I'm trying to prove that the norm $\| \circ \| = n \|A\|_{l_\infty}$ is a matrix norm. Reminder that the $l_\infty$-norm of a matrix $A \in \mathbb{R}^{n \times n }$ is defined as: \begin{equation} \|A\|_{l_\infty} = \max \left\{\left|a_{i j}\right|: i, j=1,2, \ldots, n\right\} \end{equation}
What I need to prove is that the sub-multiplicative property holds (the other four conditions can be proven fairly simply), that is, for every $A,B \in \mathbb{R}^{n \times n}$ the following is true: \begin{align} \|A B\| &\leq \|A\| \|B\| \Longleftrightarrow \\ n \|A B\|_{l_\infty} &\leq \left(n\|A\|_{l_\infty} \right) \left(n|B\|_{l_\infty}\right) \end{align}
My effort so far: Knowing that $(AB)_{i j} = \sum_k^n a_{i k} b_{k j}$, \begin{align*} \|AB\| &= n \|AB\|_{l_\infty} = n \max \left\{ \left| \sum_k^n a_{i k} b_{k j} \right|: i,j = 1, \ldots, n \right\} \leq n \max \left\{ \sum_k^n \left| a_{i k} b_{k j} \right|: i,j = 1, \ldots, n \right\} \\ &= n \max \left\{ \sum_k^n \left| a_{i k} | |b_{k j} \right|: i,j = 1, \ldots, n \right\} \leq n \max \left\{ \sum_k^n \left| a_{i k} \right|: i = 1, \ldots, n \right\} \max \left\{ \sum_k^n \left| b_{k j} \right|: j = 1, \ldots, n \right\} \end{align*}
Don't know if the last inequality holds, I do not know how to proceed.
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