Proposition 1

Tag 09IK, permanent. Cite it as [Bourbaki, Tag 09IK].

Integration, chapter VIII, § 2. Linear representations of groups

— If $G$ is a locally compact group and $E$ is barreled, then every separately continuous linear representation $U$ of $G$ in $E$ is continuous.

For every compact subset $K$ of $G$ , $U(K)$ is compact for the topology of pointwise convergence (Remark 1), therefore is equicontinuous (TVS, III, §4, No. 2, Th. 1); one then applies Remark 2.

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

Cited by

Remark 2, § 2
Proposition 4, § 2
Corollary 1, § 4

Cites

III, §4, No. 2, Th. 1
Remark 1
Remark 2