The metric theory of tensor products: Grothendieck's résumé revisited
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2008]
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 278 Seiten Diagramme |
| ISBN: | 9780821844403 |
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MARC
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| 245 | 1 | 0 | |a The metric theory of tensor products |b Grothendieck's résumé revisited |c Joe Diestel ; Jan H. Fourie ; Johan Swart |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c [2008] | |
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Record in the Search Index
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Contents
Preface vii
Chapter 1. Basics on tensor norms 1
The algebraic preliminaries 1
1.1. Reasonable crossnorms, including the norms A and V 5
1.1.1. Definitions 5
1.1.2. Injectivity of V and projectivity of A 11
1.1.3. The universal mapping property of g and the dual of X ® Y 13
1.1.4. Examples: C(K) g X and L1^) ® X 14
v
1.1.5. Integral bilinear forms and the dual of X 8 Y 22
1.2. Definition of ®-norms 25
1.2.1. Fundamental operations on g -norms 26
1.2.2. Order relations among ®-norms 28
1.3. Extension of ®-norms to spaces of infinite dimensions 29
1.3.1. Metric accessibility and accessibility 32
1.4. Bilinear forms and linear operators of type a 40
1.4.1. General properties of a-forms 42
1.4.2. General properties of a-integral operators 47
1.4.3. Composition of a-integral and a-integral operators 48
1.4.4. Accessibility and metric accessibility (continued) 50
1.5. a-nuclear forms and operators 54
1.6. The Dvoretzky-Rogers theorem, Grothendieck-style 59
1.6.1. The fundamental lemma 59
1.6.2. Consequences 63
Chapter 2. The role of C(ii')-spaces and Lx-spaces 67
2.1. Complements on A and V 67
2.1.1. Representability, equimeasurability and nuclearity 72
2.2. Fundamental linear topological properties of C- and L-spaces 76
Notes 81
2.3. Injective and projective ®-norms 84
2.4. Formation of new ®-norms 90
2.5. Complements on /A, A\, / A \, \V, V/, \ V / 101
2.6. A table of natural g -norms 106
Chapter 3. ®-norms related to Hilbert space 111
3.1. Definitions and generalities about H and H* 111
3.2. Hermitian H-forms 119
3.3. Hermitian H*-forms 122
iii
iv CONTENTS
3.4. Basic relations between H, H*, etc. 131
3.5. The "little" Grothendieck inequality 132
3.6. The classes of a-integral operators between Hilbert spaces 142
Chapter 4. The Fundamental Theorem and its consequences 149
4.1. Functions of type a 149
4.2. The Fundamental Theorem (Grothendieck's inequality)
and its variants 152
4.3. Consequences to the theory of linear operators 159
4.3.1. Compositions of operators between spaces of type C, L and H 159
4.3.2. Linear topological characterizations of Hilbert space 160
4.3.3. A theorem of Littlewood 161
4.4. A table of the fourteen natural g -norms 163
4.4.1. A summary with regards to the characterizations and factorization
schemes of the various classes of integral operators 164
4.4.2. There are at most 14 natural ®-norms 165
4.4.3. There are exactly 14 natural g -norms 166
Notes and remarks on the complexification of tensor norms 168
4.5. Notes and remarks on the natural tensor norms and
Banach Algebras 168
Further notes and remarks 175
Glossary of terms 177
Appendix A. The problems of the Resume 183
A.I. Problem 1: The approximation problem 183
A.2. Problem 2: The possible reduction of the table
of "natural" tensor norms 186
A.3. Problem 3: Grothendieck's inequality and the "best" constant 191
A.4. Problem 4: Algebraic-topological properties of C*-algebras 201
A.5. Problem 5: Characterizing classes of spaces by the behavior of
tensor products and the action of operators on the spaces 208
A.6. Problem 6: Comparison of the projective and
injective tensor products 209
Appendix B. The Blaschke selection principle and compact convex sets in
