Geodesic
Regular methods of summability in some locally convex spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 50 (2009) no. 3, pp. 401-411.

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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.
Classification : 05D10, 46A04, 46B15
Keywords: Fréchet space; regular method of summability; summable sequence; Galvin-Prikry theorem; Erdös-Magidor theorem 
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     title = {Regular methods of summability in some locally convex spaces},
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Poulios, Costas. Regular methods of summability in some locally convex spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 50 (2009) no. 3, pp. 401-411. http://geodesic.mathdoc.fr/item/CMUC_2009__50_3_a6/