Functional Analysis

Preface to the Second EditionFrom the Preface to the First EditionPart I Linear Operators and Functionals Chapter I. Topological and Metric Spaces § 1. General Information on Sets. Ordered Sets § 2. Topological Spaces § 3. Metric Spaces § 4. Completeness and Separability. Sets of the First and Second Categories § 5. Compactness in Metric Spaces § 6. Measure Spaces Chapter II. Vector Spaces § 1. Basic Definitions § 2. Linear Operators and Functional § 3. Convex Sets and Seminorms § 4. The Hahn-Banach Theorem Chapter III. Topological Vector Spaces § 1. General Definitions § 2. Locally Convex Spaces § 3. Duality Chapter IV. Normed Spaces § 1. Basic Definitions and Simplest Properties of Normed Spaces § 2. Auxiliary Inequalities § 3. Normed Spaces of Measurable Functions and Sequences § 4. Other Normed Spaces of Functions § 5. Hilbert Space Chapter V. Linear Operators and Functionals § 1. Spaces of Operators and Dual Spaces § 2. Some Functionals and Operators on Specific Spaces § 3. Linear Functionals and Operators on Hilbert Space § 4. Rings of Operators § 5. The Method of Successive Approximations § 6. The Ring of Operators on a Hilbert Space § 7. The Weak Topology and Reflexive Spaces § 8. Extensions of Linear Operators Chapter VI. The Analytic Representation of Functionals § 1. Integral Representations for Functionals on Spaces of Measurable Functions § 2. The Spaces Lp(T,S,µ) § 3. A General Form for Linear Functionals on the Space C(K) Chapter VII. Sequences of Linear Operators § 1. Basic Theorems § 2. Some Applications to the Theory of Functions Chapter VIII. The Weak Topology in a Banach Space § 1. Weakly Bounded Sets § 2. Eberlein-ShmuTyan Theory § 3. Weak Convergence in Specific Spaces § 4. The Problem of Translocation of Mass and the Normed Space it Generates Chapter IX. Compact and Adjoint Operators § 1. Compact Sets in Normed Spaces § 2. Compact Operators § 3. Adjoint Operators § 4. Compact Self-Adjoint Operators on Hilbert Space § 5. Integral Representations of Self-Adjoint Operators Chapter X. Ordered Normed Spaces § 1. Vector Lattices § 2. Linear Operators and Functionals § 3. Normed Lattices § 4. XB-Spaces § 5. Convex Sets That are Closed with Respect to Convergence in Measure Chapter XI. Integral Operators § 1. Integral Representations of Operators § 2. Operators on Sequence Spaces § 3. Integral Operators on Function Spaces § 4. Sobolev's Embedding TheoremsPart II Functional Equations Chapter XII. The Adjoint Equation § 1. Theorems on Inverse Operators § 2. The Connection between an Equation and its Adjoint Chapter XIII. Functional Equations of the Second Kind § 1. Equations with Compact Kernels § 2. Complex Normed Spaces § 3. The Spectrum § 4. Resolvents § 5. The Fredholm Alternative § 6. Applications to Integral Equations § 7. Invariant Subspaces of Compact Operators. The Approximation Problem Chapter XIV. A General Theory of Approximation Methods § 1. A General Theory for Equations of the Second Kind § 2. Equations Reducible to Equations of the Second Kind § 3. Applications to Infinite Systems of Equations § 4. Applications to Integral Equations § 5. Applications to Ordinary Differential Equations § 6. Applications to Boundary-Value Problems for Equations of Elliptic Type Chapter XV.