Fréchetdifferentiability

Fréchet differentiability is a concept in real analysis and functional analysis that generalizes the idea of differentiability to functions defined on normed vector spaces. A function f mapping from a normed vector space V to a normed vector space W is said to be Fréchet differentiable at a point x in V if there exists a linear map L from V to W such that the limit of the difference between f(x+h) and f(x) minus L(h), divided by the norm of h, approaches zero as h approaches zero. Mathematically, this is expressed as: lim_{h->0} ||f(x+h) - f(x) - L(h)||_W / ||h||_V = 0. The linear map L is called the Fréchet derivative of f at x, and it is unique if it exists. If the function is differentiable at every point in an open set, it is said to be Fréchet differentiable on that set. This definition is stronger than the concept of Gâteaux differentiability, which requires the limit to exist along every direction, but not necessarily uniformly for all directions. In finite-dimensional spaces, Fréchet differentiability is equivalent to the existence of all partial derivatives and their continuity. The Fréchet derivative, when it exists, is linear and continuous.