23

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?

Tim van Beek
  • 1,544

1 Answers1

26

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).

Andrey Rekalo
  • 21,997
  • 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 07:52
  • 1
    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 07:56
  • 1
    @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 08:02
  • 13
    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. – Andrew Stacey Apr 29 '11 at 08:45
  • @Andrew Stacey: That's really nice and intuitively appealing. – Andrey Rekalo Apr 29 '11 at 08:55
  • @Andrew Stacey: Where can one find a proof and detailed discussion of your statement? Thanks! – Tobias Diez Oct 26 '12 at 15:42