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 $(Q_{\gamma }:\gamma \in \Gamma )$ on $X$ such that $(Q_{\gamma }^{\ast }:\gamma \in \Gamma )$ is a commutative 1-projectional skeleton on $X^{\ast }$. We consider both, real and also complex, Banach spaces.

In the old version we constructed "shrinking projectional skeleton" in Asplund + WCG spaces. In this new version we observe that actually we have an equivalence (as mentioned in the abstract)

References in corpus (2)

• Separable reduction of Frechet subdifferentiability in Asplund spaces
• On Markushevich bases in preduals of von Neumann algebras

Cited by in corpus (4)

• Projectional skeletons and Markushevich bases
• Characterizations of weakly $\mathcal{K}$-analytic and Vašák spaces using projectional skeletons and separable PRI
• Sequential and exact formulae for the subdifferential of nonconvex integral functionals
• Characterization of (semi-)Eberlein compacta using retractional skeletons