Functional Analysis

Contents......Page v
Foreword......Page xvii
1. Linear Spaces......Page 1
2.1 Algebra of linear maps......Page 8
2.2. Index of a linear map......Page 12
3.1 The extension theorem......Page 19
3.2 Geometric Hahn-Banach theorem......Page 21
3.3 Extensions of the Hahn-Banach theorem......Page 24
4.1 Extension of positive linear functionals......Page 29
4.2 Banach limits......Page 31
4.3 Finitely additive invariant set functions......Page 33
Historical note......Page 34
5.1 Norms......Page 36
5.2 Noncompactness of the unit ball......Page 43
5.3 Isometries......Page 47
6.1 Scalar product......Page 52
6.2 Closest point in a closed convex subset......Page 54
6.3 Linear functionals......Page 56
6.4 Linear span......Page 58
7.1 Radon-Nikodym theorem......Page 63
7.2 Dirichlet\'s problem......Page 65
8.1 Bounded linear functionals......Page 72
8.2 Extension of bounded linear functionals......Page 74
8.3 Reflexive spaces......Page 78
8.4 Support function of a set......Page 83
9.1 Completeness of weighted powers......Page 87
9.2 The Muntz approximation theorem......Page 88
9.4 Dual variational problems in function theory......Page 91
9.5 Existence of Green\'s function......Page 94
10. Weak Convergence......Page 99
10.1 Uniform boundedness of weakly convergent sequences......Page 101
10.2 Weak sequential compactness......Page 104
10.3 Weak* convergence......Page 105
11.1 Approximation of the 8 function by continuous functions......Page 108
11.2 Divergence of Fourier series......Page 109
11.3 Approximate quadrature......Page 110
11.4 Weak and strong analyticity of vector-valued functions......Page 111
11.5 Existence of solutions of partial differential equations......Page 112
11.6 The representation of analytic functions with positive real part......Page 115
12. The Weak and Weak* Topologies......Page 118
13. Locally Convex Topologies and the Krein-Milman Theorem......Page 122
13.1 Separation of points by linear functionals......Page 123
13.2 The Krein-Milman theorem......Page 124
13.3 The Stone-Weierstrass theorem......Page 126
13.4 Choquet\'s theorem......Page 128
14.1 Positive functionals......Page 133
14.2 Convex functions......Page 135
14.3 Completely monotone functions......Page 137
14.4 Theorems of Caratheodory and Bochner......Page 141
14.5 A theorem of Krein......Page 147
14.6 Positive harmonic functions......Page 148
14.7 The Hamburger moment problem......Page 150
14.8 G. Birkhoff \'s conjecture......Page 151
14.9 DeFinetti\'s theorem......Page 156
14.10 Measure-preserving mappings......Page 157
Historical note......Page 159
15.1 Boundedness and continuity......Page 160
15.2 Strong and weak topologies......Page 165
15.3 Principle of uniform boundedness......Page 166
15.4 Composition of bounded maps......Page 167
15.5 The open mapping principle......Page 168
Historical note......Page 172
16.1 Boundedness of integral operators......Page 173
16.2 The convexity theorem of Marcel Riesz......Page 177
16.3 Examples of bounded integral operators......Page 180
16.4 Solution operators for hyperbolic equations......Page 186
16.5 Solution operator for the heat equation......Page 188
16.6 Singular integral operators pseudodifferential operators and Fourier integral operators......Page 190
17.1 Normed algebras......Page 192
17.2 Functional calculus......Page 197
18. Gelfand\'s Theory of Commutative Banach Algebras......Page 202
19.2 Gelfand compactification......Page 210
19.3 Absolutely convergent Fourier series......Page 212
19.4 Analytic functions in the closed unit disk......Page 213
19.5 Analytic functions in the open unit disk......Page 214
19.6 Wiener\'s Tauberian theorem......Page 215
19.7 Commutative B*-algebras......Page 221
Historical note......Page 224
20.1 Invertible maps......Page 226
20.2 Shifts......Page 229
20.3 Volterra integral operators......Page 230
20.4 The Fourier transform......Page 231
21.1 Basic properties of compact maps......Page 233
21.2 The spectral theory of compact maps......Page 238
Historical note......Page 244
22.1 Compactness criteria......Page 245
22.2 Integral operators......Page 246
22.3 The inverse of elliptic partial differential operators......Page 249
22.4 Operators defined by parabolic equations......Page 250
22.5 Almost orthogonal bases......Page 251
23.1 The spectrum of compact positive operators......Page 253
23.2 Stochastic integral operators......Page 256
23.3 Inverse of a second order elliptic operator......Page 258
