Corollary 2
Tag 07OL, permanent. Cite it as [Bourbaki, Tag 07OL].
Topological Vector Spaces, chapter III, § 4. The Banach-Steinhaus theorem
— Every semi-complete bornological space is barrelled.
Every simply bounded subset of the dual of such a space is strongly bounded (cor. 1), hence equicontinuous (III, p. 22, prop. 10).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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