Corollary 1 (Hilbert–Schmidt)

Let $k\in \mathscr{L}^2(X\times Y)$. The linear mapping $u_k$ is compact from $L^2(X)$ into $L^2(Y)$.

By prop. 5, one has $k\in \mathscr{N}^{2,2}(X\times Y)$, hence the linear mapping $u_k$ is defined (prop. 3). One has $k\in \mathscr{L}^{2,2}(X\times Y)$ (prop. 6, a)), hence $u_k$ is compact (prop. 7).