Corollary 1

— A family $(a_i)$ of points of $F$ is total for the topology $\sigma(F, G)$ if, and only if, for every $y \neq 0$ in $G$, there exists an index $i$ such that $\langle a_i, y \rangle \neq 0$.

For using prop. 3 and I, p. 13, th. 1, the property expresses the fact that for $\sigma(F, G)$ no closed hyperplane contains all the $a_i$; the corollary follows therefore from cor. 3 of II, p. 38.