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