View source for 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 (mathematics)|manifold]] is modeled on a [[Euclidean space]].

More precisely, a Fréchet manifold consists of a [[Hausdorff space]] <math>X</math> with an atlas of coordinate charts over Fréchet spaces whose transitions are [[Differentiation in Fréchet spaces|smooth mappings]].  Thus <math>X</math> has an [[open cover]] <math>\left\{ U_{\alpha} \right\}_{\alpha \in I},</math> and a collection of [[homeomorphism]]s <math>\phi_{\alpha} : U_{\alpha} \to F_{\alpha}</math> onto their images, where <math>F_{\alpha}</math> are [[Fréchet space]]s, such that 
<math display=block>\phi_{\alpha\beta} := \phi_\alpha \circ \phi_\beta^{-1}|_{\phi_\beta\left(U_\beta\cap U_\alpha\right)}</math> is smooth for all pairs of indices <math>\alpha, \beta.</math>

== Classification up to homeomorphism ==

It is by no means true that a finite-dimensional manifold of dimension <math>n</math> is {{em|globally}} homeomorphic to <math>\R^n</math> or even an open subset of <math>\R^n.</math>  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 <math>X</math> can be [[Embedding|embedded]] as an open subset of the infinite-dimensional, separable [[Hilbert space]], <math>H</math> (up to linear isomorphism, there is only one such space).

The embedding homeomorphism can be used as a global chart for <math>X.</math>  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 {{em|differentiable}} or {{em|smooth}} Fréchet manifolds (up to the appropriate notion of diffeomorphism) this fails{{cn|date=July 2020}}.

== See also ==

* {{annotated link|Banach manifold}}, of which a Fréchet manifold is a generalization
* {{annotated link|Convenient vector space#Application: Manifolds of mappings between finite dimensional manifolds|Manifolds of mappings}}
* {{annotated link|Differentiation in Fréchet spaces}}
* {{annotated link|Hilbert manifold}}

== References ==

{{reflist}}
{{reflist|group=note}}

* {{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&ndash;222
| issn = 0273-0979
| doi = 10.1090/S0273-0979-1982-15004-2
| doi-access = free
}} {{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&ndash;762
| doi = 10.1090/S0002-9904-1969-12276-7
| issue = 4
| doi-access = free
}} {{MathSciNet|id=0247634}}

{{Manifolds}}
{{Analysis in topological vector spaces}}
{{Topological vector spaces}}
{{Functional Analysis}}

{{DEFAULTSORT:Frechet Manifold}}
[[Category:Generalized manifolds]]
[[Category:Manifolds]]
[[Category:Nonlinear functional analysis]]
[[Category:Structures on manifolds]]