Problem: Let $V, W$ be two finite dimensional vector spaces. Show that $L(W) \otimes L(V) \cong L(W \otimes V)$ where $L(X)$ denotes the vector space of all linear transformations $X \to X$.
Context: I have a map $N: L(V) \to L(V)$ which I am supposed to show is completely positive. For that I need to show that $Id_W \otimes N: L(W \otimes V) \to L(W \otimes V)$ is a positive map. I showed that for any PSD (Positive Semi-Definite) operator $\sigma \otimes \rho : W \otimes V \to W \otimes V$, we get $Id_W \otimes N (\sigma \otimes \rho) = \sigma \otimes N(\rho)$ is also a PSD operator. For the general case, I said let $\theta: W \otimes V \to W \otimes V$ be any PSD operator and I decomposed it as sum of elementary tensors as $\theta = \sum_{i,j} c_{ij} (\sigma_i \otimes \rho_j)$ and using this I managed to showed that $Id_W \otimes N(\theta)$ is also PSD.
But then I realized that I am assuming $L(W \otimes V) \cong L(W) \otimes L(V)$ when decomposing $\theta$ as sum of elementary tensors. So I tried to explicitly construct this isomorphism (for my own satisfaction).
Attempt: First I checked that dimension of both the objects is $(\text{dim }V.\text{dim }W)^2$ so they should be isomorphic right?
Define a map $L(W) \otimes L(V) \to L(W \otimes V)$ as $f \otimes g \mapsto h$ where $h(w \otimes v) = f(w) \otimes g(v)$ and extend $h$ linearly.
Having defined the above map on the elementary tensors, we can extend it again linearly to obtain a homomorphism between $L(W) \otimes L(V)$ and $L(W \otimes V)$. Now I will attempt to show that it is injective. Suffices to show that $\ker = \{0\}$.
Fix a basis $\{f_i\} \subset L(W)$ and $\{g_j\} \subset L(V)$. Let $\sum_{i,j} c_{ij} (f_i \otimes g_j) \in \ker \implies \sum_{i,j} c_{ij} (f_i(w) \otimes g_j(v)) = 0$, $\forall w \in W, v \in V$. Suppose some $c_{ij} \neq 0$ then I claim either $f_i$ or $g_j$ are identically $0$. If not, then find $w,v$ such that $c_{ij} (f_i(w) \otimes g_j(v)) \neq 0$. I am not sure how to proceed from here.
Then I instead tried showing that the map will be surjective.
My idea is something like this: first fix a basis $\{w_i\} \subset W$ and $\{v_j\} \subseteq V$, then for $h \in L(W \otimes V)$, and for $x = w_i \otimes v_j \in V \otimes W$,
$h(x) = \sum_{i,j} c_{ij} (w_i \otimes v_j) = \sum_{i,j} c_{ij} h_{ij}(x)$ where $h_{ij}(w_i \otimes v_j) = w_i \otimes v_j$ and $h_{ij}(w_k \otimes v_l) = 0$ otherwise.
I can see that $f_i \otimes g_j \mapsto h_{ij}$ where $f(w_i) = w_i$ and $f(w_k) = 0$ otherwise, $g(v_j) = v_j$ and $g(v_l) = 0$ otherwise and so it should be that $c_{ij}(f_i \otimes g_j) \mapsto h$. But these constants $c_{ij}$ only depend on $x (= w_i \otimes v_j)$ which I began with, so something is wrong here as well, I guess?
I think I have muddled myself too much in notation/indices which is why am not able to make progress. Is there a better way to do this? Any help would be appreciated.