On the injective tensor product of distinguished Frechet spaces

An example of two distinguished Frechet spaces E, F is given (even more, E is quasinormable and F is normable) such that their completed injective tensor product E (X) over cap(epsilon)F is not distinguished. On the other hand, it is proved that for arbitrary reflexive Frechet space E and ;arbitrary compact set K the space of E-valued continuous functions C(K, E) is distinguished and its strong dual is naturally isomorphic to L-1 (mu)(X) over cap(pi)E(b)' where L-1(mu) = C(K)'.