# 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 of a normed space into a normed space at a point is the linear continuous operator satisfying the condition

where

The operator satisfying these conditions is unique (if it exists) and is denoted by ; the linear mapping is called the Fréchet differential. If has a Fréchet derivative at , 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 is continuously Fréchet differentiable in a neighbourhood of a point and if the Fréchet derivative at is a homeomorphism of the Banach spaces and , then the inverse mapping theorem holds. See also Differentiation of a mapping.

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#### References

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

**How to Cite This Entry:**

Fréchet derivative.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Fr%C3%A9chet_derivative&oldid=23280