The monotonicity of Orlicz–Lorentz spaces equipped with the F-norm

In this paper, we introduce a new F-normed space, namely Orlicz–Lorentz spaces equipped with the Mazur–Orlicz F-norm. Some basic properties in Orlicz–Lorentz spaces equipped with the Mazur–Orlicz F-norm are given. We find a tool to study the geometry property of Orlicz–Lorentz function spaces, the necessary and sufficient conditions for strict monotonicity, lower local uniform monotonicity, upper local uniform monotonicity in Orlicz–Lorentz spaces endowed with the Mazur–Orlicz F-norm are obtained without any assumptions. The tool also can simplify the proof of the corresponding results of Orlicz–Lorentz spaces equipped with the Luxemburg norm without condition (+).