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)