Topological Groups of Bounded Homomorphisms on a Topological Group

TL;DR

This paper investigates various classes of bounded homomorphisms on topological groups, demonstrating that each class forms a complete topological group under suitable topologies, even when weaker than continuous homomorphisms.

Contribution

It introduces and analyzes different classes of bounded homomorphisms, establishing their completeness as topological groups under specific topologies.

Findings

Each class of bounded homomorphisms forms a complete topological group.

The classes considered are weaker than continuous homomorphisms.

Completeness is achieved with appropriate topologies.

Abstract

We consider a few types of bounded homomorphisms on a topological group. These classes of bounded homomorphisms are, in a sense, weaker than the class of continuous homomorphisms. We show that with appropriate topologies each class of these homomorphisms on a complete topological group forms a complete topological group.