History of Banach spaces and linear operators

877 Pages·2007·4.209 MB·English

by  Albrecht Pietsch

Albrecht Pietsch
History of Banach Spaces
and Linear Operators
Birkha¨user
Boston • Basel • Berlin
AlbrechtPietsch
MathematischesInstitut
Friedrich-Schiller-Universita¨t
D-07740Jena
Germany
MathematicsSubjectClassification:Primary:01A05,01A60,46Bxx,46Exx,46Gxx,46Lxx,46Mxx,
47Axx,47Bxx,47Dxx;Secondary:04A25,06E15,15A60,28B05,28C20,30D15,34L10,35P10,
40A05,41A15,41A65,42A20,42A55,58Bxx,60B11,60B12,60J65
LibraryofCongressControlNumber:2007922147
ISBN-10:0-8176-4367-2 e-ISBN-10:0-8176-4596-9
ISBN-13:978-0-8176-4367-6 e-ISBN-13:978-0-8176-4596-0
Printedonacid-freepaper.
(cid:2)c2007Birkha¨userBoston
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Contents
PREFACE ...................................................... xiv
NOTATION AND TERMINOLOGY ............................... xviii
INTRODUCTION ............................................... xix
1 THE BIRTH OF BANACH SPACES ............................ 1
1.1 Completenormedlinearspaces ............................... 1
1.2 Linearspaces .............................................. 4
1.3 Metricspaces ............................................. 6
1.4 Minkowskispaces .......................................... 8
1.5 Hilbertspaces ............................................. 9
1.6 AlbertA.BennettandKennethW.Lamson ...................... 14
1.7 NorbertWiener ............................................ 18
1.8 EduardHellyandHansHahn ................................. 21
1.9 Summary ................................................. 22
2 HISTORICAL ROOTS AND BASIC RESULTS .................. 25
2.1 Operators ................................................. 25
2.2 Functionalsanddualoperators ................................ 30
2.3 ThemomentproblemandtheHahn–Banachtheorem .............. 36
2.4 Theuniformboundednessprinciple ............................ 40
2.5 Theclosedgraphtheoremandtheopenmappingtheorem .......... 43
2.6 Riesz–Schaudertheory ...................................... 45
2.6.1 Completelycontinuousoperators .......................... 45
2.6.2 Finiterankoperators ................................... 46
2.6.3 Approximableoperators ................................ 47
2.6.4 Compactoperators .................................... 49
2.6.5 Resolvents,spectra,andeigenvalues ........................ 52
2.6.6 Classicaloperatorideals ................................ 53
2.7 Banach’smonograph ........................................ 54
vii
viii Contents
3 TOPOLOGICAL CONCEPTS – WEAK TOPOLOGIES ........... 56
3.1 Weaklyconvergentsequences ................................. 56
3.2 Topologicalspacesandtopologicallinearspaces .................. 58
3.2.1 Topologicalspaces .................................... 58
3.2.2 Netsandfilters ....................................... 60
3.2.3 Compactness ........................................ 63
3.2.4 Topologicallinearspaces ................................ 65
3.2.5 Locallyboundedlinearspaces ............................ 66
3.3 Locallyconvexlinearspacesandduality ........................ 68
3.3.1 Locallyconvexlinearspaces ............................. 68
3.3.2 Weaktopologiesanddualsystems ......................... 69
3.3.3 Separationofconvexsets ................................ 71
3.3.4 TopologiesonL(X,Y) .................................. 73
3.4 Weak∗andweakcompactness ................................ 75
3.4.1 Tychonoff’stheorem ................................... 75
3.4.2 Weak∗compactnesstheorem ............................. 76
3.4.3 Weakcompactnessandreflexivity ......................... 78
3.5 WeaksequentialcompletenessandtheSchurproperty ............. 81
3.6 Transfinitelyclosedsets ..................................... 82
4 CLASSICAL BANACH SPACES ............................... 86
4.1 Banachlattices ............................................ 86
4.2 Measuresandintegralsonabstractsets ......................... 92
4.2.1 Set-theoreticoperations ................................. 92
4.2.2 Measures ........................................... 95
4.2.3 Frommeasurestointegrals .............................. 98
4.2.4 IntegralsandtheBanachspacesL ......................... 99
1
4.2.5 Banachspacesofadditivesetfunctions ...................... 101
4.3 ThedualitybetweenL1andL∞ ................................ 103
4.4 TheBanachspacesLp ....................................... 106
4.5 Banachspacesofcontinuousfunctions ......................... 109
4.6 Measuresandintegralsontopologicalspaces .................... 111
4.7 Measuresversusintegrals .................................... 117
4.8 AbstractLp-andM-spaces ................................... 118
4.8.1 Booleanalgebras ...................................... 118
