Testing compactness of linear operators

Let $(F_i)$ be a sequence of sets in a Banach space $X$. For what sequences does the condition $$ \lim_{i\to \infty} \sup_{f_i\in F_i} \|Tf_i\|_Y=0 $$ 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 su