Proposition 5

— Let $(e_i)_{i \in I}$ be an orthonormal family in a Hausdorff prehilbertian space E. The following properties are equivalent : a) the family $(e_i)$ is total ; b) for every $x \in E$, the family $\langle e_i | x \rangle \cdot e_i$ is summable in E, and we have $x = \sum_{i \in I} \langle e_i | x \rangle \cdot e_i$; c) for every $x \in E$,

(Parseval’s relation).

When E is hilbertian these conditions are also equivalent to : d) the relations $\langle e_i | x \rangle = 0$ for all $i \in I$ imply that $x = 0$.

The equivalence of conditions a), b), c) follows immediately from prop. 4. When E is hilbertian, the equivalence of conditions a) and d) follows from cor. 1 of V, p. 16.