Regularly ordered

In mathematics, specifically in order theory and functional analysis, an ordered vector space $X$ is said to be regularly ordered and its order is called regular if $X$ is Archimedean ordered and the order dual of $X$ distinguishes points in $X$.Template:Sfn 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.Template:Sfn The canonical orderings of subspaces, products, and direct sums of regularly ordered vector spaces are again regularly ordered.Template:Sfn

Properties

If $X$ is a regularly ordered vector lattice then the order topology on $X$ is the finest topology on $X$ making $X$ into a locally convex topological vector lattice.Template:Sfn

See also

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Bibliography

• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces