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. 393394 in the English edition).

$\begingroup$ Thanks, so one has to combine §21.5 (3): "For a Fréchet space, the original topology is equal to the strong topology", with §29.1 (7), which is what you quoted. $\endgroup$ – Tim van Beek Apr 29 '11 at 7:52

1$\begingroup$ SO what does this mean for the question, for someone who doesn't have instant recall about the lattice of properties of topological vector spaces? $\endgroup$ – David Roberts Apr 29 '11 at 7:56

1$\begingroup$ @David Roberts: This implies that the claim stated by the OP holds true. The observation is probably due to Grothendieck. $\endgroup$ – Andrey Rekalo Apr 29 '11 at 8:02

12$\begingroup$ A nice way to think of this is as the observation that a LCTVS cannot be a (nontrivial) projective limit and an inductive limit of countably infinite families of Banach spaces at the same time. Either one family has to be uncountable, or both have to be finite. $\endgroup$ – Loop Space Apr 29 '11 at 8:45

$\begingroup$ @Andrew Stacey: That's really nice and intuitively appealing. $\endgroup$ – Andrey Rekalo Apr 29 '11 at 8:55