Corollary 2
Tag 07N2, permanent. Cite it as [Bourbaki, Tag 07N2].
Topological Vector Spaces, chapter III, § 3. Spaces of continuous linear mappings
— Every closed ball in the dual E' of a normed space satisfying the first axiom of countability, is a compact metrizable space for the weak topology σ(E', E), and for this topology there exists a countable dense subset in E'.
This follows from prop. 6 and from III, p. 17, cor. 3.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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