Regularly ordered
In mathematics, specifically in order theory and functional analysis, an ordered vector space [math]\displaystyle{ X }[/math] is said to be regularly ordered and its order is called regular if [math]\displaystyle{ X }[/math] is Archimedean ordered and the order dual of [math]\displaystyle{ X }[/math] distinguishes points in [math]\displaystyle{ X }[/math].[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 [math]\displaystyle{ X }[/math] is a regularly ordered vector lattice then the order topology on [math]\displaystyle{ X }[/math] is the finest topology on [math]\displaystyle{ X }[/math] making [math]\displaystyle{ X }[/math] into a locally convex topological vector lattice.[3]
See also
References
- ↑ Schaefer & Wolff 1999, pp. 204–214.
- ↑ 2.0 2.1 Schaefer & Wolff 1999, pp. 222–225.
- ↑ 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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
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