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
with an atlas of coordinate charts over Fréchet spaces whose transitions are smooth mappings.
α β
is smooth for all pairs of indices
It is by no means true that a finite-dimensional manifold of dimension
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
can be embedded as an open subset of the infinite-dimensional, separable Hilbert space,
(up to linear isomorphism, there is only one such space).
The embedding homeomorphism can be used as a global chart for
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].