Let $T$ be a linear operator in a vector space $V$ such as $T$ admits an ajoint. Prove that if $T^*T=0$ then $T=0$.

Question

Let $T$ be a linear operator in a vector space $V$ such as $T$ admits an ajoint. Prove that if $T^*T=0$ then $T=0$.

I tried to do the following:

If $v\in V$ then $0=\left< v,T^*T(v)\right>=\left< T(v),T(v)\right>$ then $||T(v)||=0$ and so $T(v)=0$ for every $v\in V$.

Is this correct?