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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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up vote 17 down vote accepted

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).

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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. –  Tim van Beek Apr 29 '11 at 7:52
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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? –  David Roberts Apr 29 '11 at 7:56
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@David Roberts: This implies that the claim stated by the OP holds true. The observation is probably due to Grothendieck. –  Andrey Rekalo Apr 29 '11 at 8:02
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A nice way to think of this is as the observation 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. Either one family has to be uncountable, or both have to be finite. –  Loop Space Apr 29 '11 at 8:45
    
@Andrew Stacey: That's really nice and intuitively appealing. –  Andrey Rekalo Apr 29 '11 at 8:55

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