Dual (mathematics) | Wikipedia audio article
Channel: wikipedia tts
Date: 07/10/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Duality_(mathematics)


00:00:19 1 Introductory examples
00:00:38 1.1 Complement of a subset
00:00:58 1.2 Dual cone
00:01:37 1.3 Dual vector space
00:01:56 1.4 Galois theory
00:02:16 2 Order-reversing dualities
00:02:26 3 Dimension-reversing dualities
00:02:45 4 Duality in logic and set theory
00:03:14 5 Dual objects
00:03:34 5.1 Dual vector spaces revisited
00:03:53 5.1.1 Isomorphisms of spaniV/i
00:04:13 5.1.2 Duality in projective geometry
00:04:32 5.2 Topological vector spaces and Hilbert spaces
00:04:52 5.3 Further dual objects
00:05:31 6 Dual categories
00:05:50 6.1 Opposite category and adjoint functors
00:06:10 6.2 Spaces and functions
00:06:20 6.3 Galois connections
00:06:39 6.4 Pontryagin duality
00:06:58 7 Analytic dualities
00:07:18 8 Homology and cohomology
00:07:37 8.1 Poincaré duality
00:08:07 8.2 Duality in algebraic and arithmetic geometry
00:08:26 9 See also
00:08:46 10 Notes
00:09:05 11 References



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SUMMARY
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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 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. 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∗.