finite dimensional Banach spaces 211
B.I. Blaschke's Selection Principle 211
B.2. Compact sets in Euclidean spaces 212
B.3. Ellipsoids in finite dimensional Banach spaces 216
Appendix C. A short introduction to Banach lattices 217
C.I. The facts, ma'm, just the facts 217
C.2. Some basics about duality in Banach lattices 218
C.3. Lattice homomorphisms 222
C.4. AM-spaces and AL-spaces 225
C.5. Kakutani's vector lattice version of the Stone-Weierstrass theorem 227
C.6. Kakutani's characterization of AM-spaces with unit 228
C.7. AL-spaces: The Freudenthal-Kakutani theorem 229
C.8. Kakutani's characterization of .AL-spaces 234
CONTENTS v
C.9. Grothendieck's inequality for Banach lattices 235
Notes and remarks 240
Appendix D. Stonean spaces and injectivity 241
D.I. The Nakano Stone Theorem 241
D.2. Injective Banach spaces 242
Notes and remarks 253
Epilogue 255
References to operator spaces 259
Monographs 259
Papers 260
Bibliography 261
Author Index 271
Index of Notation 273
Index 277 |
| adam_txt |
Contents
Preface vii
Chapter 1. Basics on tensor norms 1
The algebraic preliminaries 1
1.1. Reasonable crossnorms, including the norms A and V 5
1.1.1. Definitions 5
1.1.2. Injectivity of V and projectivity of A 11
1.1.3. The universal mapping property of g and the dual of X ® Y 13
1.1.4. Examples: C(K) g X and L1^) ® X 14
v
1.1.5. Integral bilinear forms and the dual of X 8 Y 22
1.2. Definition of ®-norms 25
1.2.1. Fundamental operations on g -norms 26
1.2.2. Order relations among ®-norms 28
1.3. Extension of ®-norms to spaces of infinite dimensions 29
1.3.1. Metric accessibility and accessibility 32
1.4. Bilinear forms and linear operators of type a 40
1.4.1. General properties of a-forms 42
1.4.2. General properties of a-integral operators 47
1.4.3. Composition of a-integral and a-integral operators 48
1.4.4. Accessibility and metric accessibility (continued) 50
1.5. a-nuclear forms and operators 54
1.6. The Dvoretzky-Rogers theorem, Grothendieck-style 59
1.6.1. The fundamental lemma 59
1.6.2. Consequences 63
Chapter 2. The role of C(ii')-spaces and Lx-spaces 67
2.1. Complements on A and V 67
2.1.1. Representability, equimeasurability and nuclearity 72
2.2. Fundamental linear topological properties of C- and L-spaces 76
Notes 81
2.3. Injective and projective ®-norms 84
2.4. Formation of new ®-norms 90
2.5. Complements on /A, A\, / A \, \V, V/, \ V / 101
2.6. A table of natural g -norms 106
Chapter 3. ®-norms related to Hilbert space 111
3.1. Definitions and generalities about H and H* 111
3.2. Hermitian H-forms 119
3.3. Hermitian H*-forms 122
iii
iv CONTENTS
3.4. Basic relations between H, H*, etc. 131
3.5. The "little" Grothendieck inequality 132
3.6. The classes of a-integral operators between Hilbert spaces 142
Chapter 4. The Fundamental Theorem and its consequences 149
4.1. Functions of type a 149
4.2. The Fundamental Theorem (Grothendieck's inequality)
and its variants 152
4.3. Consequences to the theory of linear operators 159
4.3.1. Compositions of operators between spaces of type C, L and H 159
4.3.2. Linear topological characterizations of Hilbert space 160
4.3.3. A theorem of Littlewood 161
4.4. A table of the fourteen natural g -norms 163
4.4.1. A summary with regards to the characterizations and factorization
schemes of the various classes of integral operators 164
4.4.2. There are at most 14 natural ®-norms 165
4.4.3. There are exactly 14 natural g -norms 166
Notes and remarks on the complexification of tensor norms 168
4.5. Notes and remarks on the natural tensor norms and
Banach Algebras 168
Further notes and remarks 175
Glossary of terms 177
Appendix A. The problems of the Resume 183
A.I. Problem 1: The approximation problem 183
A.2. Problem 2: The possible reduction of the table
of "natural" tensor norms 186
A.3. Problem 3: Grothendieck's inequality and the "best" constant 191
A.4. Problem 4: Algebraic-topological properties of C*-algebras 201
