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

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

Published $\text {1981}$, 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