Boundaries for Banach spaces determine weak compactness

• 成果类型: Article
• 署名作者: Pfitzner, Hermann
• 署名单位: Universite de Orleans
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-010-0267-6
• 发表日期: 2010
• 页码: 585-604
• 关键词: pointwise compactness; convexity; THEOREM; sets

摘要

A boundary for a real Banach space is a subset of the dual unit sphere with the property that each element of the Banach space attains its norm on an element of that subset. Trivially, the pointwise convergence with respect to such a boundary is coarser than the weak topology on the Banach space. The boundary problem asks whether nevertheless both topologies have the same norm bounded compact sets. The main theorem of this paper states the equivalence of countable and sequential compactness of norm bounded sets with respect to an appropriate topology. This result contains, as a special case, the positive answer to the boundary problem and it carries James' sup-characterization as a corollary.