Department of Mathematics, Harbin University of Science and Technology, Xuefu Road 52, 150080 Harbin, China, yunancui@public.hr.hl.cn and Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Ul. Matejki 48/49, 60-769 Poznan, Poland, hudzik@amu.edu.pl and Institute of Mathematics, T. Kotarbinski Pedagogical University, Pl. Slowianski 9, 65-069 Zielona Gora, Poland, nowakmar@omega.im.wsp.zgora.pl and Institute of Mathematics, Poznan University of Technology, Piotrowo 3A, 60-965 Poznan, Poland, rplucien@math.put.poznan.pl
Abstract: 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 $(0_1)$ and $(\infty_1)$. All our criteria except Corollary 2.15 are given for arbitrary Orlicz functions. Criteria for smoothness of $l_\Phi^0$ in Corollary 2.15 are given for any finitely valued Orlicz function satisfying condition $(\infty_1)$, extending the respective result of [2] proved only for Orlicz functions vanishing only at zero.
Keywords: Orlicz sequence space, rotundity, local uniform rotundity, compact local uniform rotundity, property H, smoothness, copy of $l_\infty$
Classification (MSC2000): 46E30, 46E40, 46B20
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