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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 consisting of two vector spaces and over the same field and a bilinear map : with : and : We call the duality pairing, and say that it puts and in duality. When the two spaces are a vector space (or a module over a ring in general) and its dual , we call the canonical duality pairing the natural pairing. We call two elements and orthogonal if : We call two sets and orthogonal if each pair of elements from and 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
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