Fréchet Derivative

The Fréchet derivative, named after Maurice Fréchet, is a derivative defined on normed spaces that extends the ordinary derivative of a real valued function of one real variable to vector valued functions of several variables and, more generally, to functions between normed spaces. It also underlies the functional derivative used in the calculus of variations. Because it is defined on normed spaces rather than only on finite dimensional Euclidean space, the Fréchet derivative allows differentiation to be carried out in infinite dimensional settings where ordinary calculus techniques do not directly apply, and it provides a stronger, more rigorous notion of differentiability than the related Gateaux derivative. It has significant applications to nonlinear problems throughout mathematical analysis and the physical sciences, especially in the calculus of variations and nonlinear functional analysis.