Difference between revisions of "Pairing"
From Encyclopedia of Mathematics
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| − | A mapping defined on the Cartesian product of two sets. The requirements of bilinearity, continuity and others may be imposed upon this mapping, according to the context. A pairing $X\times Y\to Z$ defines a mapping from $X$ into the set of functions acting from $Y$ into $Z$ (or into some subset of the latter, for example consisting of homomorphisms, continuous mappings, etc.). Statements about the properties of the mapping thus obtained form the essence of various duality theorems in algebra, topology and functional analysis. | + | A mapping defined on the Cartesian product of two sets. The requirements of bilinearity, continuity and others may be imposed upon this mapping, according to the context. A pairing $X\times Y\to Z$ defines a mapping from $X$ into the set of functions acting from $Y$ into $Z$ (or into some subset of the latter, for example consisting of homomorphisms, continuous mappings, etc.), cf. [[Exponential law for sets]]. Statements about the properties of the mapping thus obtained form the essence of various duality theorems in algebra, topology and functional analysis. |
Latest revision as of 19:24, 4 January 2015
A mapping defined on the Cartesian product of two sets. The requirements of bilinearity, continuity and others may be imposed upon this mapping, according to the context. A pairing $X\times Y\to Z$ defines a mapping from $X$ into the set of functions acting from $Y$ into $Z$ (or into some subset of the latter, for example consisting of homomorphisms, continuous mappings, etc.), cf. Exponential law for sets. Statements about the properties of the mapping thus obtained form the essence of various duality theorems in algebra, topology and functional analysis.
How to Cite This Entry:
Pairing. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pairing&oldid=36085
Pairing. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pairing&oldid=36085
This article was adapted from an original article by M.Sh. Farber (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article