H-derivative

In mathematics, the H-derivative is a notion of derivative in the study of abstract Wiener spaces and the Malliavin calculus.

Definition
Let $$i : H \to E$$ be an abstract Wiener space, and suppose that $$F : E \to \mathbb{R}$$ is differentiable. Then the Fréchet derivative is a map
 * $$\mathrm{D} F : E \to \mathrm{Lin} (E; \mathbb{R})$$;

i.e., for $$x \in E$$, $$\mathrm{D} F (x)$$ is an element of $$E^{*}$$, the dual space to $$E$$.

Therefore, define the $$H$$-derivative $$\mathrm{D}_{H} F$$ at $$x \in E$$ by
 * $$\mathrm{D}_{H} F (x) := \mathrm{D} F (x) \circ i : H \to \R$$,

a continuous linear map on $$H$$.

Define the $$H$$-gradient $$\nabla_{H} F : E \to H$$ by
 * $$\langle \nabla_{H} F (x), h \rangle_{H} = \left( \mathrm{D}_{H} F \right) (x) (h) = \lim_{t \to 0} \frac{F (x + t i(h)) - F(x)}{t}$$.

That is, if $$j : E^{*} \to H$$ denotes the adjoint of $$i : H \to E$$, we have $$\nabla_{H} F (x) := j \left( \mathrm{D} F (x) \right)$$.