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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