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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. 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∗.
References
- Atiyah 2007, p. 1
- The complement is also denoted as S \ A.
- More precisely, [math]\displaystyle{ C^{**} }[/math] is the smallest closed convex cone containing [math]\displaystyle{ C }[/math].
- Artstein-Avidan & Milman 2007
- Artstein-Avidan & Milman 2008
- Veblen & Young 1965.
- (Veblen & Young 1965, Ch. I, Theorem 11)
- More generally, one can consider the projective planes over any field, such as the complex numbers or finite fields or even division rings.
- See elliptic regularity.
- (Edwards 1965).
- Fulton 1993
- Mac Lane 1998, Ch. II.1.
- (Lam 1999, §19C)
- Jiří Adámek; J. Rosicky (1994). Locally Presentable and Accessible Categories. Cambridge University Press. p. 62. ISBN 978-0-521-42261-1. https://books.google.com/books?id=iXh6rOd7of0C&pg=PA62.
- Weibel (1994)
- Dwyer and Spaliński (1995)
- Negrepontis 1971.
- Hartshorne 1966, Ch. II.2, esp. Prop. II.2.3
- Joyal and Street (1991)
- See (Lang 2002, Theorem VI.1.1) for finite Galois extensions.
- (Loomis 1953, p. 151, section 37D)
- Griffiths & Harris 1994, p. 56
- Milne 1980, Ch. VI.11
- Iversen 1986, Ch. VII.3, VII.5
- Hartshorne 1966, Ch. III.7
- Milne (2006, Example I.1.10)
- Mazur (1973); Milne (2006)
