Definition 3

Suppose that E is of finite dimension and that Φ is a non-degenerate ε-hermitian form satisfying (T) (resp. the bilinear form associated with a non-degenerate quadratic form Q). We call the index of Φ (resp. Q) the common dimension of the maximal totally isotropic (resp. totally singular) subspaces of E.

If n is the dimension of E and ν the index of Φ (resp. Q), Prop. 3 shows that

(4)

Moreover, since every totally isotropic (resp. totally singular) subspace is contained in a maximal totally isotropic (resp. totally singular) subspace, the totally isotropic (resp. totally singular) subspaces which are maximal are those which are of dimension $v$. The assertion that $\Phi$ (resp. Q) is of index 0 means that every isotropic (resp. singular) vector of E is zero. In a space of even dimension $n$, the neutral forms are those of index $\frac{1}{2}n$; there is no neutral form in a space of odd dimension. Prop. 3 shows that every form is a direct sum of a neutral form and a form of index 0.