Regular methods of summability in some locally convex spaces

Costas Poulios

Commentationes Mathematicae Universitatis Carolinae (2009)

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

Abstract

top
Suppose that X is a Fréchet space, 〈 a i j 〉 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 〈 a i j 〉 ; or (b) no subsequence of ( y i ) is summable with respect to 〈 a i j 〉 . 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 ω 1 -locally convex spaces are consistent to ZFC.

How to cite

top

Poulios, 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 -

References

top
  1. Balcar B., Pelant J., Simon P., The space of ultrafilters on ℕ covered by nowhere dense sets, Fund. Math. 110 (1980), 11--24. MR0600576
  2. Dunford N., Schwartz J.T., Linear Operators I: General Theory, Pure and Applied Mathematics, vol. 7, Interscience, New York, 1958. Zbl0084.10402MR0117523
  3. Ellentuck E., 10.2307/2272356, J. Symbolic Logic 39 (1974), 163--165. Zbl0292.02054MR0349393DOI10.2307/2272356
  4. Erdös P., Magidor M., A note on regular methods of summability and the Banach-Saks property, Proc. Amer. Math. Soc. 59 (1976), 232--234. MR0430596
  5. Galvin F., Prikry K., 10.2307/2272055, J. Symbolic Logic 38 (1973), 193--198. Zbl0276.04003MR0337630DOI10.2307/2272055
  6. Kechris A., Classical Descriptive Set Theory, Springer, New York, 1995. Zbl0819.04002MR1321597
  7. Plewik S., On completely Ramsey sets, Fund. Math. 127 (1986), 127--132. Zbl0632.04005MR0882622
  8. Rudin W., Functional Analysis, McGraw-Hill, New York, 1973. Zbl0867.46001MR0365062
  9. Tsarpalias A., 10.1090/S0002-9939-99-04518-9, Proc. Amer. Math. Soc. 127 (1999), 583--587. Zbl0953.03058MR1458267DOI10.1090/S0002-9939-99-04518-9

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.