Proof that f : V* ⊗ W −→ HomK (V, W ) is an isomorphism

Question

If K is a field and V,W vector spaces how to prove that there is a function f : V* ⊗ W −→ HomK (V, W ) uniquely defined by the equation f (α ⊗ w)(v) = α(v)w and f is an isomorphism (if V and W are finite-dimensional)

Answer 1

Note that the dimension of $Hom_k(V,W)$ is $dim V.dim W$

Further, since $V$ is finite dimensional, hence $V^{*} \cong V$.

Thus, $V^{*} \otimes_K W \cong K^{(dim V)} \otimes K^{(dim W)} \cong K^{(dim V.dim W)}$

since tensoring with any module commutes with direct sums.

Thus, both the domain and the codomain of $f$ have the same dimension.

Now, fix bases of V and W. It is easy to see that $f$ is surjective.

Rank Nullity theorem says that $f$ is an isomorphism.

(Side Remark : Note that $f$ is natural in both $V$ and $W$)