# 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 ). 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.

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