Example 1

Let I and J be two sets. By the proof given above, in order that a mapping $u$ from $\ell^2(I)$ into $\ell^2(J)$ be a Hilbert-Schmidt mapping, it is necessary and sufficient that there exists a matrix $(a_{ji})$ in $\ell^2(J \times I)$ such that $u(\xi)_j = \sum_{i \in I} a_{ji} \xi_i$ for all $\xi = (\xi_i)$ in $\ell^2(I)$.