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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 Tensor product of Hilbert spaces ： ウィキペディア英語版 Tensor product of Hilbert spaces In mathematics, and in particular functional analysis, the tensor product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert spaces is another Hilbert space. Roughly speaking, the tensor product is the metric space completion of the ordinary tensor product. This is a special case of a topological tensor product. The tensor product allows the Hilbert space to be described by a symmetric monoidal category.〔B. Coecke and E. O. Paquette, Categories for the practising physicist, in: New Structures for Physics, B. Coecke (ed.), Springer Lecture Notes in Physics, 2009. (arXiv:0905.3010 )〕 ==Definition== Since Hilbert spaces have inner products, one would like to introduce an inner product, and therefore a topology, on the tensor product that arise naturally from those of the factors. Let ''H''1 and ''H''2 be two Hilbert spaces with inner products $\backslash langle\; \backslash cdot,\backslash cdot\backslash rangle\_1$ and $\backslash langle\; \backslash cdot,\backslash cdot\backslash rangle\_2$, respectively. Construct the tensor product of ''H''1 and ''H''2 as vector spaces as explained in the article on tensor products. We can turn this vector space tensor product into an inner product space by defining :$\backslash langle\backslash phi\_1\backslash otimes\backslash phi\_2,\backslash psi\_1\backslash otimes\backslash psi\_2\backslash rangle\; =\; \backslash langle\backslash phi\_1,\backslash psi\_1\backslash rangle\_1\; \backslash ,\; \backslash langle\backslash phi\_2,\backslash psi\_2\backslash rangle\_2\; \backslash quad\; \backslash mbox\; \backslash phi\_1,\backslash psi\_1\; \backslash in\; H\_1\; \backslash mbox\; \backslash phi\_2,\backslash psi\_2\; \backslash in\; H\_2$ and extending by linearity. That this inner product is the natural one is justified by the identification of scalar-valued bilinear maps on ''H''1 × ''H''2 and linear functionals on their vector space tensor product. Finally, take the completion under this inner product. The resulting Hilbert space is the tensor product of  ''H''1 and ''H''2. 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「Tensor product of Hilbert spaces」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.