History of Banach spaces and linear operators:
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| Format: | Book |
| Language: | English |
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Boston, Mass. [u.a.]
Birkhäuser
2007
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| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 683 - 829 |
| Physical Description: | XXIII, 855 S. graph. Darst. |
| ISBN: | 9780817643676 0817643672 9780817645960 0817645969 |
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| 100 | 1 | |a Pietsch, Albrecht |d 1934-2024 |e Verfasser |0 (DE-588)104515996 |4 aut | |
| 245 | 1 | 0 | |a History of Banach spaces and linear operators |c Albrecht Pietsch |
| 264 | 1 | |a Boston, Mass. [u.a.] |b Birkhäuser |c 2007 | |
| 300 | |a XXIII, 855 S. |b graph. Darst. | ||
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| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Literaturverz. S. 683 - 829 | ||
| 648 | 7 | |a Geschichte 1950-2006 |2 gnd |9 rswk-swf | |
| 650 | 4 | |a Geschichte | |
| 650 | 4 | |a Banach spaces |x History | |
| 650 | 4 | |a Linear operators |x History | |
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Record in the Search Index
| _version_ | 1805085844776681472 |
|---|---|
| adam_text |
PREFACE xiv
Notation and Terminology xvm
INTRODUCTION xix
1 The Birth of Banach Spaces 1
1.1 Complete normed linear Spaces 1
1.2 Linear Spaces 4
1.3 Metric spaces 6
1.4 Minkowski spaces 8
1.5 Hubert spaces 9
1.6 Albert A. Bennett and Kenneth W. Lamson 14
1.7 Norbert Wiener 18
1.8 Eduard Helly and Hans Hahn 21
1.9 Summary 22
2 HlSTORICAL ROOTS AND BASIC RESULTS 25
2.1 Operators 25
2.2 Functionals and dual Operators 30
2.3 The moment problem and the Hahn Banach theorem 36
2.4 The uniform boundedness principle 40
2.5 The closed graph theorem and the open mapping theorem 43
2.6 Riesz Schauder theory 45
2.6.1 Completely continuous Operators 45
2.6.2 Finite rank Operators 46
2.6.3 Approximable Operators 47
2.6.4 Compact Operators 49
2.6.5 Resolvents, spectra, and eigenvalues 52
2.6.6 Classical Operator ideals 53
2.7 Banach's monograph 54
vii
3 TOPOLOGICAL CONCEPTS WEAK TOPOLOGIES 56
3.1 Weakly convergent sequences 56
3.2 Topological Spaces and topological linear Spaces 58
3.2.1 Topological Spaces 58
3.2.2 Nets and filters 60
3.2.3 Compactness 63
3.2.4 Topological linear Spaces 65
3.2.5 Locally bounded linear Spaces 66
3.3 Locally convex linear Spaces and duality 68
3.3.1 Locally convex linear Spaces 68
3.3.2 Weak topologies and dual Systems 69
3.3.3 Separation of convex sets 71
3.3.4 Topologies on £{X,Y) 73
3.4 Weak* and weak compactness 75
3.4.1 Tychonoff's theorem 75
3.4.2 Weak* compactness theorem 76
3.4.3 Weak compactness and reflexivity 78
3.5 Weak sequential completeness and the Schur property 81
3.6 Transfinitely closed sets 82
4 Classical Banach Spaces 86
4.1 Banach lattices 86
4.2 Measures and integrals on abstract sets 92
4.2.1 Set theoretic operations 92
4.2.2 Measures 95
4.2.3 From measures to integrals 98
4.2.4 Integrals and the Banach Spaces L\ 99
4.2.5 Banach Spaces of additive set functions 101
4.3 The duality between L\ and L«, 103
4.4 The Banach Spaces Lp 106
4.5 Banach spaces of continuous functions 109
4.6 Measures and integrals on topological Spaces 111
4.7 Measures versus integrals 117
4.8 Abstract Lp and M spaces 118
4.8.1 Boolean algebras 118
4.8.2 Measure algebras 120
4.8.3 Abstract Lp spaces 124
4.8.4 Abstract A/ spaces 127
4.8.5 The Dunford Pettis property 128
4.9 Structure theory 129
4.9.1 Isomorphisms, injections, surjections, and projections 129
4.9.2 Extensions and liftings 133
4.9.3 Isometric and isomorphic classification 134
4.10 Operator ideals and Operator algebras 137
4.10.1 Schatten von Neumann ideals 137
4.10.2 Banach algebras 139
4.10.3 B* algebras = C* algebras 142
4.10.4 W* algebras 144
4.11 Complexification 155
5 Basic Results from the Post Banach Period 158
5.1 Analysis in Banach Spaces 158
5.1.1 Convergence of series 158
5.1.2 Integration of vector valued functions 160
