Remark 3

Tag 07QW, permanent. Cite it as [Bourbaki, Tag 07QW].

Topological Vector Spaces, chapter IV, § 1. Duality

Suppose E is Hausdorff (for the initial topology). Let A be a subset of E which is closed and bounded for $\sigma(E, E')$ , hence also for every topology compatible with the duality between E and $E'$ . Since A is precompact for $\sigma(E, E')$ (III, p. 3, Remark 5), assuming that A is compact for $\sigma(E, E')$ is equivalent to A being complete for $\sigma(E, E')$ .

Therefore, on account of remark 2, we see that :

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Corollary, § 1

Cites

Nothing.