  Let $A$ be a $C^*$-algebra, $H$ be a Hilbert $A$-module and $K(H)$ be the
closure of the set of finite rank module maps. We show that the $W^*$-algebra
of all bounded $A^{**}$-module maps on the smallest self-dual Hilbert
$A^{**}$-module containing $H$ is isomorphic to $K(H)^{**}$ as $W^*$-algebras.
We also show that the unit ball of $H$ is closed in $H^\sharp,$ the dual of
$H,$ in an $A$-weak topology of $H^\sharp$ as well as dense in the unit ball of
$H^\sharp$ in a weak*-topology and some versions of Kaplansky density theorem
for Hilbert $C^*$-modules.
