The generalized Banach sequence spaces and their dual spaces based on permutation symmetric gauge norms

Abstract

In the present paper, we introduce and study a class $\mathcal{N}$ of norms α on the sequence space ${\mathbb{C}}_{00}$, called permutation symmetric gauge norms, which properly contains the classical class of $\{\Vert \cdot {\Vert }_p:1\le p\le \mathrm{\infty }\}$. For each $\alpha \in \mathcal{N}$, we define the generalized sequence space ${\ell }^{\alpha }$, which is proved to be a Banach space, and then we obtain a characterization of ${\ell }^{\alpha }$ in terms of the projection onto ${\mathbb{C}}_{00}$. For the duality, the expected results in the classical sequence spaces ${\ell }^p$ ($1\le p\le \mathrm{\infty }$) are still valid in these new settings, including the characterization of dual space and the existence of Hölder's inequality. It is worthy pointing out that the generalized sequence space ${\ell }^{\alpha }$ is not a reflexive space, which is different from the usual sequence spaces.

Keywords:

• Sequence space
• permutation symmetric gauge norm
• completeness
• dual space
• Hölder's inequality

2010 Mathematics Subject Classifications:

• Primary: 40A05
• 46B10
• 46B45
• Secondary:46B50

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was completed with the support of the National Natural Science Foundation of China [grant number 11971283] and the Central University Basic Research Fund of China [grant numbers GK201903008 and GK202107014].