Quasi-interior point

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In mathematics, specifically in order theory and functional analysis, an element ${\displaystyle x}$ of an ordered topological vector space ${\displaystyle X}$ is called a quasi-interior point of the positive cone ${\displaystyle C}$ of ${\displaystyle X}$ if ${\displaystyle x\geq 0}$ and if the order interval ${\displaystyle [0,x]:=\{z\in Z:0\leq z{\text{ and }}z\leq x\}}$ is a total subset of ${\displaystyle X}$; that is, if the linear span of ${\displaystyle [0,x]}$ is a dense subset of ${\displaystyle X.}$^[1]

Properties[]

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

Examples[]

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

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

See also[]

• Weak order unit
• Vector lattice

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.