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

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 $C.$Template:Sfn

Examples

If $1\le p<\mathrm{\infty }$ 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 $\mu$).Template:Sfn

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

See also

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Bibliography

• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces