Measure and integration theory on infinite-dimensional spaces, Volume 48: Abstract harmonic analysis (Pure and Applied Mathematics)

✍ Scribed by Xia Dao-Xing

• Publisher: Academic Press
• Year: 1972
• Tongue: English
• Leaves: 439
• Category: Library

✦ Synopsis

Text: English, Chinese (translation)

✦ Table of Contents

Measure and Integration Theory on Infinite-Dimensional Spaces
Copyright Page
Contents
Foreword
Preface
Chapter I. Some Supplementary Background in Measure Theory
§1.1. Some Measure-Theoretic Concepts
§1.2. Localizable Measure Spaces
§1.3. The Kolmogorov Theorem
§1.4. Kakutani Distance
Chapter II. Representation of Positive Functionals and Operator Rings
§2.1. Topological Algebras with Involution: Fundamental Concepts
§2.2. Representation of Positive Functionals on Seminormed Algebras
§2.3. Weakly Closed Operator Algebras: Fundamental Concepts
§2.4. Representation of Commutative Weakly Closed Operator Rings
Chapter III. Harmonic Analysis on Groups with Quasi-Invariant Measures
§3.1. Basic Properties of Quasi-Invariant Measures
§3.2. Characters and Quasi-Characters
§3.3. Integral Representation of Positive Definite Functions on Groups
§3.4. L2-Fourier Transforms
Chapter IV. Quasi-Invariant Measures and Harmonic Analysis on Linear Topological Spaces
§4.1. Quasi-Invariant Measures on Linear Topological Spaces
§4.2. Linear and Quasi-Linear Functionals on Linear Spaces
§4.3. Continuous Positive Definite Functions on Linear Topological Spaces
Chapter V. Gaussian Measures
§5.1. Some Properties of Gaussian Measures
§5.2. Equivalence and Perpendicularity of Gaussian Measures
55.3. Gaussian Measures on Linear Spaces
§5.4. Fourier-Gauss Transforms
Chapter VI. Representation of Commutation Relations in Bose-Einstein Fields
§6.1. Representations of the Commutation Relations in Quantum Mechanics
§6.2. Quasi-Invariant Measures Applied to Representations of the Commutation Relations in Bose-Einstein Fields
§6.3. The Relation of Gaussian Measures and Rotationally Invariant Measures to Conventional Free-Field Systems
Appendix I. Background Material on Topological Groups and Linear Topological Spaces
§1.1. Pseudometrics, Convex Functions, and Pseudonorms
§1.2. Some Properties of Semicontinuous Functions
§1.3. Countably Hilbert Spaces and Rigged Hilbert Spaces
Appendix II. Background Material on Functional Analysis in Hilbert Spaces
§11.1. Operators of Hilbert–Schmidt Type, Nuclear Operators, and Equivalence Operators
§11.2. Tensor Products of Hilbert Spaces
§11.3. Unitary Representations of Groups
Notes and References to the Literature
Bibliography
Index