$n$-dual spaces associated to a normed space

For a real normed space $X$, we study the $n$-dual space of $\left(X,\left\Vert \cdot \right\Vert \right)$ and show that the space is a Banach space. Meanwhile, for a real normed space $X$ of dimension $d\geq n$ which satisfies property (G), we discuss the $n$-dual space of $\left(X,\left\Vert \cdot,\ldots,\cdot \right\Vert _{G}\right)$, where $$ is the G\"ahler $n$% -norm. We then investigate the relationship between the $n$-dual space of $$ and the $n$-dual space of $$. We use this relationship to determine the $n$-dual space of $\left(X,\left\Vert \cdot,\ldots,\cdot \right\Vert _{G}\right) ~$and show that the space is also a Banach space.