On the characterization of least upper bound norms in matrix space

of the origin in C ' , and vice versa. But clearly, not every convex compact neighbourhood of the origin in C ~' belongs to a lub norm. The principal problem to be considered here is the geometric characterizat ion of those convex bodies in C ~' which are associated with lub norms. The question arises because of the fact tha t m a n y of the properties of vector norms fall quite easily and natura l ly out of a consideration of the associated convex bodies, and it is hoped tha t these results will th row some light on the ra ther more difficult questions related with multiplicative mat r ix norms and lub norms.