泛函分析

本書是一部數學經典教材，初版於1965年，以作者在東京大學任教十餘年所用的講義為基礎寫成的。經過幾次修訂和增補，1980年出了第5版，本版（第6版）實際上是第5版的重印版。全書論述了泛函空間的線性算子理論及其在現代分析和經典分析各領域中的許多應用。目次：預備知識；半範數；Baire-Hausdorff定理的應用；正交射影和riesz表示定理；Hahn-Banach定理；強收斂和弱收斂；傅里葉變換和微分方程；對偶算子；預解和譜；半羣的解析理論；緊緻算子；賦範環和譜表示；線性空間中的其他表示定理；遍歷性理論和擴散理論；發展方程的積分。 ^[1] 

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 (Hormander's Theorem)

Ⅲ. The Orthogonal Projection and F. Riesz' Representation Theorem

1. The Orthogonal Projection

2. "Nearly Orthogonal" Elements

3. The Ascoli-Arzela 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 Cosvergence 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. Mikusinski's Operational Calculus

7. Sobolev's Lemma

8. Garding's Inequality

9. Friedrichs' ThEorem

10. The Malgrange-Ehrenpreis Theorem

11. Differential Operators with Uniform Strength

12. The I-Iypoellipticity (Hormander'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 (Co)

2. The Equi-continuous Semi-group of Class (Co) in Locally Convex Spaces, Examples of Semi-groups

3. The Infinitesimal Generator of an Equi-continuous Semi-group of Class (Co)

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-con-tinuous Semi-groups of Class (Co) in Terms of the Corre-sponding Infinitesimal Generators

8. Contraction Semi-groups and Dissipative Operators

9. Equi-continuous Groups of Class (Co). Stone's Theorem

10. Holomorphic Semi-groups

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-Garding 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 Prin-ciple 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 ldentity

16. The Group-ring L' 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 Homogeneous Riemannian Space

6. The Generalized Laplacian of W. FELLER

7. An Extension of the Diffusion Operator

8. Markov Processes and Potentials

9. Abstract Potential Operators and Semi-groups

ⅩⅣ. The Integration of the Equation of Evolution

1. Integration of Diffusion Equationsin LS(Rm)

2. Integration of Diffusion Equations in a Compact Riemannian Space

3. Integration of Wave Equations in a Euclidean Space Rm

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

5. The Method of TANABE and SOBOI.EVSKI

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

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

Supplementary Notes

Bibliography

Index

Notation of Spaces ^[2]