Past exam of the linear algebra course of the University of Oxford / 2023 / 3 / a / ii / Solution

If $R$ and $S$ both satisfy the defining identity of an adjoint operator, thenfor every $u,w\in V$. Choosing $u=(R-S)w$ gives $\Vert (R-S)w{\Vert }^2=0$. The positive definiteness of the inner product forces $(R-S)w=0$ for every $w$, proving