Remark

— 1) The hyperplane $H$ does not meet $A$ (II, p. 15, prop. 1; if $A$ and $B$ are two convex non-empty open sets that do not meet then there exists a closed hyperplane that separates $A$ strictly from $B$. 2) However, when $B$ is not open, it is not necessarily the case that there exists a closed hyperplane that separates $A$ strictly from $B$, even if $E$ is of finite dimension, and even if $\overline{A}$ does not meet $\overline{B}$ (II, p. 78, exerc. 12).