Mathematical Introduction to Signals and Systems

Content: 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 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. Garding\'s Inequality.- 9. Friedrichs\' Theorem.- 10. The Malgrange-Ehrenpreis Theorem.- 11. Differential Operators with Uniform Strength.- 12. The Hypoellipticity (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 (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-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 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 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 Rie-mannian 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).- Supplementary Notes.- Notation of Spaces.