Some Geometrical Properties Of Generalized Orlicz Spaces

It is well known that monotonicity is important in various spaces. Various monotonicity properties are important in best approximation theory. The monotonicity and the locally uniformly monotonicity are important properties of Banach lattices. Roughly speaking, monotonicity properties of Banach lattices play similar role as rotundity properties of Banach spaces. A lot of people got many results through studied monotonicity in various spaces. Upper monotone, lower monotonicity and upper locally uniformly monotone are important properties of Banach space, they are closely linked. So, it is necessary to lauch the research relationships between them.In this paper, the monotonicity of Musielak-Orlicz spaces are discussed. And points of lower monotonicity, upper monotonicity and upper local uniform monotonicity in Musielak-Orlicz function spaces are discussed.First, the relationship of Musielak-Orlicz function equipped with Luxemburg norms, Orlicz norms and Amemiya norms have been given. We get the relationship of Luxemburg norms, p-Amemiya norms and Amemiya norms. According to the property of Musielak-Orlicz spaces equipped with Amemiya norms, we get the property of Musielak-Orlicz spaces equipped with p-Amemiya norms. LM.p is strictly monotone launched by M>0, M>0is launched by EM.p is strictly monotone.Second, according to the points of monotonicity in Musielak-Orlicz function spaces equipped with Orlicz norms. For Musielak-Orlicz function spaces equipped with p-Amemiya norms, we get the property of upper monotone, lower monotonicity, upper locally uniformly monotone. Further more, we get the necessary and sufficient conditions of a point x∈S((LMp)+) is lower monotone point.