of the galley proof, correcting errors and improving the presentation. To all of them, the author expresses his warmest gratitude. Thanks are also due to Professor F. K. SCHMIDT of Heidelberg Uni­ Yersity and to Professor T. KATO of the University of California at Berkeley who constantly encouraged the author to write up the present book. Finally, the author wishes to express his appreciation to Springer­ Verlag for their most efficient handling of the publication of this book. Tokyo, September 1964 I{oSAKu YOSIDA Preface to the Second Edition In the preparation of this edition, the author is indebted to Mr. FLORET of Heidelberg who kindly did the task of enlarging the Index to make the book more useful. The errors in the second printing are cor­ rected thanks to the remarks of many friends. In order to make the book more up-to-date, Section 4 of Chapter XIV has been rewritten entirely for this new edition. Tokyo, September 1967 KOSAKU YOSIDA Preface to the Third Edition A new Section (9. Abstract Potential Operators and Semi-groups) pertaining to G. HUNT'S theory of potentials is inserted in Chapter XIII of this edition. The errors in the second edition are corrected thanks to kind remarks of many friends, especially of Mr. KLAUS-DIETER BIER­ STEDT.

Inhalt

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 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. Int…