= Solution
The exact solution is feasible because
$$
\|Au^\dagger-f^\delta\|_Y=\|f-f^\delta\|_Y\leq\delta\leq c\delta.
$$
The feasible set $E_\delta$ is convex and weakly closed. A minimizing sequence has bounded residual and bounded $J$; the coercivity assumption from part (a), applied to a fixed positive weighted objective, makes it bounded in $X$. Reflexivity gives a weakly convergent subsequence, and weak lower semicontinuity of the residual and $J$ keeps its limit feasible and minimizing.
Since $\widehat u_\delta$ minimizes $J$ over $E_\delta$,
$$
J(\widehat u_\delta)\leq J(u^\dagger).
$$
The <source condition in variational regularization> and feasibility then yield
$$
\begin{aligned}
D_J^{p^\dagger}(\widehat u_\delta,u^\dagger)
&\leq-\langle w^\dagger,A\widehat u_\delta-f\rangle\\
&\leq\|w^\dagger\|_{Y^*}
\left(\|A\widehat u_\delta-f^\delta\|_Y
+\|f^\delta-f\|_Y\right)\\
&\leq(c+1)\|w^\dagger\|_{Y^*}\delta.
\end{aligned}
$$
Thus the claimed constant is $\boxed{C=(c+1)\|w^\dagger\|_{Y^*}}$.
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