5.1.3 Representation of Operators from L\ intoX 165
5.1.4 The Radon Nikodym property: analytic aspects 171
5.1.5 Representation of Operators from Lp irAoX 174
5.1.6 Representation of Operators fromC(Ä') intoX 175
5.1.7 Vector valued analytic functions on the complex plane 176
5.1.8 Gäteaux and Frechet derivatives 177
5.1.9 Polynomials and derivatives of higher order 180
5.1.10 Analytic functions on Banach Spaces 185
5.2 Spectral theory 189
5.2.1 Operational calculus 189
5.2.2 Fredholm Operators 192
5.2.3 Riesz Operators 198
5.2.4 Invariant subspaces 202
5.2.5 Spectral Operators 206
5.3 Semi groups of Operators 209
5.3.1 Deterministic and stochastic processes 209
5.3.2 The Hille Yosida theorem 211
5.3.3 Analytic semi groups 214
5.3.4 The abstract Cauchy problem 215
5.3.5 Ergodic theory 216
5.4 Convexity, extreme points, and related topics 219
5.4.1 The Krein Milman theorem 219
5.4.2 Integral representations 222
5.4.3 Gelfand Naimark Segal representations 226
5.4.4 The Radon Nikodym property: geometric aspects 227
5.4.5 Convex and concave functions 231
5.4.6 Lyapunov's theorem and the bang bang principle 232
5.5 Geometry of the unit ball 233
5.5.1 Strict convexity and smoothness 233
5.5.2 Uniform convexity and uniform smoothness 236
5.5.3 Further concepts related to convexity and smoothness 239
5.5.4 Applications of convexity and smoothness 242
5.5.5 Complex convexity 245
3 TOPOLOGICAL CONCEPTS WEAK TOPOLOGIES 56
3.1 Weakly convergent sequences 56
3.2 Topological Spaces and topological linear Spaces 58
3.2.1 Topological spaces 58
3.2.2 Nets and filtere 60
3.2.3 Compactness 63
3.2.4 Topological linear Spaces 65
3.2.5 Locally bounded linear Spaces 66
3.3 Locally convex linear spaces and duality 68
3.3.1 Locally convex linear spaces 68
3.3.2 Weak topologies and dual Systems 69
3.3.3 Separation of convex sets 71
3.3.4 Topologies on £(X,Y) 73
3.4 Weak* and weak compactness 75
3.4.1 Tychonoff's theorem 75
3.4.2 Weak* compactness theorem 76
3.4.3 Weak compactness and reflexivity 78
3.5 Weak sequential completeness and the Schur property 81
3.6 Transfinitely closed sets 82
4 Classical Banach Spaces 86
4.1 Banach lattices 86
4.2 Measures and integrals on abstract sets 92
4.2.1 Set theoretic operations 92
4.2.2 Measures 95
4.2.3 From measures to integrals 98
4.2.4 Integrals and the Banach spaces L\ 99
4.2.5 Banach Spaces of additive set functions 101
4.3 The duality between L\ and Lx 103
4.4 The Banach spaces Lp 106
4.5 Banach spaces of continuous functions 109
4.6 Measures and integrals on topological spaces 111
4.7 Measures versus integrals 117
4.8 Abstract Lp and M spaces 118
4.8.1 Boolean algebras 118
4.8.2 Measure algebras 120
4.8.3 Abstract Lp spaces 124
4.8.4 Abstract M spaces 127
4.8.5 The Dunford Pettis property 128
4.9 Structure theory 129
4.9.1 Isomorphisms, injections, surjections, and projections 129
4.9.2 Extensions and liftings 133
4.9.3 Isometric and isomorphic classification 134
4.10 Operator ideals and Operator algebras 137
4.10.1 Schatten von Neumann ideals 137
4.10.2 Banach algebras 139
4.10.3 ß* algebras = C* algebras 142
4.10.4 W* algebras 144
4.11 Complexification 155
5 Basic Results from the Post Banach Period 158
5.1 Analysis in Banach spaces 158
5.1.1 Convergence of series 158
5.1.2 Integration of vector valued functions 160
5.1.3 Representation of Operators from Lj intoX 165
5.1.4 The Radon Nikodym property: analytic aspects 171
5.1.5 Representation of Operators from Lp intoX 174
5.1.6 Representation of Operators fromC(^) intoX 175
5.1.7 Vector valued analytic functions on the complex plane 176
5.1.8 Gäteaux and Frechet derivatives 177
5.1.9 Polynomials and derivatives of higher order 180
5.1.10 Analytic functions on Banach Spaces 185
5.2 Spectral theory 189
5.2.1 Operational calculus 189
5.2.2 Fredholm Operators 192
5.2.3 Riesz Operators 198
5.2.4 Invariant subspaces 202
5.2.5 Spectral Operators 206
5.3 Semi groups of Operators 209
5.3.1 Deterministic and stochastic processes 209
5.3.2 The Hille Yosida theorem 211
5.3.3 Analytic semi groups 214
