Consider the entrywise $L_1$ norm on matrices, given by
$$\|M\|_1 = \sum_{i,j} |M_{i,j}|.$$
I'm looking for useful properties this norm might have. Is there anything we can say about $\|A \cdot B \|_1$, in terms of the matrices $A,B$? (e.g., upper-bound $\|A \cdot B\|_1$, based on some quantities about $A$ and $B$?)
$$ \| AB \|_1 = \sum_{i,j} |(AB)_{ij}| \le \sum_{i,j,k} |A_{ik}| |B_{kj}| \le \sum_{i,j,k,l} |A_{ik}| |B_{lj}| = \|A\|_1 \|B\|_1 $$
The following property holds for most matrix norms as long as $A$ and $B$ are square $$||AB||\leq||A||\cdot||B||$$ If the matrices are not square, then this submultiplicative property still holds for any vector-induced matrix norm.
