Fréchet manifold
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In mathematics, in particular in
nonlinear analysis Nonlinear functional analysis is a branch of mathematical analysis that deals with nonlinear mappings. Topics Its subject matter includes: * generalizations of calculus to Banach spaces * implicit function theorems * fixed-point theorems (Br ...
, 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 po ...
modeled on a
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to th ...
in much the same way as a manifold 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'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
. 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 m ...
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_\alph ...
\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 isomor ...
s \phi_ : U_ \to F_ onto their images, where F_ are
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to th ...
s, 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 In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved. Th ...
" 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 mathem ...
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 or Fréchet manifolds (up to the appropriate notion of diffeomorphism) this fails.


See also

* , of which a Fréchet manifold is a generalization * * *


References

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