 Solution

The proposed stability property is false. Set $u^{\ast }=0$, $f^{\ast }=0$, and take $u_n=e_n$. Thenwhile $\Vert u_n-u^{\ast }{\Vert }_2=1$ for every $n$. Hence the inverse is not continuous on its range and the recovery problem is not stable with respect to perturbations. This is the standard unbounded inverse on a nonclosed operator range.