Cui, Yunan; Hudzik, Henryk; Zhang, Tao On some geometric properties of certain Köthe sequence spaces. (English) Zbl 0941.46005 Math. Bohem. 124, No. 2-3, 303-314 (1999). Summary: It is proved that if a Köthe 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 Köthe 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 \((\bigoplus _{i=1}^{\infty }X_{i})_{\Phi }\) 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 \(\text{WCS}((\bigoplus _{i=1}^{n}X_{i})_{\Phi })=\min \{\text{WCS}(X_{i})\:i=1,\dots ,n\}\) and that if \(\Phi \) does not satisfy the \(\Delta _2\)-condition, then WCS\(((\bigoplus _{i=1}^{\infty }X_{i}) _{\Phi })=1\) for any infinite sequence \((X_{i})\) of Banach spaces. Cited in 1 Document MSC: 46A45 Sequence spaces (including Köthe sequence spaces) 46B25 Classical Banach spaces in the general theory 46E40 Spaces of vector- and operator-valued functions 46B20 Geometry and structure of normed linear spaces 46E20 Hilbert spaces of continuous, differentiable or analytic functions 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 PDFBibTeX XMLCite \textit{Y. Cui} et al., Math. Bohem. 124, No. 2--3, 303--314 (1999; Zbl 0941.46005) Full Text: EuDML