Functional analysis:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Acad. Press
1972
|
| Ausgabe: | 5. print. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 530 S. |
Internformat
MARC
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| 264 | 1 | |a New York [u.a.] |b Acad. Press |c 1972 | |
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Datensatz im Suchindex
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|---|---|
| adam_text | Contents
Preface v
chapter 1. Introduction to Inner Product Spaces I
1.1 Some Prerequisite Material and Conventions 3
1.2 Inner Product Spaces 7
1.3 Linear Functionals, the Riesz Representation Theorem, and
Adjoints 15
Exercises 1 18
References 19
chapter 2. Orthogonal Projections and the Spectral Theorem for
Normal Transformations 20
2.1 The Complexification 21
2.2 Orthogonal Projections and Orthogonal Direct Sums 25
2.3 Unitary and Orthogonal Transformations 33
Exercises 2 36
References 37
chapter 3. Normed Spaces and Metric Spaces 38
3.1 Norms and Normed Linear Spaces 39
3.2 Metrics and Metric Spaces 40
3.3 Topological Notions in Metric Spaces 43
3.4 Closed and Open Sets, Continuity, and Homeomorphisms 45
Exercises 3 48
Reference 49
chapter 4. Isometries and Completion of a Metric Space 50
4.1 Isometries and Homeomorphisms 51
4.2 Cauchy Sequences and Complete Metric Spaces 52
Exercises 4 58
Reference 59
vii
viii CONTENTS
chapter 5. Compactness in Metric Spaces 60
5.1 Nested Sequences and Complete Spaces 61
5.2 Relative Compactness e Nets and Totally Bounded Sets 64
5.3 Countable Compactness and Sequential Compactness 67
Exercises 5 72
References 73
chapter 6. Category and Separable Spaces 74
6.1 Fa Sets and Ga Sets 75
6.2 Nowhere Dense Sets and Category 76
6.3 The Existence of Functions Continuous Everywhere,
Differentiable Nowhere 80
6.4 Separable Spaces 82
Exercises 6 84
References 84
chapter 7. Topological Spaces 85
7.1 Definitions and Examples 86
7.2 Bases 90
7.3 Weak Topologies 93
7.4 Separation 96
7.5 Compactness 98
Exercises 7 103
References 107
chapter 8. Banach Spaces, Equivalent Norms, and Factor Spaces 108
8.1 The Holder and Minkowski Inequalities 109
8.2 Banach Spaces and Examples 112
8.3 The Completion of a Normed Linear Space 118
8.4 Generated Subspaces and Closed Subspaces 121
8.5 Equivalent Norms and a Theorem of Riesz 122
8.6 Factor Spaces 126
8.7 Completeness in the Factor Space 129
8.8 Convexity 131
Exercises 8 133
References 135
CONTENTS ix
chapter 9. Commutative Convergence, Hilbert Spaces, and Bessel s
Inequality 136
9.1 Commutative Convergence 138
9.2 Norms and Inner Products on Cartesian Products of Normed
and Inner Product Spaces 140
9.3 Hilbert Spaces 141
9.4 A Nonseparable Hilbert Space 143
9.5 Bessel s Inequality 144
9.6 Some Results from L2(0, 2tt) and the Riesz Fischer Theorem 146
9.7 Complete Orthonormal Sets 149
9.8 Complete Orthonormal Sets and Parseval s Identity 153
9.9 A Complete Orthonormal Set for L2(0, 2tt) 155
Appendix 9 157
Exercises 9 160
References 161
chapter 10. Complete Orthonormal Sets 162
10.1 Complete Orthonormal Sets and Parseval s Identity 163
10.2 The Cardinality of Complete Orthonormal Sets 166
10.3 A Note on the Structure of Hilbert Spaces 167
10.4 Closed Subspaces and the Projection Theorem for Hilbert
Spaces 168
Exercises 10 172
References 174
chapter 11. The Hahn Banach Theorem 175
11.1 The Hahn Banach Theorem 176
11.2 Bounded Linear Functionals 182
11.3 The Conjugate Space 184
Exercises 11 187
Appendix 11. The Problem of Measure and the Hahn Banach
Theorem 188
Exercises 11 Appendix 195
References 195
chapter 12. Consequences oi the Hahn Banach Theorem 196
12.1 Some Consequences of the Hahn Banach Theorem 197
12.2 The Second Conjugate Space 203
x CONTENTS
12.3 The Conjugate Space of lp 205
12.4 The Riesz Representation Theorem for Linear Functionals on
a Hilbert Space 209
