I have the following map
$\pi: W \otimes V^* \rightarrow \textbf{Hom}(V,W)$
Where:
$\pi(w\otimes f)(v)= f(v)w$.
Both $V,W$ are vector spaces. And I need to prove that $\pi$ is well-defined and injective. What I'm trying to do is working with the following commutative diagram:
I have succesfully showed that $\tilde{\pi}$ is bilinear. But I am not sure if this is sufficient to say that $\pi$ is both inhective and well-defined.
Thanks for the help!