Diagonalizable Operator Bounded iff Value Set Bounded

Theorem
Let $H$ be a Hilbert space.

Let $A: H \to H$ be a diagonalizable operator.

Let $\left({\alpha_e}\right)_{e \in E}$ be the value set of $A$, with respect to a suitable basis $E$ for $H$.

Then $A$ is bounded iff $\left({\alpha_e}\right)_{e \in E}$ is, i.e., iff:


 * $\exists M \in \R: \forall e \in E: \left\vert{\alpha_e}\right\vert \le M$