Let $H$ be a complex [Hilbert space](/page/Hilbert%20Space), and let $A:D(A)\subset H\to H$ and $B:D(B)\subset H\to H$ be [self-adjoint operators](/page/Self-Adjoint%20Operators) with pure point spectrum. Suppose their spectral projections commute, equivalently each spectral subspace of $A$ reduces $B$ and each spectral subspace of $B$ reduces $A$. Then $H$ has an [orthonormal basis](/page/Orthonormal%20Basis) consisting of common eigenvectors of $A$ and $B$.