Compatible topologies and bornologies on modules

The Triangle of Operators, Topologies, Bornologies

This paper discusses two common techniques in functional analysis: the topological method and the bornological method. In terms of Pietsch’s operator ideals, we establish the equivalence of the notions of operators, topologies and bornologies. The approaches in the study of locally convex spaces of Grothendieck (via Banach space operators), Randtke (via continuous seminorms) and Hogbe-Nlend (vi...

ar X iv : m at h / 05 06 18 3 v 1 [ m at h . FA ] 1 0 Ju n 20 05 THE TRIANGLE OF OPERATORS , TOPOLOGIES , BORNOLOGIES

This paper discusses two common techniques in functional analysis: the topological method and the bornological method. In terms of Pietsch’s operator ideals, we establish the equivalence of the notions of operators, topologies and bornologies. The approaches in the study of locally convex spaces of Grothendieck (via Banach space operators), Randtke (via continuous seminorms) and Hogbe-Nlend (vi...

Dual Topologies on Non-abelian Groups

The notion of locally quasi-convex abelian group, introduce by Vilenkin, is extended to maximally almost-periodic non-necessarily abelian groups. For that purpose, we look at certain bornologies that can be defined on the set rep(G) of all finite dimensional continuous representations on a topological group G in order to associate well behaved group topologies (dual topologies) to them. As a co...

Examples of Convex Functions and Classiications of Normed Spaces

We study various properties of convex functions and their connections to the structure of the spaces on which they are deened. In particular, it is shown boundedness properties of convex functions on various bornologies are related to sequential convergence in dual topologies. Convex functions whose subdiierentials have range with nonconvex interior are constructed on nonreeexive spaces, and we...

Extensions of Baer and quasi-Baer modules