Corollary 2

(i) Let $(x_i^*)_{1 \leq i \leq p}$ be a finite sequence of linear forms on E. For $(x_i^*)$ to be a free system, it is necessary and sufficient that there exist a sequence $(x_i)_{1 \leq i \leq p}$ of elements of E such that $\langle x_i, x_j^* \rangle = \delta_{ij}$ (Kronecker index).

(ii) Let $(x_i)_{1 \leq i \leq p}$ be a finite sequence of elements of E. For $(x_i)$ to be a free system, it is necessary and sufficient that there exist a sequence $(x_i^*)_{1 \leq i \leq p}$ of linear forms on E such that $\langle x_i, x_j^* \rangle = \delta_{ij}$.

Clearly (ii) follows from (i) by considering E as identified with a subspace of $E^{**}$ by means of $c_E$ (Theorem 6). Let $G'$ be the subspace of $E^*$ generated by the $x_i^*$ and F its orthogonal in E; E/F and $G'$ can each be canonically identified with the dual of the other; if the family $(x_i^*)$ is free, there is in E/F a basis $(\dot{x}_i)$ the dual of $(x_i^*)$ and every representative system $(x_i)$ of the classes $\dot{x}_i$ has the required properties. Conversely, the existence of the system $(x_i)$ such that $\langle x_i, x_j^* \rangle = \delta_{ij}$ implies that for all $i$ the subspace of $E^*$ orthogonal to K.$x_i$ contains the $x_j^*$ of index $j \neq i$ but does not contain $x_i^*$, hence the system $(x_i^*)_{1 \leq i \leq p}$ is free.