at a point of a mapping of a normed space into a normed space
The mapping which is linear and continuous from into and has the property that
If a mapping admits an expansion (1) at a point , then it is said to be Fréchet differentiable, and the actual operator
is called the Fréchet derivative.
For a function in a finite number of variables, the Fréchet differential is the linear function
that has the property that
where or any other equivalent norm in . Here are the partial derivatives of at .
Definition (2), which is now commonplace, apparently first appeared in an explicit form in the lectures of K. Weierstrass (1861, see ). At the end of the 19th century this definition gradually came into the textbooks (see ,  and others). But at the time when M. Fréchet began to develop infinite-dimensional analysis, the now classical definition of the differential was so far from commonplace that even Fréchet himself supposed that his definition of the differential in an infinite-dimensional space was a new concept in the finite-dimensional case too. Nowadays the term is only used in relation to infinite-dimensional mappings. See Gâteaux differential; Differential.
|||P. Dugac, "Eléments d'analyse de Karl Weierstrass" , Paris (1972)|
|||O. Stolz, "Grundzüge der Differential- und Integralrechnung" , 1 , Teubner (1893)|
|||W. Young, "The fundamental theorems of the differential calculus" , Cambridge Univ. Press (1910)|
|||M. Fréchet, "Sur la notion de différentielle" C.R. Acad. Sci. Paris , 152 (1911) pp. 845–847; 1050–1051|
|||M. Fréchet, "Sur la notion de différentielle totale" Nouvelles Ann. Math. Sér. 4 , 12 (1912) pp. 385–403; 433–449|
|||A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)|
|||V.M. Alekseev, V.M. Tikhomirov, S.V. Fomin, "Optimal control" , Consultants Bureau (1987) (Translated from Russian)|
Fréchet differential. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Fr%C3%A9chet_differential&oldid=23282