A simpler description of the κ‐topologies on the spaces DLp, Lp, M1. Issue 9 (6th July 2020)

• Record Type: Journal Article
• Authors: Bargetz, Christian; Nigsch, Eduard A.; Ortner, Norbert
• Is Part Of: Mathematische Nachrichten. Volume 293:Issue 9(2020)
• Page Start: 1691
• Page End: 1706
• Publication Date: 2020-07-06
• Subjects: compact sets -- locally convex distribution spaces -- p‐integrable smooth functions -- topology of uniform convergence on compact sets; Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201900109
• Languages: English
• ISSNs: 0025-584X

Abstract

Abstract: For the spaces D L p, L p and M 1, we consider the topology of uniform convergence on absolutely convex compact subsets of their (pre‐)dual space. Following the notation of J. Horváth's book we call these topologies κ‐topologies. They are given 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 D L p 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 the spaces D L p ′, L p and M 1 .