Some Geometrical Properties Of Generalized Orlicz Spaces

Orlicz space is a class of specific Banach spaces. The rich model repository of Banach spaces is reserved by the study of properties of Orlicz spaces and criterion of properties in Orlicz spaces. Generalized Orlicz space is a extension of Orlicz space. The relationship between generalized Orlicz norm and Luxemburg norm in Orlicz spaces, the closed linear subspace of Orlicz spaces equipped with the generalized Orlicz norm and the distance of the point to the closed linear subspace, and the criterion of the uniform k-rotundity of Orlicz function spaces equipped with the generalized Orlicz norm are studied in the paper. This thesis consists of four parts. The main results are as follows:First of all, it is reviewed that research background, developing course and significance of Orlicz spaces theory and generalized Orlicz spaces during more than 80 years in the domestic and foreignSecond, the relationship between generalized Orlicz norm and Luxemburg norm in Orlicz spaces that are generated by N-function was discussed. We obtain a strict inequality and an important equivalent proposition.Third, we give a criterion of the closed linear subspace of Orlicz spaces equipped with the generalized Orlicz norm. And the distance of the point to the closed linear subspace were shown.As well known, uniform k-rotundity play important role in the geometry of Banach spaces. A good many scholars at home and abroad had done an in deep study of uniform k-rotund of Orlicz spaces. The criterion of the uniform k-rotund of the Orlicz function spaces generated by N-function whose derivative on the right is convex and equipped with the generalized Orlicz norm were presented.