Spaces of d.c. mappings on arbitrary intervals

Let X be a Banach space. Using derivatives in the sense of vector distributions, we show that the space DC([0, 1], X) of all d.c. mappings from [0, 1] into X, in a natural norm, is isomorphic to the space M-bv([0, 1], X) of all vector measures with bounded variation. The same is proved for the space BDCb((0,infinity), X) of all bounded d.c. mappings with a bounded control function. The result for the space DC([0, 1], R) of all continuous d.c. functions was (essentially) proved by M.Zippin (2000) by a quite different method. The space BDCb((0,infinity), R) consists of all differences of two bounded convex functions. Internal characterizations of its members were given by O. Bohme (1985), but our characterization of its Banach structure is new.