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In mathematics, a Banach manifold is a manifold modeled on Banach spaces. Thus it is a topological space in which each point has a neighbourhood homeomorphic to an open set in a Banach space (a more involved and formal definition is given below). Banach manifolds are one possibility of extending manifolds to infinite dimensions. A further generalisation is to Fréchet manifolds, replacing Banach spaces by Fréchet spaces. On the other hand, a Hilbert manifold is a special case of a Banach manifold in which the manifold is locally modelled on Hilbert spaces. ==Definition== Let ''X'' be a set. An atlas of class ''C''''r'', ''r'' ≥ 0, on ''X'' is a collection of pairs (called charts) (''U''''i'', ''φ''''i''), ''i'' ∈ ''I'', such that # each ''U''''i'' is a subset of ''X'' and the union of the ''U''''i'' is the whole of ''X''; # each ''φ''''i'' is a bijection from ''U''''i'' onto an open subset ''φ''''i''(''U''''i'') of some Banach space ''E''''i'', and for any ''i'' and ''j'', ''φ''''i''(''U''''i'' ∩ ''U''''j'') is open in ''E''''i''; # the crossover map :: : is an ''r''-times continuously differentiable function for every ''i'' and ''j'' in ''I'', i.e. the ''r''th Fréchet derivative :: : exists and is a continuous function with respect to the ''E''''i''-norm topology on subsets of ''E''''i'' and the operator norm topology on Lin(''E''''i''''r''; ''E''''j''.) One can then show that there is a unique topology on ''X'' such that each ''U''''i'' is open and each ''φ''''i'' is a homeomorphism. Very often, this topological space is assumed to be a Hausdorff space, but this is not necessary from the point of view of the formal definition. If all the Banach spaces ''E''''i'' are equal to the same space ''E'', the atlas is called an ''E''-atlas. However, it is not ''a priori'' necessary that the Banach spaces ''E''''i'' be the same space, or even isomorphic as topological vector spaces. However, if two charts (''U''''i'', ''φ''''i'') and (''U''''j'', ''φ''''j'') are such that ''U''''i'' and ''U''''j'' have a non-empty intersection, a quick examination of the derivative of the crossover map : shows that ''E''''i'' and ''E''''j'' must indeed be isomorphic as topological vector spaces. Furthermore, the set of points ''x'' ∈ ''X'' for which there is a chart (''U''''i'', ''φ''''i'') with ''x'' in ''U''''i'' and ''E''''i'' isomorphic to a given Banach space ''E'' is both open and closed. Hence, one can without loss of generality assume that, on each connected component of ''X'', the atlas is an ''E''-atlas for some fixed ''E''. A new chart (''U'', ''φ'') is called compatible with a given atlas if the crossover map : is an ''r''-times continuously differentiable function for every ''i'' ∈ ''I''. Two atlases are called compatible if every chart in one is compatible with the other atlas. Compatibility defines an equivalence relation on the class of all possible atlases on ''X''. A ''C''''r''-manifold structure on ''X'' is then defined to be a choice of equivalence class of atlases on ''X'' of class ''C''''r''. If all the Banach spaces ''E''''i'' are isomorphic as topological vector spaces (which is guaranteed to be the case if ''X'' is connected), then an equivalent atlas can be found for which they are all equal to some Banach space ''E''. ''X'' is then called an ''E''-manifold, or one says that ''X'' is modeled on ''E''. 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「Banach manifold」の詳細全文を読む スポンサード リンク
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