View source for Gâteaux derivative

{{TEX|done}}
''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|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|Differentiation of a mapping]].

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  R. Gâteaux,   "Sur les fonctionnelles continues et les fonctionnelles analytiques"  ''C.R. Acad. Sci. Paris Sér. I Math.'' , '''157'''  (1913)  pp. 325–327</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.N. Kolmogorov,   S.V. Fomin,   "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W.I. [V.I. Sobolev] Sobolew,   "Elemente der Funktionalanalysis" , H. Deutsch , Frankfurt a.M.  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  M.S. Berger,   "Nonlinearity and functional analysis" , Acad. Press  (1977)</TD></TR>
</table>