This paper is a continuation of investigations of $n$-inner product spaces given in [five, six, seven] and an extension of results given in [three] to arbitrary natural $n$. It concerns families of projections of a given linear space $L$ onto its $n$-dimensional subspaces and shows that between these families and $n$-inner products there exist interesting close relations.

We present a new method for studying the numerical radius of bounded operators on Hilbert $C^{*}$-modules. Our method enables us to obtain some new results and generalize some known theorems for bounded operators on Hilbert spaces to bounded adjointable operators on Hilbert $C^{*}$-module spaces.