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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
Summary:
It is proved that if a Kothe 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 Kothe 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 $\mathop WCS((\bigoplus\nolimits_{i=1}^nX_i)_{\Phi})=\min\{\mathop WCS(X_i) i=1,\dots,n\}$ 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.
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