On the rank of 3×3×3-tensors

Let U, V and W be finite dimensional vector spaces over the same field. The rank of a tensor ? in U?V?W is the minimum dimension of a subspace of U?V?W containing ? and spanned by fundamental tensors, i.e. tensors of the form u?v?w for some u in U, v in V and w in W. We prove that if U, V and W have dimension three, then the rank of a tensor in U?V?W is at most six, and such a bound cannot be improved in general. Moreover we discuss how the techniques employed in the proof might be extended to prove upper bounds for the rank of a tensor in U?V?W when the dimensions of U, V and W are higher.