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Kadec-Klee properties of Calderón-Lozanovskiĭ sequence spaces. (English) Zbl 1267.46028

Two Kadec-Klee properties; the first one with respect to coordinatewise convergence and the second one with respect to the uniform convergence in Calderón-Lozanovskiǐ sequence spaces are studied. Full criteria for these properties in these spaces are given. As an application of the results, characterizations of the Kadec-Klee properties in Orlicz-Lorentz spaces are deduced in more general form than those ones that were known earlier.
Let us recall that the Calderón-Lozanovskiǐ spaces which are considered in this paper are generated by a real Köthe sequence space \(E\) and an Orlicz function \(\varphi\) in the following way \[ E_\varphi =\left\{x\in \ell^0\colon \varphi \circ\lambda x \in E \text{ for some } \lambda>0\right\}, \] where \(\ell^0\) is the space of all real sequences, and it is endowed with the norm \[ \left\|x\right\|_\varphi =\inf\left\{\lambda>0\colon I_\varphi\left(\frac{x}{\lambda}\right)\leq 1 \right\}, \] where \[ I_\varphi(x)=\begin{cases} \left\|\varphi \circ x\right\|_E & \text{if \(\varphi\circ x \in E\)} \\ \infty &\text{otherwise.} \end{cases} \] Let us recall that a Köthe sequence space \(E\) has the Kadec-Klee property with respect to the coordinatewise (resp. with respect to the uniform) convergence if for all \(x\in E\) and \(\left(x_n\right)_{n=1}^\infty\) in \(E\) such that \(\left\|x_n\right\|\rightarrow \left\|x\right\|\) as \(n\rightarrow\infty\) and \(x_n\rightarrow x\) coordinatewise (resp. uniformly) as \(n\rightarrow\infty\) it holds that \(\left\|x_n-x\right\|\rightarrow 0\) as \(n\rightarrow\infty\).

MSC:

46B20 Geometry and structure of normed linear spaces
46B42 Banach lattices
46B45 Banach sequence spaces
46A45 Sequence spaces (including Köthe sequence spaces)
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