Definition 1

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

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

— Two non-empty sets $A, B$ of a real topological vector space $E$ are said to be separated by a closed hyperplane $H$ if $A$ is contained in one of the closed half-spaces determined by $H$ and $B$ is contained in the other closed half-space.

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

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