2011/11/30 by Ron C. Blei, Blei, Ron
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Limits and Structures in Graph Theory #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.1111.7304
openalex publication_date 2011/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.