翻訳と辞書 Words near each other ・ Dual module ・ Dual monarchy ・ Dual monarchy of England and France ・ Dual Mono ・ Dual naming ・ Dual narrative ・ Dual norm ・ Dual number ・ Dual object ・ Dual of BCH is an independent source ・ Dual Orb ・ Dual Orb II ・ Dual overhead rate ・ Dual oxidase 1 ・ Dual oxidase 2 ・ Dual pair ・ Dual Peppy ・ Dual Phase Steels Magnetism Modeling ・ Dual photography ・ Dual piping ・ Dual pivot steering geometry ・ Dual player ・ Dual Plover ・ Dual polygon ・ Dual polyhedron ・ Dual power ・ Dual process theory ・ Dual process theory (moral psychology) ・ Dual processor ・ Dual q-Hahn polynomials Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Dual pair ： ウィキペディア英語版 Dual pair In functional analysis and related areas of mathematics a dual pair or dual system is a pair of vector spaces with an associated bilinear map to the base field. A common method in functional analysis, when studying normed vector spaces, is to analyze the relationship of the space to its continuous dual, the vector space of all possible continuous linear forms on the original space. A dual pair generalizes this concept to arbitrary vector spaces, with the duality being expressed as a bilinear map. Using the bilinear map, semi norms can be constructed to define a polar topology on the vector spaces and turn them into locally convex spaces, generalizations of normed vector spaces. ==Definition== A dual pair is a 3-tuple $(X,Y,\backslash langle\; ,\; \backslash rangle)$ consisting of two vector spaces $X$ and $Y$ over the same field $F$ and a bilinear map :$\backslash langle\; ,\; \backslash rangle\; :\; X\; \backslash times\; Y\; \backslash to\; F$ with :$\backslash forall\; x\; \backslash in\; X\; \backslash setminus\; \backslash \; \backslash quad\; \backslash exists\; y\; \backslash in\; Y\; :\; \backslash langle\; x,y\; \backslash rangle\; \backslash neq\; 0$ and :$\backslash forall\; y\; \backslash in\; Y\; \backslash setminus\; \backslash \; \backslash quad\; \backslash exists\; x\; \backslash in\; X\; :\; \backslash langle\; x,y\; \backslash rangle\; \backslash neq\; 0$ We call $\backslash langle\; ,\; \backslash rangle$ the duality pairing, and say that it puts $X$ and $Y$ in duality. When the two spaces are a vector space $X$ (or a module over a ring in general) and its dual $X^$ *, we call the canonical duality pairing $\backslash langle\; \backslash cdot,\backslash cdot\; \backslash rangle\; :\; X^$ * \times X \rarr F : (\varphi, x) \mapsto \varphi(x) the natural pairing. We call two elements $x\; \backslash in\; X$ and $y\; \backslash in\; Y$ orthogonal if :$\backslash langle\; x,\; y\backslash rangle\; =\; 0.$ We call two sets $M\; \backslash subseteq\; X$ and $N\; \backslash subseteq\; Y$ orthogonal if each pair of elements from $M$ and $N$ are orthogonal. 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「Dual pair」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.