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Gâteaux differential

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of a mapping of a linear topological space into a linear topological space

The function

where

on the assumption that the limit exists for all , the convergence being understood in the topology of . The Gâteaux differential thus defined is homogeneous, but is not additive. Gâteaux differentials of higher orders are defined in a similar manner. The mapping 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: , . In such a case is known as the Gâteaux derivative. If the mapping is uniformly continuous in and continuous in in some domain, then the Fréchet derivative of exists in this domain and .

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)


Comments

References

[a1] M.S. Berger, "Nonlinearity and functional analysis" , Acad. Press (1977)
How to Cite This Entry:
Gâteaux differential. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=G%C3%A2teaux_differential&oldid=23302
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article