Definition
Let
be the Cameron–Martin space, and
denote classical Wiener space:
;

By the Sobolev embedding theorem,
. Let

denote the inclusion map.
Suppose that
is Fréchet differentiable. Then the Fréchet derivative is a map

i.e., for paths
,
is an element of
, the dual space to
. Denote by
the continuous linear map
defined by

sometimes known as the H-derivative. Now define
to be the adjoint of
in the sense that

Then the Malliavin derivative
is defined by

The domain of
is the set
of all Fréchet differentiable real-valued functions on
; the codomain is
.
The Skorokhod integral
is defined to be the adjoint of the Malliavin derivative:

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