Geodesic
The property ($\beta $) of Orlicz-Bochner sequence spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) no. 1, pp. 119-132 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 46B20, 46B45, 46E30, 46E40
Keywords: Orlicz-Bochner space; property $(\beta )$; Orlicz space 
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     title = {The property ($\beta $) of {Orlicz-Bochner} sequence spaces},
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Kolwicz, Paweł. The property ($\beta $) of Orlicz-Bochner sequence spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) no. 1, pp. 119-132. http://geodesic.mathdoc.fr/item/CMUC_2001_42_1_a8/