Fundamentals of Functional Analysis

Description

Cover Title page Preface to the English Translation Preface to the First Russian Edition Preface to the Second Russian Edition Chapter 1. An Excursion into Set Theory 1.1. Correspondences 1.2. Ordered Sets 1.3. Filters Exercises Chapter 2. Vector Spaces 2.1. Spaces and Subspaces 2.2. Linear Operators 2.3. Equations in Operators Exercises Chapter 3. Convex Analysis 3.1. Sets in Vector Spaces 3.2. Ordered Vector Spaces 3.3. Extension of Positive Functionals and Operators 3.4. Convex Functions and Sublinear Functionals 3.5. The Hahn-Banach Theorem 3.6. The Krein-Milman Theorem 3.7. The Balanced Hahn-Banach Theorem 3.8. The Minkowski Functional and Separation Exercises Chapter 4. An Excursion into Metric Spaces 4.1. The Uniformity and Topology of a Metric Space 4.2. Continuity and Uniform Continuity 4.3. Semicontinuity 4.4. Compactness 4.5. Completeness 4.6. Compactness and Completeness 4.7. Baire Spaces 4.8. The Jordan Curve Theorem and Rough Drafts Exercises Chapter 5. Multinormed and Banach Spaces 5.1. Seminorms and Multinorms 5.2. The Uniformity and Topology of a Multinormed Space 5.3. Comparison Between Topologies 5.4. Metrizable and Normable Spaces 5.5. Banach Spaces 5.6. The Algebra of Bounded Operators Exercises Chapter 6. Hilbert Spaces 6.1. Hermitian Forms and Inner Products 6.2. Orthoprojections 6.3. A Hilbert Basis 6.4. The Adjoint of an Operator 6.5. Hermitian Operators 6.6. Compact Hermitian Operators Exercises Chapter 7. Principles of Banach Spaces 7.1. Banach's Fundamental Principle 7.2. Boundedness Principles 7.3. The Ideal Correspondence Principle 7.4. Open Mapping and Closed Graph Theorems 7.5. The Automatic Continuity Principle 7.6. Prime Principles Exercises Chapter 8. Operators in Banach Spaces 8.1. Holomorphic Functions and Contour Integrals 8.2. The Holomorphic Functional Calculus 8.3. The Approximation Property 8.4. The Riesz-Schauder Theory 8.5. Fredholm Operators Exercises Chapter 9. An Excursion into General Topology 9.1. Pretopologies and Topologies 9.2. Continuity 9.3. Types of Topological Spaces 9.4. Compactness 9.5. Uniform and Multimetric Spaces 9.6. Covers, and Partitions of Unity Exercises Chapter 10. Duality and Its Applications 10.1. Vector Topologies 10.2. Locally Convex Topologies 10.3. Duality Between Vector Spaces 10.4. Topologies Compatible with Duality 10.5. Polars 10.6. Weakly Compact Convex Sets 10.7 . Refl exi ve S p aces 10.8. The Space C(Q,R) 10.9. Radon Measures 10.10. The Spaces D(Ω) and D'(Ω) 10.11. The Fourier Transform of a Distribution Exercises Chapter 11. Banach Algebras 11.1. The Canonical Operator Representation 11.2. The Spectrum of an Element of an Algebra 11.3. The Holomorphic Functional Calculus in Algebras 11.4. Ideals of Commutative Algebras 11.5. Ideals of the Algebra C(Q,C) 11.6. The Gelfand Transform 11.7. The Spectrum of an Element of a C*-Algebra 11.8. The Commutative Gelfand-Naimark Theorem 11.9. Operator *-Representations of a C*-Algebra Exercises References Notation Index Subject Index