Journal of Convex Analysis 24 (2017), No. 3, 969--985
Copyright Heldermann Verlag 2017
Diameter of Weak Neighborhoods and the Radon-Nikodym Property in Orlicz-Lorentz Spaces
Anna Kaminska
Dept. of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, U.S.A.
kaminska@memphis.edu
Hyung-Joon Tag
Dept. of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, U.S.A.
htag@memphis.edu
[Abstract-pdf]
Given an Orlicz $N$-function $\varphi$ and a positive decreasing weight $w$, we present criteria for the diameter two property and for the Radon-Nikod\'ym property in the Orlicz-Lorentz function and the sequence spaces $\Lambda_{\varphi,w}$ and $\lambda_{\varphi,w}$. We show that in the spaces $\Lambda_{\varphi,w}$ and $\lambda_{\varphi,w}$, equipped with the Luxemburg norm, the diameter of any relatively weakly open subset of the unit ball in these spaces is two if and only if $\varphi$ does not satisfy the appropriate $\Delta_2$-condition, while they have the Radon-Nikod\'ym property if and only if $\varphi$ satisfies the appropriate $\Delta_2$-condition.
Keywords: Diameter two property, Radon-Nikodym property, Orlicz-Lorentz space.
MSC: 46B20, 46E30, 47B38
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