Corollary 1

Let $(x_i^*)_{1 \leq i \leq p}$ be a finite sequence of linear forms on E and let F be the subspace of E consisting of the x such that

Then $\operatorname{codim}_E F$ is equal to the rank of the set of the $x_i^*$ and every linear form on E which is zero on F is a linear combination of the $x_i^*$. Then $\operatorname{codim}_E F \leq p$ and in order that $\operatorname{codim}_E F = p$, it is necessary and sufficient that the $x_i^*$ be linearly independent.

The set $G'$ of linear combinations of the $x_i^*$ is a subspace of $E^*$ and F is the orthogonal of $G'$ in E, hence $\operatorname{codim}_E F = \dim G'$ by Theorem 7; further $\dim G' \leq p$ and the relation $\dim G' = p$ means that $(x_i^*)$ is a free system (no. 2, Proposition 1); hence the corollary.