A few remarks on boundedness in topological modules and topological groups

TL;DR

This paper explores different notions of boundedness in topological modules and groups, introducing new classes of bounded homomorphisms and characterizing bounded sets in product topologies.

Contribution

It introduces three classes of bounded bigroup homomorphisms and characterizes bounded sets in product topologies of topological groups.

Findings

Spaces of bounded homomorphisms form topological modules.

Characterization of bounded sets in product and box topologies.

New insights into boundedness in topological modules and groups.

Abstract

Let X, Y, and Z be topological modules over a topological ring $R$. In the first part of the paper, we introduce three different classes of bounded bigroup homomorphisms from $X\times Y$ into $Z$ with respect to the three different uniform convergence topologies. We show that these spaces form again topological modules over $R$. In the second part, we characterize bounded sets in the arbitrary product of topological groups with respect to the both product and box topologies.