Corollary 2

Tag 07O8, permanent. Cite it as [Bourbaki, Tag 07O8].

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

— Let $(E_i)_{i \in I}$ be a family of locally convex spaces and let E be the topological direct sum of this family. For E to be barrelled, it is necessary and sufficient that each $E_i$ is barrelled.

The condition is evidently sufficient by virtue of prop. 3; it is also necessary, by cor. 1, since each $E_i$ is isomorphic to a quotient space of E (II, p. 31, prop. 8).

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

Cited by

Nothing.

Cites

II, p. 31