Functional Analysis

0. Preliminaries.- 1. Set Theory.- 2. Topological Spaces.- 3. Measure Spaces.- 4. Linear Spaces.- I. Semi-norms.- 1. Semi-norms and Locally Convex Linear Topological Spaces.- 2. Norms and Quasi-norms.- 3. Examples of Normed Linear Spaces.- 4. Examples of Quasi-normed Linear Spaces.- 5. Pre-Hilbert Spaces.- 6. Continuity of Linear Operators.- 7. Bounded Sets and Bornologic Spaces.- 8. Generalized Functions and Generalized Derivatives.- 9. B-spaces and F-spaces.- 10. The 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 Uniform 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 of the Condensation of Singularities.- 5. The Open Mapping Theorem.- 6. The Closed Graph Theorem.- 7. An Application of the Closed Graph Theorem (Hörmander s Theorem).- III. The Orthogonal Projection and F. Riesz Representation Theorem.- 1. The Orthogonal Projection.- 2. Nearly Orthogonal Elements.- 3. The Ascoli-Arzelà Theorem.- 4. The Orthogonal Base. Bessel s Inequality and Parseval s Relation.- 5. E. Schmidt s Orthogonalization.- 6. F. Riesz Representation Theorem.- 7. The Lax-Milgram Theorem.- 8. A Proof of the Lebesgue-Nikodym Theorem.- 9. The Aronszajn-Bergman Reproducing Kernel.- 10. The Negative Norm of P. Lax.- 11. Local Structures of Generalized Functions.- IV. The Hahn-Banach Theorems.- 1. The Hahn-Banach Extension Theorem in Real Linear Spaces.- 2. The Generalized Limit.- 3. Locally Convex, Complete Linear Topological Spaces.- 4. The Hahn-Banach Extension Theorem in Complex Linear Spaces.- 5. The Hahn-Banach Extension Theorem in Normed Linear Spaces.- 6. The Existence of Non-trivial Continuous Linear Functionals.- 7. Topologies of Linear Maps.- 8. The Embedding of X in its Bidual Space X .- 9. Examples of Dual Spaces.- V. Strong Convergence and Weak Convergence.- 1. The Weak Convergence and The Weak Convergence.- 2. The Local Sequential Weak Compactness of Reflexive B-spaces. The Uniform Convexity.- 3. Dunford s Theorem and The Gelfand-Mazur Theorem.- 4. The Weak and Strong Measurability. Pettis Theorem.- 5. Bochner s Integral.- Appendix to Chapter V. Weak Topologies and Duality in Locally Convex Linear Topological Spaces.- 1. Polar Sets.- 2. Barrel Spaces.- 3. Semi-reflexivity and Reflexivity.- 4. The Eberlein-Shmulyan Theorem.- VI. Fourier Transform and Differential Equations.- 1. The Fourier Transform of Rapidly Decreasing Functions.- 2. The Fourier Transform of Tempered Distributions.- 3. Convolutions.- 4. The Paley-Wiener Theorems. The One-sided Laplace Transform.- 5. Titchmarsh s Theorem.- 6. Mikusi?ski s Operational Calculus.- 7. Sobolev s Lemma.- 8. Gårding s Inequality.- 9. Friedrichs Theorem.- 10. The Malgrange-Ehrenpreis Theorem.- 11. Differential Operators with Uniform Strength.- 12. The Hypoellipticity (Hörmander s Theorem).- VII. Dual Operators.- 1. Dual Operators.- 2. Adjoint Operators.- 3. Symmetric Operators and Self-adjoint Operators.- 4. Unitary Operators. The Cayley Transform.- 5. The Closed Range Theorem.- VIII. Resolvent and Spectrum.- 1. The Resolvent and Spectrum.- 2. The Resolvent Equation and Spectral Radius.- 3. The Mean Ergodic Theorem.- 4. Ergodic Theorems of the Hille Type Concerning Pseudo-resolvents.- 5. The Mean Value of an Almost Periodic Function.- 6. The Resolvent of a Dual Operator.- 7. Dunford s Integral.- 8. The Isolated Singularities of a Resolvent.- IX. Analytical Theory of Semi-groups.- 1. The Semi-group of Class (C0).- 2. The Equi-continuous Semi-group of Class (C0) in Locally Convex Spaces. Examples of Semi-groups.- 3. The Infinitesimal Generator of an Equi-continuous Semigroup of Class (C0).- 4. The Resolvent of the Infinitesimal Generator A.- 5. Examples of Infinitesimal Generators.- 6. The Expo

Biography of Kôsaku YosidaKôsaku Yosida (7.2.1909-20.6.1990) was born in Hiroshima, Japan. After studying mathematics a the University of Tokyo he held posts at Osaka and Nagoya Universities before returning to the University of Tokyo in 1955.Yosida obtained important and fundamental results in functional analysis and probability. He is best remembered for his joint work with E. Hille which brought forth a theory of semigroups of operators successfully applied to diffusion equations, Markov processes, hyperbolic equations and potential theory. His famous textbook on Functional Analysis was published in 6 distinct editions between 1965 and 1980.