The present book is based on lectures given by the author at the University of Tokyo during the past ten years. It is intended as a textbook to be studied by students on their own or to be used in a course on Functional Analysis, i. e. , the general theory of linear operators in function spaces together with salient features of its application to diverse fields of modern and classical analysis. Necessary prerequisites for the reading of this book are summarized, with or without proof, in Chapter 0 under titles: Set Theory, Topo- logical Spaces, Measure Spaces and Linear Spaces. Then, starting with the chapter on Semi-norms, a general theory of Banach and Hilbert spaces is presented in connection with the theory of generalized functions of S. L. SOBOLEV and L. SCHWARTZ. While the book is primarily addressed to graduate students, it is hoped it might prove useful to research mathe- maticians, both pure and applied. The reader may pass, e. g. , from Chapter IX (Analytical Theory of Semi-groups) directly to Chapter XIII (Ergodic Theory and Diffusion Theory) and to Chapter XIV (Integration of the Equation of Evolution). Such materials as "Weak Topologies and Duality in Locally Convex Spaces" and "Nuclear Spaces" are presented in the form of the appendices to Chapter V and Chapter X, respectively. These might be skipped for the first reading by those who are interested rather in the application of linear operators.
Biography of Kosaku Yosida Kosaku Yosida (7.2.1909-20.6.1990) was born in Hiroshima, Japan. After studying mathematics a the University of Tokyo he held posts at Osaka and Nagoya Universities before returning to the University of Tokyo in 1955. Yosida obtained important and fundamental results in functional analysis and probability. He is best remembered for his joint work with E. Hille which brought forth a theory of semigroups of operators successfully applied to diffusion equations, Markov processes, hyperbolic equations and potential theory. His famous textbook on Functional Analysis was published in 6 distinct editions between 1965 and 1980.
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 (Hoermander'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 (Hoermander'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.