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Regular methods of summability in some locally convex spaces. (English) Zbl 1212.46005
Summary: Suppose that \(X\) is a Fréchet space, \(\langle a_{ij}\rangle \) is a regular method of summability and \((x_{i})\) is a bounded sequence in \(X\). We prove that there exists a subsequence \((y_{i})\) of \((x_{i})\) such that: either (a) all the subsequences of \((y_{i})\) are summable to a common limit with respect to \(\langle a_{ij}\rangle \); or (b) no subsequence of \((y_{i})\) is summable with respect to \(\langle a_{ij}\rangle \). This result generalizes the Erdös-Magidor theorem which refers to summability of bounded sequences in Banach spaces. We also show that two analogous results for some \(\omega _{1}\)-locally convex spaces are consistent with ZFC.

MSC:
46A04 Locally convex Fréchet spaces and (DF)-spaces
46A35 Summability and bases in topological vector spaces
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