A few remarks on boundedness in topological modules and topological groups

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 × 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.