Quasi-interior point

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]

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]

• Weak order unit
• Vector lattice

• 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.