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

• 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