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

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