Remark 1

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

Topological Vector Spaces, IV, p. 16, § 2. Bidual. Reflexive spaces

Suppose E is semi-reflexive. Every subset of E which is convex, closed and bounded for the initial topology is compact for the topology $\sigma(E, E')$ (IV, p. 1, prop. 1). On the other hand, the unit sphere (with the equation $\|x\| = 1$ ) of an infinite dimensional hilbertian space E is closed and bounded for the initial topology, but is not closed for the weakened topology, even if E is semi-reflexive. \

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

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