Let $\mathcal{F}$ be a Fréchet space (locally convex, Hausdorff, metrizable, with a family of seminorms $\{\|~\|_n\}$).
I've read that the dual $\mathcal{F}^*$ is never a Fréchet space, unless $\mathcal{F}$ is actually a Banach space. I'd like to know in which ways the dual can fail to be Fréchet in general; for example, is it always incomplete as a metric space? Or is it always non-metrizable? What are the real issues here?
If there's a relatively easy proof of this fact (that $\mathcal{F}^*$ is not Fréchet unless $\mathcal{F}$ is Banach), I would appreciate it as well.
Thanks.
[EDIT: The reference cited by Dirk contains a Theorem whose proof is inaccessible to me, so I'd upvote/accept an answer that at least sketches such a proof, or provides another way of settling the question.
I'd also be interested in any further explanations regarding the statement that "(...) a LCTVS cannot be a (non-trivial) projective limit and an inductive limit of countably infinite families of Banach spaces at the same time.", made by Andrew Stacey in that link, which is related to my question.]