Recently I've been learning Matrix perturbation theory. Many theorems deal with perturbation bound in $l_2$ norm, or more generally, unitarily invariant norm, like F-norm or others(Like Davis-Kahan's Sin$\Theta$ theorem). Here is my question, is there a direct $l_1$ bound for the eigenvectors under the assumptions that the original matrix is perturbed in terms of $l_1$ norm? Note the result of $l_2$ norm shouldn't be used, since there's equivalence in these norms.
Appreciate any hint provided!