12.5 Reflexivity of Hilbert Spaces 211
Exercises 12 214
References 215
chapter 13. The Conjugate Space of C[a, b] 216
13.1 A Representation Theorem for Bounded Linear Functionals
on C[a, b] 218
13.2 A List of Some Spaces and Their Conjugate Spaces 227
Exercises 13 227
References 229
chapter 14. Weak Convergence and Bounded Linear
Transformations 230
14.1 Weak Convergence 231
14.2 Bounded Linear Transformations 238
Exercises 14 243
References 244
chapter 15. Convergence in L(X, Y) and the Principle of Uniform
Boundedness 245
15.1 Convergence in L(X, Y) 246
15.2 The Principle of Uniform Boundedness 250
15.3 Some Consequences of the Principle of Uniform Boundedness 253
Exercises 15 257
References 258
chapter 16. Closed Transformations and the Closed Graph
Theorem 259
16.1 The Graph of a Mapping 260
16.2 Closed Linear Transformations and the Bounded Inverse
Theorem 261
16.3 Some Consequences of the Bounded Inverse Theorem 271
Appendix 16. Supplement to Theorem 16.5 273
Exercises 16 274
References 275
CONTENTS xi
chapter 17. Closures, Conjugate Transformations, and Complete
Continuity 276
17.1 The Closure of a Linear Transformation 277
17.2 A Class of Linear Transformations that Admit a Closure 279
17.3 The Conjugate Map of a Bounded Linear Transformation 281
17.4 Annihilators 283
17.5 Completely Continuous Operators; Finite Dimensional
Operators 286
17.6 Further Properties of Completely Continuous Transformations 289
Exercises 17 294
References 295
chapter 18. Spectral Notions 296
18.1 Spectra and the Resolvent Set 297
18.2 The Spectra of Two Particular Transformations 300
18.3 Approximate Proper Values 304
Exercises 18 304
References 306
chapter 19. Introduction to Banach Algebras 307
19.1 Analytic Vector Valued Functions 308
19.2 Normed and Banach Algebras 311
19.3 Banach Algebras with Identity 315
19.4 An Analytic Function — the Resolvent Operator 319
19.5 Spectral Radius and the Spectral Mapping Theorem for
Polynomials 322
19.6 The Gelfand Theory 327
19.7 Weak Topologies and the Gelfand Topology 336
19.8 Topological Vector Spaces and Operator Topologies 342
Exercises 19 348
References 350
chapter 20. Adjoints and Sesquilinear Functionals 351
20.1 The Adjoint Operator 352
20.2 Adjoints and Closures 355
20.3 Adjoints of Bounded Linear Transformations in Hilbert Spaces 362
xii CONTENTS
20.4 Sesquilinear Functional 367
Exercises 20 373
References 374
chapter 21. Some Spectral Results for Normal and Completely
Continuous Operators 375
21.1 A New Expression for the Norm of A eL(X, X) 376
21.2 Normal Transformations 379
21.3 Some Spectral Results for Completely Continuous Operators 383
21.4 Numerical Range 386
Exercises 21 389
Appendix to Chapter 21. The Fredholm Alternative Theorem
and the Spectrum of a Completely Continuous Transformation 390
A.I Motivation 391
A.2 The Fredholm Alternative Theorem 393
References 405
chapter 22. Orthogonal Projections and Positive Definite
Operators 406
22.1 Properties of Orthogonal Projections 407
22.2 Products of Projections 409
22.3 Positive Operators 411
22.4 Sums and Differences of Orthogonal Projections 412
22.5 The Product of Positive Operators 415
Exercises 22 417
References 418
chapter 23. Square Roots and a Spectral Decomposition Theorem 419
23.1 Square Root of Positive Operators 420
23.2 Spectral Theorem for Bounded, Normal, Finite Dimensional
Operators 426
Exercises 23 430
References 431
chapter 24. Spectral Theorem for Completely Continuous Normal
Operators 432
24.1 Infinite Orthogonal Direct Sums: Infinite Series of
Transformations 433
CONTENTS xiii
24.2 Spectral Decomposition Theorem for Completely Continuous
Normal Operators 438
Exercises 24 442
References 443
chapter 25. Spectral Theorem for Bounded, Self Adjoint
Operators 444
25.1 A Special Case — the Self Adjoint, Completely Continuous
Operator 445
25.2 Further Properties of the Spectrum of Bounded, Self Adjoint
