On Ultrabarrelled Spaces, their Group Analogs and Baire Spaces

Abstract

Let E and F be topological vector spaces and let G and Y be topological abelian groups. We say that E is sequentially barrelled with respect to F if every sequence \((u_n)_{n\in \mathbb {N}}\) of continuous linear maps from E to F which converges pointwise to zero is equicontinuous. We say that G is barrelled with respect to F if every set \(\mathscr {H}\) of continuous homomorphisms from G to F, for which the set \( \mathscr {H}(x)\) is bounded in F for every \(x\in E\), is equicontinuous. Finally, we say that G is g-barrelled with respect to Y if every \(\mathscr {H}\subseteq \mathrm{CHom}(G,Y)\) which is compact in the product topology of \(Y^ G\) is equicontinuous. We prove that a barrelled normed space may not be sequentially barrelled with respect to a complete metrizable locally bounded topological vector space, a topological group which is a Baire space is barrelled with respect to any topological vector space, a topological group which is a Namioka space is g-barrelled with respect to any metrizable topological group, a protodiscrete topological abelian group which is a Baire space may not be g-barrelled (with respect to \(\mathbb R/\mathbb Z\)).