Corollary
Tag 07U1, permanent. Cite it as [Bourbaki, Tag 07U1].
Topological Vector Spaces, IV, p. 24, § 3. Dual of a Fréchet space
The strong dual of a reflexive Fréchet space is bornological and barrelled.
An inductive limit of Banach spaces is bornological by definition. Further, a Banach space is barrelled (III, p. 25, corollary) and every inductive limit of barrelled spaces is a barrelled space (III, p. 25, 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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