On Some Uniform Convexities and Smoothness in Certain Sequence Spaces

Abstract

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.