Corollary 1

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

Topological Vector Spaces, chapter II, § 6. Weak topologies

With the notations of prop. 6, the relation $u(A) \subset A_1$ implies ${}^t u(A_1^{\circ}) \subset A^{\circ}$ ; if further $A_1$ is convex, closed (for $\sigma(F_1, G_1)$ ) and contains the origin, then these two relations are equivalent.

In fact, the relation $u(A) \subset A_1$ equivalent to $A \subset u^{-1}(A_1)$ , therefore implies

{}^t u(A_1^{\circ}) \subset {}^t u(A_1^{\circ}) \subset (u^{-1}(A_1))^{\circ} \subset A^{\circ}

and conversely the relation ${}^t u(A_1^{\circ}) \subset A^{\circ}$ implies

A^{\circ \circ} \subset ({}^t u(A_1^{\circ}))^{\circ} = u^{-1}(A_1^{\circ \circ})

from (2). When $A_1 = A_1^{\circ \circ}$ , we deduce that $A \subset u^{-1}(A_1)$ .

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

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