Isomorphism between vector space of all linear transformations

Question

If $V$ and $W$ are vector spaces, is there an isomorphism between $L(W, V^*)$ and $L(V,W^*)$? If so, how do I find this? ($L(V,W)$ is the vector space of linear transformation from $V$ to $W$). Any element in $L(W, V^*)$ maps elements in $w$ to a linear functional on $V$; I assume this linear functional can be further evaluated at elements in $V$, but am still very confused about how to construct an isomorphism if it exists.

Answer 1

Your intuition is correct. Take $f:W\rightarrow V^*$ linear. Then consider $\pi(f)(v)(w):=f(w)(v)$, where $\pi(f): V\rightarrow W^*$, and $\pi(f)(v)\in W^*$.

Intuitively we just exchanged the two inputs, and from this it should be clear that this "input permutation map" $\pi$ is the desired isomorphism.