Corollary 1

In order that the family $(x_i)_{i \in I}$ of points of a hilbertian space E be total, it is necessary and sufficient that the relations $\langle x_i|y \rangle = 0$ for $y \in E$ and for all indices $i \in I$ imply that $y = 0$.

In fact, this says that 0 is the only vector of E' which is orthogonal to all the $x_i$ (II, p. 43 and IV, p. 1).