Translation-invariant Linear Forms and a Formula for the Dirac Measure

Following Schwartz [2 ] we denote by £>, 8 and S the complex vector spaces of all complex-valued infinitely differentiable functions on R where the functions of 2D have compact supports, the functions of 8 have arbitrary supports, and the functions of S (along with all their derivatives) are rapidly decreasing at infinity. We equip each of these spaces with its usual locally convex topology. These spaces and their duals 3D', 8' and S' are translation-invariant in the sense that the translated function (or distribution) A (/)==$(£ — h) belongs to the space whenever 0 does. We say that a (not necessarily continuous) linear form L on any of these spaces is "translation-invariant" if L((j>h) =L(4>) for all in the domain space and for all h in R . I t is, of course, well known what the continuous translation-invariant linear forms on these spaces are like; namely, they are either identically zero or a constant multiple of integration over R. The purpose of this paper is to announce that there exists no discontinuous translation-invariant linear form on any of the six spaces 2D, 8, S, 3D', 8' or S'. That is, integration over R in the spaces 3D, S and 8' can be characterized (up to a multiplicative constant) simply as a translation-invariant linear form. Furthermore, we obtain this result as a simple consequence of a resolution of the first derivative of the Dirac measure S (on the real line R) into a sum of two finite differences of distributions of compact support. We state this as our main result.