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,