Fréchetdifferenciálhatóság

Fréchetdifferentiability, or the Fréchet derivative, is a rigorous notion of differentiability for maps between normed spaces. Let X and Y be normed spaces over the real or complex field, U ⊆ X an open set, and f: U → Y. f is Fréchet differentiable at a ∈ U if there exists a bounded linear operator A: X → Y such that

lim_{h→0} || f(a+h) − f(a) − A(h) || / ||h|| = 0.

In this case A is called the Fréchet derivative of f at a and is denoted Df(a) or f′(a). It provides the best possible linear approximation to f at a.

In finite dimensions, where X = R^n and Y = R^m with standard norms, Df(a) is represented by the Jacobian matrix Jf(a), and the defining relation becomes f(a+h) = f(a) + Jf(a)h + o(||h||). Fréchet differentiability implies the existence of directional derivatives in every direction v, with f′(a; v) = Df(a)[v]. Thus Fréchet differentiability is a strong form of differentiability that guarantees a linear approximation that works uniformly in small increments.

Key properties include the chain rule: if f: U → Y and g: V → Z are Fréchet differentiable at a and f(a) respectively, then D(g∘f)(a) = Dg(f(a)) ∘ Df(a). If Df is continuous near a, f is of class C^1. Classic examples include smooth maps between finite-dimensional spaces; the map X ↦ X^{-1} on the set GL(n) is Fréchet differentiable with derivative D(X^{-1})[H] = −X^{-1}HX^{-1}.

Originating with Maurice Fréchet in the early 20th century, Fréchet differentiability underpins much of functional analysis, optimization, and the study of maps between infinite-dimensional spaces.