Functional Analysis

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. Mikusi?ski's Operational Calculus
7. Sobolev's Lemma
8. Gårding's Inequality
9. Friedrichs' Theorem
10. The Malgrange-Ehrenpreis Theorem
11. Differential Operators with Uniform Strength
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 (C0)
2. The Equi-continuous Semi-group of Class (C0) in Locally Convex Spaces. Examples of Semi-groups
3. The Infinitesimal Generator of an Equi-continuous Semigroup of Class (C0)
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 (C0) in Terms of the Corresponding Infinitesimal Generators
8. Contraction Semi-groups and Dissipative Operators
9. Equi-continuous Groups of Class (C0). 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-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 1 dentity
16. The Group-ring L1 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 Equations in L2(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 Sobolevski
6. Non-linear Evolution Equations 1 (The K?mura-Kato Approach)
7. Non-linear Evolution Equations 2 (The Approach through the Crandall-Liggett Convergence Theorem)