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

The $\kappa$-topologies on the spaces ${\mathcal{D}}_{L^p}$, $L^p$ and ${\mathcal{M}}^1$ 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 ${\mathcal{D}}_{L^p}$ 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 ${\mathcal{D}}_{L^p}^{\mathrm{\prime }}$, $L^p$ and ${\mathcal{M}}^1$.