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.
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 ONVEX 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 VII: THE SPECTRAL THEOREM VIII: UNBOUNDED OPERATORS THE FOURIER TRANSFORM SUPPLEMENTARY MATERIAL List of Symbols Index
