Book:Kösaku Yosida/Functional Analysis/Sixth Edition

Subject Matter

 * Functional Analysis

Contents

 * 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


 * 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 Arondzajn-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. Mikusiński's Operational Calculus


 * 7. Sobolev's Lemma


 * 8. Gårding's Inequality


 * 9. Friedrichs' Theorem


 * 10. The Malgrange-Ehrenpreis Theorem


 * 11. Differential Operators with Uniform Strenght


 * 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 Pseudoresolvents


 * 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 $\paren {C_0}$


 * 2. The Equi-continuous Semi-group of Class $\paren {C_0}$ in Locally Convex Spaces. Examples of Semi-groups


 * 3. The Infinitesimal Generator of an Equi-continuous Semigroup of Class $\paren {C_0}$


 * 4. The Resolvent of the Infinitesimal Generator A


 * 5. Examples of Infinitesimal Generators


 * 6. The Exponential of a Continuous Linear Operator whose Powers are Equi-continuous


 * 7. The Representation and the Characterization of Equi-continuous Semi-groups of Class $\paren {C_0}$ in Terms of the Corresponding Infinitesimal Generators


 * 8. Contraction Semi-groups and Dissipative Operators


 * 9. Contraction Semi-groups of Class $\paren {C_0}$. Stone Theorem


 * 10. Holomorphic Semi-gorups


 * 11. Fractional Powers of Closed Operators


 * 12. The Convergence of Semi-groups. The Trotter-Kato Theorem


 * 13. Dual Semi-groups. Phillips' Theorem


 * X. Compact Operators


 * 1. Compact Sets in B-spaces


 * 2. Compact Operators and Nuclear Operators


 * 3. The Rellich-Gårding Theorem


 * 4. Schauder's Theorem


 * 5. The Riesz-Schauder Theory


 * 6. Dirichlet's Problem


 * Appendix to Chapter X. The Nuclear Space of A. Grothendieck


 * XI. Normed Rings and Spectral Representation


 * 1. Maximal Ideals of a Normed Ring


 * 2. The Radical. The Semi-simplicity


 * 3. The Spectral Resolution of Bounded Normal Operators


 * 4. The Spectral Resolution of a Unitary Operator


 * 5. The Resolution of the Identity


 * 6. The Spectral Resolution of a Self-adjoint Operator


 * 7. Real Operators and Semi-bounded Operators. Friedrichs' Theorem


 * 8. The Spectrum of a Self-adjoint Operator. Rayleigh's Principle and the Krylov-Weinstein Theorem. The Multiplicity of the Spectrum


 * 9. The General Expansion Theorem. A Condition for the Absence of the Continuous Spectrum


 * 10. The Peter-Weyl-Neumann Theorem


 * 11. Tannaka's Duality Theorem for Non-commutative Compact Groups


 * 12. Functions of a Self-adjoint Operator


 * 13. Stone's Theorem and Bochner's Theorem


 * 14. A Canonical Form of a Self-adjoint Operator with Simple Spectrum


 * 15. The Defect Indices of a Symmetric Operator. The Generalized Resolution of the Identity


 * 16. The Group-ring $L^1$ and Wiener's Tauberian Theorem


 * XII. Other Representation Theorems in Linear Spaces


 * 1. Extremal Points. The Krein-Milman Theorem


 * 2. Vector Lattices


 * 3. B-lattices and F-lattices


 * 4. A Convergence Theorem of Banach


 * 5. The Representation of a Vector Lattice as Point Functions


 * 6. The Representation of a Vector Lattice as Set Functions


 * XIII. Ergodic Theory and Diffusion Theory


 * 1. The Markov Process with an Invariant Measure


 * 2. An Individual Ergodic Theorem and Its Applications


 * 3. The Ergodic Hypothesis and the H-theorem


 * 4. The Ergodic Decomposition of a Markov Process with a Locally Compact Phase Space


 * 5. The Brownian Motion on a Homogenous Riemannian Space


 * 6. The Generalized Laplacian of W. Feller


 * 7. An Extension of the Diffusion Operator


 * 8. Markov Process and Potentials


 * 9. Abstract Potential Operators and Semi-groups


 * XIV. The Integration of the Equation of Evolution


 * 1. Integration of Diffusion Equations in $\map {L^2} {\R^m}$


 * 2. Integration of Diffusion Equations in a Compact Riemannian Space


 * 3. Integration of Wave Equations in a Euclidean Space $\R^m$


 * 4. Integration of Temporally Inhomogenous Equations of Evolution in a B-space


 * 5. The Method of Tanabe and Sobolevski


 * 6. Non-linear Evolution Equations 1 (The Komura-Kato Approach)


 * 7. Non-linear Evolution Equations 2 (The Approach through the Crandall-Liggett Convergence Theorem)

Suplementary Notes

Bibliography

Index

Notation of Spaces