Gâteaux differential

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

The function

$$h \rightarrow Df ( x _ {0} , h),$$

where

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

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

on the assumption that the limit exists for all $h \in X$, the convergence being understood in the topology of $Y$. The Gâteaux differential thus defined is homogeneous, but is not additive. Gâteaux differentials of higher orders are defined in a similar manner. The mapping $h \rightarrow Df( x _ {0} , h)$ is sometimes known as the Gâteaux variation or as the weak differential. See also Differentiation of a mapping; Variation.

Linearity and continuity are usually additionally stipulated: $Df( x, h) = f _ {G} ^ { \prime } ( x _ {0} ) h$, $f _ {G} ^ { \prime } ( x _ {0} ) \in L( X, Y)$. In such a case $f _ {G} ^ { \prime } ( x _ {0} )$ is known as the Gâteaux derivative. If the mapping $( x, h) \rightarrow Df( x, h)$ is uniformly continuous in $x$ and continuous in $h$ in some domain, then the Fréchet derivative $f ^ { \prime }$ of $f$ exists in this domain and $f ^ { \prime } ( x) h = Df ( x, h )$.

References

[1]	W.I. [V.I. Sobolev] Sobolew, "Elemente der Funktionalanalysis" , H. Deutsch , Frankfurt a.M. (1979) (Translated from Russian)
[2]	A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
[a1]	M.S. Berger, "Nonlinearity and functional analysis" , Acad. Press (1977)