Corollary

Tag 07O0, permanent. Cite it as [Bourbaki, Tag 07O0].

Topological Vector Spaces, III, p. 24, § 4. The Banach-Steinhaus theorem

The following conditions are equivalent:

(i) every weakly bounded subset of E' is equicontinuous;

(ii) every convex, balanced, closed and absorbent set in E is a neighbourhood of 0;

(iii) every lower semi-continuous semi-norm on E is continuous.

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

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