Concerning the Linearity of the adjoint of $T$

Question

While proving the Linearity of the adjoint of the linear map $T:V\to W$the authour of my text book writes that \begin{align}\langle v,T^*(w_1+w_2)\rangle&= \langle Tv,w_1+w_2\rangle\\& = \langle Tv,w_1\rangle+\langle Tv,w_2\rangle\\& = \langle v,T^*w_1\rangle+\langle v,T^*w_2\rangle\\& = \langle v,T^*w_1+T^*w_2\rangle\end{align}

But he immedietly follows this with the declartation that $T^*(w_1+w_2) = T^*w_1+T^*w_2$. How is this write?

Answer 1

If $w$ and $w'$ are vectors such that$$(\forall v\in V):\langle v,w\rangle=\langle v,w'\rangle,$$then $w=w'$. This is so because\begin{align}\|w-w'\|^2&=\langle w-w',w-w'\rangle\\&=\langle w-w',w\rangle-\langle w-w',w'\rangle\\&=0.\end{align}