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In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of ''A'' is ''B'', then the dual of ''B'' is ''A''. Such involutions sometimes have fixed points, so that the dual of ''A'' is ''A'' itself. For example, Desargues' theorem is self-dual in this sense under the ''standard duality in projective geometry''. In mathematical contexts, ''duality'' has numerous meanings〔See Atiyah (2007)〕 although it is "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics". Many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of the second type to some family of scalars. For instance, ''linear algebra duality'' corresponds in this way to bilinear maps from pairs of vector spaces to scalars, the ''duality between distributions and the associated test functions'' corresponds to the pairing in which one integrates a distribution against a test function, and ''Poincaré duality'' corresponds similarly to intersection number, viewed as a pairing between submanifolds of a given manifold. From a category theory viewpoint, duality can also be seen as a functor, at least in the realm of vector spaces. There it is allowed to assign to each space its dual space and the pullback construction allows to assign for each arrow , its dual . ==Introductory examples== In the words of Michael Atiyah, :Duality in mathematics is not a theorem, but a “principle”.〔See Atiyah (2007)〕 The following list of examples shows the common features of many dualities, but also indicates that the precise meaning of duality may vary from case to case. 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「Duality (mathematics)」の詳細全文を読む スポンサード リンク
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