For fixed $y$, the map $x\mapsto\langle Tx,y\rangle$ is a bounded functional. The <Riesz representation theorem> gives a unique vector $T^*y$ representing it. Linearity, uniqueness, and $\|T^*y\|\leq\|T\|\|y\|$ make $T^*$ the unique bounded <adjoint operator>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 22H / a / Solution

 Solution (source code)