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

Paweł Kolwicz (2001)

Commentationes Mathematicae Universitatis Carolinae

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A characterization of property $\left(\beta \right)$ of an arbitrary Banach space is given. Next it is proved that the Orlicz-Bochner sequence space ${l}_{\Phi}\left(X\right)$ has the property $\left(\beta \right)$ if and only if both spaces ${l}_{\Phi}$ and $X$ have it also. In particular the Lebesgue-Bochner sequence space ${l}_{p}\left(X\right)$ has the property $\left(\beta \right)$ iff $X$ has the property $\left(\beta \right)$. 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 $\left(\beta \right)$, nearly uniform convexity, the drop property...