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 $$x \geq 0$$ and if the order interval $$[0, x] := \{ z \in Z : 0 \leq z \text{ and } z \leq x \}$$ is a total subset of $$X$$; that is, if the linear span of $$[0, x]$$ is a dense subset of $$X.$$

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 $$C.$$

Examples
If $$1 \leq p < \infty$$ then a point in $$L^p(\mu)$$ 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 $$\mu$$).

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