Proposition 3

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

Integration, VIII, p. 11, § 2. Linear representations of groups

(i) If U is separately continuous, then $^tU$ is separately continuous when E' is equipped with the weak topology $\sigma(E', E)$ . (ii) If G is locally compact and U is continuous, then $^tU$ is continuous when E' is equipped with the topology of compact convergence.

The assertion (i) is immediate. The assertion (ii) follows from Lemma 3 where one has taken $X = G, Y = Z = E, \varphi(s, x) = U(s)x$ .

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

Cited by

Proposition 4, § 2

Cites

Lemma 3