Regularly ordered

In mathematics, specifically in order theory and functional analysis, an ordered vector space ${\displaystyle X}$ is said to be regularly ordered and its order is called regular if ${\displaystyle X}$ is Archimedean ordered and the order dual of ${\displaystyle X}$ distinguishes points in ${\displaystyle X}$.^[1] 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.^[2] The canonical orderings of subspaces, products, and direct sums of regularly ordered vector spaces are again regularly ordered.^[2]

Properties

If ${\displaystyle X}$ is a regularly ordered vector lattice then the order topology on ${\displaystyle X}$ is the finest topology on ${\displaystyle X}$ making ${\displaystyle X}$ into a locally convex topological vector lattice.^[3]

See also

• Vector lattice – Partially ordered vector space, ordered as a latticePages displaying short descriptions of redirect targets