The Grothendieck inequality revisited / Ron Blei.

Format:

Book

Author/Creator:

Blei, R. C. (Ron C.), author.

Series:

Memoirs of the American Mathematical Society ; 0065-9266. Volume 232, Number 1093.

Memoirs of the American Mathematical Society ; Volume 232, Number 1093

Language:

English

Subjects (All):

Geometry, Algebraic.

Grothendieck, A. (Alexandre).

Grothendieck, A.

Physical Description:

1 online resource (v, 90 pages).

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2014]

Language Note:

English

Summary:

The classical Grothendieck inequality is viewed as a statement about representations of functions of two variables over discrete domains by integrals of two-fold products of functions of one variable. An analogous statement is proved, concerning continuous functions of two variables over general topological domains. The main result is the construction of a continuous map \Phi from l^2(A) into L^2(\Omega_A, \mathbb{P}_A), where A is a set, \Omega_A = \{-1,1\}^A, and \mathbb{P}_A is the uniform probability measure on \Omega_A.

Contents:

Cover

Title page

Chapter 1. Introduction

1.1. The inequality

1.2. An integral representation

1.3. Parseval-like formulas

1.4. Multilinear Parseval-like formulas

1.5. Projective boundedness and projective continuity

1.6. A personal note and acknowledgements

Chapter 2. Integral representations: the case of discrete domains

2.1. First question

2.2. Second question

2.3. Third question and Grothendieck's théorème fondamental

Chapter 3. Integral representations: the case of topological domains

3.1. ²-continuous families

3.2. A "continuous"' version of le théorème fondamental

Chapter 4. Tools

4.1. The framework

4.2. Rademacher and Walsh characters

4.3. Walsh series

4.4. Riesz products

4.5. Continuity

Chapter 5. Proof of Theorem 3.5

5.1. The construction of Φ

5.2. Φ is odd

5.3. An integral representation of the dot product

5.4. Φ \ is weakly continuous

5.5. Φ \ is ( ²→ ²)-continuous

Chapter 6. Variations on a theme

6.1. The map \ Φ₂

6.2. Sharper bounds

Chapter 7. More about Φ

7.1. Spectrum

7.2. ( ² → ^{ })-continuity

7.3. ( ² → ^{∞})-continuity?

7.4. Linearization

Chapter 8. Integrability

8.1. Integrability of Φ

8.2. Integrability of Φ⊗Φ

8.3. Integrability of the inner product

Chapter 9. A Parseval-like formula for ⟨ , ⟩, \ ∈ ^{ }, \ ∈ ^{ }

Chapter 10. Grothendieck-like theorems in dimensions &gt

2?

10.1. A multidimensional version of Question 2.3

10.2. A multidimensional version of Question 2.6

Chapter 11. Fractional Cartesian products and multilinear functionals on a Hilbert space

11.1. Projective boundedness and projective continuity

11.2. Fractional Cartesian products

11.3. A characterization of projectively continuous functionals

Chapter 12. Proof of Theorem 11.11.

12.1. A multilinear Parseval-like formula

12.2. The left-side inequality in (11.81)

12.3. The right-side inequality in (11.81)

12.4. Multilinear extensions of the Grothendieck inequality

Chapter 13. Some loose ends

13.1. ₂( × ) \ vs. \ ₂( × ), \ \ ₂( × ) \ vs. \ ₂( × )

13.2. ₂( × )= ₂( × ) \ vs. \ ₂( × )= ₂( × )

13.3. Projective boundedness vs. projective continuity

13.4. A characterization of projective boundedness

Bibliography

Back Cover.

Notes:

"November 2014, volume 232, number 1093 (fifth of 6 numbers)".

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-4704-1896-7