Regular methods of summability in some locally convex spaces. (English) Zbl 1212.46005

Poulios, C.

Commentat. Math. Univ. Carol. 50, No. 3, 401-411 (2009).

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

Keywords:

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