In mathematics, specifically in order theory and functional analysis, an element x {\displaystyle x} of an ordered topological vector space X {\displaystyle X} is called a quasi-interior point of the positive cone C {\displaystyle C} of X {\displaystyle X} if x ≥ 0 {\displaystyle x\geq 0} and if the order interval [ 0 , x ] := { z ∈ X : 0 ≤ z and z ≤ x } {\displaystyle [0,x]:=\{z\in X:0\leq z{\text{ and }}z\leq x\}} is a total subset of X {\displaystyle X} ; that is, if the linear span of [ 0 , x ] {\displaystyle [0,x]} is a dense subset of X . {\displaystyle X.}
Properties If X {\displaystyle X} is a separable metrizable locally convex ordered topological vector space whose positive cone C {\displaystyle C} is a complete and total subset of X , {\displaystyle X,} then the set of quasi-interior points of C {\displaystyle C} is dense in C . {\displaystyle C.}
Examples If 1 ≤ p < ∞ {\displaystyle 1\leq p<\infty } then a point in L p ( μ ) {\displaystyle L^{p}(\mu )} is quasi-interior to the positive cone C {\displaystyle 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 {\displaystyle >\,0} almost everywhere (with respect to μ {\displaystyle \mu } ). A point in L ∞ ( μ ) {\displaystyle L^{\infty }(\mu )} is quasi-interior to the positive cone C {\displaystyle C} if and only if it is interior to C . {\displaystyle C.}
See also Weak order unit Vector lattice – Partially ordered vector space, ordered as a latticePages displaying short descriptions of redirect targets
References
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. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