5.3.4 The abstract Cauchy problem 215
5.3.5 Ergodic theory 216
5.4 Convexity, extreme points, and related topics 219
5.4.1 The Krein Milman theorem 219
5.4.2 Integral representations 222
5.4.3 Gelfand Naimark Segal representations 226
5.4.4 The Radon Nikodym property: geometric aspects 227
5.4.5 Convex and concave functions 231
5.4.6 Lyapunov's theorem and the bang bang principle 232
5.5 Geometry of the unit ball 233
5.5.1 Strict convexity and smoothness 233
5.5.2 Uniform convexity and uniform smoothness 236
5.5.3 Further concepts related to convexity and smoothness 239
5.5.4 Applications of convexity and smoothness 242
5.5.5 Complex convexity 245
5.6 Bases 245
5.6.1 Schauder bases and basic sequences 245
5.6.2 Biorthogonal Systems 248
5.6.3 Bases and structure theory 249
5.6.4 Bases in concrete Banach Spaces 256
5.6.5 Unconditional basic sequences in Hubert Spaces 261
5.6.6 Wavelets 263
5.6.7 Schauder decompositions 268
5.7 Tensor products and approximation properties 269
5.7.1 Bilinear mappings 269
5.7.2 Tensor products 271
5.7.3 Nuclear and integral Operators 275
5.7.4 Approximation properties 280
6 Modern Banach Space Theory Selected Topics . 288
6.1 Geometry of Banach spaces 288
6.1.1 Banach Mazur distance, projection, and basis constants 289
6.1.2 Dvoretzky's theorem 293
6.1.3 Finite representability, ultraproducts, and spreading modeis 296
6.1.4 ^, spaces 302
6.1.5 Local unconditional structure 304
6.1.6 Banach spaces containing Z^'s uniformly 305
6.1.7 Rademacher type and cotype, Gauss type and cotype 307
6.1.8 Fourier type and cotype, Walsh type and cotype 315
6.1.9 Superreflexivity, Haar type and cotype 318
6.1.10 UMD spaces = HT spaces 321
6.1.11 Volume ratios and Grothendieck numbers 323
6.2 s Numbers 326
6.2.1 s Numbers of Operators on Hubert space 326
6.2.2 Axiomatics of s numbers 327
6.2.3 Examples of s numbers 327
6.2.4 Entropy numbers 331
6.2.5 j Numbers of diagonal Operators 333
6.2.6 i Numbers versus widths 336
6.3 Operator ideals 336
6.3.1 Ideals on Hubert space 336
6.3.2 Basic concepts of ideal theory on Banach spaces 341
6.3.3 Ideals associated with .s numbers 345
6.3.4 Local theory of quasi Banach ideals and trace duality 348
6.3.5 p Factorable Operators 353
6.3.6 p Summing Operators 355
6.3.7 p Nuclear and p integral Operators 361
6.3.8 Specific components of Operator ideals 363
6.3.9 Grothendieck's theorem 365
6.3.10 Limit order of Operator ideals 368
6.3.11 Banach ideals and tensor products 370
6.3.12 Ideal norms computed with n vectors 374
6.3.13 Operator ideals and classes of Banach Spaces 375
6.3.14 Rademacher type and cotype, Gauss type and cotype 379
6.3.15 Fourier type and cotype, Walsh type and cotype 382
6.3.16 Super weakly corapact Operators, Haar type and cotype 383
6.3.17 UMD Operators and HT Operators 386
6.3.18 Radon Nikodym property: operator theoretic aspects 387
6.3.19 Ideal norms and parameters of Minkowski Spaces 387
6.3.20 Ideal norms of finite rank Operators 389
6.3.21 Operator ideals and classes of locally convex linear Spaces 391
6.4 Eigenvalue distributions 393
6.4.1 Eigenvalue sequences and classical results 393
6.4.2 Inequalities between .s numbers and eigenvalues 394
6.4.3 Eigenvalues of /? summing Operators 396
6.4.4 Eigenvalues of nuclear Operators 398
6.4.5 Operators of eigenvalue type lp q 399
6.5 Traces and determinants 402
6.5.1 Traces 402
6.5.2 Fredholm denominators and determinants 405
6.5.3 Regularized Fredholm denominators 412
6.5.4 The Gohberg Goldberg Krupnik approach 414
6.5.5 Eigenvalues and zeros of entire functions 415
6.5.6 Completeness of root vectors 418
6.5.7 Determinants: pros and cons 424
6.6 Interpolation theory 425
6.6.1 Classical convexity theorems 425
6.6.2 Interpolation methods 426
6.6.3 Complex and real interpolation methods 427
6.6.4 Lorentz Spaces 431
6.6.5 Applications of interpolation theory 433
6.6.6 Interpolation of Operator ideals 435
6.6.7 New trends in interpolation theory 436
6.7 Function Spaces 440
6.7.1 Hölder Lipschitz Spaces 440
6.7.2 Sobolev Spaces 442
6.7.3 Besov Spaces 445
6.7.4 Lizorkin Triebel Spaces 450
6.7.5 Interpolation of function Spaces 453
6.7.6 Spaces of smooth functions: Supplements 454
6.7.7 Bases of Besov Spaces 458
6.7.8 Embedding Operators 459
6.7.9 Spaces of vector valued functions 464
6.7.10 Integral Operators 466
6.7.11 Differential Operators 469
6.7.12 Hardy Spaces 473
6.7.13 Bergman Spaces 481
6.7.14 Orlicz spaces 485
6.8 Probability theory on Banach spaces 489
6.8.1 Baire, Borel, and Radon measures 489
6.8.2 Cylindrical measures 491
6.8.3 Characteristic functionals 493
6.8.4 Radonifying Operators 494
6.8.5 Gaussian measures 496
6.8.6 Wiener measure 497
6.8.7 Vector valued random variables 498
6.8.8 Rademacher series 501
6.8.9 The law of large numbers 502
6.8.10 The central limit theorem 503
6.8.11 Vector valued martingales 504
6.9 Further topics 506
6.9.1 Topological properties of Banach Spaces 506
6.9.2 Topological classification of Banach spaces 510
6.9.3 The analytic Radon Nikodym property 513
6.9.4 M ideals 515
6.9.5 Stable Banach spaces 519
6.9.6 Three space properties 520
6.9.7 Functors in categories of Banach Spaces 522
6.9.8 Local spectral theory 523
6.9.9 Hankel and Toeplitz Operators 526
6.9.10 Composition Operators 531
6.9.11 Methods of summability 533
6.9.12 Lacunarity 535
6.9.13 The linear group of a Banach space 537
6.9.14 Manifolds modeled on Banach spaces 540
6.9.15 Asymptotic geometric analysis 543
6.9.16 Operator Spaces 545
6.9.17 Omissions 549
7 MlSCELLANEOUS TOPICS 550
7.1 Banach Space theory as a part of mathematics 550
7.2 Spaces versus Operators 559
7.3 Modern techniques of Banach Space theory 562
7.3.1 Probabilistic methods 562
7.3.2 Combinatorial methods 564
7.4 Counterexamples 566
7.4.1 A selection of typical counterexamples 567
7.4.2 Spaces without the approximation property 569
7.4.3 Tsirelson's space 570
7.4.4 The distortion problem 572
7.4.5 The Gowers Maurey story 574
7.4.6 A few counterexamples of Operator theory 578
7.5 Banach spaces and axiomatic set theory 579
8 Mathematics Is Made by Mathematicians 589
8.1 Victims of politics 590
8.2 Scientific schools in Banach space theory 591
8.2.1 Poland 591
8.2.2 USA 596
8.2.3 The former Soviet Union, mainly Russia and Ukraine 604
8.2.4 Israel 612
8.2.5 France 614
8.2.6 United Kingdom 617
8.2.7 Further countries 619
8.2.8 Germany 630
8.3 Short biographies of some famous mathematicians 635
8.3.1 Frigyes Riesz 635
8.3.2 Eduard Helly 637
8.3.3 Stefan Banach 638
8.3.4 John von Neumann 640
8.3.5 Mark Grigorievich Krem 641
8.3.6 Alexandre Grothendieck 642
8.3.7 Nicolas Bourbaki 645
8.4 Banach space theory at the ICMs 648
8.5 The Banach space archive 650
8.6 Banach space mathematicians 651
8.7 Anniversary volumes and articles, obituaries 663
chronology 673
Original Quotations 680
BlBLIOGRAPHY 683
Textbooks and monographs 684
Historical and biographical books 709
Collected and selected works 712
Collections 716
Seminars 722
Anonymous works 723
Mathematical papers 725
Historical and biographical papers 822
Coauthors 830
Index 834 |
| adam_txt |
PREFACE xiv
Notation and Terminology xvm
INTRODUCTION xix
1 The Birth of Banach Spaces 1
1.1 Complete normed linear Spaces 1
1.2 Linear Spaces 4
1.3 Metric spaces 6
1.4 Minkowski spaces 8
1.5 Hubert spaces 9
1.6 Albert A. Bennett and Kenneth W. Lamson 14
1.7 Norbert Wiener 18
1.8 Eduard Helly and Hans Hahn 21
1.9 Summary 22
2 HlSTORICAL ROOTS AND BASIC RESULTS 25
2.1 Operators 25
2.2 Functionals and dual Operators 30
2.3 The moment problem and the Hahn Banach theorem 36
2.4 The uniform boundedness principle 40
2.5 The closed graph theorem and the open mapping theorem 43
2.6 Riesz Schauder theory 45
2.6.1 Completely continuous Operators 45
2.6.2 Finite rank Operators 46
2.6.3 Approximable Operators 47
2.6.4 Compact Operators 49
2.6.5 Resolvents, spectra, and eigenvalues 52
2.6.6 Classical Operator ideals 53
2.7 Banach's monograph 54
vii
3 TOPOLOGICAL CONCEPTS WEAK TOPOLOGIES 56
3.1 Weakly convergent sequences 56
3.2 Topological Spaces and topological linear Spaces 58
3.2.1 Topological Spaces 58
3.2.2 Nets and filters 60
