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In mathematics, the classifying space for the unitary group U(''n'') is a space BU(''n'') together with a universal bundle EU(''n'') such that any hermitian bundle on a paracompact space ''X'' is the pull-back of EU(''n'') by a map ''X'' → BU(''n'') unique up to homotopy. This space with its universal fibration may be constructed as either # the Grassmannian of ''n''-planes in an infinite-dimensional complex Hilbert space; or, # the direct limit, with the induced topology, of Grassmannians of ''n'' planes. Both constructions are detailed here. ==Construction as an infinite Grassmannian== The total space EU(''n'') of the universal bundle is given by : Here, ''H'' is an infinite-dimensional complex Hilbert space, the ''e''''i'' are vectors in ''H'', and is the Kronecker delta. The symbol is the inner product on ''H''. Thus, we have that EU(''n'') is the space of orthonormal ''n''-frames in ''H''. The group action of U(''n'') on this space is the natural one. The base space is then : and is the set of Grassmannian ''n''-dimensional subspaces (or ''n''-planes) in ''H''. That is, : so that ''V'' is an ''n''-dimensional vector 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「Classifying space for U(n)」の詳細全文を読む スポンサード リンク
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