Past exam of the analysis course of the University of Oxford / 2010 / B / 5 / i

A bounded linear operator on a complex Hilbert space is self-adjoint if $T=T^{\ast }$, equivalently $⟨Tx,y⟩=⟨x,Ty⟩$ for every $x,y$.

If $Tx=0$, then $⟨x,Ty⟩=⟨Tx,y⟩=0$ for all $y$, so $x\in (\mathrm{Ran}T{)}^{\perp }$. Conversely, membership in that orthogonal complement gives $⟨Tx,y⟩=0$ for every $y$, and choosing $y=Tx$ yields $Tx=0$. Therefore