0．Preliminaries 1
1．Set Theory 1
2．Topological Spaces 3
3．Measure Spaces 15
4．Linear Spaces 20
Ⅰ．Semi-norms 23
1．Semi-norms and Locally Convex Linear Topological Spaces 23
2．Norrns and Quasi-norms 30
3．Examples of Normed Linear Spaces 32
4．Examples of Quasi-normed Linear Spaces 38
5．Pre-Hilbert Spaces 39
6．Continuity of Linear Operators 42
7．Bounded Sets and Bornologic Spaces 44
8．Generalized Functions and Generalized Derivatives 46
9．B-spaces and F-spaces 52
10．The Completion 56
11．Factor Spaces of a B-space 59
12．The Partition of Unity 60
13．Generalized Functions with Compact Support 62
14．The Direct Product of Generalized Functions 65
Ⅱ．Applications of the Baire-Hausdorff Theorem 68
1．The Uniform Boundedness Theorem and the Resonance Theorem 68
2．The Vitali-Hahn-Saks Theorem 70
3．The Termwise Differentiability of a Sequence of Generalized Functions 72
4．The Principle of the Condensation of Singularities 72
5．The Open Mapping Theorem 75
6．The Closed Graph Theorem 77
7．An Application of the Closed Graph Theorem(H？rmander's Theorem) 80
Ⅲ．The Orthogonal Projection and F.Riesz'Representation Theo-rem 81
1．The Orthogonal Proiection 81
2．"Nearly Orthogonal"Elements 84
3．The Ascoli-ArzelàTheorem 85
4．The Orthogonal Base.Bessel's Inequality and Parseval's Relation 86
5．E.Schmidt's Orthogonalization 88
6．F.Riesz' Representation Theorem 90
7．The Lax-Milgram Theorem 92
8．A Proof of the Lebesgue-Nikodym Theorem 93
9．The Aronszajn-Bergman Reproducing Kernel 95
10．The Negative Norm of P.LAX 98
11．Local Structures of Generalized Functions 100
Ⅳ．The Hahn-Banach Theorems 102
1．The Hahn-Banach Extension Theorem in Real Linear Spaces 102
2．The Generalized Limit 103
3．Locally Convex,Complete Linear Topological Spaces 104
4．The Hahn-Banach Extension Theorem in Complex Linear Spaces 105
5．The Hahn-Banach Extension Theorem in Normed Linear Spaces 106
6．The Existence of Non-trivial Continuous Linear Functionals 107
7．Topologies of Linear Maps 110
8．The Embedding of X in its Bidual Space X″ 112
9．Examples of Dual Spaces 114
Ⅴ．Strong Convergence and Weak Convergence 119
1．The Weak Convergence and The Weak Convergence 120
2．The Local Sequential Weak Compactness of Reflexive B-spaces.The Uniform Convexity 126
3．Dunford's Theorem and The Gelfand-Mazur Theorem 128
4．The Weak and Strong Measurability.Pettis' Theorem 130
5．Bochner's Integral 132
Appendix to Chapter V.Weak Topologies and Duality in Locally Convex Linear Topological Spaces 136
1．Polar Sets 136
2．Barrel Spaces 138
3．Semi-reflexivity and Reflexivity 139
4．The Eberlein-Shmulyan Theorem 141
Ⅵ．Fourier Transform and Differential Equations 145
1．The Fourier Transform of Rapidly Decreasing Functions 146
2．The Fourier Transform of Tempered Distributions 149
3．Convolutions 156
4．The Paley-Wiener Theorems.The One-sided Laplace Trans-form 161
5．Titchmarsh's Theorem 166
6．Mikusi？ski's Operational Calculus 169
7．Sobolev'sLemma 173
8．Gārding's Inequality 175
9．Friedrichs' Theorem 177
10．The Malgrange-Ehrenpreis Theorem 182
11．Differential Operators with Uniform Strength 188
12．The Hypoellipticity(H？rmander's Theorem) 189
Ⅶ．Dual Operators 193
1．Dual Operators 193
2．Adjoint Operators 195
3．Symmetric Operators and Self-adjoint Operators 197
4．Unitary Operators.The Cayley Transform 202
5．The Closed Range Theorem 205
Ⅷ．Resolvent and Spectrum 209
1．The Resolvent and Spectrum 209
2．The Resolvent Equation and Spectral Radius 211
3．The Mean Ergodic Theorem 213
4．Ergodic Theorems of the Hille Type Concerning Pseudo-resolvents 215
5．The Mean Value of an Almost Periodic Function 218
6．The Resolvent of a Dual Operator 224
7．Dunford's Integral 225
8．The Isolated Singularities of a Resolvent 228
