Differentiation in Fréchet spaces

In mathematics, in particular in functional analysis and nonlinear analysis, it is possible to define the derivative of a function between two Fréchet spaces. This notion of differentiation is significantly weaker than the derivative in a Banach space. Nevertheless, it is the weakest notion of differentiation for which many of the familiar theorems from calculus hold. In particular, the chain rule is true. With some additional constraints on the Fréchet spaces and functions involved, there is an analog of the inverse function theorem called the Nash–Moser inverse function theorem, having wide applications in nonlinear analysis and differential geometry.

Formally, the definition of differentiation is identical to the Gâteaux derivative. Specifically, let X and Y be Fréchet spaces, U ⊂ X be an open set, and F : U → Y be a function. The directional derivative of F in the direction v ∈ X is defined by

if the limit exists. One says that F is continuously differentiable, or C¹ if the limit exists for all v ∈ X and the mapping

is a continuous map.

Higher order derivatives are defined inductively via

A function is said to be C^k if D^kF : U x X x Xx ... x X → Y is continuous. It is C^∞, or smooth if it is C^k for every k.

Let X, Y, and Z be Fréchet spaces. Suppose that U is an open subset of X, V is an open subset of Y, and F : U → V, G : V → Z are a pair of C¹ functions. Then the following properties hold:

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