The Grothendieck inequality revisited

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.