  Let K be a field of characteristic different from 2 and let V be a vector
space of dimension n over K. Let M be a non-zero subspace of symmetric bilinear
forms defined on V x V and let r=rank(M) denote the set of different positive
integers that occur as the ranks of the non-zero elements of M. The main result
of this paper is the inequality that dim M is at most rn-r(r-1)/2 provided that
|K| is at least n.
