The $n$-dual space of the space of $p$-summable sequences

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 $\ell^p$ spaces, when these spaces are not only equipped with the usual norm but also with some $n$-norms.