Fundamentals of Functional Analysis

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