# Gâteaux variation

of a mapping $f$ of a linear space $X$ into a linear topological space $Y$

The limit, in the topology of $Y$,

$$\tag{* } \delta f ( x _ {0} , h) = \left . { \frac{d}{dt } } f ( x _ {0} + th) \right | _ {t = 0 } =$$

$$= \ \lim\limits _ {t \rightarrow 0 } \frac{f ( x _ {0} + th) - f ( x _ {0} ) }{t }$$

on the assumption that it exists for all $h \in X$. This is how the first variation was introduced by R. Gâteaux in 1913–1914. This definition for functionals of the classical calculus of variations was given by J.L. Lagrange (cf. Variation of a functional).

The expression $\delta f( x _ {0} , h)$ need not necessarily be a linear functional in $h$, but it is always a homogeneous function of the first degree in $h$. The mapping $h \rightarrow \delta f ( x _ {0} , h)$ is also known as the Gâteaux differential or weak differential. Beginning with the work of P. Lévy, it is usual to stipulate the linearity and continuity of $\delta f ( x _ {0} , h)$ in $h$:

$$\delta f ( x _ {0} , h) = f _ {G} ^ { \prime } ( x _ {0} ) h,\ \ f _ {G} ^ { \prime } ( x _ {0} ) \in L ( X, Y).$$

where $f _ {G} ^ { \prime } ( x _ {0} )$ is called the Gâteaux derivative. Second, etc., variations are defined similarly to (*). See also Variation; Second variation; Differentiation of a mapping.

How to Cite This Entry:
Gâteaux variation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=G%C3%A2teaux_variation&oldid=53802
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article