  In this paper we introduce the hypo-q-norms on a Cartesian product of normed
linear spaces. A representation of these norms in terms of bounded linear
functionals of norm less than one, the equivalence with the q-norms on a
Cartesian product and some reverse inequalities obtained via the scalar
Shisha-Mond, Birnacki et al. and other Gruss type inequalities are also given.
