Lemma 13

Let I be an orthonormal family in E. The family I converges weakly to 0 with respect to the filter of complements of finite subsets of I.

Let $x\in E$. By Bessel's inequality (EVT, V, p. 21, prop. 4) and TG, IV, p. 37, th. 1, the family $(|\langle e_i|x\rangle |^2)_{i\in I}$ is summable, hence $\langle e_i|x\rangle$ tends to 0 with respect to the filter of complements of finite subsets of I (TG, III, p. 38, prop. 1).