Regularly ordered
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In mathematics, specifically in order theory and functional analysis, an ordered vector space is said to be regularly ordered and its order is called regular if is Archimedean ordered and the order dual of distinguishes points in .[1] Being a regularly ordered vector space is an important property in the theory of topological vector lattices.
Examples
[edit]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
[edit]If is a regularly ordered vector lattice then the order topology on is the finest topology on making into a locally convex topological vector lattice.[3]
See also
[edit]- Vector lattice – Partially ordered vector space, ordered as a lattice
References
[edit]- ^ Schaefer & Wolff 1999, pp. 204–214.
- ^ a b Schaefer & Wolff 1999, pp. 222–225.
- ^ Schaefer & Wolff 1999, pp. 234–242.
Bibliography
[edit]- 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.