Indeholder "Preface", "Introduction", "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", " Appendix. Weak and strong measurability", " Notes", " Problems", "V. 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", " Problems", "VI. 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", " Problems", "VII. The Spectral Theorem", " 1. The continuous functional calculus", " 2. The spectral measures", " 3. Spectral projections", " 4. Ergodic theory revisited: Koopmanism", " Notes", " Problems", "VIII. Unbounded Operators", " 1. Domains, graphs, adjoints, and spectrum", " 2. Symmetry 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 om L(R^n) and L'(R^n), convolutions", " 2. The range of tFourier transform: Classical spaces", " 3. The range of tFourier 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 = [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".

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