A.5. Problem 5: Characterizing classes of spaces by the behavior of
tensor products and the action of operators on the spaces 208
A.6. Problem 6: Comparison of the projective and
injective tensor products 209
Appendix B. The Blaschke selection principle and compact convex sets in
finite dimensional Banach spaces 211
B.I. Blaschke's Selection Principle 211
B.2. Compact sets in Euclidean spaces 212
B.3. Ellipsoids in finite dimensional Banach spaces 216
Appendix C. A short introduction to Banach lattices 217
C.I. The facts, ma'm, just the facts 217
C.2. Some basics about duality in Banach lattices 218
C.3. Lattice homomorphisms 222
C.4. AM-spaces and AL-spaces 225
C.5. Kakutani's vector lattice version of the Stone-Weierstrass theorem 227
C.6. Kakutani's characterization of AM-spaces with unit 228
C.7. AL-spaces: The Freudenthal-Kakutani theorem 229
C.8. Kakutani's characterization of .AL-spaces 234
CONTENTS v
C.9. Grothendieck's inequality for Banach lattices 235
Notes and remarks 240
Appendix D. Stonean spaces and injectivity 241
D.I. The Nakano Stone Theorem 241
D.2. Injective Banach spaces 242
Notes and remarks 253
Epilogue 255
References to operator spaces 259
Monographs 259
Papers 260
Bibliography 261
Author Index 271
Index of Notation 273
Index 277 |
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| id | DE-604.BV023338724 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T21:00:48Z |
| indexdate | 2024-07-20T08:12:47Z |
| institution | BVB |
| isbn | 9780821844403 |
| language | English |
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| physical | X, 278 Seiten Diagramme |
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| spelling | Diestel, Joseph 1943- Verfasser (DE-588)108479773 aut The metric theory of tensor products Grothendieck's résumé revisited Joe Diestel ; Jan H. Fourie ; Johan Swart Providence, Rhode Island American Mathematical Society [2008] © 2008 X, 278 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Banach-Raum (DE-588)4004402-6 gnd rswk-swf Tensorprodukt (DE-588)4059478-6 gnd rswk-swf Banach-Raum (DE-588)4004402-6 s Tensorprodukt (DE-588)4059478-6 s DE-604 Fourie, Jan H. Verfasser aut Swart, Johan Verfasser (DE-588)125964358 aut Erscheint auch als Online-Ausgabe 978-1-4704-2483-1 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016522540&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Diestel, Joseph 1943- Fourie, Jan H. Swart, Johan The metric theory of tensor products Grothendieck's résumé revisited Banach-Raum (DE-588)4004402-6 gnd Tensorprodukt (DE-588)4059478-6 gnd |
| subject_GND | (DE-588)4004402-6 (DE-588)4059478-6 |
| title | The metric theory of tensor products Grothendieck's résumé revisited |
| title_auth | The metric theory of tensor products Grothendieck's résumé revisited |
| title_exact_search | The metric theory of tensor products Grothendieck's résumé revisited |
| title_exact_search_txtP | The metric theory of tensor products Grothendieck's résumé revisited |
| title_full | The metric theory of tensor products Grothendieck's résumé revisited Joe Diestel ; Jan H. Fourie ; Johan Swart |
| title_fullStr | The metric theory of tensor products Grothendieck's résumé revisited Joe Diestel ; Jan H. Fourie ; Johan Swart |
| title_full_unstemmed | The metric theory of tensor products Grothendieck's résumé revisited Joe Diestel ; Jan H. Fourie ; Johan Swart |
| title_short | The metric theory of tensor products |
| title_sort | the metric theory of tensor products grothendieck s resume revisited |
| title_sub | Grothendieck's résumé revisited |
| topic | Banach-Raum (DE-588)4004402-6 gnd Tensorprodukt (DE-588)4059478-6 gnd |
| topic_facet | Banach-Raum Tensorprodukt |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016522540&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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