Corollary
Tag 07T3, permanent. Cite it as [Bourbaki, Tag 07T3].
Topological Vector Spaces, chapter IV, § 2. Bidual. Reflexive spaces
— Let E be a metrizable Montel space. Then there exists a countable dense set in the strong dual of E.
On the dual E' of E, the strong topology is identical with that of compact convergence, since E is a Montel space. It is now enough to apply cor. 1 of prop. 6 of III, p. 18.
We can show that the strong dual of a metrizable Montel space E is not metrizable if E is infinite dimensional (IV, p. 57, exerc. 1).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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