# Gâteaux differential

*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