On the extremal structure of least upper bound norms and their dual

Abstract

Given finite dimensional real or complex Banach spaces, E and F, with norms ν:E→R and μ:F→R, we denote by Nμν the least upper bound norm induced on L(E,F). Some results are given on the extremal structures of B, the unit ball of Nμν, of its polar B°, and of B′, which is the polar of the unit ball of the least upper bound norm Nμ°ν°.The exposed faces, the extreme points, and a large family of other faces of B° and B′ are presented. It turns out that B′ is a subset of B; the set of tangency points of the surfaces of B and B′ is completely determined and represented as the union of the exposed faces of B′ which are normal to rank-one mappings. We determine sharp bounds on the ranks of mappings in these exposed faces.