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Difference between revisions of "Gâteaux derivative"

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''weak 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
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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 $x_0$ 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*}
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f(x_0 + h) = f(x_0)+f'_G(x_0)h + \epsilon(h),
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\end{equation*}
  
<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>
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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]].
 
 
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====
 
====References====

Revision as of 05:50, 24 November 2012

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


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=28869
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article