2 1. INTRODUCTION

equipped with the usual product topology and the sigma field generated by it, and

consider the coordinate functions, rx : ΩBl2 Ñ t´1, 1u, defined by

(1.4) rxpωq “ ωpxq, ω P t´1,

1uBl2

, x P Bl2 .

We refer to RB

l2

:“ trx : x P Bl2 u as a Rademacher system indexed by Bl2 , and to

its members as Rademacher characters; see §4 of this work.

Proposition 1.1. The Grothendieck inequality (1.2) is equivalent to the exis-

tence of a complex measure

λ P M

`

ΩBl2 ˆ ΩBl2

˘

p“ complex measures on ΩBl2 ˆ ΩBl2

(

q

such that

(1.5) xx, yy “

ż

ΩB

l2

ˆΩB

l2

rxpω1qrypω2qλpdω1,dω2q, px, yq P Bl2 ˆ Bl2 ,

and the Grothendieck constant KG is the infimum of }λ}M over all the representa-

tions of the dot product by (1.5). (}λ}M “ total variation norm of λ.)

Proof. We first verify (1.5) ñ (1.2). Let a “ pajkq be a finite scalar array,

and denote by }a}F2 the supremum on the right side of (1.2). Assuming (1.5), let

pxj q and pyk q be arbitrary sequences of vectors in Bl2 , and then estimate

ˇ

ˇ

ÿ

j,k

ajkxxj,

ykyˇ

ˇ

“

ˇ

ˇ

ÿ

j,k

ajk

ż

ΩB

l2

ˆΩB

l2

rxj pω1qryk pω2qλpdω1,

dω2qˇ

ˇ

ď

ż

ΩBl2 ˆΩBl2

ˇ

ˇ

ÿ

j,k

ajkrxj pω1qryk

pω2qˇ

ˇ

|λ|pdω1, dω2q

ď }a}F2 }λ}M .

(1.6)

We thus obtain (1.2) with KG ď }λ}M .

To verify (1.2) ñ (1.5), we first associate with a finite scalar array a “ pajkq,

and sequences of vectors pxj q and pyk q in Bl2 , the Walsh polynomial

(1.7) ˆ a “

ÿ

j,k

ajk rxj b ryk .

Note that ˆ a is a continuous function on ΩB

l2

ˆ ΩB

l2

, and

(1.8) }ˆ}8 a “ }a}F2 ,

where }ˆ}8 a is the supremum of ˆ a over ΩBl2 ˆ ΩBl2 . Such polynomials are norm-

dense in the space of continuous functions on ΩBl2 ˆ ΩBl2 with spectrum in RBl2 ˆ

RB

l2

, which is denoted by CRB

l2

ˆRB

l2

pΩB

l2

ˆΩB

l2

q; e.g., see [7, Ch. VII, Corollary

9]. Then, (1.2) becomes the statement that

(1.9)

ÿ

j,k

ajk rxj b ryk Þ Ñ

ÿ

j,k

ajkxxj, yky

determines a bounded linear functional on CRBl2

ˆRBl2

pΩBl2 ˆ ΩBl2 q, with norm

bounded by KG. Therefore, by the Riesz Representation theorem and by the Hahn-

Banach theorem, there exists λ P M

`

ΩBl2 ˆ ΩBl2 q such that (1.5) holds, and

}λ}M ď KG.