The Closed Graph Theorem and the Space of Henstock-Kurzweil Integrable Functions with the Alexiewicz Norm

Abstract

We prove that the cardinality of the space $\mathscr{H}𝒦([a,b])$ is equal to the cardinality of real numbers. Based on this fact we show that there exists a norm on $\mathscr{H}𝒦([a,b])$ under which it is a Banach space. Therefore if we equip $\mathscr{H}𝒦([a,b])$ with the Alexiewicz topology then $\mathscr{H}𝒦([a,b])$ is not K-Suslin, neither infra-($u$) nor a webbed space.