3.2.3 Compactness 63
3.2.4 Topological linear Spaces 65
3.2.5 Locally bounded linear Spaces 66
3.3 Locally convex linear Spaces and duality 68
3.3.1 Locally convex linear Spaces 68
3.3.2 Weak topologies and dual Systems 69
3.3.3 Separation of convex sets 71
3.3.4 Topologies on £{X,Y) 73
3.4 Weak* and weak compactness 75
3.4.1 Tychonoff's theorem 75
3.4.2 Weak* compactness theorem 76
3.4.3 Weak compactness and reflexivity 78
3.5 Weak sequential completeness and the Schur property 81
3.6 Transfinitely closed sets 82
4 Classical Banach Spaces 86
4.1 Banach lattices 86
4.2 Measures and integrals on abstract sets 92
4.2.1 Set theoretic operations 92
4.2.2 Measures 95
4.2.3 From measures to integrals 98
4.2.4 Integrals and the Banach Spaces L\ 99
4.2.5 Banach Spaces of additive set functions 101
4.3 The duality between L\ and L«, 103
4.4 The Banach Spaces Lp 106
4.5 Banach spaces of continuous functions 109
4.6 Measures and integrals on topological Spaces 111
4.7 Measures versus integrals 117
4.8 Abstract Lp and M spaces 118
4.8.1 Boolean algebras 118
4.8.2 Measure algebras 120
4.8.3 Abstract Lp spaces 124
4.8.4 Abstract A/ spaces 127
4.8.5 The Dunford Pettis property 128
4.9 Structure theory 129
4.9.1 Isomorphisms, injections, surjections, and projections 129
4.9.2 Extensions and liftings 133
4.9.3 Isometric and isomorphic classification 134
4.10 Operator ideals and Operator algebras 137
4.10.1 Schatten von Neumann ideals 137
4.10.2 Banach algebras 139
4.10.3 B* algebras = C* algebras 142
4.10.4 W* algebras 144
4.11 Complexification 155
5 Basic Results from the Post Banach Period 158
5.1 Analysis in Banach Spaces 158
5.1.1 Convergence of series 158
5.1.2 Integration of vector valued functions 160
5.1.3 Representation of Operators from L\ intoX 165
5.1.4 The Radon Nikodym property: analytic aspects 171
5.1.5 Representation of Operators from Lp irAoX 174
5.1.6 Representation of Operators fromC(Ä') intoX 175
5.1.7 Vector valued analytic functions on the complex plane 176
5.1.8 Gäteaux and Frechet derivatives 177
5.1.9 Polynomials and derivatives of higher order 180
5.1.10 Analytic functions on Banach Spaces 185
5.2 Spectral theory 189
5.2.1 Operational calculus 189
5.2.2 Fredholm Operators 192
5.2.3 Riesz Operators 198
5.2.4 Invariant subspaces 202
5.2.5 Spectral Operators 206
5.3 Semi groups of Operators 209
5.3.1 Deterministic and stochastic processes 209
5.3.2 The Hille Yosida theorem 211
5.3.3 Analytic semi groups 214
5.3.4 The abstract Cauchy problem 215
5.3.5 Ergodic theory 216
5.4 Convexity, extreme points, and related topics 219
5.4.1 The Krein Milman theorem 219
5.4.2 Integral representations 222
5.4.3 Gelfand Naimark Segal representations 226
5.4.4 The Radon Nikodym property: geometric aspects 227
5.4.5 Convex and concave functions 231
5.4.6 Lyapunov's theorem and the bang bang principle 232
5.5 Geometry of the unit ball 233
5.5.1 Strict convexity and smoothness 233
5.5.2 Uniform convexity and uniform smoothness 236
5.5.3 Further concepts related to convexity and smoothness 239
5.5.4 Applications of convexity and smoothness 242
5.5.5 Complex convexity 245
3 TOPOLOGICAL CONCEPTS WEAK TOPOLOGIES 56
3.1 Weakly convergent sequences 56
3.2 Topological Spaces and topological linear Spaces 58
3.2.1 Topological spaces 58
3.2.2 Nets and filtere 60
3.2.3 Compactness 63
3.2.4 Topological linear Spaces 65
3.2.5 Locally bounded linear Spaces 66
3.3 Locally convex linear spaces and duality 68
3.3.1 Locally convex linear spaces 68
3.3.2 Weak topologies and dual Systems 69
3.3.3 Separation of convex sets 71
3.3.4 Topologies on £(X,Y) 73
3.4 Weak* and weak compactness 75
3.4.1 Tychonoff's theorem 75
3.4.2 Weak* compactness theorem 76
3.4.3 Weak compactness and reflexivity 78
3.5 Weak sequential completeness and the Schur property 81
3.6 Transfinitely closed sets 82
4 Classical Banach Spaces 86
4.1 Banach lattices 86
4.2 Measures and integrals on abstract sets 92
4.2.1 Set theoretic operations 92
4.2.2 Measures 95
4.2.3 From measures to integrals 98
4.2.4 Integrals and the Banach spaces L\ 99
