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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Banach bundle ： ウィキペディア英語版 Banach bundle In mathematics, a Banach bundle is a vector bundle each of whose fibres is a Banach space, i.e. a complete normed vector space, possibly of infinite dimension. ==Definition of a Banach bundle== Let ''M'' be a Banach manifold of class ''C''''p'' with ''p'' ≥ 0, called the base space; let ''E'' be a topological space, called the total space; let ''π'' : ''E'' → ''M'' be a surjective continuous map. Suppose that for each point ''x'' ∈ ''M'', the fibre ''E''''x'' = ''π''−1(''x'') has been given the structure of a Banach space. Let :$\backslash$ be an open cover of ''M''. Suppose also that for each ''i'' ∈ ''I'', there is a Banach space ''X''''i'' and a map ''τ''''i'' :$\backslash tau\_\; :\; \backslash pi^\; (U\_)\; \backslash to\; U\_\; \backslash times\; X\_$ such that * the map ''τ''''i'' is a homeomorphism commuting with the projection onto ''U''''i'', i.e. the following diagram commutes: :: : and for each ''x'' ∈ ''U''''i'' the induced map ''τ''''ix'' on the fibre ''E''''x'' ::$\backslash tau\_\; :\; \backslash pi^\; (x)\; \backslash to\; X\_$ : is an invertible continuous linear map, i.e. an isomorphism in the category of topological vector spaces; * if ''U''''i'' and ''U''''j'' are two members of the open cover, then the map ::$U\_\; \backslash cap\; U\_\; \backslash to\; \backslash mathrm(X\_;\; X\_)$ ::$x\; \backslash mapsto\; (\backslash tau\_\; \backslash circ\; \backslash tau\_^)\_$ : is a morphism (a differentiable map of class ''C''''p''), where Lin(''X''; ''Y'') denotes the space of all continuous linear maps from a topological vector space ''X'' to another topological vector space ''Y''. The collection is called a trivialising covering for ''π'' : ''E'' → ''M'', and the maps ''τ''''i'' are called trivialising maps. Two trivialising coverings are said to be equivalent if their union again satisfies the two conditions above. An equivalence class of such trivialising coverings is said to determine the structure of a Banach bundle on ''π'' : ''E'' → ''M''. If all the spaces ''X''''i'' are isomorphic as topological vector spaces, then they can be assumed all to be equal to the same space ''X''. In this case, ''π'' : ''E'' → ''M'' is said to be a Banach bundle with fibre ''X''. If ''M'' is a connected space then this is necessarily the case, since the set of points ''x'' ∈ ''M'' for which there is a trivialising map :$\backslash tau\_\; :\; \backslash pi^\; (x)\; \backslash to\; X$ for a given space ''X'' is both open and closed. In the finite-dimensional case, the second condition above is implied by the first. 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 bundle」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.