Let $H$ and $K$ be complex Hilbert spaces, let $T\in\mathcal{L}(H,K)$ be compact, and let its non-zero singular values and singular vectors be indexed as in the singular-value decomposition theorem:
with corresponding orthonormal singular systems $(v_j)_{j\in J}\subset H$ and $(u_j)_{j\in J}\subset K$. For $n\in\mathbb N$, define the Schmidt truncation $T_n\in\mathcal{L}(H,K)$ by