The triangle of operators, topologies, bornologies

TL;DR

This paper explores the relationship between operators, topologies, and bornologies in functional analysis, establishing their equivalence within Pietsch's operator ideals and comparing various approaches to locally convex spaces.

Contribution

It demonstrates the equivalence of operators, topologies, and bornologies in the context of Pietsch's operator ideals, unifying different methodological approaches in functional analysis.

Findings

Established the equivalence of operators, topologies, and bornologies.

Compared approaches by Grothendieck, Randtke, and Hogbe-Nlend.

Clarified the relationships among various techniques in locally convex spaces.

Abstract

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 (via convex bounded sets) are compared.