Geodesic
Dichotomies for ${\bf C}_0(X)$ and ${\bf C}_b(X)$ spaces
Czechoslovak Mathematical Journal, Tome 63 (2013) no. 1, pp. 91-105
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.1007/s10587-013-0006-4
Classification : 28A25, 46B25, 54C35, 54E52
Keywords: continuous function; integration; Baire category; porosity 
@article{10_1007_s10587_013_0006_4,
     author = {G{\l}\k{a}b, Szymon and Strobin, Filip},
     title = {Dichotomies for ${\bf C}_0(X)$ and ${\bf C}_b(X)$ spaces},
     journal = {Czechoslovak Mathematical Journal},
     pages = {91--105},
     year = {2013},
     volume = {63},
     number = {1},
     doi = {10.1007/s10587-013-0006-4},
     mrnumber = {3035499},
     zbl = {1274.46046},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-013-0006-4/}
}
TY  - JOUR
AU  - Głąb, Szymon
AU  - Strobin, Filip
TI  - Dichotomies for ${\bf C}_0(X)$ and ${\bf C}_b(X)$ spaces
JO  - Czechoslovak Mathematical Journal
PY  - 2013
SP  - 91
EP  - 105
VL  - 63
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-013-0006-4/
DO  - 10.1007/s10587-013-0006-4
LA  - en
ID  - 10_1007_s10587_013_0006_4
ER  - 
%0 Journal Article
%A Głąb, Szymon
%A Strobin, Filip
%T Dichotomies for ${\bf C}_0(X)$ and ${\bf C}_b(X)$ spaces
%J Czechoslovak Mathematical Journal
%D 2013
%P 91-105
%V 63
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-013-0006-4/
%R 10.1007/s10587-013-0006-4
%G en
%F 10_1007_s10587_013_0006_4
Głąb, Szymon; Strobin, Filip. Dichotomies for ${\bf C}_0(X)$ and ${\bf C}_b(X)$ spaces. Czechoslovak Mathematical Journal, Tome 63 (2013) no. 1, pp. 91-105. doi: 10.1007/s10587-013-0006-4

[1] Balcerzak, M., Wachowicz, A.: Some examples of meager sets in Banach spaces. Real Anal. Exch. 26 877-884 (2001). | DOI | MR | Zbl

[2] Engelking, R.: General Topology. Sigma Series in Pure Mathematics, 6. Berlin, Heldermann (1989). | MR

[3] Głąb, S., Strobin, F.: Descriptive properties of density preserving autohomeomorphisms of the unit interval. Cent. Eur. J. Math. 8 928-936 (2010). | DOI | MR | Zbl

[4] Halmos, P. R.: Measure Theory. New York: D. Van Nostrand London, Macmillan (1950). | MR | Zbl

[5] Jachymski, J.: A nonlinear Banach-Steinhaus theorem and some meager sets in Banach spaces. Stud. Math. 170 303-320 (2005). | DOI | MR | Zbl

[6] Strobin, F.: Porosity of convex nowhere dense subsets of normed linear spaces. Abstr. Appl. Anal. 2009 (2009), Article ID 243604, pp. 11. | MR | Zbl

[7] Zajíek, L.: On $\sigma$-porous sets in abstract spaces. Abstr. Appl. Anal. 2005 509-534 (2005). | DOI | MR

Cité par Sources :