4.8.2 Measurealgebras ..................................... 120
4.8.3 AbstractLp-spaces .................................... 124
4.8.4 AbstractM-spaces ..................................... 127
4.8.5 TheDunford–Pettisproperty ............................. 128
Contents ix
4.9 Structuretheory ............................................ 129
4.9.1 Isomorphisms,injections,surjections,andprojections ........... 129
4.9.2 Extensionsandliftings ................................. 133
4.9.3 Isometricandisomorphicclassification ...................... 134
4.10 Operatoridealsandoperatoralgebras ......................... 137
4.10.1 Schatten–vonNeumannideals ............................ 137
4.10.2 Banachalgebras ...................................... 139
4.10.3 B(cid:2)-algebras=C(cid:2)-algebras ............................... 142
4.10.4 W(cid:2)-algebras ......................................... 144
4.11 Complexification .......................................... 155
5 BASIC RESULTS FROM THE POST-BANACH PERIOD ......... 158
5.1 AnalysisinBanachspaces ................................... 158
5.1.1 Convergenceofseries .................................. 158
5.1.2 Integrationofvector-valuedfunctions ....................... 160
5.1.3 RepresentationofoperatorsfromL intoX ................... 165
1
5.1.4 TheRadon–Nikodymproperty:analyticaspects ............... 171
5.1.5 RepresentationofoperatorsfromLpintoX ................... 174
5.1.6 RepresentationofoperatorsfromC(K)intoX ................. 175
5.1.7 Vector-valuedanalyticfunctionsonthecomplexplane ........... 176
5.1.8 GaˆteauxandFre´chetderivatives ........................... 177
5.1.9 Polynomialsandderivativesofhigherorder .................. 180
5.1.10 AnalyticfunctionsonBanachspaces ....................... 185
5.2 Spectraltheory ............................................ 189
5.2.1 Operationalcalculus ................................... 189
5.2.2 Fredholmoperators .................................... 192
5.2.3 Rieszoperators ....................................... 198
5.2.4 Invariantsubspaces .................................... 202
5.2.5 Spectraloperators ..................................... 206
5.3 Semi-groupsofoperators .................................... 209
5.3.1 Deterministicandstochasticprocesses ...................... 209
5.3.2 TheHille–Yosidatheorem ............................... 211
5.3.3 Analyticsemi-groups .................................. 214
5.3.4 TheabstractCauchyproblem ............................. 215
5.3.5 Ergodictheory ....................................... 216
5.4 Convexity,extremepoints,andrelatedtopics .................... 219
5.4.1 TheKre˘ın–Milmantheorem .............................. 219
5.4.2 Integralrepresentations ................................. 222
5.4.3 Gelfand–Na˘ımark–Segalrepresentations ..................... 226
5.4.4 TheRadon–Nikodymproperty:geometricaspects .............. 227
5.4.5 Convexandconcavefunctions ............................ 231
5.4.6 Lyapunov’stheoremandthebang-bangprinciple ............... 232
5.5 Geometryoftheunitball .................................... 233
5.5.1 Strictconvexityandsmoothness ........................... 233
5.5.2 Uniformconvexityanduniformsmoothness .................. 236
5.5.3 Furtherconceptsrelatedtoconvexityandsmoothness ........... 239
5.5.4 Applicationsofconvexityandsmoothness ................... 242
5.5.5 Complexconvexity .................................... 245
x Contents
5.6 Bases .................................................... 245
5.6.1 Schauderbasesandbasicsequences ........................ 245
5.6.2 Biorthogonalsystems .................................. 248
5.6.3 Basesandstructuretheory ............................... 249
5.6.4 BasesinconcreteBanachspaces .......................... 256
5.6.5 UnconditionalbasicsequencesinHilbertspaces ............... 261
5.6.6 Wavelets ........................................... 263
5.6.7 Schauderdecompositions ................................ 268
5.7 Tensorproductsandapproximationproperties .................... 269
5.7.1 Bilinearmappings ..................................... 269
5.7.2 Tensorproducts ...................................... 271
5.7.3 Nuclearandintegraloperators ............................ 275
5.7.4 Approximationproperties ............................... 280
6 MODERN BANACH SPACE THEORY – SELECTED TOPICS .... 288
6.1 GeometryofBanachspaces .................................. 288
6.1.1 Banach–Mazurdistance,projection,andbasisconstants .......... 289
6.1.2 Dvoretzky’stheorem ................................... 293
6.1.3 Finiterepresentability,ultraproducts,andspreadingmodels ....... 296
6.1.4 Lp-spaces .......................................... 302
6.1.5 Localunconditionalstructure ............................. 304
6.1.6 Banachspacescontainingln’suniformly ..................... 305
