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Fréchet space; regular method of summability; summable sequence; Galvin-Prikry theorem; Erdös-Magidor theorem
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 to ZFC.
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