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 $\sup_j|d_j|\leq\|D\|<\infty$.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-326/1/c/i/solution