# Book:Kösaku Yosida/Functional Analysis/Sixth Edition

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## Kösaku Yosida:

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

Published $1980$, **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**