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 $\text {1981}$, Academic Press
- ISBN 978-0125850506
Subject Matter
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