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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 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 α ε I, and a collection of homeomorphisms φα : Uα → ''F''α onto their images, where ''F''α are Fréchet spaces, such that :: is smooth for all pairs of indices α, β. ==Classification up to homeomorphism== It is by no means true that a finite-dimensional manifold of dimension ''n'' is ''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” Fréchet manifolds are the open subsets of Hilbert space. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Fréchet manifold」の詳細全文を読む スポンサード リンク
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