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.
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