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Monotonicities in Orlicz spaces equipped with Mazur-Orlicz \(F\)-norm. (English) Zbl 1462.46011

Summary: Some basic properties in Orlicz spaces and Orlicz sequence spaces that are generated by monotone function equipped with the Mazur-Orlicz \(F\)-norm are studied in this paper. We give some relationships between the modulus and the Mazur-Orlicz \(F\)-norm. We obtain an interesting result that the norm of an element in line segments is formed by two elements on the unit sphere less than or equal to 1 if and only if that the monotone function is a convex function. The criterion that Orlicz spaces and Orlicz sequence spaces that are generated by monotone function equipped with the Mazur-Orlicz \(F\)-norm are strictly monotone or lower locally uniform monotone is presented.

MSC:

46B04 Isometric theory of Banach spaces
46B20 Geometry and structure of normed linear spaces
46B25 Classical Banach spaces in the general theory
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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