Corollary 1

— Let E be a locally convex space. If $x_0 \in E$ is not in the closure of $\{0\}$, then there exists a continuous linear form f defined over E with $f(x_0) \neq 0$.

Apply prop. 2 to the one dimensional vector space M generated by $x_0$ and to the linear form $\xi x_0 \mapsto \xi$ defined over M, which, by I, p. 12, prop. 2, is continuous.