4.2.5 Banach Spaces of additive set functions 101
4.3 The duality between L\ and Lx 103
4.4 The Banach spaces Lp 106
4.5 Banach spaces of continuous functions 109
4.6 Measures and integrals on topological spaces 111
4.7 Measures versus integrals 117
4.8 Abstract Lp and M spaces 118
4.8.1 Boolean algebras 118
4.8.2 Measure algebras 120
4.8.3 Abstract Lp spaces 124
4.8.4 Abstract M spaces 127
4.8.5 The Dunford Pettis property 128
4.9 Structure theory 129
4.9.1 Isomorphisms, injections, surjections, and projections 129
4.9.2 Extensions and liftings 133
4.9.3 Isometric and isomorphic classification 134
4.10 Operator ideals and Operator algebras 137
4.10.1 Schatten von Neumann ideals 137
4.10.2 Banach algebras 139
4.10.3 ß* algebras = C* algebras 142
4.10.4 W* algebras 144
4.11 Complexification 155
5 Basic Results from the Post Banach Period 158
5.1 Analysis in Banach spaces 158
5.1.1 Convergence of series 158
5.1.2 Integration of vector valued functions 160
5.1.3 Representation of Operators from Lj intoX 165
5.1.4 The Radon Nikodym property: analytic aspects 171
5.1.5 Representation of Operators from Lp intoX 174
5.1.6 Representation of Operators fromC(^) intoX 175
5.1.7 Vector valued analytic functions on the complex plane 176
5.1.8 Gäteaux and Frechet derivatives 177
5.1.9 Polynomials and derivatives of higher order 180
5.1.10 Analytic functions on Banach Spaces 185
5.2 Spectral theory 189
5.2.1 Operational calculus 189
5.2.2 Fredholm Operators 192
5.2.3 Riesz Operators 198
5.2.4 Invariant subspaces 202
5.2.5 Spectral Operators 206
5.3 Semi groups of Operators 209
5.3.1 Deterministic and stochastic processes 209
5.3.2 The Hille Yosida theorem 211
5.3.3 Analytic semi groups 214
5.3.4 The abstract Cauchy problem 215
5.3.5 Ergodic theory 216
5.4 Convexity, extreme points, and related topics 219
5.4.1 The Krein Milman theorem 219
5.4.2 Integral representations 222
5.4.3 Gelfand Naimark Segal representations 226
5.4.4 The Radon Nikodym property: geometric aspects 227
5.4.5 Convex and concave functions 231
5.4.6 Lyapunov's theorem and the bang bang principle 232
5.5 Geometry of the unit ball 233
5.5.1 Strict convexity and smoothness 233
5.5.2 Uniform convexity and uniform smoothness 236
5.5.3 Further concepts related to convexity and smoothness 239
5.5.4 Applications of convexity and smoothness 242
5.5.5 Complex convexity 245
5.6 Bases 245
5.6.1 Schauder bases and basic sequences 245
5.6.2 Biorthogonal Systems 248
5.6.3 Bases and structure theory 249
5.6.4 Bases in concrete Banach Spaces 256
5.6.5 Unconditional basic sequences in Hubert Spaces 261
5.6.6 Wavelets 263
5.6.7 Schauder decompositions 268
5.7 Tensor products and approximation properties 269
5.7.1 Bilinear mappings 269
5.7.2 Tensor products 271
5.7.3 Nuclear and integral Operators 275
5.7.4 Approximation properties 280
6 Modern Banach Space Theory Selected Topics . 288
6.1 Geometry of Banach spaces 288
6.1.1 Banach Mazur distance, projection, and basis constants 289
6.1.2 Dvoretzky's theorem 293
6.1.3 Finite representability, ultraproducts, and spreading modeis 296
6.1.4 ^, spaces 302
6.1.5 Local unconditional structure 304
6.1.6 Banach spaces containing Z^'s uniformly 305
6.1.7 Rademacher type and cotype, Gauss type and cotype 307
6.1.8 Fourier type and cotype, Walsh type and cotype 315
6.1.9 Superreflexivity, Haar type and cotype 318
6.1.10 UMD spaces = HT spaces 321
6.1.11 Volume ratios and Grothendieck numbers 323
6.2 s Numbers 326
6.2.1 s Numbers of Operators on Hubert space 326
6.2.2 Axiomatics of s numbers 327
6.2.3 Examples of s numbers 327
6.2.4 Entropy numbers 331
6.2.5 j Numbers of diagonal Operators 333
6.2.6 i Numbers versus widths 336
6.3 Operator ideals 336
6.3.1 Ideals on Hubert space 336
6.3.2 Basic concepts of ideal theory on Banach spaces 341
6.3.3 Ideals associated with .s numbers 345
6.3.4 Local theory of quasi Banach ideals and trace duality 348
6.3.5 p Factorable Operators 353
6.3.6 p Summing Operators 355
6.3.7 p Nuclear and p integral Operators 361
