泛函分析=FUNCTIONAL ANALYSIS 第6版2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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0．Preliminaries1

1．Set Theory1

2．Topological Spaces3

3．Measure Spaces15

4．Linear Spaces20

Ⅰ．Semi-norms23

1．Semi-norms and Locally Convex Linear Topological Spaces23

2．Norrns and Quasi-norms30

3．Examples of Normed Linear Spaces32

4．Examples of Quasi-normed Linear Spaces38

5．Pre-Hilbert Spaces39

6．Continuity of Linear Operators42

7．Bounded Sets and Bornologic Spaces44

8．Generalized Functions and Generalized Derivatives46

9．B-spaces and F-spaces52

10．The Completion56

11．Factor Spaces of a B-space59

12．The Partition of Unity60

13．Generalized Functions with Compact Support62

14．The Direct Product of Generalized Functions65

Ⅱ．Applications of the Baire-Hausdorff Theorem68

1．The Uniform Boundedness Theorem and the Resonance Theorem68

2．The Vitali-Hahn-Saks Theorem70

3．The Termwise Differentiability of a Sequence of Generalized Functions72

4．The Principle of the Condensation of Singularities72

5．The Open Mapping Theorem75

6．The Closed Graph Theorem77

7．An Application of the Closed Graph Theorem(H？rmander's Theorem)80

Ⅲ．The Orthogonal Projection and F.Riesz'Representation Theo-rem81

1．The Orthogonal Proiection81

2．"Nearly Orthogonal"Elements84

3．The Ascoli-ArzelàTheorem85

4．The Orthogonal Base.Bessel's Inequality and Parseval's Relation86

5．E.Schmidt's Orthogonalization88

6．F.Riesz' Representation Theorem90

7．The Lax-Milgram Theorem92

8．A Proof of the Lebesgue-Nikodym Theorem93

9．The Aronszajn-Bergman Reproducing Kernel95

10．The Negative Norm of P.LAX98

11．Local Structures of Generalized Functions100

Ⅳ．The Hahn-Banach Theorems102

1．The Hahn-Banach Extension Theorem in Real Linear Spaces102

2．The Generalized Limit103

3．Locally Convex,Complete Linear Topological Spaces104

4．The Hahn-Banach Extension Theorem in Complex Linear Spaces105

5．The Hahn-Banach Extension Theorem in Normed Linear Spaces106

6．The Existence of Non-trivial Continuous Linear Functionals107

7．Topologies of Linear Maps110

8．The Embedding of X in its Bidual Space X″112

9．Examples of Dual Spaces114

Ⅴ．Strong Convergence and Weak Convergence119

1．The Weak Convergence and The Weak Convergence120

2．The Local Sequential Weak Compactness of Reflexive B-spaces.The Uniform Convexity126

3．Dunford's Theorem and The Gelfand-Mazur Theorem128

4．The Weak and Strong Measurability.Pettis' Theorem130

5．Bochner's Integral132

Appendix to Chapter V.Weak Topologies and Duality in Locally Convex Linear Topological Spaces136

1．Polar Sets136

2．Barrel Spaces138

3．Semi-reflexivity and Reflexivity139

4．The Eberlein-Shmulyan Theorem141

Ⅵ．Fourier Transform and Differential Equations145

1．The Fourier Transform of Rapidly Decreasing Functions146

2．The Fourier Transform of Tempered Distributions149

3．Convolutions156

4．The Paley-Wiener Theorems.The One-sided Laplace Trans-form161

5．Titchmarsh's Theorem166

6．Mikusi？ski's Operational Calculus169

7．Sobolev'sLemma173

8．Gārding's Inequality175

9．Friedrichs' Theorem177

10．The Malgrange-Ehrenpreis Theorem182

11．Differential Operators with Uniform Strength188

12．The Hypoellipticity(H？rmander's Theorem)189

Ⅶ．Dual Operators193

1．Dual Operators193

2．Adjoint Operators195

3．Symmetric Operators and Self-adjoint Operators197

4．Unitary Operators.The Cayley Transform202

5．The Closed Range Theorem205

Ⅷ．Resolvent and Spectrum209

1．The Resolvent and Spectrum209

2．The Resolvent Equation and Spectral Radius211

3．The Mean Ergodic Theorem213

4．Ergodic Theorems of the Hille Type Concerning Pseudo-resolvents215

5．The Mean Value of an Almost Periodic Function218

6．The Resolvent of a Dual Operator224

7．Dunford's Integral225

8．The Isolated Singularities of a Resolvent228

Ⅸ．Analytical Theory of Semi-groups231

1．The Semi-group of Class(C0)232

2．The Equi-continuous Semi-group of Class(C0)in Locally Convex Spaces.Examples of Semi-groups234

