Structure of spaces of \(C^{\infty}-functions\) on nuclear spaces (Q796050)

Let E be a real nuclear locally convex space. The authors prove that the space \({\mathcal E}_{ub}(E)\) of \(C^{\infty}\)-functions of uniform bounded type on E (the inductive limit, when V ranges over a base of convex balanced O-neighbourhoods of E, of the spaces \({\mathcal E}_ b(E_ V)\) of \(C^{\infty}\) functions on \(E_ V\) which are bounded with all their derivatives on each bounded subset of \(E_ V)\), coincides with the inductive limit of the space of Nachbin-Dineen \({\mathcal E}_{Nbc}(E_ V).\) If E is a real nuclear bornological vector space, they prove that the space \({\mathcal E}(E)\) coincides with the projective limit of the spaces \({\mathcal E}_{Nbc}(E_ B)\), where B is a closed convex balanced bounded subset of E. As a consequence, a version of the Paley-Wiener-Schwartz theorem is obtained.