= Schur test
{c}
{title2=$\|A\|_{\ell^2\to\ell^2}\le\sqrt{RC}$}
{wiki}
For a matrix with nonnegative entries, row sums at most $R$ and column sums at most $C$ imply an $\ell^2$ operator <norm> at most $\sqrt{RC}$. Indeed the <Cauchy-Schwarz inequality> gives $|\sum_j A_{ij}x_j|^2\le(\sum_jA_{ij})\sum_jA_{ij}|x_j|^2$. Summing over $i$ bounds the result by $RC\sum_j|x_j|^2$. Integral kernels and positive weight functions give analogous forms. The symmetric case has equal row and column bounds.
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