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MATHEMATICA BOHEMICA, Vol. 124, No. 2–3, pp. 303-314 (1999)
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# On some geometric properties of certain Köthe sequence spaces

## Yunan Cui, Henryk Hudzik, Tao Zhang

* Yunan Cui, Tao Zhang*, Department of Mathematics, Harbin University of Science and Technology, Xuefu Road 52, 150080 Harbin, P. R. China, e-mail: ` cuiya@hkd.hrbust.edu.cn`; * Henryk Hudzik*, Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Matejki 48/49, 60-769 Poznan, Poland, e-mail: ` hudzik@amu.edu.pl`

**Abstract:**
It is proved that if a K\"{o}the sequence space $X$ is monotone complete and has the weakly convergent sequence coefficient WCS$(X)>1$, then $X$ is order continuous. It is shown that a weakly sequentially complete K\"{o}the sequence space $X$ is compactly locally uniformly rotund if and only if the norm in $X$ is equi-absolutely continuous. The dual of the product space $(\bigoplus\nolimits_{i=1}^{\infty}X_{i})_{\Phi}$ of a sequence of Banach spaces $(X_{i})_{i=1}^{\infty}$, which is built by using an Orlicz function $\Phi$ satisfying the $\Delta_2$-condition, is computed isometrically (i.e. the exact norm in the dual is calculated). It is also shown that for any Orlicz function $\Phi$ and any finite system $X_{1},\dots,X_{n}$ of Banach spaces, we have $\mathop WCS((\bigoplus\nolimits_{i=1}^{n}X_{i})_{\Phi})=\min\{\mathop WCS(X_{i}) i=1,\dots,n\}$ and that if $\Phi$ does not satisfy the $\Delta_2$-condition, then WCS$((\bigoplus\nolimits_{i=1}^{\infty}X_{i}) _{\Phi})=1$ for any infinite sequence $(X_{i})$ of Banach spaces.

**Keywords:** Köthe sequence space, weakly convergent sequence coefficient, order continuity of the norm, absolute continuity of the norm, compact local uniform rotundity, Orlicz sequence space, Luxemburg norm, Orlicz norm, dual space, product space

**Classification (MSC2000):** 46B20, 46E20, 46E40

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