he present book is based on lectures given by the author at the University of Tokyo during the past ten years. It is intended as a textbook to be studied by students on their own or to be used in a course on Functional Analysis, i.e., the general theory of linear operators infunction spaces together with salient features of its application to diverse fields of modem and classical analysis. Necessary prerequisites for the reading of this book are summarized,with or without proof, in Chapter 0 under titles: Set Theory, Topological Spaces, Measure Spaces and Linear Spaces. Then, starting with the chapter on Semi-norms, a general theory of Banach and Hilbert spaces is presented in connection with the theory of generalized functions of S. L. SOBOLEV and L. SCHWARTZ. While the book is primarily addressed to graduate students, it is hoped it might prove useful to research mathematicians, both pure and applied. The reader may pass, e.g., fromChapter IX (Analytical Theory. of Semi-groups) directly to Chapter XIII (Ergodic Theory and Diffusion Theory) and to Chapter XIV (Integration of the Equation of Evolution). Such materials as "Weak Topologies and Duality in Locally Convex Spaces" and "Nuclear Spaces" are presented in the form of the appendices to Chapter V and Chapter X,respectively. These might be skipped for the first reading by those who are interested rather in the application of linear operators.
目录
Contents 0. Preliminaries 1. Set Theory 2. Topological Spaces 3. Measure Spaces 4. Linear Spaces I. Semi-nonns 1. Semi-nonns and Locally Convex Linear Topological Spaces 2. Nonns and Quasi-nonns 3. Examples of Normed Linear Spaces 4. Examples of Quasi-nonned Linear Spaces 5. Pre-Hilbert Spaces 6. Continuity of Linear Operators 7. Bounded Sets and Bomologic Spaces 8. Generalized Functions and Generalized Derivatives 9. B-spaces and F-spaces 10. Tbe Completion 11. Factor Spaces of a B-space 12. The Partition of Unity 13. Generalized Functions with Compact Support 14. The Direct Product of Generalized Functions II. Applications of the Baire-Hausdorff Theorem 1. The Unifonn Boundedness Theorem and the Resonance Theorem 2. The Vitali-Hahn-Saks Theorem 3. The Termwise Differentiability of a Sequence of Generalized Functions 4. The Principle ot the Condensation of Singularities 5. The Open Mapping Theorem 6. The Closed Graph Theorem 7. An Application of the Closed Graph Theorem (Hormander's Theorem) III. The Orthogonal Projection and F. Riesz Representation Theo-rem 1. The Orthogonal Projection 2. "Nearly Orthogonal" Elements …… IV. The Hahn-Banach Theorems V. Strong Convergence and Weak Convergence VI. Fourier Transform and Differential Equations VII. Dual Operators VIII. Resolvent and Spectrum IX. Analytical Theory of Semi-groups X Compact Operators XI. Nonned Rings and Spectral Representation XII. Other Representation Theorems in Linear Spaces XIIT. Ergodic Theory and Diffusion Theory XIV The Integration of the Equation of Evolution Supplementary Notes Bibliography Index Notation of Spaces
