Some lattice and topological properties of the space of integrable Henstock-Kurzweil vector distributions

In this paper, we define a product and an order on the space \(D_{HK}([a,b],X)\) of integrable Henstock-Kurzweil vector distributions to give it the structure of a Banach lattice algebra. We give sufficient conditions for \(D_{HK}([a,b],X)\) to be an \(AM-\)space, for it to have the Dunford-Pettis property, and for its cone to be generating. We prove that the Alexiewicz norm on \(D_{HK}([a,b],X)\) is not \(\sigma-\)order continuous and that this space has no order units. We further prove that \(D_{HK}([a,b],X)\) has a complemented copy of \(c_0\) and give necessary and sufficient conditions for it to have a complemented copy of \(l_1\) and a copy of \(l_\infty\). As an application of the obtained results, we present a fixed point theorem on \(D_{HK}([a,b], X)\).