Let $H$ be a nonzero complex [Hilbert space](/page/Hilbert%20Space), let $\mathcal{L}(H)$ denote the C*-algebra of bounded linear operators $H\to H$ equipped with the operator norm and adjoint, and let $K(H)\subseteq \mathcal{L}(H)$ denote the C*-subalgebra of compact operators on $H$. Then $K(H)$ is simple; that is, every norm-closed two-sided ideal $I\trianglelefteq K(H)$ satisfies either $I=\{0\}$ or $I=K(H)$.