Quasi-interior point

In mathematics, specifically in order theory and functional analysis, an element x of an ordered topological vector space X is called a quasi-interior point of the positive cone C of X if x ≥ 0 and if the order interval [0, x] := { z ∈ X : 0 ≤ z and z ≤ x } is a total subset of X (i.e. if the linear span of [0, x] is a dense subset of X).[1]

Properties

If X is a separable metrizable locally convex ordered topological vector space whose positive cone C is a complete and total subset of X, then the set of quasi-interior points of C is dense in C.[1]

Examples

If ${\displaystyle 1\leq p<\infty }$ then a point in ${\displaystyle L^{p}(\mu )}$ is quasi-interior to the positive cone C 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 > 0 almost everywhere (with respect to ${\displaystyle \mu }$).[1]

A point in ${\displaystyle L^{\infty }(\mu )}$ is quasi-interior to the positive cone C if and only if it is interior to C.[1]

See also

• Weak order unit
• Vector lattice

References

1. 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.