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The Grothendieck inequality revisited


About this Title

Ron Blei

Publication: Memoirs of the American Mathematical Society
Publication Year: 2014; Volume 232, Number 1093
ISBNs: 978-0-8218-9855-0 (print); 978-1-4704-1896-0 (online)
DOI: http://dx.doi.org/10.1090/memo/1093
Published electronically: March 11, 2014
Keywords:The Grothendieck inequality, Parseval-like formulas, integral representations, fractional Cartesian products, projective continuity

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Integral representations: the case of discrete domains
  • Chapter 3. Integral representations: the case of topological domains
  • Chapter 4. Tools
  • Chapter 5. Proof of Theorem 3.5
  • Chapter 6. Variations on a theme
  • Chapter 7. More about $\Phi $
  • Chapter 8. Integrability
  • Chapter 9. A Parseval-like formula for $\langle {\bf {x}}, {\bf {y}}\rangle $, ${\bf {x}} \in l^p$, ${\bf {y}} \in l^q$
  • Chapter 10. Grothendieck-like theorems in dimensions $>2$?
  • Chapter 11. Fractional Cartesian products and multilinear functionals on a Hilbert space
  • Chapter 12. Proof of Theorem 11.11
  • Chapter 13. Some loose ends

Abstract


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 from into , where is a set, , and is the uniform probability measure on , such that

and

for an absolute constant . ( is non-linear, and does not commute with complex conjugation.) The Parseval-like formula in [[eqref]]ab1 is obtained by iterating the usual Parseval formula in a framework of harmonic analysis on dyadic groups. A modified construction implies a similar integral representation of the dual action between and , . Variants of the Grothendieck inequality in higher dimensions are derived. These variants involve representations of functions of variables in terms of functions of variables, Multilinear Parseval-like formulas are obtained, extending the bilinear formula in [[eqref]]ab1. The resulting formulas imply multilinear extensions of the Grothendieck inequality, and are used to characterize the feasibility of integral representations of multilinear functionals on a Hilbert space.

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