Is this a matrix operator norm?

Question

Is the max element-wise norm a matrix operator norm?

I know a matrix operator norm is defined by $$ |A|_p=\sup_{v≠0} \frac{|Av|_p}{|v|_p} $$

But how can I tell if the max norm is an operator norm?

Answer 1

Yes this is a norm.

Operator norms. Let $(X,\|\cdot\|_X)$, $(Y,\|\cdot\|_Y)$ be two normed spaces and $T:X\to Y$ a linear operator. $T$ is bounded, if there exists a $c>0$, such that $$ \|Tx\|_Y\le c\|x\|_X, \quad\text{for all $x\in X$}.\qquad (\star) $$ The infimum of all $c$ satisfying the above, also satisfies the above and it is the operator norm $\|T\|$. In fact $$ \|T\|=\inf\{c:c \,\,\,\text{satifies $(\star)$}\}=\sup_{x\ne 0}\frac{\|Tx\|_Y}{\|x\|_X} =\sup_{\|x\|_X=1}\|Tx\|_Y. $$ So an operator norm is one defined as $\sup_{x\ne 0}\frac{\|Tx\|_Y}{\|x\|_X}$.