Quasi-interior point
In mathematics, specifically in order theory and functional analysis, an element [math]\displaystyle{ x }[/math] of an ordered topological vector space [math]\displaystyle{ X }[/math] is called a quasi-interior point of the positive cone [math]\displaystyle{ C }[/math] of [math]\displaystyle{ X }[/math] if [math]\displaystyle{ x \geq 0 }[/math] and if the order interval [math]\displaystyle{ [0, x] := \{ z \in Z : 0 \leq z \text{ and } z \leq x \} }[/math] is a total subset of [math]\displaystyle{ X }[/math]; that is, if the linear span of [math]\displaystyle{ [0, x] }[/math] is a dense subset of [math]\displaystyle{ X. }[/math][1]
Properties
If [math]\displaystyle{ X }[/math] is a separable metrizable locally convex ordered topological vector space whose positive cone [math]\displaystyle{ C }[/math] is a complete and total subset of [math]\displaystyle{ X, }[/math] then the set of quasi-interior points of [math]\displaystyle{ C }[/math] is dense in [math]\displaystyle{ C. }[/math][1]
Examples
If [math]\displaystyle{ 1 \leq p \lt \infty }[/math] then a point in [math]\displaystyle{ L^p(\mu) }[/math] is quasi-interior to the positive cone [math]\displaystyle{ C }[/math] 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 [math]\displaystyle{ \gt \, 0 }[/math] almost everywhere (with respect to [math]\displaystyle{ \mu }[/math]).[1]
A point in [math]\displaystyle{ L^\infty(\mu) }[/math] is quasi-interior to the positive cone [math]\displaystyle{ C }[/math] if and only if it is interior to [math]\displaystyle{ C. }[/math][1]
See also
References
- ↑ 1.0 1.1 1.2 1.3 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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