| Title:
|
Regular methods of summability in some locally convex spaces (English) |
| Author:
|
Poulios, Costas |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
50 |
| Issue:
|
3 |
| Year:
|
2009 |
| Pages:
|
401-411 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 to ZFC. (English) |
| Keyword:
|
Fréchet space |
| Keyword:
|
regular method of summability |
| Keyword:
|
summable sequence |
| Keyword:
|
Galvin-Prikry theorem |
| Keyword:
|
Erdös-Magidor theorem |
| MSC:
|
05D10 |
| MSC:
|
46A04 |
| MSC:
|
46B15 |
| idZBL:
|
Zbl 1212.46005 |
| idMR:
|
MR2573413 |
| . |
| Date available:
|
2009-09-23T21:34:47Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134912 |
| . |
| Reference:
|
[1] Balcar B., Pelant J., Simon P.: The space of ultrafilters on $\mathbb N$ covered by nowhere dense sets.Fund. Math. 110 (1980), 11--24. MR 0600576 |
| Reference:
|
[2] Dunford N., Schwartz J.T.: Linear Operators I: General Theory.Pure and Applied Mathematics, vol. 7, Interscience, New York, 1958. Zbl 0084.10402, MR 0117523 |
| Reference:
|
[3] Ellentuck E.: A new proof that analytic sets are Ramsey.J. Symbolic Logic 39 (1974), 163--165. Zbl 0292.02054, MR 0349393, 10.2307/2272356 |
| Reference:
|
[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. MR 0430596 |
| Reference:
|
[5] Galvin F., Prikry K.: Borel sets and Ramsey's theorem.J. Symbolic Logic 38 (1973), 193--198. Zbl 0276.04003, MR 0337630, 10.2307/2272055 |
| Reference:
|
[6] Kechris A.: Classical Descriptive Set Theory.Springer, New York, 1995. Zbl 0819.04002, MR 1321597 |
| Reference:
|
[7] Plewik S.: On completely Ramsey sets.Fund. Math. 127 (1986), 127--132. Zbl 0632.04005, MR 0882622 |
| Reference:
|
[8] Rudin W.: Functional Analysis.McGraw-Hill, New York, 1973. Zbl 0867.46001, MR 0365062 |
| Reference:
|
[9] Tsarpalias A.: A note on the Ramsey property.Proc. Amer. Math. Soc. 127 (1999), 583--587. Zbl 0953.03058, MR 1458267, 10.1090/S0002-9939-99-04518-9 |
| . |
