# Fréchet differential

at a point $x _ {0}$ of a mapping $f: X \rightarrow Y$ of a normed space $X$ into a normed space $Y$

The mapping $h \rightarrow D ( x _ {0} , h)$ which is linear and continuous from $X$ into $Y$ and has the property that

$$\tag{1 } f ( x _ {0} + h) = \ f ( x _ {0} ) + D ( x _ {0} , h) + \epsilon ( h),$$

where

$$\lim\limits _ {\| h \| \rightarrow 0 } \ \frac{\| \epsilon ( h) \| }{\| h \| } = 0.$$

If a mapping $f$ admits an expansion (1) at a point $x _ {0}$, then it is said to be Fréchet differentiable, and the actual operator

$$f ^ { \prime } ( x _ {0} ) h = \ D ( x _ {0} , h),\ \ f ^ { \prime } ( x _ {0} ) \in \ L ( X, Y),$$

is called the Fréchet derivative.

For a function $f$ in a finite number of variables, the Fréchet differential is the linear function

$$h \rightarrow \ \sum _ {i = 1 } ^ { n } \alpha _ {i} h _ {i} = \ l _ {x _ {0} } h$$

that has the property that

$$\tag{2 } f ( x _ {0} + h) = \ f ( x _ {0} ) + l _ {x _ {0} } ( h) + o ( | h | ),$$

where $| h | = ( \sum _ {i = 1 } ^ {n} h _ {i} ^ {2} ) ^ {1/2}$ or any other equivalent norm in $\mathbf R ^ {n}$. Here $\alpha _ {i} = \partial f / \partial x _ {i} \mid _ {x _ {0} }$ are the partial derivatives of $f$ at $x _ {0}$.

Definition (2), which is now commonplace, apparently first appeared in an explicit form in the lectures of K. Weierstrass (1861, see [1]). At the end of the 19th century this definition gradually came into the textbooks (see [2], [3] 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.

#### References

 [1] P. Dugac, "Eléments d'analyse de Karl Weierstrass" , Paris (1972) [2] O. Stolz, "Grundzüge der Differential- und Integralrechnung" , 1 , Teubner (1893) [3] W. Young, "The fundamental theorems of the differential calculus" , Cambridge Univ. Press (1910) [4] M. Fréchet, "Sur la notion de différentielle" C.R. Acad. Sci. Paris , 152 (1911) pp. 845–847; 1050–1051 [5] M. Fréchet, "Sur la notion de différentielle totale" Nouvelles Ann. Math. Sér. 4 , 12 (1912) pp. 385–403; 433–449 [6] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) [7] V.M. Alekseev, V.M. Tikhomirov, S.V. Fomin, "Optimal control" , Consultants Bureau (1987) (Translated from Russian)
How to Cite This Entry:
Fréchet differential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fr%C3%A9chet_differential&oldid=46999
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article