Corollary 1

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

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

— In a locally convex space, every closed convex set $A$ is the intersection of the closed half-spaces which contain it.

In fact, for every point $x \notin A$ , there exists a closed hyperplane that separates $x$ strictly from $A$ (using prop. 4).

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

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