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

✍ Scribed by E. Marques de Sá; Virgı´nia Santos

• Publisher: Elsevier Science
• Year: 2008
• Tongue: English
• Weight: 167 KB
• Volume: 428
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2007.10.038

✦ Synopsis

Given finite dimensional real or complex Banach spaces, E and F, with norms ν : E → R and μ : F → R, we denote by N μν the least upper bound norm induced on L(E, F ). Some results are given on the extremal structures of B, the unit ball of N μν , of its polar B • , and of B , which is the polar of the unit ball of the least upper bound norm N μ • 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.