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Keywords:
continuous function; integration; Baire category; porosity
continuous function; integration; Baire category; porosity
Summary:
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.
References:
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