Fréchet manifold
# Fréchet manifold

In mathematics, in particular in nonlinear analysis, a Fréchet manifold is a topological space modeled on a Fréchet space in much the same way as a manifold is modeled on a Euclidean space.

More precisely, a Fréchet manifold consists of a Hausdorff space $X$ with an atlas of coordinate charts over Fréchet spaces whose transitions are smooth mappings. Thus $X$ has an open cover $\left\{U_{\alpha }\right\}_{\alpha \in I},$ and a collection of homeomorphisms $\phi _{\alpha }:U_{\alpha }\to F_{\alpha }$ onto their images, where $F_{\alpha }$ are Fréchet spaces, such that

$\phi _{\alpha \beta }:=\phi _{\alpha }\circ \phi _{\beta }^{-1}|_{\phi _{\beta }\left(U_{\beta }\cap U_{\alpha }\right)}$

is smooth for all pairs of indices $\alpha ,\beta .$

## Classification up to homeomorphism

It is by no means true that a finite-dimensional manifold of dimension $n$ is globally homeomorphic to $\mathbb {R} ^{n}$ or even an open subset of $\mathbb {R} ^{n}.$ However, in an infinite-dimensional setting, it is possible to classify "well-behaved" Fréchet manifolds up to homeomorphism quite nicely. A 1969 theorem of David Henderson states that every infinite-dimensional, separable, metric Fréchet manifold $X$ can be embedded as an open subset of the infinite-dimensional, separable Hilbert space, $H$ (up to linear isomorphism, there is only one such space).

The embedding homeomorphism can be used as a global chart for $X.$ Thus, in the infinite-dimensional, separable, metric case, up to homeomorphism, the "only" topological Fréchet manifolds are the open subsets of the separable infinite-dimensional Hilbert space. But in the case of differentiable or smooth Fréchet manifolds (up to the appropriate notion of diffeomorphism) this fails.

