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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 $n\rightarrow\infty$.

46B20Geometry and structure of normed linear spaces
46B42Banach lattices
46B45Banach sequence spaces
46A45Sequence spaces
Full Text: DOI
[1] Castaing C., Płuciennik R.: The property (H) in Köthe-Bochner spaces. C. R. Acad. Sci. Paris, Serie I 319, 1159--1163 (1994) · Zbl 0824.46042
[2] Cerda J., Hudzik H., Kamińska A., Mastyło M.: Geometric properties of symmetric spaces with applications to Orlicz-Lorentz spaces. Positivity 2, 311--337 (1998) · Zbl 0920.46022 · doi:10.1023/A:1009728519669
[3] Cerdá J., Hudzik H., Mastyło M.: On the geometry of some Calderón-Lozanovskiĭ interpolation spaces. Indag. Math. N.S. 6(1), 35--49 (1995) · Zbl 0831.46016 · doi:10.1016/0019-3577(95)98199-L
[4] Chen S.: Geometry of Orlicz spaces. Dissertationes Math. 356, 1--204 (1996) · Zbl 1089.46500
[5] Chilin V.I., Dodds P.G., Sedaev A.A., Sukochev F.A.: Characterizations of Kadec-Klee properties in symmetric spaces of measurable functions. Trans. Amer. Math. Soc. 348(12), 4895--4918 (1996) · Zbl 0862.46015 · doi:10.1090/S0002-9947-96-01782-5
[6] Dominguez T., Hudzik H., López G., Mastyło M., Sims B.: Complete characterization of Kadec-Klee properties in Orlicz spaces. Houston J. Math. 29(4), 1027--1044 (2003) · Zbl 1155.46305
[7] Foralewski P., Hudzik H.: On some geometrical and topological properties of generalized Calderón-Lozanovskiĭ sequence spaces. Houston J. Math 25(3), 523--542 (1999) · Zbl 0997.46022
[8] Hudzik H., Kamińska A., Mastyło M.: Geometric properties of some Calderón-Lozanovskiĭ spaces and Orlicz-Lorentz spaces. Houston J. Math. 22, 639--663 (1996) · Zbl 0867.46025
[9] Hudzik H., Mastyło M.: Strongly extreme points in Köthe-Bochner spaces. Rocky Mountain J. Math. 23(3), 899--909 (1993) · Zbl 0795.46017 · doi:10.1216/rmjm/1181072531
[10] Hudzik H., Narloch A.: Relationships between monotonicity and complex rotundity properties with some consequences. Math. Scand. 96(2), 289--306 (2005) · Zbl 1085.46011
[11] Kantorovich, L.V., Akilov, G.P.: Functional Analysis, Nauka, Moscow (1984) (in Russian) · Zbl 0555.46001
[12] Kolwicz P.: Rotundity properties in Calderón-Lozanovskiĭ spaces. Houston J. Math. 31.3, 883--912 (2005) · Zbl 1084.46011
[13] Kolwicz, P.: Kadec-Klee properties of Calderón-Lozanovskiĭ function spaces, submitted to Mathematical Inequalities and Applications
[14] Kolwicz P., Płuciennik R.: Local $${$\backslash$Delta_{2}{E}(x)}$$ condition as a crucial tool for local structure of Calderón-Lozanovskiĭ spaces. J. Math. Anal. Appl. 356, 605--614 (2009) · Zbl 1211.46012 · doi:10.1016/j.jmaa.2009.03.030
[15] Krassowska D., Płuciennik R.: A note on property (H) in Köthe-Bochner sequence spaces. Math. Japonica 46(3), 407--412 (1997) · Zbl 0911.46003
[16] Krein, S.G., Petunin, Yu.I., Semenov, E.M.: Interpolation of linear operators, Nauka, Moscow (1978) (in Russian)
[17] Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces I, Springer (1979) · Zbl 0403.46022