Question

Let \(V\) and \(W\) be finite-dimensional vector spaces, let \(\omega\) be a contravariant \(1\)-tensor on \(V\), and let \(\eta\) be a contravariant \(1\)-tensor on \(W\). Then I know that \(\omega\otimes\eta\) is in \(V\otimes W\), which (I think) is isomorphic to \(T^2\left(V\times W\right)\). Why does this not hold if \(V\) or \(W\) has an infinite basis?