Frechetdifferenciál

Frechetdifferenciál refers to a generalization of the concept of a derivative for functions that map between Banach spaces. In essence, it provides a way to define differentiability for functions whose inputs and outputs are elements of abstract vector spaces. The Frechetdifferenciál at a point is a linear operator that best approximates the function's change near that point. It is a more stringent condition than directional differentiability.

A function f mapping from a Banach space X to a Banach space Y is said to be Frechet differentiable at a point x in X if there exists a bounded linear operator T from X to Y such that the limit of ||f(x+h) - f(x) - T(h)|| / ||h|| as h approaches zero is zero. This linear operator T is called the Frechetdifferenciál of f at x, often denoted by df(x) or f'(x). If such an operator exists, it is unique.

The existence of a Frechetdifferenciál implies that the function is locally "linearizable" in a precise sense. This concept is fundamental in various areas of mathematics, including differential geometry, functional analysis, and optimization theory, allowing for the study of calculus on infinite-dimensional spaces. Properties like the chain rule can be extended to Frechet differentiable functions.