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

A few remarks on boundedness in topological modules and topological groups