# Book:Kösaku Yosida/Functional Analysis/Sixth Edition

## Kösaku Yosida: Functional Analysis (6th Edition)

Published $\text {1980}$, Springer

ISBN 978-3540102108.

### 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
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
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