Operator norm of an infinite diagonal matrix

Question

Say, $D=\operatorname{diag}(d_1,d_2,d_3,\ldots )$ is an infinite diagonal matrix. I was able to prove that

$$\|D\|\leq \sup_{k\geq 1}\{|d_k|\}.$$

Question: Does equality hold?

I can see how trivial the equality is in finite dimension because you can pick a vector that attains the sup(max in that case). I am not quite sure about the infinite dimension though. Any hint is appreciated! Or if it's true at all!

Answer 1

Equality holds! Let $\{e_j\}$ be an orthonormal basis of the given Hilbert space.

Then $|d_k|=|\langle De_k,e_k\rangle|\leq ||D||$ for all $k$ by Cauchy-Schwarz.