  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.
