On the Theory of Kernels of Schwartz1

Let 3D denote the space of indefinitely differentiable functions of compact carrier on Euclidean space R of dimension p, and denote by 3D' the dual of 3D, that is, 30' is the space of distributions on R; we give 3D and 3D' their usual topologies (see [2]). By 23D, 23D' we denote respectively the spaces corresponding to 3D and 3D' on AXA. Let L be any continuous linear map of 3D—»3D'; then L. Schwartz has shown (see [3 ] for a summary of Schwartz' results, the details will appear in a series of articles in the Journal D'Analyse (Jerusalem)) that L can be represented by a kernel, i.e. there exists a distribution S on AXA so that we may write (symbolically)