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We pursue the study of the multiscale spaces $S^\nu$ introduced by Jaffard inthe context of multifractal analysis. We give the necessary and sufficientcondition 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 $\nu$, endowedwith an inductive limit topology. As a particular case, we describe the dual ofa countable intersection of Besov spaces.

Advanced topology on the multiscale sequence spaces S^\nu