H-property Of Generalized Orlicz Spaces

Generalized Orlicz space is a extension of Orlicz space and it is a class ofspecific Banach spaces. The research of the properties and criteria of propertiesof generalizd Orlicz space offers us many kinds of models of Banach space and itmakes easy for further research. In this paper, we research the Orlicz spaceequipped with the generalized Orlicz norm, the relationship between the H-point and the extreme point and the criterion of the H-property of Orlicz spaceequipped with the generalizd Orlicz norm. There are three parts in this paper. Theimportant consequece of this paper are summarized as following:First, this paper reviewed that research background, developing course andsignificance of Orlicz space theory and generalized Orlicz space during morethan80years in the domestic and foreign.Second, we discuss the relationship between H-point and extreme point inthe Orlicz which space equipped with the generalizd orlicz norm and we alsoconsider the relationship between the Orlicz function M which satisfies2—condition and H-point on the Orlicz space surface which equipped with thegeneralized Orlicz norm.We obtain one lemma and two theorems.At last, we discuss the sufficient and necessary criteria of the H-propertyof Orlicz space equipped with the generalized orlicz norm. We discuss therelationship between the Orlicz space equipped with the generalized Orlicz normhas H-property, the Orlicz function satisfies2—condition and the conditionof strict convexity. During the process of proof, we also research the relationshipbetween the boundary of sequence and weak convergence, and the relationshipbetween norm convergence and convergence in measure by Orlicz functionsatisfies2—condition.