$n$-Dual Spaces Associated to a Normed Space

Document Type : Original Article

Author

Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand

Abstract

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 $% \left\Vert \cdot ,\ldots ,\cdot \right\Vert _{G}$ is the Gähler $n$-norm. We then investigate the relationship between the $n$-dual space of $% \left( X,\left\Vert \cdot \right\Vert \right)$ and the $n$-dual space of $% \left( X,\left\Vert \cdot,\ldots ,\cdot \right\Vert _{G}\right)$. 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.

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