Let $l\in \mathcal{H}^*$ be a real linear functional on a hilbert space $\mathcal{H}$.
I want to compute the adjoint $l^*$ in $\mathcal{H}$. Using the Riesz-Fischer representation, the unique existence of a $h_0\in \mathcal{H}$ with $l(h)=\langle h,h_0\rangle $ for every $h\in\mathcal{H}$ is guaranteed.
Hence I obtain with the definition of the adjoint operator via inner product,
$\langle \langle h,h_0\rangle u, v\rangle=\langle h,h_0\rangle\langle u, v\rangle =\langle u,\langle h,h_0\rangle v\rangle$ for all $v,u\in \mathcal{H}$.
I have my doubts that this is correct.