A note on Fréchet-Montel paces

Let E be a Fréchet space and let Cb(E) denote the vector space of all bounded continuous functions on E. It is shown that the following statements are equivalent: (i) E is Montel. (ii) Every bounded continuous function from E into Co maps every absolutely convex closed bounded subset of E into a relatively compact subset c₀. (iii) Every sequence in C^b(E) that converges to zero in the compact-open topology also converges uniformly to zero on absolutely convex closed bounded subsets of E.