Barrelled Weakly Köthe–Orlicz Summable Sequence Spaces

Abstract

Let E be a Hausdorff locally convex space. We investigate the space Λφ[E] of weakly Köthe–Orlicz summable sequences in E with respect to an Orlicz function φ and a perfect sequence space Λ. We endow Λφ[E] with a Hausdorff locally convex topology and determine the continuous dual of the so-obtained space in terms of strongly Köthe–Orlicz summable sequences from the dual space E′ of E. Next, we give necessary and sufficient conditions for Λφ[E] to be barrelled or quasi-barrelled. This contributes to the understanding of different spaces of vector-valued sequences and their topological properties.