Proposition 11

For every hyperplane H in a vector space E, there exists a linear form $x_0^*$ on E such that H = $x_0^{*-1}(0)$. Given such a form $x_0^*$, for a linear form $x^*$ on E to satisfy H = $x^{*-1}(0)$, it is necessary and sufficient that $x^* = x_0^* \alpha$, where $\alpha$ is a scalar ≠ 0. Conversely, for every linear form $x^* \neq 0$ on E, the subspace $x^{*-1}(0)$ is a hyperplane of E.

This statement merely expresses Theorem 7 for subspaces of E of codimension 1 and subspaces of E* of dimension 1.

If H is a hyperplane and $x_0^*$ a linear form such that H = $x^{*-1}(0)$, the relation

which characterizes the elements $x \in H$, is called an equation of H.

More generally, if $(x_i^*)$ is a family of linear forms on E and F denotes the vector subspace the intersection of the hyperplanes $x_i^{*-1}(0)$, the relation "for all $i, \langle x, x_i^* \rangle = 0$" characterizes the elements of F; the relations

form a system of equations of the subspace F. Theorem 7 (ii) expresses the fact that every vector subspace of E can be defined by a system of equations.

Theorem 7 (i) and (ii) proves moreover that a subspace F of finite codimension $p$ can be defined by a system of $p$ equations

(19)

where the forms $x_i^*$ are linearly independent. Conversely, Corollary 1 to Theorem 7 shows that a subspace F defined by a system of $p$ equations (19) is of co-dimension $\leq p$ and that it is of codimension $p$ if and only if the $x_i^*$ are linearly independent; it amounts to the same to say that F cannot be defined by a system consisting of at most $p - 1$ of the equations (19).