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Erschienen in:
2019 | OriginalPaper | Buchkapitel
On Ultrabarrelled Spaces, their Group Analogs and Baire Spaces
In Honour of Manuel López-Pellicer, Loyal Friend and Indefatigable Mathematician
verfasst von : Xabier Domínguez, Elena Martín-Peinador, Vaja Tarieladze
Erschienen in: Descriptive Topology and Functional Analysis II
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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
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a barrelled normed space may not be sequentially barrelled with respect to a complete metrizable locally bounded topological vector space,
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a topological group which is a Baire space is barrelled with respect to any topological vector space,
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a topological group which is a Namioka space is g-barrelled with respect to any metrizable topological group,
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a protodiscrete topological abelian group which is a Baire space may not be g-barrelled (with respect to \(\mathbb R/\mathbb Z\)).
We also formulate some open questions.