Lemma 1

Let $u\in \mathscr{D}_B(E)$ and $\lambda = (\lambda_i)_{i\in I}$ the family of its eigenvalues. The family $\lambda$ is bounded and one has sup$_i|\lambda_i|=\|u\|$.

One has $|\lambda_i|\leqslant \|u\|$ for every $i$, whence sup$_{i\in I}|\lambda_i|\leqslant \|u\|$. Since moreover

$\|u(x)\|^2=\sum_{i\in I}|\lambda_i|^2|\langle e_i|x\rangle |^2\leqslant ($sup$_{i\in I}|\lambda_i|^2)\|x\|^2$

for every $x\in E$ (EVT, V, p. 22, prop. 5), it follows that $\|u\|=$ sup$|\lambda_i|$.