sets, metric and topological linear spaces, reflexivity and weak convergence. Additional material on the Weiner algebra of absolutely convergent Fourier series and on weak topologies is included. A final chapter includes elementary applications of functional analysis to differential and integral equations.…mehr
The second edition of this successful textbook, first published in 1970, retains the aims of the first, namely to provide a truly introductory course in functional analysis, but the opportunity has been taken to add more detail and worked examples. The main changes are complete revisons of the work on convex sets, metric and topological linear spaces, reflexivity and weak convergence. Additional material on the Weiner algebra of absolutely convergent Fourier series and on weak topologies is included. A final chapter includes elementary applications of functional analysis to differential and integral equations.
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.
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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