Large spaces of symmetric matrices of bounded rank are decomposable

Let k and n be positive integers such that k less than or equal to n. Let S-n(F) denote the space of all n x n symmetric matrices over the field F with char F not equal 2. A subspace L of S-n(F) is said to be a k-subspace if rank A less than or equal to k for every A is an element of L. Now suppose that k is even, and write k=2r. We say a (k) over bar -subspace of S-n(F) is decomposable if there exists in F-n a subspace W of dimension n - r such that x(t)Ax = 0 for every x is an element of W, A is an element of L. We show here, under some mild assumptions on k, n and F, that every Tc-subspace of S-n(F) of sufficiently large dimension must be decomposable. This is an analogue of a result obtained by Atkinson and Lloyd for corresponding subspaces of F-m,F-n.