Quasi-interior point
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In mathematics, specifically in order theory and functional analysis, an element of an ordered topological vector space is called a quasi-interior point of the positive cone of if and if the order interval is a total subset of ; that is, if the linear span of is a dense subset of [1]
Properties
[edit]If is a separable metrizable locally convex ordered topological vector space whose positive cone is a complete and total subset of then the set of quasi-interior points of is dense in [1]
Examples
[edit]If then a point in is quasi-interior to the positive cone if and only it is a weak order unit, which happens if and only if the element (which recall is an equivalence class of functions) contains a function that is almost everywhere (with respect to ).[1]
A point in is quasi-interior to the positive cone if and only if it is interior to [1]
See also
[edit]- Weak order unit
- Vector lattice – Partially ordered vector space, ordered as a lattice
References
[edit]- ^ a b c d 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.