Uniform boundedness for groups

Let G and H be locally compact abelian groups with character groups G*, H*, and let denote the pairing between a group and its dual. In 1952 Kaplansky proved the following result, using the structure of locally compact abelian groups and category arguments. Theorem 1.1. Let τ: G → H be an algebraic homomorphism for which there is a dual τ* : H* → G* (so that = for all g in G, h* in H*). Then τ is continuous. The result bears a striking similarity to a well-known fact about Banach spaces which is a consequence of uniform boundedness; the present note is devoted to an analogous “uniform boundedness” for groups, which yields a non-structural proof of Kaplansky's theorem.