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 wit
h 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-subs
pace of S-n(F) of sufficiently large dimension must be decomposable. This i
s an analogue of a result obtained by Atkinson and Lloyd for corresponding
subspaces of F-m,F-n.