Corollary

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

Topological Vector Spaces, II, p. 17, § 2. Convex sets

If $f$ is convex in $X$ , then for all $\alpha \in \mathbf{R}$ , the set of $x \in X$ such that $f(x) \leqslant \alpha$ (resp. $f(x) < \alpha$ ) is convex.

In fact, it is the projection on $E$ of the intersection of $F$ (resp. $F'$ ) and the hyperplane $E \times \{\alpha\}$ in $E \times \mathbf{R}$ .

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

Cited by

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Cites

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