Advanced topology on the multiscale sequence spaces S^\nu

TL;DR

This paper investigates the topological properties of multiscale sequence spaces $S^ u$, providing conditions for local p-convexity, defining their topology via p-norms, and characterizing their dual spaces, including special cases like intersections of Besov spaces.

Contribution

It establishes necessary and sufficient conditions for local p-convexity of $S^ u$, constructs its topology with p-norms, and characterizes its dual space, extending to intersections of Besov spaces.

Findings

Characterization of local p-convexity conditions.

Construction of the natural topology via p-norms.

Identification of the strong dual space, including Besov space intersections.

Abstract

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.