Methods of Modern Mathematical Physics
Functional Analysis
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Methods of Modern Mathematical Physics, Volume I: Functional Analysis discusses the fundamental principles of functional analysis in modern mathematical physics. This book also analyzes the influence of mathematics on physics, such as the Newtonian mechanics used to interpret all physical phenomena. Organized into eight chapters, this volume starts with an overview of the functional analysis in the study of several concrete models. This book then discusses how to generalize the Lebesgue integral to work with functions on the real line and with Borel sets. This text also explores the properties of finite-dimensional vector spaces. Other chapters discuss the normed linear spaces, which have the property of being complete. This monograph further examines the general class of topologized vector spaces and the spaces of distributions that arise in a wide variety of physical problems and functional situations. This book is a valuable resource for mathematicians and physicists. Students and researchers in the field of geometry will also find this book extremely useful.
Table of Contents
Preface
Introduction
Contents of Volumes II and III
I: Preliminaries
1. Sets and Functions
2. Metric and Normed Linear Spaces
Appendix Lim Sup and Lim Inf
3. The Lebesgue Integral
4. Abstract Measure Theory
5. Two Convergence Arguments
6. Equicontinuity
Notes
Problems
II: Hilbert Spaces
7. The Geometry of Hilbert Space
2. The Riesz Lemma
3. Orthonormal Bases
4. Tensor Products of Hilbert Spaces
5. Ergodic Theory: An Introduction
Notes
Problems
III: Banach Spaces
1. Definition and Examples
2. Duals and Double Duals
3. The Hahn-Banach Theorem
4. Operations on Banach Spaces
5. The Baire Category Theorem and Its Consequences
Notes
Problems
Iv: Topological Spaces
1. General Notions
2. Nets and Convergence
3. Compactness
Appendix The Stone-Weierstrass Theorem
4. Measure Theory on Compact Spaces
5. Weak Topologies on Banach Spaces
Appendix Weak and Strong Measurability
Notes
Problems
V: Locally Convex Spaces
1. General Properties
2. Fréchet Spaces
3. Functions of Rapid Decrease and The Tempered Distributions
Appendix The N-Representation for L and L'
4. Inductive Limits: Generalized Functions and Weak Solutions of Partial Differential Equations
5. Fixed Point Theorems
6. Applications of Fixed Point Theorems
7. Topologies on Locally Convex Spaces: Duality Theory and the Strong Dual Topology
Appendix Polars and the Mackey-Arens Theorem
Notes
Problems
Vi: Bounded Operators
1. Topologies on Bounded Operators
2. Adjoints
3. The Spectrum
4. Positive Operators and the Polar Decomposition
5. Compact Operators
6. The Trace Class and Hilbert-Schmidt Ideals
Notes
Problems
Vii: The Spectral Theorem
1. The Continuous Functional Calculus
2. The Spectral Measures
3. Spectral Projections
4. Ergodic Theory Revisited: Koopmanism
Notes
Problems
Viii: Unbounded Operators
1. Domains, Graphs, Adjoints, and Spectrum
2. Symmetric and Self-Adjoint Operators: The Basic Criterion for Self-Adjointness
3. The Spectral Theorem
4. Stone's Theorem
5. Formal Manipulation is a Touchy Business: Nelson's Example
6. Quadratic Forms
7. Convergence of Unbounded Operators
8. The Trotter Product Formula
9. The Polar Decomposition for Closed Operators
10. Tensor Products
11. Three Mathematical Problems in Quantum Mechanics
Notes
Problems
List of Symbols
Index
Product details
- No. of pages: 344
- Language: English
- Copyright: © Academic Press 1972
- Published: January 1, 1972
- Imprint: Academic Press
- eBook ISBN: 9780323155007
About the Author
Michael Reed
Affiliations and Expertise
Training Director, CEYT, UK
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