Remark 2
Tag 07JA, permanent. Cite it as [Bourbaki, Tag 07JA].
Topological Vector Spaces, II, p. 59, § 7. Extremal points and extremal generators
There exist proper convex cones that are weakly complete and which have no extremal generator (II, p. 92, exerc. 31).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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