Difference between revisions of "Gâteaux derivative"
From Encyclopedia of Mathematics
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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 | + | 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*} | ||
| + | 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]]. | |
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====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
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