Let $A \subseteq \mathcal{L}(H)$ be a unital self-adjoint norm-closed algebra, and let $M=\overline{A}^{\mathrm{SOT}}$. Then the unit ball of $A$ is strongly dense in the unit ball of $M$. The same statement holds for self-adjoint parts: the self-adjoint unit ball of $A$ is strongly dense in the self-adjoint unit ball of $M$.