= Row-column bound for the matrix 2-norm
{title2=$\|B\|_2\leq\sqrt{\|B\|_1\|B\|_\infty}$}
For each row, the weighted <Cauchy-Schwarz inequality> gives $|(Bx)_i|^2\leq(\sum_j|b_{ij}|)(\sum_j|b_{ij}||x_j|^2)$. Summing over rows bounds the result by the maximum absolute row sum times the maximum absolute column sum times $\|x\|_2^2$. Taking square roots and the supremum proves the displayed <induced matrix norm> inequality. It gives a Euclidean estimate directly from entries without computing <singular values>.
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