The Grothendieck Inequality Revisited

الأصل · EN

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 Φ from l²(A) into L²(Ωₐ, Pₐ), where A is a set, Ωₐ = -1,1ᵃ, and Pₐ is the uniform probability measure on Ωₐ, such that ∑α∈ ₐ x(α) y(α) = ∫Ωₐ Φ(x)Φ(y)dPₐ, x ∈ l²(A), y ∈ l²(A), and |Φ(x)|ₗ∞ ≤ K |x|₂, x ∈ l²(A), for an absolute constant K > 1. (Φ 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 lᵖ and lq, 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.