Corollary 1

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

Topological Vector Spaces, II, p. 45, § 6. Weak topologies

For any family $(M_\alpha)$ of closed convex sets of $F$ (in the topology $\sigma(F, G)$ *), each containing* 0, the polar set of the intersections $M = \bigcap \limits_{\alpha} M_\alpha$ is the convex closed envelope (for $\sigma(G, F)$ ) of the union of the $M_\alpha^\circ$ .

For, if $N$ is this convex closed envelope, then

N^\circ = \bigcap \limits_{\alpha} M_\alpha^{\circ\circ} = \bigcap \limits_{\alpha} M_\alpha = M

whence $N = N^{\circ\circ} = M^\circ$ .

The conclusion of cor. 1 does not necessarily hold if the $M_\alpha$ are not convex.

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

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Cites

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