dual

dual (ˈdjuːəl)
—adj
1.	relating to or denoting two
2.	twofold; double
3.	(in the grammar of Old English, Ancient Greek, and certain other languages) denoting a form of a word indicating that exactly two referents are being referred to
4.	maths, logic (of structures or expressions) having the property that the interchange of certain pairs of terms, and usually the distribution of negation, yields equivalent structures or expressions
—n
5.	grammar
	a.  the dual number
	b.  a dual form of a word
—vb , duals, dualling, dualled
6.	(Brit) (tr) to make (a road) into a dual carriageway
[C17: from Latin duālis concerning two, from duo two]
'dually
—adv

dual

dual definition

mathematics
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.
(1997-03-16)