Linear bornologies and associated topologies

Abstract

This paper is basically devoted to the study of the relation between a linear homology and the associated M-closure topology, introduced in (3) and (18). This topology is invariant in translations and dilations, and it has a base of absorbing and balanced neighbourhoods of zero. In Section I we study the class of topologies on linear spaces having these properties, namely, quasilinear topologies. In Section 2, we associate in a natural way a homology to every quasilinear topology, and, in Section 3, we introduce, under the same point of view, the M-closure topology relative to a linear homology. We study the quasilinear topo­logies whlch can be obtained as M-closurc topologies (q-bomological topologies), and the linear homologies obtained from a quasilinear topology by the procedure of Section 2 (infratopological bornologics), also considered by B. Perrot in (18). ln Section 4, we study the stability of the preceding classes of topologies and bornologics in passing to initial and final structures, and in forming spaces of bounded linear mappings. finally, in Section 5, we sec another way to obtain the M-closure topology. We use, if the contrary is not specilicd, the usual terminology on homologies, which can be found in (3), (9) and (IO).