Fréchet manifold

Summary

A Fréchet manifold is a space that looks locally like an open subset of a Fréchet space, with smooth changes of coordinates. It is the natural setting for spaces of smooth maps, such as the smooth loops in a manifold or the diffeomorphisms of a compact manifold. Calculus on Fréchet manifolds is harder than on Banach manifolds, because the inverse function theorem fails in general Fréchet spaces. The Nash–Moser theorem gives a replacement under extra hypotheses.