Methods of modern mathematical physics. 1: functional analysis
| Автор(ы): | Reed M., Simon B.
06.10.2007
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| Год изд.: | 1980 |
| Описание: | This book is the first of a multivolume series devoted to an exposition of functional analysis methods in modern mathematical physics. It describes the fundamental principles of functional analysis and is essentially self-contained, although there are occasional references to later volumes. The authors have included a few applications when they think that they would provide motivation for the reader. |
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Обложка книги.
1. Sets and functions [1] 2. Metric and normed linear spaces [3] Appendix Lim sup and lim inf [11] 3. The Lebesgue integral [12] 4. Abstract measure theory [19] 5. Two convergence arguments [26] 6. Equicontinuity [28] Notes [31] Problems [32] II. HUBERT SPACES 1. The geometry of Hilbert space [36] 2. The Riesz lemma [41] 3. Orthonormal bases [44] 4. Tensor products of Hilbert spaces [49] 5. Ergodic theory: an introduction [54] Notes [60] Problems [63] III. BANACH SPACES 1. Definition and examples [67] 2. Duals and double duals [72] 3. The Hahn-Banach theorem [75] 4. Operations on Banach spaces [78] 5. The Baire category theorem and its consequences [79] Notes [84] Problems [86] IV. TOPOLOGICAL SPACES 1. General notions [90] 2. Nets and convergence [95] 3. Compactness [97] Appendix The Stone—Weierstrass theorem [103] 4. Measure theory on compact spaces [104] 5. Weak topologies on Banach spaces [111] Appendix Weak and strong measurability [115] Notes [117] Problems [119] V. LOCALLY CONVEX SPACES 1. General properties [124] 2. Frechet spaces [131] 3. Functions of rapid decease and the tempered distributions [133] Appendix The N-representation for (?) and (?) [141] 4. Inductive limits: generalized functions and weak solutions of partial differential equations [145] 5. Fixed point theorems [150] 6. Applications affixed point theorems [153] 7. Topologies on locally convex spaces: duality theory and the strong dual topology [162] Appendix Polars and the Mackey-Arens theorem [167] Notes [169] Problems [173] VI. BOUNDED OPERATORS 1. Topologies on bounded operators [182] 2. Adjoints [185] 3. The spectrum [188] 4. Positive operators and the polar decomposition [195] 5. Compact operators [198] 6. The trace class and Hilbert-Schmidt ideals [206] Notes [213] Problems [216] VII. THE SPECTRAL THEOREM 1. The continuous functional calculus [221] 2. The spectral measures [224] 3. Spectral projections [234] 4. Ergodic theory revisited: Koopmanism [237] Notes [243] Problems [245] VIII. UNBOUNDED OPERATORS 1. Domains, graphs, adjoints, and spectrum [249] 2. Symmetric and self-adjoint operators: the basic criterion for self-adjointness [255] 3. The spectral theorem [259] 4. Stone's theorem [264] 5. Formal manipulation is a touchy business: Nelson's example [270] 6. Quadratic forms [276] 7. Convergence of unbounded operators [283] 8. The Trotter product formula [295] 9. The polar decomposition for closed operators [297] 10. Tensor products [298] 11. Three mathematical problems in quantum mechanics [302] Notes [305] Problems [312] THE FOURIER TRANSFORM 1. The Fourier transform on (?) and (?) convolutions [318] 2. The range of the Fourier transform: Classical spaces [326] 3. The range of the Fourier transform: Analyticity [332] Notes [338] Problems [339] SUPPLEMENTARY MATERIAL II.2. Applications of the Riesz lemma [344] III.1. Basic properties of (?) spaces [348] IV.3. Proof of Tychonoff s theorem [351] IV.4. The Riesz-Markov theorem for X=[0,1] [353] IV.5. Minimization of'functional [354] V.5. Proofs of some theorems in nonlinear functional analysis [363] VI.5. Applications of compact operators [368] VIII.7. Monotone convergence for forms [372] VIII.8. More on the Trotter product formula [377] Uses of the maximum principle [382] Notes [385] Problems [387] List of Symbols [393] Index [395] |
| Формат: | djvu |
| Размер: | 9102604 байт |
| Язык: | ENG |
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