Transformations 448
25.3 Spectral Theorem for Bounded, Self Adjoint Operators 450
Exercises 25 458
References 459
chapter 26. A Second Approach to the Spectral Theorem for
Bounded, Self Adjoint Operators 460
26.1 A Second Approach to the Spectral Theorem for Bounded,
Self Adjoint Operators 461
Exercises 26 469
References 470
chapter 27. A Third Approach to the Spectral Theorem for
Bounded, Self Adjoint Operators and Some
Consequences 471
27.1 A Third Approach to the Spectral Theorem for Bounded,
Self Adjoint Operators 472
27.2 Two Consequences of the Spectral Theorem 478
Exercises 27 483
References 483
chapter 28. Spectral Theorem for Bounded, Normal Operators 484
28.1 The Spectral Theorem for Bounded, Normal Operators on
a Hilbert Space 485
28.2 Spectral Measures; Unitary Transformations 488
Exercises 28 493
References 493
xiv CONTENTS
chapter 29. Spectral Theorem for Unbounded, Self Adjoint
Operators 494
29.1 Permutativity 495
29.2 The Spectral Theorem for Unbounded, Self Adjoint Operators 498
29.3 A Proof of the Spectral Theorem Using the Cayley Transform 516
29.4 A Note on the Spectral Theorem for Unbounded Normal
Operators 520
Exercises 29 521
References 522
Bibliography 523
Index of Symbols 525
Subject Index 527
|
| adam_txt |
Contents
Preface v
chapter 1. Introduction to Inner Product Spaces I
1.1 Some Prerequisite Material and Conventions 3
1.2 Inner Product Spaces 7
1.3 Linear Functionals, the Riesz Representation Theorem, and
Adjoints 15
Exercises 1 18
References 19
chapter 2. Orthogonal Projections and the Spectral Theorem for
Normal Transformations 20
2.1 The Complexification 21
2.2 Orthogonal Projections and Orthogonal Direct Sums 25
2.3 Unitary and Orthogonal Transformations 33
Exercises 2 36
References 37
chapter 3. Normed Spaces and Metric Spaces 38
3.1 Norms and Normed Linear Spaces 39
3.2 Metrics and Metric Spaces 40
3.3 Topological Notions in Metric Spaces 43
3.4 Closed and Open Sets, Continuity, and Homeomorphisms 45
Exercises 3 48
Reference 49
chapter 4. Isometries and Completion of a Metric Space 50
4.1 Isometries and Homeomorphisms 51
4.2 Cauchy Sequences and Complete Metric Spaces 52
Exercises 4 58
Reference 59
vii
viii CONTENTS
chapter 5. Compactness in Metric Spaces 60
5.1 Nested Sequences and Complete Spaces 61
5.2 Relative Compactness e Nets and Totally Bounded Sets 64
5.3 Countable Compactness and Sequential Compactness 67
Exercises 5 72
References 73
chapter 6. Category and Separable Spaces 74
6.1 Fa Sets and Ga Sets 75
6.2 Nowhere Dense Sets and Category 76
6.3 The Existence of Functions Continuous Everywhere,
Differentiable Nowhere 80
6.4 Separable Spaces 82
Exercises 6 84
References 84
chapter 7. Topological Spaces 85
7.1 Definitions and Examples 86
7.2 Bases 90
7.3 Weak Topologies 93
7.4 Separation 96
7.5 Compactness 98
Exercises 7 103
References 107
chapter 8. Banach Spaces, Equivalent Norms, and Factor Spaces 108
8.1 The Holder and Minkowski Inequalities 109
8.2 Banach Spaces and Examples 112
8.3 The Completion of a Normed Linear Space 118
8.4 Generated Subspaces and Closed Subspaces 121
8.5 Equivalent Norms and a Theorem of Riesz 122
8.6 Factor Spaces 126
8.7 Completeness in the Factor Space 129
8.8 Convexity 131
Exercises 8 133
References 135
CONTENTS ix
chapter 9. Commutative Convergence, Hilbert Spaces, and Bessel's
Inequality 136
9.1 Commutative Convergence 138
9.2 Norms and Inner Products on Cartesian Products of Normed
and Inner Product Spaces 140
9.3 Hilbert Spaces 141
9.4 A Nonseparable Hilbert Space 143
9.5 Bessel's Inequality 144
9.6 Some Results from L2(0, 2tt) and the Riesz Fischer Theorem 146
9.7 Complete Orthonormal Sets 149
9.8 Complete Orthonormal Sets and Parseval's Identity 153
9.9 A Complete Orthonormal Set for L2(0, 2tt) 155
Appendix 9 157
Exercises 9 160
References 161
chapter 10. Complete Orthonormal Sets 162
