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.
(1997-03-16)