Functional analysis

Title page......Page 1
Prefaces......Page 3
1. Set Theory......Page 11
2. Topological Spaces......Page 13
3. Measure Spaces......Page 25
4. Linear Spaces......Page 30
1. Semi-norms and Locally Convex Linear Topological Spaces......Page 33
2. Norms and Quasi-norms......Page 40
3. Examples of Normed Linear Spaces......Page 42
4. Examples of Quasi-normed Linear Spaces......Page 48
5. Pre-Hilbert Spaces......Page 49
6. Continuity of Linear Operators......Page 52
7. Bounded Sets and Bornologic Spaces......Page 54
8. Generalized Functions and Generalized Derivatives......Page 56
9. B-spaces and F-spaces......Page 62
10. The Completion......Page 66
11. Factor Spaces of a B-space......Page 69
12. The Partition of Unity......Page 70
13. Generalized Functions with Compact Support......Page 72
14. The Direct Product of Generalized Functions......Page 75
1. The Uniform Boundedness Theorem and the Resonance Theorem......Page 78
2. The Vitali-Hahn-Saks Theorem......Page 80
4. The Principle of the Condensation of Singularities......Page 82
5. The Open Mapping Theorem......Page 85
6. The Closed Graph Theorem......Page 87
7. An Application of the Closed Graph Theorem (Hörmander's Theorem)......Page 90
1. The Orthogonal Projection......Page 91
2. Linearly Orthogonal Elements......Page 94
3. The Ascoli-Arzelà Theorem......Page 95
4. The Orthogonal Base. Bessel's Inequality and Parseval's Relation......Page 96
5. E. Schmidt's Orthogonalization......Page 98
6. F. Riesz' Representation Theorem......Page 100
7. The Lax-Milgram Theorem......Page 102
8. A Proof of the Lebesgue-Nikodym Theorem......Page 103
9. The Aronszajn-Bergman Reproducing Kernel......Page 105
10. The Negative Norm of P. LAX......Page 108
11. Local Structures of Generalized Functions......Page 110
1. The Hahn-Banach Extension Theorem in Real Linear Spaces......Page 112
2. The Generalized Limit......Page 113
3. Locally Convex, Complete Linear Topological Spaces......Page 114
4. The Hahn-Banach Extension Theorem in Complex Linear Spaces......Page 115
5. The Hahn-Banach Extension Theorem in Normed Linear Spaces......Page 116
6. The Existence of Non-trivial Continuous Linear Functionals......Page 117
7. Topologies of Linear Maps......Page 120
8. The Embedding of X in its Bidual Space X"......Page 122
9. Examples of Dual Spaces......Page 124
V. Strong Convergence and Weak Convergence......Page 129
1. The Weak Convergence and The Weak* Convergence......Page 130
2. The Local Sequential Weak Compactness of Reflexive E-spaces. The Uniform Convexity......Page 136
3. Dunford's Theorem and The Gelfand-Mazur Theorem......Page 138
4. The Weak and Strong Measurability. Pettis' Theorem......Page 140
5. Bochner's Integral......Page 142
1. Polar Sets......Page 146
2. Barrel Spaces......Page 148
3. Semi-reflexivity and Reflexivity......Page 149
4. The Eberlein-Shmulyan Theorem......Page 151
VI. Fourier Transform and Differential Equations......Page 155
1. The Fourier Transform of Rapidly Decreasing Functions......Page 156
2. The Fourier Transform of Tempered Distributions......Page 159
3. Convolutions......Page 166
4. The Paley-Wiener Theorems. The One-sided Laplace Transform......Page 171
5. Titchmarsh's Theorem......Page 176
6. Mikusinski's Operational Calculus......Page 179
7. Sobolev's Lemma......Page 183
8. Garding's Inequality......Page 185
9. Friedrichs' Theorem......Page 187
10. The Malgrange-Ehrenpreis Theorem......Page 192
11. Differential Operators with Uniform Strength......Page 198
12. The Hypoellipticity (Hörmander's Theorem)......Page 199
1. Dual Operators......Page 203
2. Adjoint Operators......Page 205
3. Symmetric Operators and Self-adjoint Operators......Page 207
4. Unitary Operators. The Cayley Transform......Page 212
5. The Closed Range Theorem......Page 215
1. The Resolvent and Spectrum......Page 219
2. The Resolvent Equation and Spectral Radius......Page 221
3. The Mean Ergodic Theorem......Page 223
4. Ergodic Theorems of the Hille Type Concerning Pseudo-resolvents......Page 225
5. The Mean Value of an Almost Periodic Function......Page 228
6. The Resolvent of a Dual Operator......Page 234
7. Dunford's Integral......Page 235
8. The Isolated Singularities of a Resolvent......Page 238
