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In mathematics, specifically in order theory and functional analysis, the order dual of an ordered vector space X {\displaystyle X} is the set Pos − Pos {\displaystyle \operatorname {Pos} \left-\operatorname {Pos} \left} where Pos {\displaystyle \operatorname {Pos} \left} denotes the set of all positive linear functionals on X {\displaystyle X} , where a linear function f {\displaystyle f} on X {\displaystyle X} is called positive if for all x ∈ X , {\displaystyle x\in X,} x ≥ 0 {\displaystyle x\geq 0} implies f ≥ 0. {\displaystyle f\geq 0.} The order dual of X {\displaystyle X} is denoted by X + {\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.