 Solution

For fixed $y$, the map $x\mapsto ⟨Tx,y⟩$ is a bounded functional. The Riesz representation theorem gives a unique vector $T^{\ast }y$ representing it. Linearity, uniqueness, and $\Vert T^{\ast }y\Vert \le \Vert T\Vert \Vert y\Vert$ make $T^{\ast }$ the unique bounded adjoint operator.