Weak order unit

In mathematics, specifically in order theory and functional analysis, an element [math]\displaystyle{ x }[/math] of a vector lattice [math]\displaystyle{ X }[/math] is called a weak order unit in [math]\displaystyle{ X }[/math] if [math]\displaystyle{ x \geq 0 }[/math] and also for all [math]\displaystyle{ y \in X, }[/math] [math]\displaystyle{ \inf \{ x, |y| \} = 0 \text{ implies } y = 0. }[/math]

Examples

• If [math]\displaystyle{ X }[/math] is a separable Fréchet topological vector lattice then the set of weak order units is dense in the positive cone of [math]\displaystyle{ X. }[/math]

See also

• Quasi-interior point

References

• 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.