 Solution

Part i gives $\Vert D\Vert \le B$. Conversely, choose indices $j_k$ with $|d_{j_k}|\to B$. Since $\Vert e_{j_k}{\Vert }_2=1$,Taking the limit proves $\boxed{\Vert D\Vert =B}$, whether or not the supremum is attained.