Previous |  Up |  Next


continuous function; integration; Baire category; 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.
[1] Balcerzak, M., Wachowicz, A.: Some examples of meager sets in Banach spaces. Real Anal. Exch. 26 877-884 (2001). MR 1844401 | Zbl 1046.46013
[2] Engelking, R.: General Topology. Sigma Series in Pure Mathematics, 6. Berlin, Heldermann (1989). MR 1039321
[3] Głąb, S., Strobin, F.: Descriptive properties of density preserving autohomeomorphisms of the unit interval. Cent. Eur. J. Math. 8 928-936 (2010). DOI 10.2478/s11533-010-0054-z | MR 2727440 | Zbl 1217.28001
[4] Halmos, P. R.: Measure Theory. New York: D. Van Nostrand London, Macmillan (1950). MR 0033869 | Zbl 0040.16802
[5] Jachymski, J.: A nonlinear Banach-Steinhaus theorem and some meager sets in Banach spaces. Stud. Math. 170 303-320 (2005). DOI 10.4064/sm170-3-7 | MR 2185961 | Zbl 1090.46015
[6] Strobin, F.: Porosity of convex nowhere dense subsets of normed linear spaces. Abstr. Appl. Anal. 2009 (2009), Article ID 243604, pp. 11. MR 2576578 | Zbl 1192.46020
[7] Zajíek, L.: On $\sigma$-porous sets in abstract spaces. Abstr. Appl. Anal. 2005 509-534 (2005). DOI 10.1155/AAA.2005.509 | MR 2201041
Partner of
EuDML logo