10.1 Complete Orthonormal Sets and Parseval's Identity 163
10.2 The Cardinality of Complete Orthonormal Sets 166
10.3 A Note on the Structure of Hilbert Spaces 167
10.4 Closed Subspaces and the Projection Theorem for Hilbert
Spaces 168
Exercises 10 172
References 174
chapter 11. The Hahn Banach Theorem 175
11.1 The Hahn Banach Theorem 176
11.2 Bounded Linear Functionals 182
11.3 The Conjugate Space 184
Exercises 11 187
Appendix 11. The Problem of Measure and the Hahn Banach
Theorem 188
Exercises 11 Appendix 195
References 195
chapter 12. Consequences oi the Hahn Banach Theorem 196
12.1 Some Consequences of the Hahn Banach Theorem 197
12.2 The Second Conjugate Space 203
x CONTENTS
12.3 The Conjugate Space of lp 205
12.4 The Riesz Representation Theorem for Linear Functionals on
a Hilbert Space 209
12.5 Reflexivity of Hilbert Spaces 211
Exercises 12 214
References 215
chapter 13. The Conjugate Space of C[a, b] 216
13.1 A Representation Theorem for Bounded Linear Functionals
on C[a, b] 218
13.2 A List of Some Spaces and Their Conjugate Spaces 227
Exercises 13 227
References 229
chapter 14. Weak Convergence and Bounded Linear
Transformations 230
14.1 Weak Convergence 231
14.2 Bounded Linear Transformations 238
Exercises 14 243
References 244
chapter 15. Convergence in L(X, Y) and the Principle of Uniform
Boundedness 245
15.1 Convergence in L(X, Y) 246
15.2 The Principle of Uniform Boundedness 250
15.3 Some Consequences of the Principle of Uniform Boundedness 253
Exercises 15 257
References 258
chapter 16. Closed Transformations and the Closed Graph
Theorem 259
16.1 The Graph of a Mapping 260
16.2 Closed Linear Transformations and the Bounded Inverse
Theorem 261
16.3 Some Consequences of the Bounded Inverse Theorem 271
Appendix 16. Supplement to Theorem 16.5 273
Exercises 16 274
References 275
CONTENTS xi
chapter 17. Closures, Conjugate Transformations, and Complete
Continuity 276
17.1 The Closure of a Linear Transformation 277
17.2 A Class of Linear Transformations that Admit a Closure 279
17.3 The Conjugate Map of a Bounded Linear Transformation 281
17.4 Annihilators 283
17.5 Completely Continuous Operators; Finite Dimensional
Operators 286
17.6 Further Properties of Completely Continuous Transformations 289
Exercises 17 294
References 295
chapter 18. Spectral Notions 296
18.1 Spectra and the Resolvent Set 297
18.2 The Spectra of Two Particular Transformations 300
18.3 Approximate Proper Values 304
Exercises 18 304
References 306
chapter 19. Introduction to Banach Algebras 307
19.1 Analytic Vector Valued Functions 308
19.2 Normed and Banach Algebras 311
19.3 Banach Algebras with Identity 315
19.4 An Analytic Function — the Resolvent Operator 319
19.5 Spectral Radius and the Spectral Mapping Theorem for
Polynomials 322
19.6 The Gelfand Theory 327
19.7 Weak Topologies and the Gelfand Topology 336
19.8 Topological Vector Spaces and Operator Topologies 342
Exercises 19 348
References 350
chapter 20. Adjoints and Sesquilinear Functionals 351
20.1 The Adjoint Operator 352
20.2 Adjoints and Closures 355
20.3 Adjoints of Bounded Linear Transformations in Hilbert Spaces 362
xii CONTENTS
20.4 Sesquilinear Functional 367
Exercises 20 373
References 374
chapter 21. Some Spectral Results for Normal and Completely
Continuous Operators 375
21.1 A New Expression for the Norm of A eL(X, X) 376
21.2 Normal Transformations 379
21.3 Some Spectral Results for Completely Continuous Operators 383
21.4 Numerical Range 386
Exercises 21 389
Appendix to Chapter 21. The Fredholm Alternative Theorem
and the Spectrum of a Completely Continuous Transformation 390
A.I Motivation 391
A.2 The Fredholm Alternative Theorem 393
References 405
chapter 22. Orthogonal Projections and Positive Definite
Operators 406
22.1 Properties of Orthogonal Projections 407
22.2 Products of Projections 409
22.3 Positive Operators 411
22.4 Sums and Differences of Orthogonal Projections 412
