Corollary 2

Tag 07C1, permanent. Cite it as [Bourbaki, Tag 07C1].

Topological Vector Spaces, chapter II, § 2. Convex sets

— In a topological vector space E, the convex envelope of a finite set is precompact.

In fact, let j be the canonical mapping of E in its Hausdorff completion $\hat{E}$ ; if C is the convex envelope of A, then $j(C)$ is the convex envelope of the finite set $j(A)$ in $\hat{E}$ , hence $j(C)$ is compact (cor. 1) and therefore C is precompact (GT, II, § 4.2).

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

Cited by

Nothing.

Cites

II, § 4