CHAPTER 1
Introduction
1.1. The inequality
We start with an infinite-dimensional Euclidean space, whose coordinates are
indexed by a set A,
(1.1)
l2

l2pAq
:“ x
`
xpαq
˘
αPA
P
CA
:
ÿ
αPA
|xpαq|2
ă 8
(
,
equipped with the usual dot product
xx, yy :“
ÿ
αPA
xpαqypαq, x P
l2,
y P
l2,
and the Euclidean norm
}x}2 :“
a
xx, xy
˜
ÿ
αPA
|xpαq|2
¸
1
2
, x P
l2.
We let Bl2 denote the closed unit ball,
Bl2 :“ x P
l2
: }x}2 ď 1
(
.
The Grothendieck inequality in this setting is the assertion that there exists
1 ă K ă 8 such that for every finite scalar array pajk q,
(1.2)
sup
"
ˇ
ˇ
ÿ
j,k
ajkxxj,
ykyˇ:
ˇ
pxj , ykq P pBl2
q2
*
ď K sup
"
ˇ
ˇ
ÿ
j,k
ajksjtkˇ
ˇ
: psj , tkq P r´1,
1s2
*
.
An assertion equivalent to (1.2), couched in a setting of topological tensor prod-
ucts, had appeared first in Alexandre Grothendieck’s landmark Resum´ e [19], and
had remained largely unnoticed until it was deconstructed and reformulated in [28]
another classic as the inequality above. Since its reformulation, which became
known as the Grothendieck inequality, it has been duly recognized as a fundamental
statement, with diverse appearances and applications in functional, harmonic, and
stochastic analysis, and recently also in theoretical physics and theoretical computer
science. (See [31], and also Remark 2.9.ii in this work.).
The numerical value of the ”smallest” K in (1.2), denoted by KG and dubbed
the Grothendieck constant, is an open problem that to this day continues to attract
interest. (For the latest on KG, see [13].)
1.2. An integral representation
Consider the infinite product space
(1.3) ΩBl2 :“ t´1,
1uBl2
,
1
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