Schwartz groups and convergence of characters

It is proved that a locally quasi-convex group is a Schwartz group if and only if every continuously convergent filter on its dual group converges locally uniformly. We also show that for metrizable separable groups a similar result remains true when filters are replaced by sequences. As an ingredient in the proofs of these results, we obtain a Schauder-type theorem on compact homomorphisms acting between the natural group analogues of normed spaces.