1 jul. PDF | On Jul 1, , Rogério de Aguiar and others published Considerações sobre as derivadas de Gâteaux e Fréchet. In particular, then, Fréchet differentiability is stronger than differentiability in the Gâteaux sense, meaning that every function which is Fréchet differentiable is. 3, , no. 19, – A Note on the Derivation of Fréchet and Gâteaux. Oswaldo González-Gaxiola. 1. Departamento de Matemáticas Aplicadas y Sistemas. Author: Tygom Jutaur Country: Rwanda Language: English (Spanish) Genre: Life Published (Last): 28 December 2004 Pages: 456 PDF File Size: 13.51 Mb ePub File Size: 7.96 Mb ISBN: 664-1-45606-636-1 Downloads: 19076 Price: Free* [*Free Regsitration Required] Uploader: Nigrel Generally, it extends the idea of the derivative from real-valued functions of one real variable to functions on Banach spaces. The limit here is meant in the usual sense of a limit of a function defined on a metric space see Functions on metric spacesusing V and W as the two metric spaces, and the above expression as the function of argument h in V. Note that this is not the same as requiring that freceht map D f x: This notion of derivative is a generalization of the ordinary derivative of a function on the real numbers f: Differentiation is a linear operation in the following sense: The chain rule is also valid in this context: In particular, it is represented in coordinates by the Jacobian matrix. Frexhet that f is a map, f: The converse is not true: This means that there exists a function g: This is analogous to the fact that the existence of all directional derivatives at a point does not guarantee total differentiability or even dw at that point.

The following example only works in infinite dimensions. This function may also have a derivative, the second order derivative of fwhich, by the definition of derivative, will be a map.

The n -th derivative will be a function. Letting U be an open subset of X that contains the origin and given a function f: The chain rule also holds as does the Leibniz rule whenever Y is an algebra and a TVS in which multiplication frechey continuous.

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