Homomorphisms on C∞ (E) and C∞-bounding sets

We prove, for the class of real locally convex spaces E that are continuously and linearly injectable into some c0(Γ), that every non-zero homomorphism on the algebra C∞ (E) of C∞-functions on E is given by a point evaluation at some point of E. Furthermore, if every real-valued C∞-function on the weak topology of a quasi-complete locally convex space E is bounded on a subset A of E, then A is relatively weakly compact. © 1993 Springer-Verlag.