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:
- :=\left(\mathrm {D} _{t}\right)^{*}:\operatorname {image} \left(\mathrm {D} _{t}\right)\subseteq L^{2}([0,T];\mathbb {R} ^{n})\to \mathbf {F} ^{*}=\mathrm {Lin} (\mathbf {F} ;\mathbb {R} ).}