Fréchet derivative

In mathematics, the Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued function of multiple real variables, and to define the functional derivative used widely in the calculus of variations.

Generally, it extends the idea of the derivative from real-valued functions of one real variable to functions on Banach spaces. The Fréchet derivative should be contrasted to the more general Gâteaux derivative which is a generalization of the classical directional derivative.

The Fréchet derivative has applications to nonlinear problems throughout mathematical analysis and physical sciences, particularly to the calculus of variations and much of nonlinear analysis and nonlinear functional analysis.

Let V and W be Banach spaces, and ${\displaystyle U\subset V}$ be an open subset of V. A function f : U → W is called Fréchet differentiable at ${\displaystyle x\in U}$ if there exists a bounded linear operator ${\displaystyle A:V\to W}$ such that

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