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We show that the dual to any subspace of 
$c_{0}({\rm\Gamma})$ (
${\rm\Gamma}$ is an arbitrary index set) has the strongest possible quantitative version of the Schur property. Further, we establish a relationship between the quantitative Schur property and quantitative versions of the Dunford–Pettis property. Finally, we apply these results to show, in particular, that any subspace of the space of compact operators on 
$\ell _{p}$ (
$1<p<\infty$) with the Dunford–Pettis property automatically satisfies both its quantitative versions.
