On super-weakly compact sets and uniformly convexifiable sets

Lixin Cheng¹ ; Qingjin Cheng¹ ; Bo Wang¹ ; Wen Zhang¹

¹ Department of Mathematics Xiamen University Xiamen 361005, China

Studia Mathematica, Tome 199 (2010) no. 2, pp. 145-169

Résumé

This paper mainly concerns the topological nature of uniformly convexifiable sets in general Banach spaces: A sufficient and necessary condition for a bounded closed convex set $C$ of a Banach space $X$ to be uniformly convexifiable (i.e. there exists an equivalent norm on $X$ which is uniformly convex on $C$) is that the set $C$ is super-weakly compact, which is defined using a generalization of finite representability. The proofs use appropriate versions of classical theorems, such as James' finite tree theorem, Enflo's renorming technique, Grothendieck's lemma and the Davis–Figiel–Johnson–Pełczyński lemma.

Détail

DOI : 10.4064/sm199-2-2

Keywords: paper mainly concerns topological nature uniformly convexifiable sets general banach spaces sufficient necessary condition bounded closed convex set banach space uniformly convexifiable there exists equivalent norm which uniformly convex set super weakly compact which defined using generalization finite representability proofs appropriate versions classical theorems james finite tree theorem enflos renorming technique grothendiecks lemma davis figiel johnson czy ski lemma