1.3. PARSEVAL-LIKE FORMULAS 3
The implication (1.5) ñ (1.2) provides a template for direct proofs of the
Grothendieck inequality: first establish an integral representation of the dot prod-
uct, like the one in (1.5), and then verify the inequality by an ”averaging” argument,
as in (1.6); e.g., [28], [33], [4], [27]. Whereas there are other equivalent formula-
tions of the inequality (e.g., see [31, §1, §2]), a representation of the dot product
by an integral with uniformly bounded integrands is, arguably, the ”closest” to it.
Building on ideas in [4], [5], and [6], we establish here integral representations
that go a little further than (1.5), and imply a little more than (1.2).
1.3. Parseval-like formulas
Without a requirement that integrands be uniformly bounded, integral repre-
sentations of the l2-dot product are ubiquitous and indeed easy to produce: let
tfα : α P Au be an orthonormal system in L2pΩ,μq, where pΩ, μq is a probability
space, and let U be the map from l2 into L2pΩ,μq given by
(1.10) U pxq
ÿ
α
xpαqfα, x P
l2,
whence (Parseval’s formula),
(1.11) xx, yy
ż
Ω
U pxqU pyqdμ, x P
l2,
y P
l2,
and
(1.12) }U pxq}L2 }x}2, x P
l2.
The map U is linear and (therefore) continuous.
The Grothendieck inequality (via Proposition 1.1) is ostensibly the more strin-
gent assertion, that there exist a probability space pΩ, μq and a map Φ from l2 into
L8pΩ,μq,
such that
(1.13)
ÿ
α
xpαqypαq
ż
Ω
ΦpxqΦpyqdμ, x P
l2,
y P
l2,
and
(1.14) }Φpxq}L8 ď K}x}2, x P
l2,
for some K ą 1. Specifically, assuming (1.2), we take λ P M pΩBl2 ˆ ΩBl2 q supplied
by Proposition 1.1, and let ψ be the Radon-Nikodym derivative of λ with respect
to its total variation measure |λ|. For x P
l2,
define
(1.15) σx :“
"
x{}x}2 if x - 0
0 if x 0,
and let πi (i 1, 2) denote the canonical projections from ΩB
l2
ˆ ΩB
l2
onto ΩB
l2
,
πipω1,ω2q ωi, pω1, ω2q P ΩBl2 ˆ ΩBl2 .
Define
φi :
l2
Ñ
L8
`
ΩBl2 ˆ ΩBl2 , |λ|{}λ}M
˘
by
(1.16) φipxq p}λ}M ψq
1
2
}x}2 rσx ˝ πi, x P
l2,
i 1, 2,
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