Let $H$ and $K$ be complex Hilbert spaces whose inner products $(\cdot,\cdot)_H$ and $(\cdot,\cdot)_K$ are linear in the first argument. Let $\mathcal{L}(H,K)$ denote the [Banach space](/page/Banach%20Space) of bounded linear maps from $H$ to $K$, let $\mathcal{L}(K,H)$ denote the Banach space of bounded linear maps from $K$ to $H$, let $\mathcal{L}(H):=\mathcal{L}(H,H)$, and let $T \in \mathcal{L}(H,K)$ be compact with Hilbert-space adjoint $T^*\in\mathcal{L}(K,H)$. Then there exist an index set $J$, where either $J=\varnothing$, $J=\{1,\dots,m\}$ for some $m\in\mathbb N$, or $J=\mathbb N$, positive numbers $(s_j)_{j\in J}$ listed with multiplicity and ordered so that $s_1\ge s_2\ge \cdots$ when $J\ne\varnothing$, an orthonormal family $(v_j)_{j\in J}$ in $(\ker T)^\perp\subset H$, and an orthonormal family $(u_j)_{j\in J}$ in $\overline{\operatorname{Range}(T)}\subset K$, such that

\begin{align*} \overline{\operatorname{span}}\{v_j:j\in J\}=(\ker T)^\perp \end{align*}

and

\begin{align*} \overline{\operatorname{span}}\{u_j:j\in J\}=\overline{\operatorname{Range}(T)}. \end{align*}

For every $j\in J$,

\begin{align*} Tv_j=s_j u_j \end{align*}

and

\begin{align*} T^*u_j=s_j v_j. \end{align*}

For every $x\in H$,

\begin{align*} Tx=\sum_{j\in J}s_j(x,v_j)_H u_j, \end{align*}

where the sum is interpreted as $0$ when $J=\varnothing$ and as a finite sum when $J=\{1,\dots,m\}$. If $J=\mathbb N$, then $s_j\to 0$, the series converges in $K$ for every $x\in H$, and the finite-rank operators $T_n:H\to K$ defined by

\begin{align*} T_nx=\sum_{j=1}^n s_j(x,v_j)_H u_j \end{align*}

satisfy

\begin{align*} \|T-T_n\|_{\mathcal{L}(H,K)}\to 0. \end{align*}