I stumbled on the following simple matrix norm, which I haven't seen elsewhere. I wonder if it is well known, has a name, and has been studied elsewhere. The definition of this norm for a matrix A with real entries is as follows:
$||A|| = \sum_i ||A_i||_\infty$,
where $A_i$ is the $i$'th column of A. Obviously, an analogous norm may be defined for $p \neq \infty$.