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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 $$I_\varphi(x)=\cases \left\|\varphi \circ x\right\|_E & \text{if $\varphi\circ x \in E$} \\ \infty &\text{otherwise.} \endcases$$ 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:
46B20Geometry and structure of normed linear spaces
46B42Banach lattices
46B45Banach sequence spaces
46A45Sequence spaces
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References:
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