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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 Real structure ： ウィキペディア英語版 Real structure In mathematics, a real structure on a complex vector space is a way to decompose the complex vector space in the direct sum of two real vector spaces. The prototype of such a structure is the field of complex numbers itself, considered as a complex vector space over itself and with the conjugation map $\backslash sigma:\; \backslash to\; \backslash ,$, with $\backslash sigma\; (z)=$, giving the "canonical" real structure on $\backslash ,$, that is $=\backslash oplus\; i\backslash ,$. The conjugation map is antilinear: $\backslash sigma\; (\backslash lambda\; z)=\backslash sigma(z)\backslash ,$ and $\backslash sigma\; (z\_1+z\_2)=\backslash sigma(z\_1)+\backslash sigma(z\_2)\backslash ,$. ==Vector space== A real structure on a complex vector space ''V'' is an antilinear involution $\backslash sigma:\; V\; \backslash to\; V$. A real structure defines a real subspace $V\_\; \backslash otimes\_\; \backslash to\; V$ is an isomorphism. Conversely any vector space that is the complexification of a real vector space has a natural real structure. One first notes that every complex space ''V'' has a real form obtained by taking the same vectors as in the original set and restricting the scalars to be real. If $t\backslash in\; V\backslash ,$ and $t\backslash neq\; 0$ then the vectors $t\backslash ,$ and $it\backslash ,$ are linearly independent in the real form of ''V''. Hence: :$\backslash dim\_V\; =\; 2\backslash dim\_V$ Naturally, one would wish to represent ''V'' as the direct sum of two real vector spaces, the "real and imaginary parts of ''V''". There is no canonical way of doing this: such a splitting is an additional real structure in ''V''. It may be introduced as follows.〔Budinich, P. and Trautman, A. ''The Spinorial Chessboard''. Springer-Verlag, 1988, p. 29.〕 Let $\backslash sigma:\; V\; \backslash to\; V\backslash ,$ be an antilinear map such that $\backslash sigma\backslash circ\backslash sigma=id\_\backslash ,$, that is an antilinear involution of the complex space ''V''. Any vector $t\backslash in\; V\backslash ,$ can be written $\}\backslash ,$, where $t^+\; =\}(t-\backslash sigma\; t)\backslash ,$. Therefore, one gets a direct sum of vector spaces $V=V^\backslash oplus\; V^\backslash ,$ where: :$V^=\backslash$ and $V^=\backslash \backslash ,$. Both sets $V^+\backslash ,$ and $V^-\backslash ,$ are real vector spaces. The linear map $K:\; V^+\; \backslash to\; V^-\backslash ,$, where $K(t)=it\backslash ,$, is an isomorphism of real vector spaces, whence: :$\backslash dim\_V^+\; =\; \backslash dim\_V^-\; =\; \backslash dim\_V\backslash ,$. The first factor $V^+\backslash ,$ is also denoted by $V\_\})\backslash subset\; V\_\}\backslash ,$. The direct sum $V=V^\backslash oplus\; V^\backslash ,$ reads now as: :$V=V\_\}\backslash ,$, i.e. as the direct sum of the "real" $V\_\}\backslash ,$ parts of ''V''. This construction strongly depends on the choice of an antilinear involution of the complex vector space ''V''. The complexification of the real vector space $V\_\}=\; V\_\; \backslash otimes\_\backslash ,$ admits a natural real structure and hence is canonically isomorphic to the direct sum of two copies of $V\_\backslash ,$: :$V\_\; \backslash otimes\_=\; V\_\}\backslash ,$. It follows a natural linear isomorphism $V\_\; \backslash otimes\_\; \backslash to\; V\backslash ,$ between complex vector spaces with a given real structure. A real structure on a complex vector space ''V'', that is an antilinear involution $\backslash sigma:\; V\; \backslash to\; V\backslash ,$, may be equivalently described in terms of the linear map $\backslash hat\; \backslash sigma:V\backslash to\backslash bar\; V\backslash ,$ from the vector space $V\backslash ,$ to the complex conjugate vector space $\backslash bar\; V\backslash ,$ defined by :$v\; \backslash mapsto\; \backslash hat\backslash sigma\; (v):=\backslash overline\backslash ,$.〔Budinich, P. and Trautman, A. ''The Spinorial Chessboard''. Springer-Verlag, 1988, p. 29.〕 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「Real structure」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.