eBook ISBN:  9781470418960 
Product Code:  MEMO/232/1093.E 
List Price:  $71.00 
MAA Member Price:  $63.90 
AMS Member Price:  $42.60 
eBook ISBN:  9781470418960 
Product Code:  MEMO/232/1093.E 
List Price:  $71.00 
MAA Member Price:  $63.90 
AMS Member Price:  $42.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 232; 2014; 90 ppMSC: Primary 46; Secondary 47; 42
The classical Grothendieck inequality is viewed as a statement about representations of functions of two variables over discrete domains by integrals of twofold 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 \(\Omega_A\).

Table of Contents

Chapters

1. Introduction

2. Integral representations: the case of discrete domains

3. Integral representations: the case of topological domains

4. Tools

5. Proof of Theorem 3.5

6. Variations on a theme

7. More about $\Phi $

8. Integrability

9. A Parsevallike formula for $\langle {\bf {x}}, {\bf {y}}\rangle $, ${\bf {x}} \in l^p$, ${\bf {y}} \in l^q$

10. Grothendiecklike theorems in dimensions $>2$?

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

12. Proof of Theorem 11.11

13. Some loose ends


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The classical Grothendieck inequality is viewed as a statement about representations of functions of two variables over discrete domains by integrals of twofold 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 \(\Omega_A\).

Chapters

1. Introduction

2. Integral representations: the case of discrete domains

3. Integral representations: the case of topological domains

4. Tools

5. Proof of Theorem 3.5

6. Variations on a theme

7. More about $\Phi $

8. Integrability

9. A Parsevallike formula for $\langle {\bf {x}}, {\bf {y}}\rangle $, ${\bf {x}} \in l^p$, ${\bf {y}} \in l^q$

10. Grothendiecklike theorems in dimensions $>2$?

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

12. Proof of Theorem 11.11

13. Some loose ends