# Fréchet derivative

strong derivative

The most widespread (together with the Gâteaux derivative, which is sometimes called the weak derivative) derivative of a functional or a mapping. The Fréchet derivative of a mapping $f: X \rightarrow Y$ of a normed space $X$ into a normed space $Y$ at a point $x _ {0}$ is the linear continuous operator $\Lambda : X \rightarrow Y$ satisfying the condition

$$f ( x _ {0} + h) = \ f ( x _ {0} ) + \Lambda h + \epsilon ( h),$$

where

$$\lim\limits _ {\| h \| \rightarrow 0 } \ \frac{\| \epsilon ( h) \| }{\| h \| } = 0.$$

The operator $\Lambda$ satisfying these conditions is unique (if it exists) and is denoted by $f ^ { \prime } ( x _ {0} )$; the linear mapping $h \rightarrow f ^ { \prime } ( x _ {0} ) h$ is called the Fréchet differential. If $f$ has a Fréchet derivative at $x _ {0}$, it is said to be Fréchet differentiable. The most important theorems of differential calculus hold for Fréchet derivatives — the theorem on the differentiation of a composite function and the mean value theorem. If $f$ is continuously Fréchet differentiable in a neighbourhood of a point $x _ {0}$ and if the Fréchet derivative $f ^ { \prime } ( x _ {0} )$ at $x _ {0}$ is a homeomorphism of the Banach spaces $X$ and $Y$, then the inverse mapping theorem holds. See also Differentiation of a mapping.