22.5 The Product of Positive Operators 415
Exercises 22 417
References 418
chapter 23. Square Roots and a Spectral Decomposition Theorem 419
23.1 Square Root of Positive Operators 420
23.2 Spectral Theorem for Bounded, Normal, Finite Dimensional
Operators 426
Exercises 23 430
References 431
chapter 24. Spectral Theorem for Completely Continuous Normal
Operators 432
24.1 Infinite Orthogonal Direct Sums: Infinite Series of
Transformations 433
CONTENTS xiii
24.2 Spectral Decomposition Theorem for Completely Continuous
Normal Operators 438
Exercises 24 442
References 443
chapter 25. Spectral Theorem for Bounded, Self Adjoint
Operators 444
25.1 A Special Case — the Self Adjoint, Completely Continuous
Operator 445
25.2 Further Properties of the Spectrum of Bounded, Self Adjoint
Transformations 448
25.3 Spectral Theorem for Bounded, Self Adjoint Operators 450
Exercises 25 458
References 459
chapter 26. A Second Approach to the Spectral Theorem for
Bounded, Self Adjoint Operators 460
26.1 A Second Approach to the Spectral Theorem for Bounded,
Self Adjoint Operators 461
Exercises 26 469
References 470
chapter 27. A Third Approach to the Spectral Theorem for
Bounded, Self Adjoint Operators and Some
Consequences 471
27.1 A Third Approach to the Spectral Theorem for Bounded,
Self Adjoint Operators 472
27.2 Two Consequences of the Spectral Theorem 478
Exercises 27 483
References 483
chapter 28. Spectral Theorem for Bounded, Normal Operators 484
28.1 The Spectral Theorem for Bounded, Normal Operators on
a Hilbert Space 485
28.2 Spectral Measures; Unitary Transformations 488
Exercises 28 493
References 493
xiv CONTENTS
chapter 29. Spectral Theorem for Unbounded, Self Adjoint
Operators 494
29.1 Permutativity 495
29.2 The Spectral Theorem for Unbounded, Self Adjoint Operators 498
29.3 A Proof of the Spectral Theorem Using the Cayley Transform 516
29.4 A Note on the Spectral Theorem for Unbounded Normal
Operators 520
Exercises 29 521
References 522
Bibliography 523
Index of Symbols 525
Subject Index 527 |
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| author | Bachman, George Narici, Lawrence 1941- |
| author_GND | (DE-588)121796361 |
| author_facet | Bachman, George Narici, Lawrence 1941- |
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| index_date | 2024-07-02T16:05:47Z |
| indexdate | 2024-07-09T20:47:28Z |
| institution | BVB |
| language | English |
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| spelling | Bachman, George Verfasser aut Functional analysis George Bachman and Lawrence Narici 5. print. New York [u.a.] Acad. Press 1972 XIV, 530 S. txt rdacontent n rdamedia nc rdacarrier Funktionalanalysis Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf Funktionalanalysis (DE-588)4018916-8 s DE-604 Narici, Lawrence 1941- Verfasser (DE-588)121796361 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015137505&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bachman, George Narici, Lawrence 1941- Functional analysis Funktionalanalysis Funktionalanalysis (DE-588)4018916-8 gnd |
| subject_GND | (DE-588)4018916-8 |
| title | Functional analysis |
| title_auth | Functional analysis |
| title_exact_search | Functional analysis |
| title_exact_search_txtP | Functional analysis |
| title_full | Functional analysis George Bachman and Lawrence Narici |
| title_fullStr | Functional analysis George Bachman and Lawrence Narici |
| title_full_unstemmed | Functional analysis George Bachman and Lawrence Narici |
| title_short | Functional analysis |
| title_sort | functional analysis |
| topic | Funktionalanalysis Funktionalanalysis (DE-588)4018916-8 gnd |
| topic_facet | Funktionalanalysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015137505&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT bachmangeorge functionalanalysis AT naricilawrence functionalanalysis |