If yes, how can you prove it?
We have $\langle T(v),w\rangle =\langle v,T^*(w)\rangle$. If $T(v)=0$, then what?
If yes, how can you prove it?
We have $\langle T(v),w\rangle =\langle v,T^*(w)\rangle$. If $T(v)=0$, then what?
Using the definition of adjoint, $ \langle 0^*v,w\rangle = \langle v,0w\rangle=\langle v,0\rangle =0. $ As this happens for every $w$, $0^*v=0$ for all $v$. i.e. $0^*=0$.
If $T(v)=0$ $\forall v$, this implies $\langle v, T^\ast(w)\rangle=0$ $\forall v$.
In particular, you can choose $v=T^\ast(w)$ and find $\langle T^\ast(w), T^\ast(w)\rangle=0$ and thus $T^\ast(w)=0$.