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 $X$. 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 $X$ 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.

See also

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

Bibliography

• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.