On the rank of 3x3x3-tensors

Let $U$, $V$ and $W$ be finite dimensional vector spaces over the same field. The rank of a tensor $\tau$ in $U \otimes V \otimes W$ is the minimum dimension of a subspace of $U \otimes V \otimes W$ containing $\tau$ and spanned by fundamental tensors, i.e.\ tensors of the form $u \otimes v \otimes w$ 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 $U \otimes V \otimes W$ 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 $U \otimes V \otimes W$ when the dimensions of $U$, $V$ and $W$ are higher.