Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 326 / 1 / c / iv / Solution

For $d_j=1/j$, the equation $Du=f$ forces $u_j=j{f}_j$. It has a solution in ${\ell }^2$ exactly whenFor example, $f_j=1/j$ defines an element of ${\ell }^2$, but its forced preimage is the constant sequence $u_j=1$, which is not in ${\ell }^2$. Thus existence can fail. Since every $d_j$ is nonzero, $D$ has trivial kernel and any solution is unique.