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Duality (mathematics)
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Duality (mathematics)
In mathematics, a duality 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. In other cases the dual of the dual – the double dual or bidual – is not necessarily identical to the original (also called primal). 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. It has been described as "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. This functor assigns to each space its dual space, and the pullback construction assigns to each arrow f: V → W its dual f∗: W∗ → V∗.
In the words of Michael Atiyah,
Duality in mathematics is not a theorem, but a "principle".
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.
A simple duality arises from considering subsets of a fixed set S. To any subset A ⊆ S, the complement A∁ consists of all those elements in S that are not contained in A. It is again a subset of S. Taking the complement has the following properties:
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Duality (mathematics)
In mathematics, a duality 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. In other cases the dual of the dual – the double dual or bidual – is not necessarily identical to the original (also called primal). 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. It has been described as "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. This functor assigns to each space its dual space, and the pullback construction assigns to each arrow f: V → W its dual f∗: W∗ → V∗.
In the words of Michael Atiyah,
Duality in mathematics is not a theorem, but a "principle".
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.
A simple duality arises from considering subsets of a fixed set S. To any subset A ⊆ S, the complement A∁ consists of all those elements in S that are not contained in A. It is again a subset of S. Taking the complement has the following properties: