Corollary

Tag 07O5, permanent. Cite it as [Bourbaki, Tag 07O5].

Topological Vector Spaces, chapter III, § 4. The Banach-Steinhaus theorem

— Every Fréchet space (and in particular, every Banach space) is barrelled. This follows from Baire’s theorem (GT, IX, § 5, No. 3, th. 1).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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