6.3.8 Specific components of Operator ideals 363
6.3.9 Grothendieck's theorem 365
6.3.10 Limit order of Operator ideals 368
6.3.11 Banach ideals and tensor products 370
6.3.12 Ideal norms computed with n vectors 374
6.3.13 Operator ideals and classes of Banach Spaces 375
6.3.14 Rademacher type and cotype, Gauss type and cotype 379
6.3.15 Fourier type and cotype, Walsh type and cotype 382
6.3.16 Super weakly corapact Operators, Haar type and cotype 383
6.3.17 UMD Operators and HT Operators 386
6.3.18 Radon Nikodym property: operator theoretic aspects 387
6.3.19 Ideal norms and parameters of Minkowski Spaces 387
6.3.20 Ideal norms of finite rank Operators 389
6.3.21 Operator ideals and classes of locally convex linear Spaces 391
6.4 Eigenvalue distributions 393
6.4.1 Eigenvalue sequences and classical results 393
6.4.2 Inequalities between .s numbers and eigenvalues 394
6.4.3 Eigenvalues of /? summing Operators 396
6.4.4 Eigenvalues of nuclear Operators 398
6.4.5 Operators of eigenvalue type lp q 399
6.5 Traces and determinants 402
6.5.1 Traces 402
6.5.2 Fredholm denominators and determinants 405
6.5.3 Regularized Fredholm denominators 412
6.5.4 The Gohberg Goldberg Krupnik approach 414
6.5.5 Eigenvalues and zeros of entire functions 415
6.5.6 Completeness of root vectors 418
6.5.7 Determinants: pros and cons 424
6.6 Interpolation theory 425
6.6.1 Classical convexity theorems 425
6.6.2 Interpolation methods 426
6.6.3 Complex and real interpolation methods 427
6.6.4 Lorentz Spaces 431
6.6.5 Applications of interpolation theory 433
6.6.6 Interpolation of Operator ideals 435
6.6.7 New trends in interpolation theory 436
6.7 Function Spaces 440
6.7.1 Hölder Lipschitz Spaces 440
6.7.2 Sobolev Spaces 442
6.7.3 Besov Spaces 445
6.7.4 Lizorkin Triebel Spaces 450
6.7.5 Interpolation of function Spaces 453
6.7.6 Spaces of smooth functions: Supplements 454
6.7.7 Bases of Besov Spaces 458
6.7.8 Embedding Operators 459
6.7.9 Spaces of vector valued functions 464
6.7.10 Integral Operators 466
6.7.11 Differential Operators 469
6.7.12 Hardy Spaces 473
6.7.13 Bergman Spaces 481
6.7.14 Orlicz spaces 485
6.8 Probability theory on Banach spaces 489
6.8.1 Baire, Borel, and Radon measures 489
6.8.2 Cylindrical measures 491
6.8.3 Characteristic functionals 493
6.8.4 Radonifying Operators 494
6.8.5 Gaussian measures 496
6.8.6 Wiener measure 497
6.8.7 Vector valued random variables 498
6.8.8 Rademacher series 501
6.8.9 The law of large numbers 502
6.8.10 The central limit theorem 503
6.8.11 Vector valued martingales 504
6.9 Further topics 506
6.9.1 Topological properties of Banach Spaces 506
6.9.2 Topological classification of Banach spaces 510
6.9.3 The analytic Radon Nikodym property 513
6.9.4 M ideals 515
6.9.5 Stable Banach spaces 519
6.9.6 Three space properties 520
6.9.7 Functors in categories of Banach Spaces 522
6.9.8 Local spectral theory 523
6.9.9 Hankel and Toeplitz Operators 526
6.9.10 Composition Operators 531
6.9.11 Methods of summability 533
6.9.12 Lacunarity 535
6.9.13 The linear group of a Banach space 537
6.9.14 Manifolds modeled on Banach spaces 540
6.9.15 Asymptotic geometric analysis 543
6.9.16 Operator Spaces 545
6.9.17 Omissions 549
7 MlSCELLANEOUS TOPICS 550
7.1 Banach Space theory as a part of mathematics 550
7.2 Spaces versus Operators 559
7.3 Modern techniques of Banach Space theory 562
7.3.1 Probabilistic methods 562
7.3.2 Combinatorial methods 564
7.4 Counterexamples 566
7.4.1 A selection of typical counterexamples 567
7.4.2 Spaces without the approximation property 569
7.4.3 Tsirelson's space 570
7.4.4 The distortion problem 572
7.4.5 The Gowers Maurey story 574
7.4.6 A few counterexamples of Operator theory 578
7.5 Banach spaces and axiomatic set theory 579
8 Mathematics Is Made by Mathematicians 589
8.1 Victims of politics 590
8.2 Scientific schools in Banach space theory 591
8.2.1 Poland 591
8.2.2 USA 596
8.2.3 The former Soviet Union, mainly Russia and Ukraine 604
8.2.4 Israel 612
8.2.5 France 614
8.2.6 United Kingdom 617
8.2.7 Further countries 619
8.2.8 Germany 630
