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?
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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). 

