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We study property (β) in Köthe-Bochner sequence spaces E(X), where E is any Köthe sequence space and X is an arbitrary Banach space. The question of whether or not this geometric property lifts from X and E to E(X) is examined. We prove that if dim X = ∞, then E(X) has property (β) if and only if X has property (β) and E is orthogonally uniformly convex. It is also showed that if dim X < ∞, then E(X) has property (β) if and only if E has property (β). Our results essentially extend and improve those from [14] and [15].
Henryk Hudzik, and Paweł Kolwicz. "On property (β) of Rolewicz in Köthe-Bochner sequence spaces." Studia Mathematica 162.3 (2004): 195-212. <http://eudml.org/doc/285157>.
@article{HenrykHudzik2004, abstract = {We study property (β) in Köthe-Bochner sequence spaces E(X), where E is any Köthe sequence space and X is an arbitrary Banach space. The question of whether or not this geometric property lifts from X and E to E(X) is examined. We prove that if dim X = ∞, then E(X) has property (β) if and only if X has property (β) and E is orthogonally uniformly convex. It is also showed that if dim X < ∞, then E(X) has property (β) if and only if E has property (β). Our results essentially extend and improve those from [14] and [15].}, author = {Henryk Hudzik, Paweł Kolwicz}, journal = {Studia Mathematica}, keywords = {Köthe-Bochner spaces; orthogonal uniform convexity; Orlicz space; uniform monotonicity}, language = {eng}, number = {3}, pages = {195-212}, title = {On property (β) of Rolewicz in Köthe-Bochner sequence spaces}, url = {http://eudml.org/doc/285157}, volume = {162}, year = {2004}, }
TY - JOUR AU - Henryk Hudzik AU - Paweł Kolwicz TI - On property (β) of Rolewicz in Köthe-Bochner sequence spaces JO - Studia Mathematica PY - 2004 VL - 162 IS - 3 SP - 195 EP - 212 AB - We study property (β) in Köthe-Bochner sequence spaces E(X), where E is any Köthe sequence space and X is an arbitrary Banach space. The question of whether or not this geometric property lifts from X and E to E(X) is examined. We prove that if dim X = ∞, then E(X) has property (β) if and only if X has property (β) and E is orthogonally uniformly convex. It is also showed that if dim X < ∞, then E(X) has property (β) if and only if E has property (β). Our results essentially extend and improve those from [14] and [15]. LA - eng KW - Köthe-Bochner spaces; orthogonal uniform convexity; Orlicz space; uniform monotonicity UR - http://eudml.org/doc/285157 ER -