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| 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 (mathematics)|manifold]] is modeled on a [[Euclidean space]].
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| More precisely, a Fréchet manifold consists of a [[Hausdorff space]] ''X'' with an atlas of coordinate charts over Fréchet spaces whose transitions are [[differentiation in Fréchet spaces|smooth mappings]]. Thus ''X'' has an [[open cover]] {''U''<sub>α</sub>}<sub>α ε I</sub>, and a collection of [[homeomorphism]]s φ<sub>α</sub> : U<sub>α</sub> → ''F''<sub>α</sub> onto their images, where ''F''<sub>α</sub> are Fréchet spaces, such that
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| ::<math>\phi_{\alpha\beta} := \phi_\alpha \circ \phi_\beta^{-1}|_{\phi_\beta(U_\beta\cap U_\alpha)}</math> is smooth for all pairs of indices α, β.
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| ==Classification up to homeomorphism==
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| It is by no means true that a finite-dimensional manifold of dimension ''n'' is ''globally'' homeomorphic to '''R'''<sup>''n''</sup>, or even an open subset of '''R'''<sup>''n''</sup>. 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 space|separable]], [[metric space|metric]] Fréchet manifold ''X'' can be [[embedding|embedded]] as an open subset of the infinite-dimensional, separable [[Hilbert space]], ''H'' (up to linear isomorphism, there is only one such space).
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| 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.
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| ==See also==
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| * [[Banach manifold]], of which a Fréchet manifold is a generalization
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| ==References==
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| * {{cite journal
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| | last = Hamilton
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| | first = Richard S.
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| | title = The inverse function theorem of Nash and Moser
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| | journal = Bull. Amer. Math. Soc. (N.S.)
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| | volume = 7
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| | year = 1982
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| | issue = 1
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| | pages = 65–222
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| | issn = 0273-0979
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| | doi = 10.1090/S0273-0979-1982-15004-2
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| }} {{MathSciNet|id=656198}}
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| * {{cite journal
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| | last = Henderson
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| | first = David W.
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| | title = Infinite-dimensional manifolds are open subsets of Hilbert space
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| | journal = Bull. Amer. Math. Soc.
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| | volume = 75
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| | year = 1969
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| | pages = 759–762
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| | doi = 10.1090/S0002-9904-1969-12276-7
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| | issue = 4
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| }} {{MathSciNet|id=0247634}}
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| [[Category:Nonlinear functional analysis]]
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| [[Category:Structures on manifolds]]
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| [[Category:Manifolds]]
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| {{DEFAULTSORT:Frechet Manifold}}
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