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Journal of Convex Analysis 06 (1999), No. 1, 091--114
Copyright Heldermann Verlag 1999

Some Geometric Properties in Orlicz Sequence Spaces Equipped with Orlicz Norm

Yunan Cui
Dept. of Mathematics, Harbin University of Science and Technology, Xuefu Road 52, 150080 Harbin, P. R. China

Henryk Hudzik
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, U. Matejki 48/49, 60-769 Pozna´n, Poland

Marian Nowak
Inst. of Mathematics, T. Kotarbinski Pedagogical University, Pl. Slowianski 9, 65-069 Zielona Góra, Poland

Ryszard Pluciennik
Inst. of Mathematics, University of Technology, Piotrowo 3A, 60-965 Poznan, Poland

It is proved that for any reflexive Banach space X, both X and X* are CLUR if and only if both X and X* have property H. Criteria for rotundity, local uniform rotundity, compact local uniform rotundity and property H in Orlicz sequence spaces equipped with the Orlicz norm are given. Criteria for property H, rotundity and LUR were already known in the literature only for finitely valued Orlicz functions which vanish only at zero and are N-functions (i.e. they satisfy conditions (01) and (infty1). All our criteria except Corollary 2.15 are given for arbitrary Orlicz functions. Criteria for smoothness of lΦ0 in Corollary 2.15 are given for any finitely valued Orlicz function satisfying condition (infty1), extending the respective result of Chen, Hudzik and Wisla proved only for Orlicz functions vanishing only at zero.

Keywords: Orlicz sequence space, compact local uniform rotundity, property H, smoothness, copy of l-sub-infinity.

MSC: 46E30; 46E40, 46B20

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