$V^*\otimes W^*\cong(V\otimes W)^*$ when one factor is finite-dimensional

Let \(k\) be a field and let \(V, W\) be \(k\)-vector spaces, at least one of which is finite-dimensional. The \(k\)-linear map \[ \begin{aligned} \dualof{V} \tensor_k \dualof{W} &\to \dualof{(V \tensor_k W)} \\ v \tensor w &\mapsto \big(x \tensor y \mapsto v(x)\, w(y)\big) \end{aligned} \] is an isomorphism.