Topologies Associated with Nachbin Topology

Abstract

Publisher Summary This chapter presents a study on topologies associated with Nachbin topology. If E is a Banach or a Frechet space, then the compact-open topology T O and the toplogy T ω introduced by Nachbin induce the same bounded sets on the space H (E) of analytic functions on E (that is, continuous with analytic restrictions to finite dimensional subspace of E). This is the reason for the interest of people in the bornological topology associated with with T ω. For a Banach space E, the space H (E), T ω is barelled if only if it is bornological. The chapter discusses the barrelled topology associated with T ω and discusses that this topology coincides with T δ for any locally convex spaces (lcs) E. This result is a consequence of properties of the barelled topology associated with a lcs possessing a Schauder decomposition. The chapter also discusses various topologies associated with H(E),T ω, and discusses that these topologies can be divided into two classes: one related with barreledness, the other with infrabarelledness. The chapter also presents a characterization of the bornological topology associated with the pointwise and the compact-open topologies.