Geodesic
Dichotomies for ${\bf C}_0(X)$ and ${\bf C}_b(X)$ spaces
Czechoslovak Mathematical Journal, Tome 63 (2013) no. 1, pp. 91-105

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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 
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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

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