p
6.1.7 Rademachertypeandcotype,Gausstypeandcotype ............ 307
6.1.8 Fouriertypeandcotype,Walshtypeandcotype ................ 315
6.1.9 Superreflexivity,Haartypeandcotype ...................... 318
6.1.10 UMDspaces=HT spaces ............................... 321
6.1.11 VolumeratiosandGrothendiecknumbers .................... 323
6.2 s-Numbers ................................................ 326
6.2.1 s-NumbersofoperatorsonHilbertspace ..................... 326
6.2.2 Axiomaticsofs-numbers ................................ 327
6.2.3 Examplesofs-numbers ................................. 327
6.2.4 Entropynumbers ...................................... 331
6.2.5 s-Numbersofdiagonaloperators .......................... 333
6.2.6 s-Numbersversuswidths ................................ 336
6.3 Operatorideals ............................................ 336
6.3.1 IdealsonHilbertspace ................................. 336
6.3.2 BasicconceptsofidealtheoryonBanachspaces ............... 341
6.3.3 Idealsassociatedwiths-numbers .......................... 345
6.3.4 Localtheoryofquasi-Banachidealsandtraceduality ........... 348
6.3.5 p-Factorableoperators .................................. 353
6.3.6 p-Summingoperators .................................. 355
6.3.7 p-Nuclearandp-integraloperators ......................... 361
6.3.8 Specificcomponentsofoperatorideals ...................... 363
6.3.9 Grothendieck’stheorem ................................. 365
6.3.10 Limitorderofoperatorideals ............................. 368
6.3.11 Banachidealsandtensorproducts ......................... 370
6.3.12 Idealnormscomputedwithnvectors ....................... 374
6.3.13 OperatoridealsandclassesofBanachspaces ................. 375
Contents xi
6.3.14 Rademachertypeandcotype,Gausstypeandcotype ............ 379
6.3.15 Fouriertypeandcotype,Walshtypeandcotype ................ 382
6.3.16 Superweaklycompactoperators,Haartypeandcotype .......... 383
6.3.17 UMDoperatorsandHT operators .......................... 386
6.3.18 Radon–Nikodymproperty:operator-theoreticaspects ........... 387
6.3.19 IdealnormsandparametersofMinkowskispaces .............. 387
6.3.20 Idealnormsoffiniterankoperators ......................... 389
6.3.21 Operatoridealsandclassesoflocallyconvexlinearspaces ........ 391
6.4 Eigenvaluedistributions ..................................... 393
6.4.1 Eigenvaluesequencesandclassicalresults ................... 393
6.4.2 Inequalitiesbetweens-numbersandeigenvalues ............... 394
6.4.3 Eigenvaluesofp-summingoperators ....................... 396
6.4.4 Eigenvaluesofnuclearoperators .......................... 398
6.4.5 Operatorsofeigenvaluetypelp,q .......................... 399
6.5 Tracesanddeterminants ..................................... 402
6.5.1 Traces ............................................. 402
6.5.2 Fredholmdenominatorsanddeterminants .................... 405
6.5.3 RegularizedFredholmdenominators ........................ 412
6.5.4 TheGohberg–Goldberg–Krupnikapproach ................... 414
6.5.5 Eigenvaluesandzerosofentirefunctions .................... 415
6.5.6 Completenessofrootvectors ............................. 418
6.5.7 Determinants:prosandcons ............................. 424
6.6 Interpolationtheory ......................................... 425
6.6.1 Classicalconvexitytheorems ............................. 425
6.6.2 Interpolationmethods .................................. 426
6.6.3 Complexandrealinterpolationmethods ..................... 427
6.6.4 Lorentzspaces ....................................... 431
6.6.5 Applicationsofinterpolationtheory ........................ 433
6.6.6 Interpolationofoperatorideals ............................ 435
6.6.7 Newtrendsininterpolationtheory ......................... 436
6.7 Functionspaces ............................................ 440
6.7.1 Ho¨lder–Lipschitzspaces ................................ 440
6.7.2 Sobolevspaces ....................................... 442
6.7.3 Besovspaces ........................................ 445
6.7.4 Lizorkin–Triebelspaces ................................. 450
6.7.5 Interpolationoffunctionspaces ........................... 453
6.7.6 Spacesofsmoothfunctions:supplements .................... 454
6.7.7 BasesofBesovspaces .................................. 458
6.7.8 Embeddingoperators ................................... 459
6.7.9 Spacesofvector-valuedfunctions .......................... 464
6.7.10 Integraloperators ..................................... 466
6.7.11 Differentialoperators .................................. 469
6.7.12 Hardyspaces ........................................ 473
6.7.13 Bergmanspaces ...................................... 481
6.7.14 Orliczspaces ........................................ 485