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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 Equations in 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