Remark

— With the notations of th. 2, let J be the set of all $i \in I$ such that $u(e_i) \neq 0$. For all $i \in J$, let $\lambda_i = \| u(e_i) \|$ and $f_i = \lambda_i^{-1} u(e_i)$. Then $(e_i)_{i \in J}$ (resp. $(f_i)_{i \in J}$) is an orthonormal basis of the initial (resp. final) subspace of u, we have $u(e_i) = \lambda_i f_i$ for all $i \in J$ and $\sum_{i \in J} \lambda_i^2 = \| u \|_2^2$ is finite.