Fréchet Schwartz spaces and approximation properties (Q1345953)

We show that there is a Fréchet Schwartz space with the approximation property which cannot be written as a reduced projective limit of Banach spaces with approximable linking maps. This also yields the first example of a complete locally convex space with the approximation property which is not a reduced projective limit of Banach spaces with the approximation property and thus gives a negative answer to an old problem [see also \textit{M. S. Ramanujan}, p. 124 in Proc. of the international conference on operator algebras, ideals, and their applications in theoretical physics, Leipzig, September (1977)]. Some consequences on projective tensor products of Fréchet spaces are indicated.