Mathematics – Functional Analysis
Scientific paper
2011-11-30
Mathematics
Functional Analysis
Scientific paper
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 representation of the inner product in a Hilbert space by an integral with uniformly bounded and continuous integrands. A Parseval-like formula 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 $l^q$, \ $\frac{1}{p} + \frac{1}{q} = 1$. Variants of the Grothendieck inequality in higher dimensions are derived. These variants involve representations of functions of $n$ variables in terms of functions of $k$ variables, $0 < k < n.$ Multilinear Parseval-like formulas are obtained. The resulting formulas yield 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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