Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 22I / c / Solution

The adjoint is the unique $T^{\ast }$ satisfying $⟨Tx,y⟩=⟨x,T^{\ast }y⟩$. Apply Riesz to $x\mapsto ⟨Tx,y⟩$. The preceding norm calculation gives $\Vert T^{\ast }\Vert \le \Vert T\Vert$, while $\Vert Tx\Vert ={\sup }_{\Vert y\Vert =1}|⟨x,T^{\ast }y⟩|$ gives the reverse inequality.