Regular methods of summability in some locally convex spaces

@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 -