Rich families and projectional skeletons in Asplund WCG spaces

Abstract

We show a way of constructing projectional skeletons using the concept of rich families in Banach spaces which admit a projectional generator. Our next result is that a Banach space X is Asplund and weakly compactly generated if and only if there exists a commutative 1-projectional skeleton (Q(gamma) : gamma is an element of Gamma) on X such that (Q(gamma)* : gamma is an element of Gamma) is a commutative 1-projectional skeleton on X*.

We consider both, real and also complex, Banach spaces.

Keywords

• Asplund space
• Weakly compactly generated space
• Weakly Lindelof determined space
• 1-Plichko space
• Commutative projectional skeleton
• Rich family