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.
This paper is closely related to the paper of Harry I. Miller: Measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc. 347 (1995), 1811–1819 and contains a general investigation of statistical convergence of subsequences of an arbitrary sequence from the point of view of Lebesgue measure, Hausdorff dimensions and Baire’s categories.