Proposition 6
Tag 07SQ, permanent. Cite it as [Bourbaki, Tag 07SQ].
Topological Vector Spaces, chapter IV, § 2. Bidual. Reflexive spaces
— In order that a normed space E be reflexive, it is necessary and sufficient that the unit ball in E be compact for the weakened topology .
We observe that a reflexive normed space is complete hence a Banach space, and that its dual is a reflexive Banach space by prop. 4 of IV, p. 16.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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Nothing.