Fréchet derivative

Set

 context $X,Y$ … Banach spaces with topology context $\mathcal O$ … open in $X$ definiendum $D:\mathrm{Continuous}(\mathcal O,Y)\to \mathrm{Continuous}(\mathcal O,\mathrm{BoundedLinOp}(X,Y))$ definiendum $Df:=x\mapsto J_x^f$

For $J_x f$, see Linear approximation.

Discussion

This definition does nothing more than emphasizing the functionality of $L_x^f$ in $f$.