Regularly ordered

Mathematical definition From Wikipedia, the free encyclopedia

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 $X$ .^[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 $X$ 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.^[3]

See also

Vector lattice – Partially ordered vector space, ordered as a lattice

References

Bibliography

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