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• W2950579917 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 a construction of a continuous map $Phi$ from $l^2(A)$ into $L^2(Omega_A, P_A)$, where $A$ is a set, $Omega_A = {-1,1}^A$, and $P_A$ is the uniform probability measure on $Omega_A$, such that $$sum_{alpha in A} x(alpha) bar{y}(alpha)} = int_{Omega_A} Phi(x)Phi(bar{y})dP_A, x in l^2(A), y in l^2(A),$$ and $$ |Phi(x)|_{L^{infty}} leq K |x|_2, x in l^2(A),$$ for an absolute constant $K > 1$. ($Phi$ is non-linear, and does not commute with complex conjugation.) The bilinear Parseval-like formula above 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 $l^p$ and $l^q$, $1/p + 1/q= 1$. Parseval-like formulas are derived in higher dimensions. These variants involve representations of functions of $n$ variables in terms of functions of $k$ variables, $0 < k < n$. Multilinear extensions of the Grothendieck inequality are obtained, and are used to characterize the feasibility of integral representations of multilinear functionals on a Hilbert space." @default.
• W2950579917 created "2019-06-27" @default.
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• W2950579917 date "2011-11-30" @default.
• W2950579917 modified "2023-09-27" @default.
• W2950579917 title "The Grothendieck Inequality Revisited" @default.
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