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Czechoslovak Mathematical Journal, Vol. 63, No. 1, pp. 91-105, 2013
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Dichotomies for $** C**_0(X)$ and $** C**_b(X)$ spaces

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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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