Banach-Hilbert Spaces, Vector Measures and Group Representations

Front Cover......Page 1
Title......Page 4
Copyright Information......Page 5
Preface......Page 6
Contents ......Page 10
Introduction......Page 16
1-1 Standard Finite Dimensional Vector Spaces......Page 26
1-2 Convergent Sequences in Metric Spaces......Page 28
1-3 Continuous Maps......Page 29
1-4 Open Sets......Page 32
1-5 Closures of Sets......Page 33
1-6 Characterization of Continuity ......Page 35
1-7 Duality of Closure-Interior Operators ......Page 37
1-8 Partition of Unity ......Page 39
2-1 Cauchy Sequences......Page 42
2-2 Bounded Sets ......Page 43
2-3 Upper and Lower Limits ......Page 44
2-4 Complete Sets ......Page 46
2-5 Precompact Sets ......Page 48
2-6 Compactness ......Page 51
2-7 Continuous Maps on Compact Spaces ......Page 54
2-8 Uniform Continuity ......Page 56
2-9 Connected Sets  ......Page 58
2-10. Components  ......Page 61
3-1 Uniform Convergence......Page 63
3-2 Bounded Continuous Functions ......Page 64
3-3 Sequence Spaces ......Page 67
3-4 Continuous Linear Maps ......Page 70
3-5 Examples of Continuous Linear Maps ......Page 74
3-6 Finite Dimensional Normed Spaces ......Page 76
3-7 Infinite Dimensional Compact Sets ......Page 79
3-8 Approximation in Function Spaces ......Page 82
4-1 Geometrically Independent Sets......Page 86
4-2 Convex Sets in Normed Spaces ......Page 90
4-3 Simplexes ......Page 92
4-4 Affine Maps ......Page 93
4-5 Simplicial Complexes ......Page 95
4-6 Small Simplexes ......Page 98
4-7 Barycentric Subdivisions ......Page 100
4-8 Simplicial Approximations ......Page 102
4-9 Existence of Simplicial Approximations ......Page 104
4-10. A Combinatorial Lemma with Application ......Page 106
5-1 Antipodal Maps......Page 110
5-2 Retracts and Fixed Points ......Page 113
5-3 Fixed Points of Compact Maps......Page 116
5-4 Compact Fields and their Homotopies......Page 117
5-5 Extension Property......Page 120
5-6 Properties of Compact Fields in Normed Spaces......Page 124
6-1 Transfmite Induction......Page 129
6-2 Hahn-Banach Extension Theorems......Page 131
6-3 Extension of Continuous Linear Forms......Page 133
6-4 Closed Hyperplanes......Page 135
6-5 Separation by Hyperplanes......Page 138
6-6 Extreme Points......Page 140
6-7 Baire\'s Property......Page 142
6-8 Uniform Boundedness......Page 143
6-9 Open Map and Closed Graph Theorems......Page 145
7-1 Bidual Spaces......Page 149
7-2 Quotient Spaces......Page 151
7-3 Duality of Subspaces and Quotients......Page 153
7-4 Direct Sums......Page 155
7-5 Transposes......Page 159
7-6 Reflexive Spaces......Page 162
7-7 Weak convergence......Page 164
7-8 Weak-Star Convergence......Page 167
8-1 Derivatives of Vector Maps......Page 169
8-2 Integrals of Regulated Maps......Page 170
8-3 Fundamental Theorems of Calculus......Page 173
8-4 Holomorpbic Maps of One Complex Variable......Page 176
8-5 Series Expansion......Page 180
8-6 Spectrum......Page 185
8-7 Spectral Radius......Page 189
8-8 Holomorphic Maps of an Operator......Page 191
9-1 Differentiable Maps......Page 197
9-2 Mean-Value Theorem......Page 200
9-3 Partial Derivatives......Page 203
9-4 Fixed Points of Contractions......Page 207
9-5 Inverse and Implicit Mapping Theorems......Page 208
9-6 Local Properties of Differentiable Maps......Page 212
10-1 Multilinear Maps on Banach Spaces......Page 216
10-2 Polynomials on Banach Spaces ......Page 219
10-3 Higher Derivatives ......Page 223
10-4 Cn-Maps ......Page 226
10-5 Taylor\'s Expansion ......Page 230
10-6 Higher Chain Formula and Higher Product Formula ......Page 234
11-1 Local Existence and Uniqueness......Page 238
11-2 Integral Curves ......Page 241
11-3 Linear Equations ......Page 243
11-4 Exponential Functions of Matrices ......Page 248
11-5 Global Dependence on Initial Conditions ......Page 250
11-6 Solutions without Uniqueness ......Page 257
12-1 Basic Properties......Page 260
12-2 Riesz-Schauder Theory ......Page 264
12-3 Spectrum of a Compact Operator ......Page 268
12-4 Existence of Invariant Subspaces ......Page 269
12-5 Fredholm Operators ......Page 271
13-1 Complex Inner Product Spaces......Page 276
13-2 Orthogonality in Inner Product Spaces ......Page 278
13-3 Orthonormal Bases of Hubert Spaces ......Page 280
13-4 Orthogonal Complements ......Page 283
13-5 Adjoints ......Page 285
13-6 Quadratic Forms ......Page 289
13-7 Normal Operators ......Page 291
