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Let S S be a physical system whose state at any time is described by an n n -dimensional vector x ( t ) x\left ( t \right ) , where x ( t ) x\left ( t \right ) is determined by a linear differential equation d z / d t = A z dz/dt = Az , with A A a constant matrix. Application of external influences will yield an inhomogeneous equation, d z / d t = A z + f dz/dt = Az + f , where f f , the “forcing term", represents the control. A problem of some importance in the theory of control circuits is that of choosing f f so as to reduce z z to 0 in minimum time. If f f is restricted to belong to the class of vectors whose i i th components can assume only the values ± b i \pm {b_i} , the control is said to be of the “bang-bang” type.

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.