On vectorial norms and pseudonorms

Abstract

A vectorial pseudonorm (norm) of order k k on the vector space C n {C^n} of all n n -tuples of complex numbers is a mapping from C n {C^n} into the positive cone of R k {R^k} which satisfies the usual axioms of a pseudonorm (norm). The vector space R k {R^k} of the k k -tupies of real numbers is partially ordered componentwise. Vectorial norms have been introduced by Kantorovitch. Recently they have been studied by Robert and Stoer. In the present paper different properties of vectorial pseudonorms are investigated. They deal mainly with the following topics: regularity of pseudonorms, the dual of a vectorial norm, inequality between vectorial pseudonorms and the G G -transform of a vectorial pseudonorm.