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, the “only” Fréchet manifolds are the open subsets of Hilbert space.

ee also

* Banach manifold, of which a Fréchet manifold is a generalization

References

* cite journal
last = Hamilton
first = Richard S.
title = The inverse function theorem of Nash and Moser
journal = Bull. Amer. Math. Soc. (N.S.)
volume = 7
year = 1982
issue = 1
pages = 65–222
issn = 0273-0979 MathSciNet|id=656198
* cite journal
last = Henderson
first = David W.
title = Infinite-dimensional manifolds are open subsets of Hilbert space
journal = Bull. Amer. Math. Soc.
volume = 75
year = 1969
pages = 759–762 MathSciNet|id=0247634