It is proved that any Banach space with property has property and that a Banach space is nearly uniformly smooth if and only if it is nearly uniformly *- smooth and weakly sequentially complete. It is shown that if is a Köthe sequence space the dual of which contains no isomorphic copy of and has property , then 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 is given.
Cite this article
Yunan Cui, Henryk Hudzik, Ryszard Pluciennik, On Some Uniform Convexities and Smoothness in Certain Sequence Spaces. Z. Anal. Anwend. 17 (1998), no. 4, pp. 893–905DOI 10.4171/ZAA/857