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)