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.

References

[a1]	M.S. Berger, "Nonlinearity and functional analysis" , Acad. Press (1977)