- Fréchet manifold
In
mathematics , in particular innonlinear analysis , a Fréchet manifold is atopological space modeled on aFréchet space in much the same way as a manifold is modeled on aEuclidean 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 anopen cover {"U"α}α ε I, and a collection ofhomeomorphism s φα : Uα → "F"α onto their images, where "F"α are Fréchet spaces, such that ::phi_{alphaeta} := phi_alpha circ phi_eta^{-1}|_{phi_eta(U_etacap 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 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, separableHilbert 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 generalizationReferences
* 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
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