A conjecture of Palamodov about the functors Extk in the category of locally convex spaces

In 1971 Palamodov proved that in the category of locally convex spaces the derived functors Extk(E,X) of Hom(E,•) all vanish if E is a (DF)-space, X is a Fréchet space, and one of them is nuclear. He conjectured a “dual result”, namely that Extk(E,X)=0 for all k∈N if E is a metrizable locally convex space, X is a complete (DF)-space, and one of them is nuclear. Assuming the continuum hypothesis we give a complete answer to this conjecture: If X is an infinite-dimensional nuclear (DF)-space, then