Functional Analysis. This item is unavailable.Functional Analysis

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 (Hörmander's 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 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 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 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 (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 G