13-8 Self-Adjoint Operators ......Page 293
13-9 Projectors and Closed Vector Subspaces ......Page 295
13-10 Partial Order of Operators ......Page 299
13-11 Eigenvalues ......Page 303
14-1 Spectrum of an Operator......Page 306
14-2 Approximate Spectrum ......Page 307
14-3 Weak Convergence ......Page 310
14-4 Diagonal Operators ......Page 312
14-5 Compact Operators ......Page 314
14-6 Functional Calculus of Self-Adjoint Operators ......Page 320
14-7 Polar Decomposition ......Page 325
15-1 Algebraic Tensor Products of Vector Spaces......Page 328
15-2 Tensor Products of Linear Maps ......Page 330
15-3 Independent Sets in Tensor Products ......Page 332
15-4 Matrix Representations ......Page 334
15-5 Projective Norms on Tensor Products ......Page 338
15-6 Inductive Norms ......Page 342
15-7 Tensor Product of Hilbert Spaces ......Page 344
16-1 Ordered Vector Spaces......Page 350
16-2 Lattice Structure ......Page 351
16-3 Decomposition Property ......Page 354
16-4 Extension of Positive Linear Forms ......Page 356
16-5 Order Bounded Linear Forms ......Page 358
17-1 Semirings......Page 361
17-2 Charges and Associated Integrals ......Page 362
17-3 Finite Variation ......Page 365
17-4 Absolutely Convergent Charges ......Page 367
17-5 Countable Additivity on Rings ......Page 370
17-6 Vector Measures ......Page 373
17-7 Lebesgue-Stieltjes Measures ......Page 375
18-1 Uniqueness of Extension......Page 379
18-2 Outer Measures ......Page 381
18-3 Extension to Decent Sets ......Page 385
19-1 Measurable Sets......Page 387
19-2 Measurable Functions ......Page 389
19-3 Limits of Measurable Functions ......Page 392
19-4 Approximations by Simple Functions ......Page 393
19-5 Measurable Maps ......Page 395
19-6 More Properties ......Page 397
20-1 Upper Functions......Page 401
20-2 Almost Everywhere ......Page 404
20-3 Seeds of the Theory ......Page 406
20-4 Sigma Finiteness ......Page 407
20-5 Comparison of Two Positive Measures ......Page 409
21-1 Extension to Integrable Sets......Page 412
21-2 Integrals of Vector Maps ......Page 414
21-3 Lp-Spaces for 121-4 Mean Convergence ......Page 420
21-5 L_infty-Spaces ......Page 423
21-6 Convergence in Measure ......Page 425
21-7 Almost Uniform Convergence ......Page 427
21-8 More Than One Measure ......Page 431
21-9 Integration on Subspaces ......Page 432
22-1 Product Measurable Spaces......Page 435
22-2 Product Measures ......Page 437
22-3 Repeated Integrals ......Page 439
23-1 Decent Sets of Rn......Page 447
23-2 Regularity ......Page 448
23-3 Translation Invariance ......Page 450
23-4 Relation to Outer Measures ......Page 453
23-5 Change Variables in Rn ......Page 454
24-1 Derivatives......Page 460
24-2 Absolute Continuity ......Page 463
24-3 Positive and Negative Sets ......Page 466
24-4 Existence of Derivatives ......Page 467
24-5 Hahn and Lebesgue Decompositions ......Page 471
24-6 Duality of Classical Spaces ......Page 473
24-7 Spaces with Radon-Nilcodym Property ......Page 481
25-1 Geometrical Expression of Radon-Nikodym Derivatives......Page 488
25-2 Jumps of Increasing Functions ......Page 492
25-3 Fundamental Theorems of Real Analysis ......Page 495
25-4 Cantor Set and Function ......Page 499
26-1 Construction from Self-Adjoint Operators......Page 502
26-2 Extension of Spectral Measures ......Page 505
26-3 Spectral Integration ......Page 510
26-4 Null Sets of Spectral Measures ......Page 513
26-5 Product Spectral Measures ......Page 516
26-6 Spectral Measures of Normal Operators ......Page 518
27-1 Regular Measures......Page 524
27-2 Construction from Positive Linear Forms ......Page 527
27-3 Representations of Order-Bounded Linear Forms ......Page 530
28-1 Almost Periodicity......Page 533
28-2 Mean Values ......Page 537
28-3 Convolutions ......Page 542
28-4 Eigen Expansion ......Page 547
29-1 Matrix Representations......Page 553
29-2 Characterization of Projectors ......Page 558
29-3 Fourier Matrices ......Page 562
30-1 Dual Objects......Page 568
30-2 Characters ......Page 570
30-3 Saturated Dual Objects ......Page 575
30-4 Separating Points ......Page 581
31-1 Representations of Product Groups......Page 586
31-2 Means on Groups ......Page 590
31-3 Order Structure on Mean Spaces ......Page 593
31-4 Identification of Functions as Means ......Page 595
31-5 Fourier Matrices of Means ......Page 598
References ......Page 602
Index ......Page 614
Back Cover......Page 622