Book:Michael Reed/Methods of Modern Mathematical Physics I: Functional Analysis/Revised Edition

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Michael Reed and Barry Simon: Methods of Modern Mathematical Physics I: Functional Analysis (Revised Edition)

Published $1981$, Academic Press

ISBN 978-0125850506.


Subject Matter

Functional Analysis

Contents

Preface

Introduction

Contents of Other Volumes

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
1. 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 decease and the tempered distributions
Appendix. The N-representation of $\mathscr S$ and $\mathscr S'$
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. Polar and the Mackey-Arens theorem
Notes
Problems
VI. Bounded Operators
1. Topologies on bounded operators
2. Adjoints
3. The spectrum
4. Positive operators and 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 revisite: 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
The Fourier Transform
1. The Fourier transform on $\map {\mathscr S} {\R^n}$ and $\map {\mathscr S'} {\R^n}$, convolutions
2. The range of the Fourier transform: Classical spaces
3. The range of the Fourier transform: Analyticity
Notes
Problems
Supplementary Material
II.2 Applications of the Riesz lemma
III.1 Basic properties of $L^p$ spaces
IV.3 Proof of Tychonoff's theorem
IV.4 The Riesz-Markov theorem for $X = \sqbrk {0,1}$
IV.5 Minimization of functionals
V.5 Proofs of some theorems in nonlinear functional analysis
VI.5 Applications of compact operators
VIII.7 Monotone convergence for forms
VIII.8 More on the Trotter product formula
Uses of the maximum principle
Notes
Problems

List of Symbols

Index