Geodesic
On some geometric properties of certain Köthe sequence spaces
Mathematica Bohemica, Tome 124 (1999) no. 2-3, pp. 303-314
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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.
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.
DOI : 10.21136/MB.1999.126253
Classification : 46A45, 46B20, 46B25, 46B45, 46E20, 46E40
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 
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Cui, Yunan; Hudzik, Henryk; Zhang, Tao. On some geometric properties of certain Köthe sequence spaces. Mathematica Bohemica, Tome 124 (1999) no. 2-3, pp. 303-314. doi: 10.21136/MB.1999.126253

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