Uniformly Nonsquare in Orlicz Space Equipped with the Mazur-Orlicz F-Norm

Abstract

The definition of uniformly nonsquareness in Banach spaces is extended to F-normed spaces. Most of the results from this paper concern (uniformly) nonsquareness in the sense of James or in the sense of Schäffer in Orlicz spaces equipped with the Mazur-Orlicz F-norm. It is well known that uniform nonsquareness in the sense of Schäffer and in the sense of James are equivalent in Banach spaces. In this paper, we found that uniform nonsquareness in the sense of James and in the sense of Schäffer are not equivalent for F-normed spaces. Criteria for Orlicz spaces equipped with the Mazur-Orlicz F-norm to be nonsquare and uniformly nonsquare in the sense of James or in the sense of Schäffer are given.