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# Zeitschrift für Analysis und ihre Anwendungen

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**Volume 17, Issue 4, 1998, pp. 893–905**

**DOI: 10.4171/ZAA/857**

Published online: 1998-12-31

On Some Uniform Convexities and Smoothness in Certain Sequence Spaces

Yunan Cui^{[1]}, Henryk Hudzik

^{[2]}and Ryszard Pluciennik

^{[3]}(1) University of Science and Technology, Harbin, China

(2) Adam Mickiewicz University, Poznan, Poland

(3) Adam Mickiewicz University, Poznan, Poland

It is proved that any Banach space $X$ with property $A^e_2$ has property $A_2$ and that a Banach space $X$ is nearly uniformly smooth if and only if it is nearly uniformly *- smooth and weakly sequentially complete. It is shown that if $X$ is a Köthe sequence space the dual of which contains no isomorphic copy of $l_1$ and has property $A^e_2$, then $X$ has the uniform Kadec-Klee property. Criteria for nearly uniformly convexity of Musielak-Orlicz spaces equipped with the Orlicz norm are presented. It is also proved that both properties nearly uniformly smoothness and nearly uniformly convexity for Musielak-Orlicz spaces equipped with the Luxemburg norm coincide with reflexivity. Finally, an interpretation of those results for Nakano spaces $l^{(p_i)} (1 < p_i < \infty)$ is given.

*Keywords: *Fotou property, order continuity, nearly uniformly convexity, nearly uniformly smoothness, nearly uniformly *-smoothness, Musielak- Orlicz sequence spaces

Cui Yunan, Hudzik Henryk, Pluciennik Ryszard: On Some Uniform Convexities and Smoothness in Certain Sequence Spaces. *Z. Anal. Anwend.* 17 (1998), 893-905. doi: 10.4171/ZAA/857