Gâteaux derivative

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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.


[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



[a1] M.S. Berger, "Nonlinearity and functional analysis" , Acad. Press (1977)
How to Cite This Entry:
Gâteaux derivative. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article