On the Index of a Quadratic Form

Extract

Given a vector space V = {x, y, ...} over an arbitrary field. In V a symmetric bilinear form (x,y) i s given. A subspace W is called totally isotropic [t.i.] if (x,y) = 0 for every pair x W, y W.

Let V_n and V_m be two t.i. subspaces of V; n < m. Lower indices always indicate dimensions. It is a well known and fundamental fact of analytic geometry that there exists a t.i. subspace W_m of V containing V_n [cf. Dieudonné: Les Groupes classiques , P. 18]. As no simple direct proof seems to be available, we propose to supply one.