Given a matrix $X$, define the $\ell_{1,\infty}$ norm:
$$ \|X\|_{1,\infty} = \sum_i \max_j |X_{ij}| $$
Can I get its dual norm with the following definition of dual norm?
$$ \|Z\|_{1,\infty}^*: = \max_X \left\{ \left<Z,X\right>, \text{s.t.} \|X\|_{1,\infty} \leq 1 \right\}.$$
Does its dual norm have an explicit expression? Say, like the dual norm of the $\ell_1$ norm is the $\ell_\infty$ norm?