Corollary 1

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

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

— Let E and F be two Banach spaces, H a family of continuous linear mapping from E into F; if, for all $x \in E$ , we have $\sup_{u \in H} \|u(x)\| < +\infty$ , we also have $\sup_{u \in H} \|u\| < +\infty$ .

In fact, the hypothesis says that H is simply bounded and the conclusion that it is equicontinuous. In addition, every Banach space is barrelled (III, p. 25).

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

Cited by

Nothing.

Cites

III, p. 25