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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 Krein space ： ウィキペディア英語版 Indefinite inner product space In mathematics, in the field of functional analysis, an indefinite inner product space :$(K,\; \backslash langle\; \backslash cdot,\backslash ,\backslash cdot\; \backslash rangle,\; J)$ is an infinite-dimensional complex vector space $K$ equipped with both an indefinite inner product :$\backslash langle\; \backslash cdot,\backslash ,\backslash cdot\; \backslash rangle\; \backslash ,$ and a positive semi-definite inner product :$(x,\backslash ,y)\; \backslash \; \backslash stackrel\backslash \; \backslash langle\; x,\backslash ,Jy\; \backslash rangle,$ where the metric operator $J$ is an endomorphism of $K$ obeying :$J^3\; =\; J.\; \backslash ,$ The indefinite inner product space itself is not necessarily a Hilbert space; but the existence of a positive semi-definite inner product on $K$ implies that one can form a quotient space on which there is a positive definite inner product. Given a strong enough topology on this quotient space, it has the structure of a Hilbert space, and many objects of interest in typical applications fall into this quotient space. An indefinite inner product space is called a Krein space (or $J$''-space'') if $(x,\backslash ,y)$ is positive definite and $K$ possesses a majorant topology. Krein spaces are named in honor of the Ukrainian mathematician Mark Grigorievich Krein (3 April 1907 – 17 October 1989). == Inner products and the metric operator == Consider a complex vector space $K$ equipped with an indefinite hermitian form $\backslash langle\; \backslash cdot\; ,\backslash ,\; \backslash cdot\; \backslash rangle$. In the theory of Krein spaces it is common to call such an hermitian form an indefinite inner product. The following subsets are defined in terms of the square norm induced by the indefinite inner product: :$K\_\; \backslash \; \backslash stackrel\backslash \; \backslash$ ("neutral") :$K\_\; \backslash \; \backslash stackrel\backslash \; \backslash$ ("positive") :$K\_\; \backslash \; \backslash stackrel\backslash \; \backslash$ ("negative") :$K\_\; \backslash \; \backslash stackrel\backslash \; K\_\; \backslash cup\; K\_$ ("non-negative") :$K\_\; \backslash \; \backslash stackrel\backslash \; K\_\; \backslash cup\; K\_$ ("non-positive") A subspace $L\; \backslash subset\; K$ lying within $K\_$ is called a ''neutral subspace''. Similarly, a subspace lying within $K\_$ ($K\_$) is called ''positive'' (''negative'') ''semi-definite'', and a subspace lying within $K\_\; \backslash cup\; \backslash$ ($K\_\; \backslash cup\; \backslash$) is called ''positive'' (''negative'') ''definite''. A subspace in any of the above categories may be called ''semi-definite'', and any subspace that is not semi-definite is called ''indefinite''. Let our indefinite inner product space also be equipped with a decomposition into a pair of subspaces $K\; =\; K\_+\; \backslash oplus\; K\_-$, called the ''fundamental decomposition'', which respects the complex structure on $K$. Hence the corresponding linear projection operators $P\_\backslash pm$ coincide with the identity on $K\_\backslash pm$ and annihilate $K\_\backslash mp$, and they commute with multiplication by the $i$ of the complex structure. If this decomposition is such that $K\_+\; \backslash subset\; K\_$ and $K\_-\; \backslash subset\; K\_$, then $K$ is called an indefinite inner product space; if $K\_\backslash pm\; \backslash subset\; K\_\; \backslash cup\; \backslash$, then $K$ is called a Krein space, subject to the existence of a majorant topology on $K$. The operator $J\; \backslash \; \backslash stackrel\backslash \; P\_+\; -\; P\_-$ is called the (real phase) ''metric operator'' or ''fundamental symmetry'', and may be used to define the ''Hilbert inner product'' $(\backslash cdot,\backslash ,\backslash cdot)$: :$(x,\backslash ,y)\; \backslash \; \backslash stackrel\backslash \; \backslash langle\; x,\backslash ,Jy\; \backslash rangle\; =\; \backslash langle\; x,\backslash ,P\_+\; y\; \backslash rangle\; -\; \backslash langle\; x,\backslash ,P\_-\; y\; \backslash rangle$ On a Krein space, the Hilbert inner product is positive definite, giving $K$ the structure of a Hilbert space (under a suitable topology). Under the weaker constraint $K\_pm\; \backslash subset\; K\_$, some elements of the neutral subspace $K\_0$ may still be neutral in the Hilbert inner product, but many are not. For instance, the subspaces $K\_0\; \backslash cap\; K\_\backslash pm$ are part of the neutral subspace of the Hilbert inner product, because an element $k\; \backslash in\; K\_0\; \backslash cap\; K\_\backslash pm$ obeys $(k,\backslash ,k)\; \backslash \; \backslash stackrel\backslash \; \backslash langle\; k,\backslash ,Jk\; \backslash rangle\; =\; \backslash pm\; \backslash langle\; k,\backslash ,k\; \backslash rangle\; =\; 0$. But an element $k\; =\; k\_+\; +\; k\_-$ ($k\_\backslash pm\; \backslash in\; K\_\backslash pm$) which happens to lie in $K\_0$ because $\backslash langle\; k\_-,\backslash ,k\_-\; \backslash rangle\; =\; -\; \backslash langle\; k\_+,\backslash ,k\_+\; \backslash rangle$ will have a positive square norm under the Hilbert inner product. We note that the definition of the indefinite inner product as a Hermitian form implies that: :$\backslash langle\; x,\backslash ,y\; \backslash rangle\; =\; \backslash frac\; (\backslash langle\; x+y,\backslash ,x+y\; \backslash rangle\; -\; \backslash langle\; x-y,\backslash ,x-y\; \backslash rangle)$ Therefore the indefinite inner product of any two elements $x,\backslash ,y\; \backslash in\; K$ which differ only by an element $x-y\; \backslash in\; K\_0$ is equal to the square norm of their average $\backslash frac$. Consequently, the inner product of any non-zero element $k\_0\; \backslash in\; (K\_0\; \backslash cap\; K\_\backslash pm)$ with any other element $k\_\backslash pm\; \backslash in\; K\_\backslash pm$ must be zero, lest we should be able to construct some $k\_\backslash pm\; +\; 2\; \backslash lambda\; k\_0$ whose inner product with $k\_\backslash pm$ has the wrong sign to be the square norm of $k\_\backslash pm\; +\; \backslash lambda\; k\_0\; \backslash in\; K\_\backslash pm$. Similar arguments about the Hilbert inner product (which can be demonstrated to be a Hermitian form, therefore justifying the name "inner product") lead to the conclusion that its neutral space is precisely $K\_\; =\; (K\_0\; \backslash cap\; K\_+)\; \backslash oplus\; (K\_0\; \backslash cap\; K\_-)$, that elements of this neutral space have zero Hilbert inner product with any element of $K$, and that the Hilbert inner product is positive semi-definite. It therefore induces a positive definite inner product (also denoted $(\backslash cdot,\backslash ,\backslash cdot)$) on the quotient space $\backslash tilde\; \backslash \; \backslash stackrel\backslash \; K\; /\; K\_$, which is the direct sum of $\backslash tilde\_\backslash pm\; \backslash \; \backslash stackrel\backslash \; K\_\backslash pm\; /\; (K\_0\; \backslash cap\; K\_\backslash pm)$. Thus $(\backslash tilde,\backslash ,(\backslash cdot,\backslash ,\backslash cdot))$ is a Hilbert space (given a suitable topology). 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「Indefinite inner product space」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.