Solution
= Solution
If $B=\sup_j|d_j|<\infty$, then for every $f\in\ell^2$,
$$
\|Df\|_2^2=\sum_j|d_j|^2|f_j|^2
\leq B^2\|f\|_2^2,
$$
so the <diagonal operator on sequence space> is bounded. Conversely, for the standard unit vector $e_j$,
$$
|d_j|=\|De_j\|_2\leq\|D\|,
$$
so boundedness of $D$ implies $\sup_j|d_j|\leq\|D\|<\infty$.