8.3 Short biographies of some famous mathematicians 635
8.3.1 Frigyes Riesz 635
8.3.2 Eduard Helly 637
8.3.3 Stefan Banach 638
8.3.4 John von Neumann 640
8.3.5 Mark Grigorievich Krem 641
8.3.6 Alexandre Grothendieck 642
8.3.7 Nicolas Bourbaki 645
8.4 Banach space theory at the ICMs 648
8.5 The Banach space archive 650
8.6 Banach space mathematicians 651
8.7 Anniversary volumes and articles, obituaries 663
chronology 673
Original Quotations 680
BlBLIOGRAPHY 683
Textbooks and monographs 684
Historical and biographical books 709
Collected and selected works 712
Collections 716
Seminars 722
Anonymous works 723
Mathematical papers 725
Historical and biographical papers 822
Coauthors 830
Index 834 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Pietsch, Albrecht 1934-2024 |
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| callnumber-first | Q - Science |
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| callnumber-raw | QA322.2 |
| callnumber-search | QA322.2 |
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| callnumber-subject | QA - Mathematics |
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| ctrlnum | (OCoLC)180884634 (DE-599)DNB971880808 |
| dewey-full | 515.732 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.732 |
| dewey-search | 515.732 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| era | Geschichte 1950-2006 gnd |
| era_facet | Geschichte 1950-2006 |
| format | Book |
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| id | DE-604.BV022460762 |
| illustrated | Illustrated |
| index_date | 2024-07-02T17:40:05Z |
| indexdate | 2024-07-20T08:26:48Z |
| institution | BVB |
| isbn | 9780817643676 0817643672 9780817645960 0817645969 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015668447 |
| oclc_num | 180884634 |
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| owner_facet | DE-12 DE-91G DE-BY-TUM DE-824 DE-19 DE-BY-UBM DE-83 DE-20 DE-188 |
| physical | XXIII, 855 S. graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Birkhäuser |
| record_format | marc |
| spelling | Pietsch, Albrecht 1934-2024 Verfasser (DE-588)104515996 aut History of Banach spaces and linear operators Albrecht Pietsch Boston, Mass. [u.a.] Birkhäuser 2007 XXIII, 855 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Literaturverz. S. 683 - 829 Geschichte 1950-2006 gnd rswk-swf Geschichte Banach spaces History Linear operators History Banach-Raum (DE-588)4004402-6 gnd rswk-swf Linearer Operator (DE-588)4167721-3 gnd rswk-swf Banach-Raum (DE-588)4004402-6 s Geschichte 1950-2006 z DE-604 Linearer Operator (DE-588)4167721-3 s b DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015668447&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Pietsch, Albrecht 1934-2024 History of Banach spaces and linear operators Geschichte Banach spaces History Linear operators History Banach-Raum (DE-588)4004402-6 gnd Linearer Operator (DE-588)4167721-3 gnd |
| subject_GND | (DE-588)4004402-6 (DE-588)4167721-3 |
| title | History of Banach spaces and linear operators |
| title_auth | History of Banach spaces and linear operators |
| title_exact_search | History of Banach spaces and linear operators |
| title_exact_search_txtP | History of Banach spaces and linear operators |
| title_full | History of Banach spaces and linear operators Albrecht Pietsch |
| title_fullStr | History of Banach spaces and linear operators Albrecht Pietsch |
| title_full_unstemmed | History of Banach spaces and linear operators Albrecht Pietsch |
| title_short | History of Banach spaces and linear operators |
| title_sort | history of banach spaces and linear operators |
| topic | Geschichte Banach spaces History Linear operators History Banach-Raum (DE-588)4004402-6 gnd Linearer Operator (DE-588)4167721-3 gnd |
| topic_facet | Geschichte Banach spaces History Linear operators History Banach-Raum Linearer Operator |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015668447&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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