This book is the first of a multivolume series devoted to an exposition of functional analysis methods in modem mathematical physics.It describes the fundamental principles of functional analysis and is essentially self-contained,although there are occasional references to later volumes.We have included a few applications when we thought that they would provide motivation for the reader.Later volumes describe various advanced topics in functional analysis and give numerous applications in classical physics,modem physics,and partial differenrial equations.
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| 电子书 | 现代数学物理方法(第1卷) |
| 分类 | 电子书下载 |
| 作者 | M.Reed//B.Simon |
| 出版社 | 世界图书出版公司 |
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| 介绍 |
编辑推荐 目录 Preface Introduction Contents of Other Volumes Ⅰ: PRELIMINARIES 1. Sets and functions 2. Metric and normed linear spaces Appendix Lira sup and lim inf 3. The Lebesgue integral 4. Abstract measure theory 5. Two conrergence 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. Frdchet spaces 3. Functions of rapid decease and the tempered distributions Appendix The N-representation for and 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. 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 (Rn) and (Rn),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 Lp spaces Ⅳ.3. Proof of Tychonoff s theorem Ⅳ.4. The Riesz-Markov theoremJor X = [0,1] Ⅳ.5. Minimization of Junctionals 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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