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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[disambiguation needed] of , we have .

See also[edit]