The property ($\beta $) of Orlicz-Bochner sequence spaces

summary:A characterization of property $(\beta )$ of an arbitrary Banach space is given. Next it is proved that the Orlicz-Bochner sequence space $l_\Phi (X)$ has the property $(\beta )$ if and only if both spaces $l_\Phi $ and $X$ have it also. In particular the Lebesgue-Bochner sequence space $l_p(X)$ has the property $(\beta )$ iff $X$ has the property $(\beta )$. As a corollary we also obtain a theorem proved directly in [5] which states that in Orlicz sequence spaces equipped with the Luxemburg norm the property $(\beta )$, nearly uniform convexity, the drop property and reflexivity are in pairs equivalent