# 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 {*U*_{α}}_{α ε I}, and a collection of homeomorphisms φ_{α} : U_{α} → *F*_{α} onto their images, where *F*_{α} are Fréchet spaces, such that

- 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 **R**^{n}, or even an open subset of **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” Fréchet manifolds are the open subsets of Hilbert space.

## See also

- Banach manifold, of which a Fréchet manifold is a generalization
- Manifolds of mappings

## References

- Hamilton, Richard S. (1982). "The inverse function theorem of Nash and Moser".
*Bull. Amer. Math. Soc. (N.S.)***7**(1): 65–222. doi:10.1090/S0273-0979-1982-15004-2. ISSN 0273-0979. MR 656198 - Henderson, David W. (1969). "Infinite-dimensional manifolds are open subsets of Hilbert space".
*Bull. Amer. Math. Soc.***75**(4): 759–762. doi:10.1090/S0002-9904-1969-12276-7. MR 0247634