Czechoslovak Mathematical Journal, Vol. 63, No. 1, pp. 91-105, 2013

# Dichotomies for $C_0(X)$ and $C_b(X)$ spaces

## Szymon Gląb, Filip Strobin

Szymon Gląb, Institute of Mathematics, Łódź University of Technology, Wólczańska 215, 93-005 Łódź, Poland, e-mail: szymon.glab@p.lodz.pl; Filip Strobin, Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warszawa, Poland. Institute of Mathematics, Łódź University of Technology, Wólczańska 215, 93-005 Łódź, Poland, e-mail: filip.strobin@p.lodz.pl

Abstract: Jachymski showed that the set
\bigg\{(x,y)\in c_0\times c_0 \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 $c_0\times c_0$ or is equal to $c_0\times c_0$. In the paper we generalize this result by considering more general spaces than $c_0$, namely $C_0(X)$, the space of all continuous functions which vanish at infinity, and $C_b(X)$, the space of all continuous bounded functions. Moreover, we replace the meagerness by $\sigma$-porosity.

Keywords: continuous function, integration, Baire category, porosity

Classification (MSC 2010): 46B25, 28A25, 54E52

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