It is well known that when $E$ is a $DF$-space and $F$ is a Fréchet space, the space $\mathcal{L}_{b} (E,F)$ is Fréchet. The converse, that is the fact that $\mathcal{L}_{b} (F,E)$ would be $DF$, is indicated as an open question in Grothendieck course on Topological vector spaces (Chapt 4, Part 3, Section 4, example e) ). Has it been answered since ?
Thank you everyone for any reference on that matter !
This question is closely related to Grothendieck's "problème des topologies" which was solved by Jari Taskinen in 1986. As far as I have heard, Susanne Dierolf (the advisor of my thesis) then observed that his construction also yields a negative solution of the problem you mention. I do not know if Dierolf published this somewhere. The following reference should be helpful: Jari Taskinen, The projective tensor product of Frechet-Montel spaces. Studia Math. 91 (1988), 17–30.
