The upper bound for supreme of product I am wondering if the following inequality for supreme norm holds? If not, can we have a upper bound for $||Ax||_{\infty}$? Thanks. $$ ||Ax||_{\infty} \leq||A||_{\infty}||x||_{\infty} $$ Where A is a matrix and x is a vector.

the inequality does hold.

$$ ||A||_{\infty} = sup_{x}\frac{||Ax||_{\infty}}{||x||_{\infty}} $$

by definition of $||A||_{\infty}$