Corollary
Tag 07EC, permanent. Cite it as [Bourbaki, Tag 07EC].
Topological Vector Spaces, chapter II, § 4. Locally convex spaces
— If, in a locally convex Hausdorff space E, a compact set X is contained in a complete convex set (complete in the uniform structure induced by that of E) then the convex closed envelope of X is compact.
For this envelope is a closed subset of a complete space, therefore it is complete, but it is also precompact and Hausdorff.
However in a non complete locally convex Hausdorff space, the convex closed envelope of a compact set need not be compact (II, p. 87, exerc. 2).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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