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泛函分析pdf电子书版本下载
- （美）拉克斯（Lax，P.D.）著 著
- 出版社： 高等教育出版社
- ISBN：7040216493
- 出版时间：2007
- 标注页数：580页
- 文件大小：105MB
- 文件页数：40196916页
- 主题词：泛函分析－高等学校－教材－英文
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图书目录
1. Linear Spaces 1
2. Linear Maps 8
2.1 Algebra of linear maps 8
2.2. Index of a linear map 12
3. The Hahn-Banach Theorem 19
3.1 The extension theorem 19
3.2 Geometric Hahn-Banach theorem 21
3.3 Extensions of the Hahn-Banach theorem 24
4. Applications of the Hahn-Banach theorem 29
4.1 Extension of positive linear functionals 29
4.2 Banach limits 31
4.3 Finitely additive invariant set functions 33
Historical note 34
5. Normed Linear Spaces 36
5.1 Norms 36
5.2 Noncompactness of the unit ball 43
5.3 Isometries 47
6. Hilbert Space 52
6.1 Scalar product 52
6.2 Closest point in a closed convex subset 54
6.3 Linear functionals 56
6.4 Linear span 58
7. Applications of Hilbert Space Results 63
7.1 Radon-Nikodym theorem 63
7.2 Dirichlet's problem 65
8. Duals of Normed Linear Spaces 72
8.1 Bounded linear functionals 72
8.2 Extension of bounded linear functionals 74
8.3 Reflexive spaces 78
8.4 Support function of a set 83
9. Applications of Duality 87
9.1 Completeness of weighted powers 87
9.2 The Muntz approximation theorem 88
9.3 Runge's theorem 91
9.4 Dual variational problems in function theory 91
9.5 Existence of Green's function 94
10. Weak Convergence 99
10.1 Uniform boundedness of weakly convergent sequences 101
10.2 Weak sequential compactness 104
10.3 Weak convergence 105
11. Applications of Weak Convergence 108
11.1 Approximation of the?? function by continuous functions 108
11.2 Divergence of Fourier series 109
11.3 Approximate quadrature 110
11.4 Weak and strong analyticity of vector-valued functions 111
11.5 Existence of solutions of partial differential equations 112
11.6 The representation of analytic functions with positive real part 115
12. The Weak and Weak Topologies 118
13. Locally Convex Topologies and the Krein-Milman Theorem 122
13.1 Separation of points by linear functional 123
13.2 The Krein-Milman theorem 124
13.3 The Stone-Weierstrass theorem 126
13.4 Choquet's theorem 128
14. Examples of Convex Sets and Their Extreme Points 133
14.1 Positive functionals 133
14.2 Convex functions 135
14.3 Completely monotone functions 137
14.4 Theorems of Caratheodory and Bochner 141
14.5 A theorem of Krein 147
14.6 Positive harmonic functions 148
14.7 The Hamburger moment problem 150
14.8 G. Birkhoff's conjecture 151
14.9 De Finetti's theorem 156
14.10 Measure-preserving mappings 157
Historical note 159
15. Bounded Linear Maps 160
15.1 Boundedness and continuity 160
15.2 Strong and weak topologies 165
15.3 Principle of uniform boundedness 166
15.4 Composition of bounded maps 167
15.5 The open mapping principle 168
Historical note 172
16. Examples of Bounded Linear Maps 173
16.1 Boundedness of integral operators 173
16.2 The convexity theorem of Marcel Riesz 177
16.3 Examples of bounded integral operators 180
16.4 Solution operators for hyperbolic equations 186
16.5 Solution operator for the heat equation 188
16.6 Singular integral operators pseudodifferential operators and Fourier integral operators 190
17. Banach Algebras and their Elementary Spectral Theory 192
17.1 Normed algebras 192
17.2 Functional calculus 197
18. Gelfand's Theory of Commutative Banach Algebras 202
19. Applications of Gelfand's Theory of Commutative Banach Algebras 210
19.1 The algebra C(S) 210
19.2 Gelfand compactification 210
19.3 Absolutely convergent Fourier series 212
19.4 Analytic functions in the closed unit disk 213
19.5 Analytic functions in the open unit disk 214
19.6 Wiener's Tauberian theorem 215
19.7 Commutative B-algebras 221
Historical note 224
20. Examples of Operators and Their Spectra 226
20.1 Invertible maps 226
20.2 Shifts 229
20.3 Volterra integral operators 230
20.4 The Fourier transform 231
21. Compact Maps 233
21.1 Basic properties of compact maps 233
21.2 The spectral theory of compact maps 238
Historical note 244
22. Examples of Compact Operators 245
22.1 Compactness criteria 245
22.2 Integral operators 246
22.3 The inverse of elliptic partial differential operators 249
22.4 Operators defined by parabolic equations 250
22.5 Almost orthogonal bases 251
23. Positive compact operators 253
