Regularly ordered

In mathematics, specifically in order theory and functional analysis, an ordered [vector space] is said to be regularly ordered and its order is called regular if is Archimedean ordered and the order dual of distinguishes points in.
Being a regularly ordered vector space is an important property in the theory of topological vector lattices.

Examples

Every ordered locally convex space is regularly ordered.
The canonical orderings of subspaces, products, and direct sums of regularly ordered vector spaces are again regularly ordered.

Properties

If is a regularly ordered vector lattice then the Order [topology (functional analysis)|order topology] on is the finest topology on making into a locally convex topological vector lattice.