On ultrabarrelled spaces, their group analogs and baire spaces: In honour of manuel lópez-pellicer, loyal friend and indefatigable mathematician

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 (formula presented) 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 (formula presented), is equicontinuous. Finally, we say that G is g-barrelled with respect to Y if every (formula presented) 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. We also formulate some open questions.