×

On metrizable Abelian groups with a completeness-type property. (English) Zbl 0552.46001

We say that an Abelian topological group X has property (K) if every sequence \((x_ n)\) in X with \(x_ n\to 0\) contains a subsequence \((x_{n_ k})\) such that the series \(\sum^{\infty}_{k=1}x_{n_ k}\) is convergent. This property was first isolated by S. Mazur and W. Orlicz [Stud. Math. 13, 137-179 (1953; Zbl 0052.111)], who realized that it can be used as a substitute for completeness in some basic theorems of functional analysis. The present paper yields the following explanation for this: Every metrizable group with property (K) is a Baire space [see L. Foged, Topology, Proc. Conf., Vol. 8, 259- 266 (1983) for a nonmetrizable generalization]. The reverse implication is shown to fail in some general sense. We also construct, by transfinite induction, for a nondiscrete metrizable complete Abelian group of cardinality \(2^{\aleph_ 0}\) satisfying some algebraic assumption (which holds, e.g., in \({\mathbb{R}}\) or \({\mathbb{T}})\), a dense proper subgroup with property (K).

MSC:

46A04 Locally convex Fréchet spaces and (DF)-spaces
54E50 Complete metric spaces
54E52 Baire category, Baire spaces
22A10 Analysis on general topological groups
PDFBibTeX XMLCite
Full Text: DOI