24.1 The Fredholm determinant and the Fredholm resolvent......Page 260
24.2 The multiplicative property of the Fredholm determinant......Page 268
24.3 The Gelfand-Levitan-Marchenko equation and Dyson\'s formula......Page 271
25.1 Invariant subspaces of compact maps......Page 275
25.2 Nested invariant subspaces......Page 277
26.1 The Phragmen-Lindelof principle for harmonic functions......Page 284
26.2 An abstract Pragmen-Lindelof principle......Page 285
26.3 Asymptotic expansion......Page 297
27. Index Theory......Page 300
27.1 The Noether index......Page 301
27.2 Toeplitz operators......Page 305
27.3 Hankel operators......Page 312
28. Compact Symmetric Operators in Hilbert Space......Page 315
29.1 Convolution......Page 323
29.2 The inverse of a differential operator......Page 326
29.3 The inverse of partial differential operators......Page 327
30.1 Polar decomposition and singular values......Page 329
30.2 Trace class trace norm and trace......Page 330
30.3 The trace formula......Page 334
30.4 The determinant......Page 341
30.5 Examples and counterexamples of trace class operators......Page 342
30.6 The Poisson summation formula......Page 348
30.7 How to express the index of an operator as a difference of traces......Page 349
30.8 The Hilbert-Schmidt class......Page 352
30.9 Determinant and trace for operator in Banach spaces......Page 353
31. Spectral Theory of Symmetric Normal and Unitary Operators......Page 354
31.1 The spectrum of symmetric operators......Page 356
31.2 Functional calculus for symmetric operators......Page 358
31.3 Spectral resolution of symmetric operators......Page 361
31.5 The spectral representation of symmetric operators......Page 364
31.6 Spectral resolution of normal operators......Page 370
31.7 Spectral resolution of unitary operators......Page 372
Historical note......Page 375
32. Spectral Theory of Self-Adjoint Operators......Page 377
32.1 Spectral resolution......Page 378
32.2 Spectral resolution using the Cayley transform......Page 389
32.3 A functional calculus for self-adjoint operators......Page 390
33.1 The extension of unbounded symmetric operators......Page 394
33.2 Examples of the extension of symmetric operators; deficiency indices......Page 397
33.3 The Friedrichs extension......Page 402
33.4 The Rellich perturbation theorem......Page 406
33.5 The moment problem......Page 410
Historical note......Page 414
34. Semigroups of Operators......Page 416
34.1 Strongly continuous one-parameter semigroups......Page 418
34.2 The generation of semigroups......Page 424
34.3 The approximation of semigroups......Page 427
34.4 Perturbation of semigroups......Page 432
34.5 The spectral theory of semigroups......Page 434
35.1 Stone\'s theorem......Page 440
35.2 Ergodic theory......Page 443
35.3 The Koopman group......Page 445
35.4 The wave equation......Page 447
35.5 Translation representation......Page 448
35.6 The Heisenberg commutation relation......Page 455
Historical note......Page 459
36.1 Semigroups defined by parabolic equations......Page 461
36.2 Semigroups defined by elliptic equations......Page 462
36.3 Exponential decay of semigroups......Page 465
36.4 The Lax-Phillips semigroup......Page 470
36.5 The wave equation in the exterior of an obstacle......Page 472
37.1 Perturbation theory......Page 477
37.2 The wave operators......Page 480
37.3 Existence of the wave operators......Page 482
37.5 Potential scattering......Page 490
37.6 The scattering operator......Page 491
Historical note......Page 492
37.7 The Lax-Phillips scattering theory......Page 493
37.8 The zeros of the scattering matrix......Page 499
37.9 The automorphic wave equation......Page 500
38.1 The Hardy space......Page 513
38.2 Beurling\'s theorem......Page 515
38.3 The Titchmarsh convolution theorem......Page 523
Historical note......Page 525
Texts......Page 527
A.1 Positive linear functionals......Page 529
A.2 Volume......Page 532
A.3 L as a space of functions......Page 535
A.4 Measurable sets and measure......Page 538
A.5 The Lebesgue measure and integral......Page 541
B.l Definitions and examples......Page 543
B.2 Operations on distributions......Page 544
B.3 Local properties of distributions......Page 547
B.4 Applications to partial differential equations......Page 554
B.5 The Fourier transform......Page 558
B.6 Applications of the Fourier transform......Page 568
B.7 Fourier series......Page 569
C. Zorn\'s Lemma......Page 571
Author Index......Page 573
Subject Index......Page 577