Mixed topologies on Saks spaces of vector-valued functions

TL;DR

This paper investigates Saks spaces of vector-valued functions with mixed topologies, focusing on their properties like completeness and semi-Montelness, with applications to bi-continuous semigroups.

Contribution

It provides new insights into the properties of Saks spaces with mixed topologies, including conditions for completeness and the relationship between mixed and submixed topologies.

Findings

Saks spaces can be complete or semi-Montel under certain conditions.

The paper characterizes when mixed and submixed topologies coincide.

Explicit systems of seminorms generating the mixed topology are identified.

Abstract

We study Saks spaces of functions with values in a normed space and the associated mixed topologies. We are interested in properties of such Saks spaces and mixed topologies which are relevant for applications in the theory of bi-continuous semigroups. In particular, we are interested if such Saks spaces are complete, semi-Montel, C-sequential or a (strong) Mackey space with respect to the mixed topology. Further, we consider the question whether the mixed and the submixed topology coincide on such Saks spaces and seek for explicit systems of seminorms that generate the mixed topology.