Order bound dual

In mathematics, specifically in order theory and functional analysis, the order bound dual of an ordered vector space ${\displaystyle X}$ is the set of all linear functionals on ${\displaystyle X}$ that map order intervals, which are sets of the form ${\displaystyle [a,b]:=\{x\in X:a\leq x{\text{ and }}x\leq b\},}$ to bounded sets.^[1] The order bound dual of ${\displaystyle X}$ is denoted by ${\displaystyle X^{\operatorname {b} }.}$ This space plays an important role in the theory of ordered topological vector spaces.