Uniform Boundedness for Groups

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Abstract
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 < rg, h* > = < g, τ*h* > 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.

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