2008;
278 pp;
Hardcover

MSC: Primary 46; 47;

Print ISBN: 978-0-8218-4440-3

Product Code: MBK/52

List Price: $86.00

AMS Member Price: $68.80

MAA Member Price: $77.40

**Electronic ISBN: 978-1-4704-2483-1
Product Code: MBK/52.E**

List Price: $86.00

AMS Member Price: $68.80

MAA Member Price: $77.40

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#### Supplemental Materials

# The Metric Theory of Tensor Products: Grothendieck’s Résumé Revisited

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*Joe Diestel; Jan H. Fourie; Johan Swart*

Grothendieck's Resumé is a landmark in functional analysis. Despite having appeared more than a half century ago, its techniques and results are still not widely known nor appreciated. This is due, no doubt, to the fact that Grothendieck included practically no proofs, and the presentation is based on the theory of the very abstract notion of tensor products. This book aims at providing the details of Grothendieck's constructions and laying bare how the important classes of operators are a consequence of the abstract operations on tensor norms. Particular attention is paid to how the classical Banach spaces (\(C(K)\)'s, Hilbert spaces, and the spaces of integrable functions) fit naturally within the mosaic that Grothendieck constructed.

#### Readership

Graduate students and research mathematicians interested in abstract analysis, Banach space theory, functional analysis, and operator theory.

#### Reviews & Endorsements

(The book) scores on several counts, not just as a serious scholarly contribution to functional analysis, but as a tribute to Grothendiecks incomparable gifts in the area of innovation and originality.

-- MAA Reviews

The exposition is clear, well-motivated and reasonably self-contained.

-- Mathematical Reviews

# Table of Contents

## The Metric Theory of Tensor Products: Grothendieck's Resume Revisited

- Cover Cover11 free
- Title i2 free
- Copyright ii3 free
- Contents iii4 free
- Preface vii8 free
- Chapter 1. Basics on tensor norms 112 free
- The algebraic preliminaries 112
- 1.1. Reasonable crossnorms, including the norms ∧ and ∨ 516
- 1.2. Definition of ⊗-norms 2536
- 1.3. Extension of ⊗-norms to spaces of infinite dimensions 2940
- 1.4. Bilinear forms and linear operators of type α 4051
- 1.5. α-nuclear forms and operators 5465
- 1.6. The Dvoretzky-Rogers theorem, Grothendieck-style 5970

- Chapter 2. The role of C(K)-spaces and L[sup(1)]-spaces 6778
- Chapter 3. ⊗-norms related to Hilbert space 111122
- Chapter 4. The Fundamental Theorem and its consequences 149160
- 4.1. Functions of type α 149160
- 4.2. The Fundamental Theorem (Grothendieck's inequality) and its variants 152163
- 4.3. Consequences to the theory of linear operators 159170
- 4.4. A table of the fourteen natural ⊗-norms 163174
- 4.5. Notes and remarks on the natural tensor norms and Banach Algebras 168179

- Glossary of terms 177188
- Appendix A. The problems of the Résumé 183194
- A.1. Problem 1: The approximation problem 183194
- A.2. Problem 2: The possible reduction of the table of "natural" tensor norms 186197
- A.3. Problem 3: Grothendieck's inequality and the "best" constant 191202
- A.4. Problem 4: Algebraic-topological properties of C*-algebras 201212
- A.5. Problem 5: Characterizing classes of spaces by the behavior of tensor products and the action of operators on the spaces 208219
- A.6. Problem 6: Comparison of the projective and injective tensor products 209220

- Appendix B. The Blaschke selection principle and compact convex sets in finite dimensional Banach spaces 211222
- Appendix C. A short introduction to Banach lattices 217228
- C.1. The facts, ma'm, just the facts 217228
- C.2. Some basics about duality in Banach lattices 218229
- C.3. Lattice homomorphisms 222233
- C.4. AM-spaces and AL-spaces 225236
- C.5. Kakutani's vector lattice version of the Stone-Weierstrass theorem 227238
- C.6. Kakutani's characterization of AM-spaces with unit 228239
- C.7. AL-spaces: The Freudenthal-Kakutani theorem 229240
- C.8. Kakutani's characterization of AL-spaces 234245
- C.9. Grothendieck's inequality for Banach lattices 235246
- Notes and remarks 240251

- Appendix D. Stonean spaces and injectivity 241252
- Epilogue 255266
- Bibliography 261272
- Author Index 271282
- Index of Notation 273284 free
- Index 277288 free
- Back Cover Back Cover1290