Advanced topology on the multiscale sequence spaces Sν

We pursue the study of the multiscale spaces Sν introduced by Jaffard in the context of multifractal analysis. We give the necessary and sufficient condition for Sν to be locally p-convex, and exhibit a sequence of p-norms that defines its natural topology. The strong topological dual of Sν is identified to another sequence space depending on ν, endowed with an inductive limit topology. As a particular case, we describe the dual of a countable intersection of Besov spaces.