Weak order unit

In mathematics, specifically in order theory and functional analysis, an element ${\displaystyle x}$ of a vector lattice ${\displaystyle X}$ is called a weak order unit in ${\displaystyle X}$ if ${\displaystyle x\geq 0}$ and also for all ${\displaystyle y\in X,}$ ${\displaystyle \inf\{x,|y|\}=0{\text{ implies }}y=0.}$^[1]

Examples

• If ${\displaystyle X}$ is a separable Fréchet topological vector lattice then the set of weak order units is dense in the positive cone of ${\displaystyle X.}$^[2]

See also

• Quasi-interior point
• Vector lattice – Partially ordered vector space, ordered as a latticePages displaying short descriptions of redirect targets

Citations

1. Schaefer & Wolff 1999, pp. 234–242.
2. Schaefer & Wolff 1999, pp. 204–214.

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. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.