The Set of Divergent Infinite Products in a Banach Space is $\sigma$-Porous

Abstract

Let $K$ be a bounded closed convex subset of a Banach space. We study several convergence properties of infinite products of non-expansive self-mappings of $K$. In our recent work we have considered several spaces of sequences of such self-mappings. Endowing them with appropriate topologies, we have shown that the infinite products corresponding to generic sequences converge. In the present paper we prove that the subsets consisting of all sequences of mappings with divergent infinite products are not only of the first Baire category, but also $\sigma$-porous.