Proposition 13

For the system (21) to be of finite rank r, it is necessary and sufficient that the linear mapping $u : x \mapsto (\langle x, x_i^* \rangle)$ of E into $K_s^I$ be of rank r.

If F' is the subspace of $E^*$ generated by the $x_i^*$, the kernel of u is the orthogonal F of F' in E; if F' is of dimension r, F is of codimension r and conversely (no. 5, Theorem 7) and $\mathrm{rg}(u) = \mathrm{codim}_E F$ (no. 4, formula (11)).