Exercise 6, § 2

Let E be a hilbertian space and $(e_i)_{i \in I}$ an orthonormal basis of E. Show that if $(a_i)_{i \in I}$ is a topologically independent family in E such that $\sum_{i \in I} \|e_i - a_i\|^2 < + \infty$, then the family $(a_i)$ is total. (Let J be a finite subset of I; show that there is a continuous linear mapping $u$ from E into itself such that $u(e_i) = e_i$ for $i \in J$, $u(e_i) = a_i$ for $i \notin J$, and that the norm of $u - 1_E$ can be made arbitrarily small by a suitable choice of J; then use IV, p. 65, exerc. 17.)