Remark

Let $N\in L^2(X\times Y)$. By corollary 2 of IV, p. 150, there exist a countable set I, orthonormal families $(f_i)_{i\in I}$ in $L^2(X)$ and $(g_i)_{i\in I}$ in $L^2$(Y), as well as a family $(\alpha_i)_{i\in I}$ in $\mathbf{R}^*_+$, such that

for all $f\in L^2$(X), where the series converges in $L^2(Y)$. By cor. 4 of IV, p. 165 and the above corollary, one has

$\sum_{i\in I}\alpha^2_i=$ Tr($u^*_Nu_N$) $=\|u_N\|^2_2=\int_{X\times Y}|N(x, y)|^2d(\mu\otimes \nu )(x, y)$.

Let us further set $h_{i,j}=\overline{f}_i\otimes g_j\in L^2(X\times Y)$ for all $(i, j)\in I\times I$. One then has

in $L^2(X\times Y)$.

In fact, let $(f_j)_{j\in J}$ and $(g_k)_{k\in K}$ be orthonormal bases of $L^2(X)$ and $L^2$(Y), respectively, extending the families $(f_i)_{i\in I}$ and $(g_i)_{i\in I}$. Set $h_{j,k}=\overline{f}_j\otimes g_k$ for all $(j, k)\in J\times K$. By lemma 10, the family $(h_{j,k})_{(j,k)\in J\times K}$ is an orthonormal basis of $L^2(X\times Y)$. One has $\langle h_{j,k}|N\rangle =\langle g_k|u_N(f_j)\rangle$ for all $(j, k)\in J\times K$. If $j \notin I$, this quantity is zero. If $j\in I$, it is equal to $\alpha_j\langle g_k|g_j\rangle$, hence is zero unless $k=j$, in which case $\langle h_{j,j}|N\rangle =\alpha_j$. Consequently