3．The Infinitesimal Generator of an Equi-continuous Semi-group of Class(C0)237

4．The Resolvent of the Infinitesimal Generator A240

5．Examples of Infinitesimal Generators242

6．The Exponential of a Continuous Linear Operator whose Powers are Equi-continuous244

7．The Representation and the Characterization of Equi-con-tinuous Semi-groups of Class(C0)in Terms of the Corre-sponding Infinitesimal Generators246

8．Contraction Semi-groups and Dissipative Operators250

9．Equi-continuous Groups of Class(C0).Stone's Theorem251

10．Holomorphic Semi-groups254

11．Fractional Powers of Closed Operators259

12．The Convergence of Semi-groups.The Trotter-Kato Theorem269

13．Dual Semi-groups.Phillips'Theorem272

Ⅹ．Compact Operators274

1．Compact Sets in B-spaces274

2．Compact Operators and Nuclear Operators277

3．The Rellich-Gārding Theorem281

4．Schauder's Theorem282

5．The Riesz-Schauder Theory283

6．Dirichlet's Problem286

Appendix to Chapter X.The Nuclear Space of A.GROTHENDIECK289

Ⅺ．Normed Rings and Spectral Representation294

1．Maximal Ideals of a Normed Ring295

2．The Radical.The Semi-simplicity298

3．The Spectral Resolution of Bounded Normal Operators302

4．The Spectral Resolution of a Unitary Operator306

5．The Resolution of the Identity309

6．The Spectral Resolution of a Self-adjoint Operator313

7．Real Operators and Semi-bounded Operators.Friedrichs' Theorem316

8．The Spectrum of a Self-adjoint Operator.Rayleigh's Prin-ciple and the Krylov-Weinstein Theorem.The Multiplicity of the Spectrum319

9．The General Expansion Theorem.A Condition for the Absence of the Continuous Spectrum323

10．The Peter-Weyl-Neumann Theorem326

11．Tannaka's Duality Theorem for Non-commutative Compact Groups332

12．Functions of a Self-adjoint Operator338

13．Stone's Theorem and Bochner's Theorem345

14．A Canonical Form of a Self-adjoint Operator with Simple Spectrum347

15．The Defect Indices of a Symmetric Operator.The Generalized Resolution of the Identity349

16．The Group-ring L1 and Wiener's Tauberian Theorem354

Ⅻ．Other Representation Theorems in Linear Spaces362

1．Extremal Points.The Krein-Milman Theorem362

2．Vector Lattices364

3．B-lattices and F-lattices369

4．A Convergence Theorem of BANACH370

5．The Representation of a Vector Lattice as Point Functions372

6．The Representation of a Vector Lattice as Set Functions375

ⅩⅢ．Ergodic Theory and Diffusion Theory379

1．The Markov Process with an Invariant Measure379

2．An Individual Ergodic Theorem and Its Applications383

3．The Ergodic Hypothesis and the H-theorem389

4．The Ergodic Decomposition of a Markov Process with a Locally Compact Phase Space393

5．The Brownian Motion on a Homogeneous Riemannian Space398

6．The Generalized Laplacian of W.FELLER403

7．An Extension of the Diffusion Operator408

8．Markov Processes and Potentials410

9．Abstract Potential Operators and Semi-groups411

ⅩⅣ．The Integration of the Equation of Evolution418

1．Integration of Diffusion Equations in L2(Rm)419

2．Integration of Diffusion Equations in a Compact Rie-mannian Space425

3．Integration of Wave Equations in a Euclidean Space Rm427

4．Integration of Temporally Inhomogeneous Equations of Evolution in a B-space430

5．The Method of TANABE and SOBOLEVsKI438

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 Notes466

Bibliography469

Index487

Notation of Spaces501