 The set of all linear functionals on a preordered vector space "X" that map every order interval into a bounded set is called the order bound dual of "X" and denoted by "X""b" 
If a space is ordered then its order bound dual is a vector subspace of its algebraic dual.A subset "A" of an ordered vector space "X" is called order complete if for every non-empty subset "B" ⊆ "A" such that "B" is order bounded in "A", both formula_5 and formula_6 exist and are elements of "A".We say that an ordered vector space "X" is order complete is "X" is an order complete subset of "X".