|a.||1.||Expressing, or consisting of, the number two; belonging to two; |
|Adj.||1.||dual - consisting of or involving two parts or components usually in pairs; "an egg with a double yolk"; "a double (binary) star"; "double doors"; "dual controls for pilot and copilot"; "duple (or double) time consists of two (or a multiple of two) beats to a measure"|
|2.||dual - having more than one decidedly dissimilar aspects or qualities; "a double (or dual) role for an actor"; "the office of a clergyman is twofold; public preaching and private influence"- R.W.Emerson; "every episode has its double and treble meaning"-Frederick Harrison|
|3.||dual - a grammatical number category referring to two items or units as opposed to one item (singular) or more than two items (plural); "ancient Greek had the dual form but it has merged with the plural form in modern Greek"|
|(mathematics)||dual - Every field of mathematics has a different
meaning of dual. Loosely, where there is some binary symmetry
of a theory, the image of what you look at normally under this
symmetry is referred to as the dual of your normal things.|
In linear algebra for example, for any vector space V, over a field, F, the vector space of linear maps from V to F is known as the dual of V. It can be shown that if V is finite-dimensional, V and its dual are isomorphic (though no isomorphism between them is any more natural than any other).
There is a natural embedding of any vector space in the dual of its dual:
V -> V'': v -> (V': w -> wv : F)
(x' is normally written as x with a horizontal bar above it). I.e. v'' is the linear map, from V' to F, which maps any w to the scalar obtained by applying w to v. In short, this double-dual mapping simply exchanges the roles of function and argument.
It is conventional, when talking about vectors in V, to refer to the members of V' as covectors.