Book:Kösaku Yosida/Functional Analysis/Sixth Edition
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Kösaku Yosida: Functional Analysis (6th Edition)
Published $\text {1980}$, Springer
- ISBN 978-3540102108
Subject Matter
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