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.

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### calculus – A New Definition of Derivative – Mathematics Stack Exchange

It’s correct, and this method is similar to user ‘s. This page was last edited on 6 Octoberat One notion of continuous differentiability in U requires that the mapping on the product space. It requires the use of the Euclidean norm, which isn’t very desirable. And you have that.

## Fréchet derivative

Note that this already presupposes the linearity of DF u. The n -th derivative will be a function.

This definition is discussed in the finite-dimensional case in: As a matter of technical convenience, this latter notion of continuous differentiability is typical but not universal when the spaces X and Y are Banach, since L XY is also Banach and standard results from functional analysis can then be employed.

The following example only works in infinite dimensions. This is analogous to the result from basic complex analysis that a function is analytic if it is complex differentiable in an open set, and is a fundamental result in the study of infinite dimensional holomorphy.

BenCrowell 4 is the standard definition. Use dmy dates from July You can use this method in an arbitrary normed vector space, even an infinite-dimensional one, but you need to replace the use of the inner product by an appeal to the Hahn-Banach theorem. By virtue of the bilinearity, the polarization identity holds.