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.
Highlights
Introduction and PreliminariesAs well known, Orlicz space is a generalization of classical Lebesgue space
In 2018, Cui et al discussed the monotonicity of Orlicz space that generated by the monotone continuous function equipped with Mazur-Orlicz F-norm
In 2020, Bai et al given criteria that Orlicz spaces that generated by the monotone function equipped with Mazur-Orlicz F-norm have strictly monotonicity and upper locally uniform monotonicity, and they get the conclusion that kλx + ð1 − λÞykF ≤ 1 for each x, y ∈ SðLΦðμÞÞ and λ ∈ ð0, 1Þ if and only if Φ is convex function on R
Summary
Introduction and PreliminariesAs well known, Orlicz space is a generalization of classical Lebesgue space. Inspired B-convex spaces, in 1964, the definition of uniformly nonsquare in normed linear space was introduced by James (see [4]). In 1976, the concept of uniformly nonsquare in normed linear space was introduced by Schäffer (see [5]).
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