Lemma 2

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

Topological Vector Spaces, IV, p. 37, § 5. Compactness criteria

Let X be a compact space. If a sequence $(f_n)_{n \in \mathbf{N}}$ is bounded in $C(X)$ and converges simply on X to a continuous function f, then $\mu(f) = \lim_{n \to \infty} \mu(f_n)$ for every $\mu$ in $C(X)'$ .

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

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