I've read the claim that Fréchet spaces that are not Banach spaces never have a dual that is a Fréchet space, but have not been able to find a proof of this statement. Is it trivial or does someone have a reference?
For any locally convex and metrizable space $E$, its strong dual is metrizable if and only if $E$ is normable.
This and related properties of (F)-spaces are discussed in detail in Topological Vector Spaces I by Köthe (see §29.1, pp. 393-394 in the English edition).