  The concept of a $ C $*-algebra-valued metric space was introduced in 2014.
It is a generalization of a metric space by replacing the set of real numbers
by a $ C $*-algebra. In this paper, we show that $ C $*-algebra-valued metric
spaces are cone metric spaces in some point of view which is useful to extend
results of the cone case to $ C $*-algebra-valued metric spaces. Then the
completion theorem of $ C $*-algebra-valued metric spaces is obtained.
Moreover, the completion theorem of $ C $*-algebra-valued normed spaces is
verified and the connection with Hilbert $ C $*-modules, generalized inner
product spaces, is also provided.
