Dichotomies for C0(X) and C b (X) spaces

Jachymski showed that the set $$left{ {(x,y) in {{rm{c}}_0} times {{rm{c}}_0}:left( {sumlimits_{i = 1}^n {alpha (i)x(i)y(i)} } right)_{n = 1}^infty {rm{ is bounded}}} right}$$ is either a meager subset of c ₀ × c ₀ or is equal to c ₀ × 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 σ-porosity.