Abstract
The classical Grothendieck inequality is viewed as a statement about represen-
tations 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 Φ from
l2pAq
into
L2pΩA,
PAq, where
A is a set, ΩA t´1,
1uA,
and PA is the uniform probability measure on ΩA,
such that
(1)
ÿ
αPA
xpαqypαq
ż
ΩA
ΦpxqΦpyqdPA, x P
l2pAq,
y P
l2pAq,
and
(2) }Φpxq}L8 ď K}x}2, x P
l2pAq,
for an absolute constant K ą 1. is non-linear, and does not commute with
complex conjugation.) The Parseval-like formula in (1) 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.
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 ă k ă n. Multilinear Parseval-like formulas are obtained, extending
the bilinear formula in (1). The resulting formulas imply multilinear extensions of
the Grothendieck inequality, and are used to characterize the feasibility of integral
representations of multilinear functionals on a Hilbert space.
Received by the editor November 30, 2011, and, in revised form, November 5, 2012.
Article electronically published on March 11, 2014.
DOI: http://dx.doi.org/10.1090/memo/1093
2010 Mathematics Subject Classification. Primary 46C05, 46E30, Secondary 47A30, 42C10.
Key words and phrases. The Grothendieck inequality, Parseval-like formulas, integral repre-
sentations, fractional Cartesian products, projective continuity, projective boundedness.
Affiliation at time of publication: Department of Mathematics, University of Connecticut,
Storrs, Connecticut 06269, email: blei@math.uconn.edu.
c
2014 American Mathematical Society
v
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