Orlicz–Lorentz function spaces equipped with the Orlicz norm

We investigate Orlicz–Lorentz function spaces equipped with the Orlicz norm generated by any Orlicz function and any non-increasing weight function. As far as we know, this is the first time such a general research is conducted. First we show some basic properties of the Orlicz norm, including its equality to the Amemiya norm, the problem of attainability of infimum in the definition of the Amemiya norm and a formula for the norm of a characteristic function. Then we find criteria for the order continuity and strict monotonicity and we study the problem of existence of order linearly isometric copies of l∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l^{\infty }$$\end{document}.