Structure of spaces ofC ∞-functions on nuclear spaces

Abstract

LetE be a real nuclear locally convex space; we prove that the space ℰub(E), of allC ∞-functions of uniform bounded type onE, coincides with the inductive limit of the spaces ℰNbc(E v) (introduced by Nachbin-Dineen), whenV ranges over a basis of convex balanced 0-neighbourhoods inE. LetE be a real nuclear bornological vector space; we prove that the space ℰ(E) of allC ∞-functions onE coincides with the projective limit of the spaces ℰNbc(E B), whenB is a closed convex balanced bounded subset ofE. As a consequence we obtain some density results and a version of the Paley-Wiener-Schwartz theorem.