Duality (mathematics) - Dual Objects

Dual Objects

A group of dualities can be described by endowing, for any mathematical object X, the set of morphisms Hom(X, D) into some fixed object D, with a structure similar to the one of X. This is sometimes called internal Hom. In general, this yields a true duality only for specific choices of D, in which case X∗=Hom(X, D) is referred to as the dual of X. It may or may not be true that the bidual, that is to say, the dual of the dual, X∗∗ = (X∗)∗ is isomorphic to X, as the following example, which is underlying many other dualities, shows: the dual vector space V∗ of a K-vector space V is defined as

V∗ = Hom (V, K).

The set of morphisms, i.e., linear maps, is a vector space in its own right. There is always a natural, injective map V → V∗∗ given by v ↦ (f ↦ f(v)), where f is an element of the dual space. That map is an isomorphism if and only if the dimension of V is finite.

In the realm of topological vector spaces, a similar construction exists, replacing the dual by the topological dual vector space. A topological vector space that is canonically isomorphic to its bidual is called reflexive space.

The dual lattice of a lattice L is given by

Hom(L, Z),

which is used in the construction of toric varieties. The Pontryagin dual of locally compact topological groups G is given by

Hom(G, S1),

continuous group homomorphisms with values in the circle (with multiplication of complex numbers as group operation).