A simpler description of the κ-topologies on p 1 the spaces DLp, L , M Christian Bargetz,∗ Eduard A. Nigsch,† Norbert Ortner‡ November 2017 Abstract p 1 The κ-topologies on the spaces DLp , L and M 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 κ-topologies on the above spaces of functions and measures defined by integrability properties. In addition, we present a sequence-space representation of the spaces DLp equipped with the κ-topology, which complements a result of J. Bonet and M. Maestre. As a byproduct, we give a characterisation of the compact subsets of 1 p 1 the spaces DLp , L and M . Keywords: locally convex distribution spaces, topology of uniform conver- gence on compact sets, p-integrable smooth functions, compact sets arXiv:1711.06577v2 [math.FA] 27 Nov 2018 MSC2010 Classification: 46F05, 46E10, 46A13, 46E35, 46A50, 46B50 ∗Institut f¨ur Mathematik, Universit¨at Innsbruck, Technikerstraße 13, 6020 Innsbruck, Austria. e-mail:
[email protected]. †Wolfgang Pauli Institut, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria. e-mail:
[email protected]. ‡Institut f¨ur Mathematik, Universit¨at Innsbruck, Technikerstraße 13, 6020 Innsbruck, Austria. e-mail:
[email protected]. 1 1 Introduction In the context of the convolution of distributions, L.