Let $H$ be a [Hilbert space](/page/Hilbert%20Space), and let $p,q\in \mathcal{L}(H)$ be projections, meaning self-adjoint idempotents. Then $p$ and $q$ are Murray-von Neumann equivalent in $\mathcal{L}(H)$ if and only if the Hilbert spaces $pH$ and $qH$ have the same Hilbert-space dimension, equivalently the same cardinality of an [orthonormal basis](/page/Orthonormal%20Basis).