$n$-Dual Spaces Associated to a Normed Space

Abstract

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