Advanced topology on the multiscale sequence spaces S^\nu

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