$n$-Dual Spaces Associated to a Normed Space

[摘要] 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.

[学科分类] 公共、环境与职业健康