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 of norms α on the sequence space , called permutation symmetric gauge norms, which properly contains the classical class of . For each , we define the generalized sequence space , which is proved to be a Banach space, and then we obtain a characterization of in terms of the projection onto . For the duality, the expected results in the classical sequence spaces () 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 is not a reflexive space, which is different from the usual sequence spaces.