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.[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
References
- Schaefer & Wolff 1999, pp. 204–214.
- Schaefer & Wolff 1999, pp. 222–225.
- Schaefer & Wolff 1999, pp. 234–242.
- 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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)
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