Remark 2

If we apply theorem 1 to the case where $E$ carries the finest locally convex topology (II, p. 25, Example 2), and if, for the sake of simplicity, we suppose that $0 \in A$, then we get the following result (that superficially does not involve topology) : if $A$ is an absorbent convex set in the real vector space $E$ and if $M$ is a non-empty linear variety that does not meet $A$, then there exists a hyperplane $H$ containing $M$ and such that $A$ lies on one side of $H$. This result is not valid for every convex set $A$ (II, p. 65, exerc. 5).