On Ultrabarrelled Spaces, their Group Analogs and Baire Spaces

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 (un)n∈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 H of continuous homomorphisms from G to F, for which the set H(x) is bounded in F for every x∈E, is equicontinuous. Finally, we say that G is g-barrelled with respect to Y if every H⊆CHom(G,Y) which is compact in the product topology of YG 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 R/Z)