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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 \left\{ U_{\alpha} \right\}_{\alpha \in I}, and a collection of homeomorphisms \phi_{\alpha} : U_{\alpha} \to F_{\alpha} onto their images, where F_{\alpha} are Fréchet spaces, such that

\phi_{\alpha\beta} := \phi_\alpha \circ \phi_\beta^{-1}|_{\phi_\beta\left(U_\beta\cap U_\alpha\right)} is smooth for all pairs of indices \alpha, \beta.

Classification up to homeomorphism

It is by no means true that a finite-dimensional manifold of dimension n is {{em|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" 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 = 65–222

| 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 = 759–762

| 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