Ⅸ．Analytical Theory of Semi-groups 231
1．The Semi-group of Class(C0) 232
2．The Equi-continuous Semi-group of Class(C0)in Locally Convex Spaces.Examples of Semi-groups 234
3．The Infinitesimal Generator of an Equi-continuous Semi-group of Class(C0) 237
4．The Resolvent of the Infinitesimal Generator A 240
5．Examples of Infinitesimal Generators 242
6．The Exponential of a Continuous Linear Operator whose Powers are Equi-continuous 244
7．The Representation and the Characterization of Equi-con-tinuous Semi-groups of Class(C0)in Terms of the Corre-sponding Infinitesimal Generators 246
8．Contraction Semi-groups and Dissipative Operators 250
9．Equi-continuous Groups of Class(C0).Stone's Theorem 251
10．Holomorphic Semi-groups 254
11．Fractional Powers of Closed Operators 259
12．The Convergence of Semi-groups.The Trotter-Kato Theorem 269
13．Dual Semi-groups.Phillips'Theorem 272
Ⅹ．Compact Operators 274
1．Compact Sets in B-spaces 274
2．Compact Operators and Nuclear Operators 277
3．The Rellich-Gārding Theorem 281
4．Schauder's Theorem 282
5．The Riesz-Schauder Theory 283
6．Dirichlet's Problem 286
Appendix to Chapter X.The Nuclear Space of A.GROTHENDIECK 289
Ⅺ．Normed Rings and Spectral Representation 294
1．Maximal Ideals of a Normed Ring 295
2．The Radical.The Semi-simplicity 298
3．The Spectral Resolution of Bounded Normal Operators 302
4．The Spectral Resolution of a Unitary Operator 306
5．The Resolution of the Identity 309
6．The Spectral Resolution of a Self-adjoint Operator 313
7．Real Operators and Semi-bounded Operators.Friedrichs' Theorem 316
8．The Spectrum of a Self-adjoint Operator.Rayleigh's Prin-ciple and the Krylov-Weinstein Theorem.The Multiplicity of the Spectrum 319
9．The General Expansion Theorem.A Condition for the Absence of the Continuous Spectrum 323
10．The Peter-Weyl-Neumann Theorem 326
11．Tannaka's Duality Theorem for Non-commutative Compact Groups 332
12．Functions of a Self-adjoint Operator 338
13．Stone's Theorem and Bochner's Theorem 345
14．A Canonical Form of a Self-adjoint Operator with Simple Spectrum 347
15．The Defect Indices of a Symmetric Operator.The Generalized Resolution of the Identity 349
16．The Group-ring L1 and Wiener's Tauberian Theorem 354
Ⅻ．Other Representation Theorems in Linear Spaces 362
1．Extremal Points.The Krein-Milman Theorem 362
2．Vector Lattices 364
3．B-lattices and F-lattices 369
4．A Convergence Theorem of BANACH 370
5．The Representation of a Vector Lattice as Point Functions 372
6．The Representation of a Vector Lattice as Set Functions 375
ⅩⅢ．Ergodic Theory and Diffusion Theory 379
1．The Markov Process with an Invariant Measure 379
2．An Individual Ergodic Theorem and Its Applications 383
3．The Ergodic Hypothesis and the H-theorem 389
4．The Ergodic Decomposition of a Markov Process with a Locally Compact Phase Space 393
5．The Brownian Motion on a Homogeneous Riemannian Space 398
6．The Generalized Laplacian of W.FELLER 403
7．An Extension of the Diffusion Operator 408
8．Markov Processes and Potentials 410
9．Abstract Potential Operators and Semi-groups 411
ⅩⅣ．The Integration of the Equation of Evolution 418
1．Integration of Diffusion Equations in L2(Rm) 419
2．Integration of Diffusion Equations in a Compact Rie-mannian Space 425
3．Integration of Wave Equations in a Euclidean Space Rm 427
4．Integration of Temporally Inhomogeneous Equations of Evolution in a B-space 430
5．The Method of TANABE and SOBOLEVsKI 438
6．Non-linear Evolution Equations 1(The Kōmura-Kato Approach) 445
7．Non-linear Evolution Equations2(The Approach through the Crandall-Liggett Convergence Theorem) 454
Supplementary Notes 466
Bibliography 469
Index 487
Notation of Spaces 501