# Gâteaux derivative

From Encyclopedia of Mathematics

*weak derivative*

The derivative of a functional or a mapping which — together with the Fréchet derivative (the strong derivative) — is most frequently used in infinite-dimensional analysis. The Gâteaux derivative at a point of a mapping from a linear topological space into a linear topological space is the continuous linear mapping that satisfies the condition

where as in the topology of (see also Gâteaux variation). If the mapping has a Gâteaux derivative at the point , it is called Gâteaux differentiable. The theorem on differentiation of a composite function is usually invalid for the Gâteaux derivative. See also Differentiation of a mapping.

#### References

[1] | R. Gâteaux, "Sur les fonctionnelles continues et les fonctionnelles analytiques" C.R. Acad. Sci. Paris Sér. I Math. , 157 (1913) pp. 325–327 |

[2] | A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) |

[3] | W.I. [V.I. Sobolev] Sobolew, "Elemente der Funktionalanalysis" , H. Deutsch , Frankfurt a.M. (1979) (Translated from Russian) |

[4] | V.I. Averbukh, O.G. Smolyanov, "Theory of differentiation in linear topological spaces" Russian Math. Surveys , 22 : 6 (1967) pp. 201–258 Uspekhi Mat. Nauk , 22 : 6 (1967) pp. 201–260 |

#### Comments

#### References

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

**How to Cite This Entry:**

Gâteaux derivative.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=G%C3%A2teaux_derivative&oldid=13686

This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article