Testing compactness of linear operators

Let $(F_i)$ be a sequence of sets in a Banach space $X$. For what sequences does the condition $$\underset{i\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\underset{f_i\in F_i}{\sup }\mathrm{\parallel }T{f}_i{\mathrm{\parallel }}_Y=0$$ hold for every Banach space $Y$ and every compact operator $T:X\to Y$? We answer this question by giving sufficient (and necessary) criteria for such sequences. We illustrate the applicability of the criteria by examples from literature and by characterizing the $L^p\to L^p$ compactness of dyadic paraproducts on general measure spaces.