Corollary

Tag 07G5, permanent. Cite it as [Bourbaki, Tag 07G5].

Topological Vector Spaces, II, p. 40, § 5. Separation of convex sets

Let $f$ be a continuous affine function in $X$ ; then there exists a sequence $(h_n)$ of elements of $L$ which converges uniformly to $f$ in $X$ .

For, prop. 6 and Dini’s theorem (GT, X, § 4.1, th. 1) show that for all $n$ there exists $h_n \in L$ such that $f \leq h_n \leq f + 1/n$ .

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

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