Continuous Additive Functional^ on the Space (bv) and Certain Subspaces

7"o(z) being employed in general to denote the total variation of the function z(t) on /. Thus metrised, (BV) is not a Banach space(2); but it is complete, separable, and boundedly compact. Although a linear space, it is not a "linear topological space" in the sense in which that term is sometimes used, for the topology introduced by the metric (1) is non-uniform. Indeed it is easily seen that the category of a subset is not always invariant under translation. For, if the closed unit sphere K(6, 1) about the zero-element 9 as center, which is a set of second category in (B V), is subjected to the translation x where x has as a representative function x(t)=0 for 0<t^l, x(0)=2, then its translate