# Regular methods of summability in some locally convex spaces

Commentationes Mathematicae Universitatis Carolinae (2009)

- Volume: 50, Issue: 3, page 401-411
- ISSN: 0010-2628

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topPoulios, Costas. "Regular methods of summability in some locally convex spaces." Commentationes Mathematicae Universitatis Carolinae 50.3 (2009): 401-411. <http://eudml.org/doc/33323>.

@article{Poulios2009,

abstract = {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.},

author = {Poulios, Costas},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {Fréchet space; regular method of summability; summable sequence; Galvin-Prikry theorem; Erdös-Magidor theorem; Fréchet space; regular method of summability; summable sequence; Galvin-Prikry theorem; Erdös-Magidor theorem},

language = {eng},

number = {3},

pages = {401-411},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {Regular methods of summability in some locally convex spaces},

url = {http://eudml.org/doc/33323},

volume = {50},

year = {2009},

}

TY - JOUR

AU - Poulios, Costas

TI - Regular methods of summability in some locally convex spaces

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 2009

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 50

IS - 3

SP - 401

EP - 411

AB - 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.

LA - eng

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

UR - http://eudml.org/doc/33323

ER -

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