TOPOLOGICAL TENSOR-PRODUCTS OF A FRECHET-SCHWARTZ SPACE AND A BANACH-SPACE

We exhibit examples of countable injective inductive limits E of Banac h spaces with compact linking maps (i.e. (DFS)-spaces) such that E x e psilon X is not an inductive limit of normed spaces for some Banach sp ace X. This solves in the negative open questions of Bierstedt, Meise and Hollstein. As a consequence we obtain Frechet-Schwartz spaces F an d Banach spaces X such that the problem of topologies of Grothendieck has a negative answer for F x pi X. This solves in the negative a ques tion of Taskinen. We also give examples of Frechet-Schwartz spaces and (DFS)-spaces without the compact approximation property and with the compact approximation property but without the approximation property.