Definition 2
Tag 07O2, permanent. Cite it as [Bourbaki, Tag 07O2].
Topological Vector Spaces, III, p. 24, § 4. The Banach-Steinhaus theorem
A space E is said to be barrelled if it satisfies the equivalent conditions of the corollary of prop. 1.
We know (III, p. 22, prop. 9) that every equicontinuous subset of the dual E' of E is strongly and weakly bounded. We can therefore restate the definition of barrelled spaces as follows:
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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