 If ("X", ≤) is a preordered vector space over the reals with order unit "u", then the map formula_7 is a sublinear functional.If "X" is a preordered vector space then for all "x", "y" ∈ "X", 

"x" ≥ 0 and "y" ≥ 0 imply "x"+"y" ≥ 0.