This paper is concerned with the space $mathcal{K}_{w^*}(X^*,Y)$ of $weak^*$ to weak continuous compact operators from the dual space $X^*$ of a Banach space X to a Banach space Y. We show that if $X^*$ or $Y^*$ has the Radon-Nikod$acute{y}$m property, $mathcal{C}$ is a convex subset of $mathcal{K}_{w^*}(X^*,Y)$ with $0{in}mathcal{C}$ and T is a bounded linear operator from $X^*$ into Y, then $T{in}ar{mathcal{C}}^{{ au}_{mathcal{c}}}$ if and only if $T{in}ar{{S{in}mathcal{C}:{parallel}S{parallel}{leq}{parallel}T{parallel}}}^{{ au}_{mathcal{c}}}$, where ${ au}_{mathcal{c}}$ is the topology of uniform convergence on each compact subset of X, moreover, if $T{in}mathcal{K}_{w^*}(X^*, Y)$, here $mathcal{C}$ need not to contain 0, then $T{in}ar{mathcal{C}}^{{ au}_{mathcal{c}}}$ if and only if $T{in}ar{mathcal{C}}$ in the topology of the operator norm. Some properties of $mathcal{K}_{w^*}(X^*,Y)$ are presented.