Weakly Realcompact Locally Convex Spaces

Abstract

In this chapter, we continue the study of spaces in the class $\mathfrak{G}$. We prove that the weak ^∗ dual (E′,σ(E′,E)) of an lcs E in the class $\mathfrak{G}$ is K-analytic if and only if (E′,σ(E′,E)) is Lindelöf if and only if (E,σ(E,E′)) has countable tightness. We show that every quasibarrelled space in the class $\mathfrak{G}$ has countable tightness both for the weak and the original topologies. This extends a classical result of Kaplansky for a metrizable lcs. Although (DF)-spaces belong to the class $\mathfrak{G}$, concrete examples of (DF)-spaces without countable tightness are provided. On the other hand, there are many Banach spaces E for which E endowed with the weak topology is not Lindelöf. We show, however (following Khurana), that every WCG Fréchet space E is weakly K-analytic. An example due to Pol showing that there exists a Banach space C(X) over a compact scattered space X such that C(X) is weakly Lindelöf and not WCG is presented. We show (after Amir and Lindenstrauss) that every nonseparable reflexive Banach space contains a complemented separable subspace. Several consequences are provided.