Characterizations of weakly $\mathcal{K}$-analytic and Vašák spaces using projectional skeletons and separable PRI

Abstract

We find characterizations of Vašák spaces and weakly $\mathcal{K}$-analytic spaces using the notions of separable projectional resolution of the identity (SPRI) and of projectional skeleton. This in particular addresses a recent challenge suggested by M. Fabian and V. Montesinos in \cite{FM18}. Our method of proof also gives similar characterizations of WCG spaces and their subspaces (some aspects of which were known, some are new). Moreover we show that for countably many projectional skeletons $\{{\mathfrak{s}}_n:n\in \omega \}$ on a Banach space inducing the same set, there exists a projectional skeleton on the space (indexed by ranges of the corresponding projections) which is isomorphic to a subskeleton of each ${\mathfrak{s}}_n$, $n\in \omega$.

arXiv comment30 pages

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Cited by in corpus (2)

• Rotund Gateaux smooth norms which are not locally uniformly rotund
• Note on almost isometric ideals and local retracts in Banach and metric spaces