 If "S" is any set and if "X" is a vector space (over the reals) of real-valued functions on "S", then the pointwise order on "X" is given by, for all "f", "g" ∈ "X", "f" ≤ "g" if and only if "f"("s") ≤ "g"("s") for all "s" ∈ "S".Spaces that are typically assigned this order include