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 {Uα}α ε I, and a collection of homeomorphisms φα : UαFα onto their images, where Fα are Fréchet spaces, such that

${\displaystyle \phi _{\alpha \beta }:=\phi _{\alpha }\circ \phi _{\beta }^{-1}|_{\phi _{\beta }(U_{\beta }\cap U_{\alpha })}}$ is smooth for all pairs of indices α, β.

Classification up to homeomorphism

It is by no means true that a finite-dimensional manifold of dimension n is globally homeomorphic to Rn, or even an open subset of Rn. 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[citation needed].