 Then the Fréchet derivative is a map

i.e., for paths formula_9, formula_10 is an element of formula_11, the dual space to formula_12.Denote by formula_13 the continuous linear map formula_14 defined by

sometimes known as the "H"-derivative.Now define formula_16 to be the adjoint of formula_17 in the sense that

Then the Malliavin derivative formula_19 is defined by

The domain of formula_19 is the set formula_22 of all Fréchet differentiable real-valued functions on formula_12; the codomain is formula_24.