mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, in particular in nonlinear analysis, a Fréchet manifold is a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...

modeled on a Fréchet space in much the same way as a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

is modeled on a

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

. More precisely, a Fréchet manifold consists of a

Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...

X

with an atlas of coordinate charts over Fréchet spaces whose transitions are smooth mappings. Thus

X

has an

open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\s ...

\left\_,

and a collection of

homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...

\phi_ : U_ \to F_

onto their images, where

F_

are Fréchet spaces, such that

\phi_ := \phi_\alpha \circ \phi_\beta^, _

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 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 Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...

Fréchet manifold

X

can be embedded as an open subset of the infinite-dimensional, separable

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...

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 or Fréchet manifolds (up to the appropriate notion of diffeomorphism) this fails.

References

* * {{DEFAULTSORT:Frechet Manifold Generalized manifolds Manifolds Nonlinear functional analysis Structures on manifolds

Classification up to homeomorphism

See also

References