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On property (B) in Musielak-Orlicz sequence spaces of Bochner type

Kolwicz P.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2000

[Z] 40

103-126

It is proved that the Musielak-Orlicz sequence space of Bochner type l_f(X) has property (B) if and only if both spaces lip and X have it also. It is considered the case of Luxemburg and Orlicz norm as well. In particular, the Lebesgue-Bochner sequence space l_p(X) has property (B) iff X has property (B)). As a corollary we also conclude that in Musielak-Orlicz sequence spaces equipped with the Luxemburg (Orlicz) norm property (B), nearly uniform convexity, drop property and reflexivity are in pairs equivalent. Such a theorem was proved directly in [4] in the case of Orlicz sequence space.