# Gâteaux derivative

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 $x_0$ of a mapping $f:X\to Y$ from a linear topological space $X$ into a linear topological space $Y$ is the continuous linear mapping $f'_G(x_0):X\to Y$ that satisfies the condition \begin{equation*} f(x_0 + h) = f(x_0)+f'_G(x_0)h + \epsilon(h), \end{equation*}

where $\epsilon(th)/ t \to 0$ as $t\to 0$ in the topology of $Y$ (see also Gâteaux variation). If the mapping $f$ has a Gâteaux derivative at the point $x_0$, 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