$n$-inner product spaces and projections

summary:This paper is a continuation of investigations of $n$-inner product spaces given in \cite{five,six,seven} and an extension of results given in \cite{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