Let $T:V \rightarrow V$ a linear map such that $Tv = <v,u>w$ then the adjoint linear map $T^*$ is $T^*v = <v,w>u. \forall u,v,w \in V $.
My professor defined the linear map $T^*$ as follow:
For any linear operator $T:V \rightarrow V$ on a finite-dimensional inner product space $V$, there is a unique operator $T^*$ on $V$ such that:
$<Tv,u> = <v,T^*u>$ for all $u,v \in V.$ $\hspace{4cm}(1)$
So my objective is arrives in $T^*v = <v,w>u. \forall u,v,w \in V $ using the equation $(1).$
Can you give me a hint about this? (I have tried substitute $Tv = <v,u>w$ in $(1)$ but isn't enough.)