  We show that the convergence rate of $\ell^1$-regularization for linear
ill-posed equations is always $O(\delta)$ if the exact solution is sparse and
if the considered operator is injective and weak*-to-weak continuous. Under the
same assumptions convergence rates in case of non-sparse solutions are proven.
The results base on the fact that certain source-type conditions used in the
literature for proving convergence rates are automatically satisfied.
