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On some geometric properties of certain Köthe sequence spaces. (English) Zbl 0941.46005

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.

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
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