 Then the Fréchet derivative is a map
i.e., for formula_4, formula_5 is an element of formula_6, the dual space to formula_7.Therefore, define the formula_8-derivative formula_9 at formula_4 by
a continuous linear map on formula_8.Define the formula_8-gradient formula_14 by
That is, if formula_16 denotes the adjoint of formula_1, we have formula_18.