In mathematics, the H-derivative is a notion of derivative in the study of abstract Wiener spaces and the Malliavin calculus.
Let be an abstract Wiener space, and suppose that is differentiable. Then the Fréchet derivative is a map
i.e., for , is an element of , the dual space to .
Therefore, define the -derivative at by
a continuous linear map on .
Define the -gradient by
That is, if denotes the adjoint of , we have .