A dimension bound for subspaces of symmetric bilinear forms in terms of the number of different ranks

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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.