| eBook ISBN: | 978-1-4704-1896-0 |
| Product Code: | MEMO/232/1093.E |
| List Price: | $71.00 |
| MAA Member Price: | $63.90 |
| AMS Member Price: | $42.60 |
| eBook ISBN: | 978-1-4704-1896-0 |
| Product Code: | MEMO/232/1093.E |
| List Price: | $71.00 |
| MAA Member Price: | $63.90 |
| AMS Member Price: | $42.60 |
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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 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 \(\Omega_A\).
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Table of Contents
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Chapters
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1. Introduction
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2. Integral representations: the case of discrete domains
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3. Integral representations: the case of topological domains
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4. Tools
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5. Proof of Theorem 3.5
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6. Variations on a theme
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7. More about $\Phi $
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8. Integrability
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9. A Parseval-like formula for $\langle {\bf {x}}, {\bf {y}}\rangle $, ${\bf {x}} \in l^p$, ${\bf {y}} \in l^q$
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10. Grothendieck-like theorems in dimensions $>2$?
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11. Fractional Cartesian products and multilinear functionals on a Hilbert space
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12. Proof of Theorem 11.11
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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 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 \(\Omega_A\).
-
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 Parseval-like formula for $\langle {\bf {x}}, {\bf {y}}\rangle $, ${\bf {x}} \in l^p$, ${\bf {y}} \in l^q$
-
10. Grothendieck-like 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
