Order bound dual

In mathematics, specifically in order theory and functional analysis, the order bound dual of an ordered vector space is the set of all linear functionals on that map order intervals, which are sets of the form to bounded sets. The order bound dual of is denoted by This space plays an important role in the theory of ordered topological vector spaces. An element of the order bound dual of is called positive if implies The positive elements of the order bound dual form a cone that induces an ordering on called the . If is an ordered vector space whose positive cone is generating (meaning ) then the order bound dual with the canonical ordering is an ordered vector space. The order bound dual of an ordered vector spaces contains its order dual. If the positive cone of an ordered vector space is generating and if for all positive and we have then the order dual is equal to the order bound dual, which is an order complete vector lattice under its canonical ordering.