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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 Direct sum of topological groups ： ウィキペディア英語版 Direct sum of topological groups In mathematics, a topological group ''G'' is called the topological direct sum〔E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I, second edition, Grundlehren der Mathematischen Wissenschaften, 115, Springer, Berlin, 1979. MR0551496 (81k:43001)〕 of two subgroups ''H''1 and ''H''2 if the map :$\backslash begin$ H_1\times H_1 &\longrightarrow G \\ (h_1,h_2) &\longmapsto h_1 h_2 \end is a topological isomorphism. More generally, ''G'' is called the direct sum of a finite set of subgroups $H\_i,\; i=1,\backslash ldots,\; n$ of the map :$\backslash begin$ \prod^n_ H_i&\longrightarrow G \\ (h_i)_ &\longmapsto h_1 h_2 \cdots h_n \end Note that if a topological group ''G'' is the topological direct sum of the family of subgroups $H\_i$ then in particular, as an abstract group (without topology) it is also the direct sum (in the usual way) of the family $H\_i$. ==Topological direct summands== Given a topological group ''G'', we say that a subgroup ''H'' is a topological direct summand of ''G'' (or that splits topologically form ''G'' ) if and only if there exist another subgroup ''K'' ≤ ''G'' such that ''G'' is the direct sum of the subgroups ''H'' and ''K''. A the subgroup ''H'' is a topological direct summand if and only if the extension of topological groups :$0\; \backslash to\; H\backslash stackrel\; G\backslash stackrel\; G/H\backslash to\; 0$ splits, where $i$ is the natural inclusion and $\backslash pi$ is the natural projection. 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「Direct sum of topological groups」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.