How to prove that the infinity norm of a matrix is the max of row sum? I know how to prove that the 1-norm of a matrix is the max of the column sum, but not sure how to prove that the inf-norm is the max of the row sum.
Any suggestion?
Thanks

Hint: Suppose that $x$ with entries $x_j$ satisfies $\|x\|_\infty = 1$.  Then the $j$th entry of $Mx$ is given by
$$
\sum_{k=1}^m M_{jk}x_k
$$
We note that
$$
\left|\sum_{k=1}^m M_{jk}x_k\right| \leq
\sum_{k=1}^m \left|M_{jk}x_k\right| \leq
\sum_{k=1}^m \left|M_{jk}\right| \|x\|_{\infty}
$$
Then, for any $j$, we can consider the vector $x$ whose entries are given by
$$
x_k = \frac{|M_{jk}|}{M_{jk}}
$$
