Methods of Modern Mathematical Physics

PrefaceIntroductionContents of Volumes II and IIII: 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 ProblemsII: 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 ProblemsIII: 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 ProblemsIv: 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 ProblemsV: 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 ProblemsVi: 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 ProblemsVii: The Spectral Theorem 1. The Continuous Functional Calculus 2. The Spectral Measures 3. Spectral Projections 4. Ergodic Theory Revisited: Koopmanism Notes ProblemsViii: 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 ProblemsList of SymbolsIndex