IX. Analytical Theory of Semi-groups......Page 241
1. The Semi-group of Class (C₀)......Page 242
2. The Equi-continuous Semi-group of Class (C₀) in Locally Convex Spaces. Examples of Semi-groups......Page 244
3. The Infinitesimal Generator of an Equi-continuous Semi-group of Class (C₀)......Page 247
4. The Resolvent of the Infinitesimal Generator A......Page 250
5. Examples of Infinitesimal Generators......Page 252
6. The Exponential of a Continuous Linear Operator whose Powers are Equi-continuous......Page 254
7. The Representation and the Characterization of Equi-continuous Semi-groups of Class (C₀) in Terms of the Corresponding Infinitesimal Generators......Page 256
8. Contraction Semi-groups and Dissipative Operators......Page 260
9. Equi-continuous Groups of Class (C₀). Stone's Theorem......Page 261
10. Holomorphic Semi-groups......Page 264
11. Fractional Powers of Closed Operators......Page 269
12. The Convergence of Semi-groups. The Trotter-Kato Theorem......Page 279
13. Dual Semi-groups. Phillips' Theorem......Page 282
1. Compact Sets in B -spaces......Page 284
2. Compact Operators and Nuclear Operators......Page 287
3. The Rellich-Garding Theorem......Page 291
4. Schauder's Theorem......Page 292
5. The Riesz-Schauder Theory......Page 293
6. Dirichlet's Problem......Page 296
Appendix to Chapter X. The Nuclear Space of A. GROTHENDIECK......Page 299
XI. Normed Rings and Spectral Representation......Page 304
1. Maximal Ideals of a Normed Ring......Page 305
2. The Radical. The Semi-simplicity......Page 308
3. The Spectral Resolution of Bounded Normal Operators......Page 312
4. The Spectral Resolution of a Unitary Operator......Page 316
5. The Resolution of the Identity......Page 319
6. The Spectral Resolution of a Self-adjoint Operator......Page 323
7. Real Operators and Semi-bounded Operators. Friedrichs' Theorem......Page 326
8. The Spectrum of a Self-adjoint Operator. Rayleigh's Principle and the Krylov-Weinstein Theorem. The Multiplicity of the Spectrum......Page 329
9. The General Expansion Theorem. A Condition for the Absence of the Continuous Spectrum......Page 333
10. The Peter-Weyl-Neumann Theorem......Page 336
11. Tannaka's Duality Theorem for Non-commutative Compact Groups......Page 342
12. Functions of a Self-adjoint Operator......Page 348
13. Stone's Theorem and Bochner's Theorem......Page 355
14. A Canonical Form of a Self-adjoint Operator with Simple Spectrum......Page 357
15. The Defect Indices of a Symmetric Operator. The Generalized Resolution of the Identity......Page 359
16. The Group-ring L¹ and Wiener's Tauberian Theorem......Page 364
1. Extremal Points. The Krein-Milman Theorem......Page 372
2. Vector Lattices......Page 374
3. B-lattices and F-lattices......Page 379
4. A Convergence Theorem of BANACH......Page 380
5. The Representation of a Vector Lattice as Point Functions......Page 382
6. The Representation of a Vector Lattice as Set Functions......Page 385
1. The Markov Process with an Invariant Measure......Page 389
2. An Individual Ergodic Theorem and Its Applications......Page 393
3. The Ergodic Hypothesis and the H-theorem......Page 399
4. The Ergodic Decomposition of a Markov Process with a Locally Compact Phase Space......Page 403
5. The Brownian Motion on a Homogeneous Riemannian Space......Page 408
6. The Generalized Laplacian of W. FELLER......Page 413
7. An Extension of the Diffusion Operator......Page 418
8. Markov Processes and Potentials......Page 420
9. Abstract Potential Operators and Semi-groups......Page 421
XIV. The Integration of the Equation of Evolution......Page 428
1 Integration of Diffusion Equations in L²(R^m)......Page 429
2. Integration of Diffusion Equations in a Compact Riemannian Space......Page 435
3. Integration of Wave Equations in a Euclidean Space R^m......Page 437
4. Integration of Temporally Inhomogeneous Equations of Evolution in a B-space......Page 440
5. The Method of TANABE and SOBOLEVSKI......Page 448
6. Non-linear Evolution Equations 1 (The Komura-Kato Approach)......Page 455
7. Non-linear Evolution Equations 2 (The Approach through the Crandall-Liggett Convergence Theorem)......Page 464
Supplementary Notes......Page 476
Bibliography......Page 479
Index......Page 497
Notation of Spaces......Page 511