Remark 1
Tag 07SM, permanent. Cite it as [Bourbaki, Tag 07SM].
Topological Vector Spaces, IV, p. 16, § 2. Bidual. Reflexive spaces
Let E be an infinite dimensional real hilbertian space. Let F denote the space E endowed with the weakened topology. The spaces E and F have the same dual E', and E is a reflexive Banach space (V, p. 17). Consequently, F is semi-reflexive.
However, the strong topology and the weakened topology on E are distinct, hence F is not reflexive.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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