ϕ- and β-topologies in sequence spaces

Abstract

The aim of this note is to examine the relationship between ϕ-topologies and β-topologies in FK-spaces. Every ϕ-topology on an FK-space is a β-topology, the converse statement is not always true. Still, in AB-BK-spaces the statement holds, i.e., every β-topology is a
ϕ-topology. We establish that an analogous statement is true for ABFK-spaces.