Let ${\displaystyle i:H\to E}$  be an abstract Wiener space, and suppose that ${\displaystyle F:E\to \mathbb {R} }$  is differentiable. Then the Fréchet derivative is a map

${\displaystyle \mathrm {D} F:E\to \mathrm {Lin} (E;\mathbb {R} )}$ ;

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

Therefore, define the ${\displaystyle H}$ -derivative ${\displaystyle \mathrm {D} _{H}F}$  at ${\displaystyle x\in E}$  by

${\displaystyle \mathrm {D} _{H}F(x):=\mathrm {D} F(x)\circ i:H\to \mathbb {R} }$ ,

a continuous linear map on ${\displaystyle H}$ .

Define the ${\displaystyle H}$ -gradient ${\displaystyle \nabla _{H}F:E\to H}$  by

${\displaystyle \langle \nabla _{H}F(x),h\rangle _{H}=\left(\mathrm {D} _{H}F\right)(x)(h)=\lim _{t\to 0}{\frac {F(x+ti(h))-F(x)}{t}}}$ .

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

• Malliavin derivative