Proposition 17

— In a topological vector space E, let A be a set with at least one interior point, and such that all its points lie on the same side of an hyperplane H. Then H is closed, the interior points of A lie strictly on the same side of H, and the cluster points of A lie on the same side of H. In particular open (resp. closed) half spaces are determined by closed hyperplanes.

In fact suppose that H contains the origin and that $f(x) = 0$ is an equation of H; suppose, for definiteness, that $f(x) \geqslant 0$ for all $x \in A$. The half space formed by the points y such that $f(y) > -1$ contains at least one interior point, and, by translation, the same is true of the half space of points y such that $f(y) > 0$; this shows that H is closed (I, p. 11, corollary). Then we know that $f$ is a strict morphism of E on $\mathbf{R}$ (I, p. 13, corollary), therefore $f(\dot{A})$ is an open set in $\mathbf{R}$. This set cannot contain 0 or it would contain numbers $< 0$ contrary to hypothesis; it is thus contained in the open interval $]0, +\infty[$. On the other hand, the half space of those y for which $f(y) \geqslant 0$ is closed and contains A, therefore it contains $\overline{A}$.