A simpler description of the $\kappa$-topologies on the spaces $\mathscr{D}_{L^p}$, $L^p$, $\mathscr{M}^1$

TL;DR

This paper provides simplified descriptions of the $ppa$-topologies on certain function and measure spaces, using semi-norms based on multiplication and convolution, and offers a sequence-space representation and compactness characterizations.

Contribution

It introduces more convenient semi-norm descriptions of the $ppa$-topologies and extends sequence-space representations for these function and measure spaces.

Findings

Semi-norm families generating $ppa$-topologies are established.

Sequence-space representation of $ppa$-topology on $_{L^p}$ is provided.

Characterization of compact subsets in the dual and measure spaces is achieved.

Abstract

The $\kappa$-topologies on the spaces $\mathscr{D}_{L^p}$, $L^p$ and $\mathscr{M}^1$ are defined by a neighbourhood basis consisting of polars of absolutely convex and compact subsets of their (pre-)dual spaces. In many cases it is more convenient to work with a description of the topology by means of a family of semi-norms defined by multiplication and/or convolution with functions and by classical norms. We give such families of semi-norms generating the $\kappa$-topologies on the above spaces of functions and measures defined by integrability properties. In addition, we present a sequence-space representation of the spaces $\mathscr{D}_{L^p}$ equipped with the $\kappa$-topology, which complements a result of J.~Bonet and M.~Maestre. As a byproduct, we give a characterisation of the compact subsets of the spaces $\mathscr{D}'_{L^p}$, $L^p$ and $\mathscr{M}^1$.