Order dual (functional analysis)

In mathematics, specifically in order theory and functional analysis, the order dual of an ordered vector space ${\displaystyle X}$ is the set ${\displaystyle \operatorname {Pos} \left(X^{*}\right)-\operatorname {Pos} \left(X^{*}\right)}$ where ${\displaystyle \operatorname {Pos} \left(X^{*}\right)}$ denotes the set of all positive linear functionals on ${\displaystyle X}$, where a linear function ${\displaystyle f}$ on ${\displaystyle X}$ is called positive if for all ${\displaystyle x\in X,}$ ${\displaystyle x\geq 0}$ implies ${\displaystyle f(x)\geq 0.}$^[1] The order dual of ${\displaystyle X}$ is denoted by ${\displaystyle X^{+}}$. Along with the related concept of the order bound dual, this space plays an important role in the theory of ordered topological vector spaces.