For a map $T:V\rightarrow V$, we define the Hermitian adjoint to be the unique $T^*:V\rightarrow V$ such that $\langle Tu,v\rangle = \langle u, T^*v\rangle$.
There are two things I'm required to show in my Algebra course. That the Hermitian adjoint is unique, and that it exists. Showing uniqueness is easy but showing existence is not...
Here is an unusual proof that I was shown today. I hope someone can help me fill in the blanks.
If $V$ is a finite dimensional vector space, then define the following two maps
$\Gamma_1: L(V,V) \rightarrow V^* \otimes V^*$
$\Gamma_2: L(V,V) \rightarrow V^* \otimes V^*$
by $\Gamma_1(T)(u,v) = \langle u,Tv\rangle$ and $\Gamma_2=(T)(u,v) = \langle Tu,v\rangle$
Now we need to show $\Gamma_1$ and $\Gamma_2$ are isomorphisms (Need help here)
Then we can let $T^* = \Gamma_2^{-1} \circ \Gamma_1 T$
then $\langle v,Tu\rangle = \Gamma_1 T(v,u) = \Gamma_2 \circ(\Gamma_2^{-1} \circ \Gamma_1 T)(v,u) = \langle (\Gamma_2^{-1} \circ \Gamma_1 T)v,u\rangle $
Which finally would imply that $\Gamma_2^{-1} \circ \Gamma_1 T$ is a Hermitian adjoint, namely $T^*$
I would appreciate if someone could tell me the intuition behind defining $\Gamma_1$ and $\Gamma_2$ as maps to some tensor product. And also the reason why we need that they are isomorphisms (and how to prove that they are).
Thank you, this has been bothering me a bit today.