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

References

1. Schaefer & Wolff 1999, pp. 204–214.
2. Schaefer & Wolff 1999, pp. 222–225.
3. Schaefer & Wolff 1999, pp. 234–242.

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.