This paper is a continuation of investigations of n-irmer product spaces given in [5, 6, 7] and an extension of results given in [3] 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.
In the theory of normed spaces, we have the concept of bounded linear functionals and dual spaces. Now, given an n-normed space, we are interested in bounded multilinear n-functionals and n-dual spaces. The concept of bounded multilinear n-functionals on an n-normed space was initially intoduced by White (1969), and studied further by Batkunde et al., and Gozali et al. (2010). In this paper, we revisit the definition of bounded multilinear n-functionals, introduce the concept of n-dual spaces, and then determine the n-dual spaces of ℓ p spaces, when these spaces are not only equipped with the usual norm but also with some n-norms.