In mathematics, specifically in order theory and functional analysis, the order dual of an ordered vector space X {\displaystyle X} is the set Pos ( X ∗ ) − Pos ( X ∗ ) {\displaystyle \operatorname {Pos} \left(X^{*}\right)-\operatorname {Pos} \left(X^{*}\right)} where Pos ( X ∗ ) {\displaystyle \operatorname {Pos} \left(X^{*}\right)} 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 ( x ) ≥ 0. {\displaystyle f(x)\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.
Canonical ordering An element f {\displaystyle f} of the order dual of X {\displaystyle X} is called positive if x ≥ 0 {\displaystyle x\geq 0} implies Re f ( x ) ≥ 0. {\displaystyle \operatorname {Re} f(x)\geq 0.} The positive elements of the order dual form a cone that induces an ordering on X + {\displaystyle X^{+}} called the canonical ordering. If X {\displaystyle X} is an ordered vector space whose positive cone C {\displaystyle C} is generating (that is, X = C − C {\displaystyle X=C-C} ) then the order dual with the canonical ordering is an ordered vector space. The order dual is the span of the set of positive linear functionals on X {\displaystyle X} .
Properties The order dual is contained in the order bound dual. If the positive cone of an ordered vector space X {\displaystyle X} is generating and if [ 0 , x ] + [ 0 , y ] = [ 0 , x + y ] {\displaystyle [0,x]+[0,y]=[0,x+y]} holds for all positive x {\displaystyle x} and y {\displaystyle y} , then the order dual is equal to the order bound dual, which is an order complete vector lattice under its canonical ordering. The order dual of a vector lattice is an order complete vector lattice. The order dual of a vector lattice X {\displaystyle X} can be finite dimension (possibly even { 0 } {\displaystyle \{0\}} ) even if X {\displaystyle X} is infinite-dimensional.
Order bidual Suppose that X {\displaystyle X} is an ordered vector space such that the canonical order on X + {\displaystyle X^{+}} makes X + {\displaystyle X^{+}} into an ordered vector space. Then the order bidual is defined to be the order dual of X + {\displaystyle X^{+}} and is denoted by X + + {\displaystyle X^{++}} . If the positive cone of an ordered vector space X {\displaystyle X} is generating and if [ 0 , x ] + [ 0 , y ] = [ 0 , x + y ] {\displaystyle [0,x]+[0,y]=[0,x+y]} holds for all positive x {\displaystyle x} and y {\displaystyle y} , then X + + {\displaystyle X^{++}} is an order complete vector lattice and the evaluation map X → X + + {\displaystyle X\to X^{++}} is order preserving. In particular, if X {\displaystyle X} is a vector lattice then X + + {\displaystyle X^{++}} is an order complete vector lattice.
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