## 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 $f$.

#### Reference

Wikipedia: Derivative of a function, Fréchet derivative