23.1 The spectrum of compact positive operators 253
23.2 Stochastic integral operators 256
23.3 Inverse of a second order elliptic operator 258
24. Fredholm's Theory of Integral Equations 260
24.1 The Fredholm determinant and the Fredholm resolvent 260
24.2 The multiplicative property of the Fredholm determinant 268
24.3 The Gelfand-Levitan-Marchenko equation and Dyson's formula 271
25. Invariant Subspaces 275
25.1 Invariant subspaces of compact maps 275
25.2 Nested invariant subspaces 277
26. Harmonic Analysis on a Halfline 284
26.1 The Phragmen-Lindelof principle for harmonic functions 284
26.2 An abstract Pragmen-Lindelof principle 285
26.3 Asymptotic expansion 297
27. Index Theory 300
27.1 The Noether index 301
Historical note 305
27.2 Toeplitz operators 305
27.3 Hankel operators 312
28. Compact Symmetric Operators in Hilbert Space 315
29. Examples of Compact Symmetric Operators 323
29.1 Convolution 323
29.2 The inverse of a differential operator 326
29.3 The inverse of partial differential operators 327
30. Trace Class and Trace Formula 329
30.1 Polar decomposition and singular values 329
30.2 Trace class,trace norm,and trace 330
30.3 The trace formula 334
30.4 The determinant 341
30.5 Examples and counterexamples of trace class operators 342
30.6 The Poisson summation formula 348
30.7 How to express the index of an operator as a difference of traces 349
30.8 The Hilbert-Schmidt class 352
30.9 Determinant and trace for operator in Banach spaces 353
31. Spectral Theory of Symmetric,Normal,and Unitary Operators 354
31.1 The spectrum of symmetric operators 356
31.2 Functional calculus for symmetric operators 358
31.3 Spectral resolution of symmetric operators 361
31.4 Absolutely continuous,singular,and point spectra 364
31.5 The spectral representation of symmetric operators 364
31.6 Spectral resolution of normal operators 370
31.7 Spectral resolution of unitary operators 372
Historical note 375
32. Spectral Theory of Self-Adjoint Operators 377
32.1 Spectral resolution 378
32.2 Spectral resolution using the Cayley transform 389
32.3 A functional calculus for self-adjoint operators 390
33. Examples of Self-Adjoint Operators 394
33.1 The extension of unbounded symmetric operators 394
33.2 Examples of the extension of symmetric operators; deficiency indices 397
33.3 The Friedrichs extension 402
33.4 The Rellich perturbation theorem 406
33.5 The moment problem 410
Historical note 414
34. Semigroups of Operators 416
34.1 Strongly continuous one-parameter semigroups 418
34.2 The generation of semigroups 424
34.3 The approximation of semigroups 427
34.4 Perturbation of semigroups 432
34.5 The spectral theory of semigroups 434
35. Groups of Unitary Operators 440
35.1 Stone's theorem 440
35.2 Ergodic theory 443
35.3 The Koopman group 445
35.4 The wave equation 447
35.5 Translation representation 448
35.6 The Heisenberg commutation relation 455
Historical note 459
36. Examples of Strongly Continuous Semigroups 461
36.1 Semigroups denned by parabolic equations 461
36.2 Semigroups defined by elliptic equations 462
36.3 Exponential decay of semigroups 465
36.4 The Lax-Phillips semigroup 470
36.5 The wave equation in the exterior of an obstacle 472
37. Scattering Theory 477
37.1 Perturbation theory 477
37.2 The wave operators 480
37.3 Existence of the wave operators 482
37.4 The invariance of wave operators 490
37.5 Potential scattering 490
37.6 The scattering operator 491
Historical note 492
37.7 The Lax-Phillips scattering theory 493
37.8 The zeros of the scattering matrix 499
37.9 The automorphic wave equation 500
38. A Theorem of Beurling 513
38.1 The Hardy space 513
38.2 Beurling's theorem 515
38.3 The Titchmarsh convolution theorem 523
Historical note 525
Texts 527
A. Riesz-Kakutani representation theorem 529
A.l Positive linear functionals 529
A.2 Volume 532
A.3 L as a space of functions 535
A.4 Measurable sets and measure 538
A.5 The Lebesgue measure and integral 541
B. Theory of distributions 543
B.l Definitions and examples 543
B.2 Operations on distributions 545
B.3 Local properties of distributions 547
B.4 Applications to partial differential equations 554
B.5 The Fourier transform 558
B.6 Applications of the Fourier transform 568
B.7 Fourier series 569
C. Zorn's Lemma 571
Author Index 573
Subject Index 577