View source for Gâteaux derivative

''weak derivative''

The derivative of a functional or a mapping which — together with the [[Fréchet derivative|Fréchet derivative]] (the strong derivative) — is most frequently used in infinite-dimensional analysis. The Gâteaux derivative at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043360/g0433601.png" /> of a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043360/g0433602.png" /> from a linear topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043360/g0433603.png" /> into a linear topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043360/g0433604.png" /> is the continuous linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043360/g0433605.png" /> that satisfies the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043360/g0433606.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043360/g0433607.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043360/g0433608.png" /> in the topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043360/g0433609.png" /> (see also [[Gâteaux variation|Gâteaux variation]]). If the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043360/g04336010.png" /> has a Gâteaux derivative at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043360/g04336011.png" />, 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></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.S. Berger,   "Nonlinearity and functional analysis" , Acad. Press  (1977)</TD></TR></table>