| Title:
|
Dichotomies for ${\bf C}_0(X)$ and ${\bf C}_b(X)$ spaces (English) |
| Author:
|
Głąb, Szymon |
| Author:
|
Strobin, Filip |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
63 |
| Issue:
|
1 |
| Year:
|
2013 |
| Pages:
|
91-105 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Jachymski showed that the set $$ \bigg \{(x,y)\in {\bf c}_0\times {\bf c}_0\colon \bigg (\sum _{i=1}^n \alpha (i)x(i)y(i)\bigg )_{n=1}^\infty \text {is bounded}\bigg \} $$ is either a meager subset of ${\bf c}_0\times {\bf c}_0$ or is equal to ${\bf c}_0\times {\bf c}_0$. In the paper we generalize this result by considering more general spaces than ${\bf c}_0$, namely ${\bf C}_0(X)$, the space of all continuous functions which vanish at infinity, and ${\bf C}_b(X)$, the space of all continuous bounded functions. Moreover, we replace the meagerness by $\sigma $-porosity. (English) |
| Keyword:
|
continuous function |
| Keyword:
|
integration |
| Keyword:
|
Baire category |
| Keyword:
|
porosity |
| MSC:
|
28A25 |
| MSC:
|
46B25 |
| MSC:
|
54C35 |
| MSC:
|
54E52 |
| idZBL:
|
Zbl 1274.46046 |
| idMR:
|
MR3035499 |
| DOI:
|
10.1007/s10587-013-0006-4 |
| . |
| Date available:
|
2013-03-01T16:04:47Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143172 |
| . |
| Reference:
|
[1] Balcerzak, M., Wachowicz, A.: Some examples of meager sets in Banach spaces.Real Anal. Exch. 26 877-884 (2001). Zbl 1046.46013, MR 1844401, 10.2307/44154085 |
| Reference:
|
[2] Engelking, R.: General Topology. Sigma Series in Pure Mathematics, 6.Berlin, Heldermann (1989). MR 1039321 |
| Reference:
|
[3] Głąb, S., Strobin, F.: Descriptive properties of density preserving autohomeomorphisms of the unit interval.Cent. Eur. J. Math. 8 928-936 (2010). Zbl 1217.28001, MR 2727440, 10.2478/s11533-010-0054-z |
| Reference:
|
[4] Halmos, P. R.: Measure Theory.New York: D. Van Nostrand London, Macmillan (1950). Zbl 0040.16802, MR 0033869 |
| Reference:
|
[5] Jachymski, J.: A nonlinear Banach-Steinhaus theorem and some meager sets in Banach spaces.Stud. Math. 170 303-320 (2005). Zbl 1090.46015, MR 2185961, 10.4064/sm170-3-7 |
| Reference:
|
[6] Strobin, F.: Porosity of convex nowhere dense subsets of normed linear spaces.Abstr. Appl. Anal. 2009 (2009), Article ID 243604, pp. 11. Zbl 1192.46020, MR 2576578 |
| Reference:
|
[7] Zajíek, L.: On $\sigma$-porous sets in abstract spaces.Abstr. Appl. Anal. 2005 509-534 (2005). MR 2201041, 10.1155/AAA.2005.509 |
| . |
