linear operators and inner product

Question

Let $R,T$ are linear operators on $H$. If $(Rv,v)=(Tv,v)$ for all $v$ then $R=T$. How can I do that? Actually I did this directly after some calculations but I guess it is wrong since under the question there is a hint as: check that $(Rv,u)=(Tv,u)$ for all $v,u \in H$. Why do we have this hint I cannot understand.

Answer 1

Because if$$\tag{1}(\forall u,v\in H):\bigl\langle R(v),u\bigr\rangle=\bigl\langle T(v),u\bigr\rangle,$$then$$(\forall u,v\in H):\bigl\langle(R-T)(v),u\bigr\rangle=0$$Therefore$$(\forall v\in H):\bigl\langle(R-T)(v),(R-T)(v)\bigr\rangle=0,$$which means that$$(\forall v\in H):\bigl\|R(v)-T(v)\bigr\|=0.$$Therefore, $R=T$.

The fact that $(1)$ holds can be proved by polarization.