  We introduce a new type of norm for ordered vector spaces majorized by a
proper (convex) cone that generalizes the notions of order unit norm and base
norm. Then we give sufficient conditions to ensure its completeness. In the
case of normed Riesz spaces, we also present necessary completeness conditions
and describe the completed norm. Finally, we define a particular family of
well-behaving sets. Spaces majorized by the cone generated by these sets
display surprisingly interesting structural properties. In particular, we
generalize two famous standard results in Banach lattice theory. Namely, every
principal ideal in a Banach lattice is an AM-space and every Banach lattice
having a generating cone with a compact base admits an equivalent L-norm.
