Compactness and sequential completeness in some spaces of operators (Q743455)

Let \(X\) be a completely regular Hausdorff space. Denote by \(C_b(X)\) the linear space of all real-valued bounded continuous functions on \(X\). Let, further, \(E\) be a real locally convex Hausdorff space. Denote by \(\mathcal{L}_z\) the linear space of all continuous linear operators from \(C_b(X)\) into \(E\), the former space being endowed with the strict topology \(\beta_z\), where \(z=t, \tau\) or \(\sigma\). (The notation is that of \textit{R. F. Wheeler}'s survey paper [Expo. Math. 1, 97--190 (1983; Zbl 0522.28009)]). The paper under review is mainly concerned with some properties of \(\mathcal{L}_z\) endowed with the topology of simple convergence. In particular, sequential completeness of \(\mathcal{L}_z\) and relative compactness of its subsets are studied. The latter is characterized in terms of Baire and Borel vector measures representing the elements of \(\mathcal{L}_z\).