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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
It is proved that if a Kothe 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 Kothe 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 $(\bigoplus\nolimits_{i=1}^{\infty}X_i)_{\Phi}$ 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 $\mathop WCS((\bigoplus\nolimits_{i=1}^nX_i)_{\Phi})=\min\{\mathop WCS(X_i) i=1,\dots,n\}$ and that if $\Phi$ does not satisfy the $\Delta_2$-condition, then WCS$((\bigoplus\nolimits_{i=1}^{\infty}X_i) _{\Phi})=1$ for any infinite sequence $(X_i)$ of Banach spaces.
[1] J. M. Ayerbe T. Dominguez Benavides G. Lopez Acedo: Compactness Conditions in Metric Fixed Point Theory. OTAA, vol. 99, Birkhäuser, Basel, 1997.
[2] M. S. Brodskij D. P. Milman: On the center of convex set. Dokl. Akad. Nauk 59 (1948), 837-840. (In Russian.) MR 0024073
[3] W. I. Bynum: Normal structure coefficient for Banach spaces. Pacific J. Math. 86 (1980), 427-436. DOI 10.2140/pjm.1980.86.427 | MR 0590555
[4] S. T. Chen: Geometry of Orlicz Spaces. Dissertationes Math. 356, 1996. MR 1410390 | Zbl 1089.46500
[5] T. Dominguez Benavides: Weak uniform normal structure in direct sum spaces. Studia Math. 103 (1992), no. 3, 283-290. DOI 10.4064/sm-103-3-283-290 | MR 1202012 | Zbl 0810.46015
[6] P. Foralewski H. Hudzik: Some basic properties of generalized Calderón-Lozanovskij spaces. Collect. Math. 48 (1997), no. 4-6. 523-538. MR 1602584
[7] K. Goebel W. Kirk: Topics in Metrix Fixed Point Theory. Cambridge University Press, Cambridge, 1991. MR 1074005
[8] A. Kamińska: Flat Orlicz-Musielak sequence spaces. Bull. Polish Acad. Sci. Math. 30 (1982), no. 7-8, 347-352. MR 0707748
[9] L. V. Kantorovich G. P. Akilov: Functional Analysis. Nauka, Moscow, 1977. (In Russian.) MR 0511615
[10] T. Landes: Permanence properties of normal structure. Pacific J. Math. 110 (1984), 125-143. DOI 10.2140/pjm.1984.110.125 | MR 0722744 | Zbl 0534.46015
[11] T. C. Lin: On normal structure coefficient and the bounded sequence coefficient. Proc. Amer. Math. Soc. 88 (1983), 262-267. DOI 10.1090/S0002-9939-1983-0695255-2 | MR 0695255
[12] J. Lindenstrauss L. Tzafriri: Classical Banach Spaces I. Springer-Verlag, Berlin, 1977. MR 0500056
[13] W. A. J. Luxemburg: Banach Function Spaces. Thesis, Delft, 1955. MR 0072440 | Zbl 0068.09204
[14] L. Maligranda: Orlicz Spaces and Interpolation. Seminars in Math. 5, Campinas, 1989. MR 2264389 | Zbl 0874.46022
[15] E. Maluta: Uniformly normal structure and related coefficients. Pacific J. Math. 111 (1984), 357-369. DOI 10.2140/pjm.1984.111.357 | MR 0734861 | Zbl 0495.46012
[16] J. Musielak: Orlicz Spaces and Modular Spaces. Lecture Notes in Math. 1034, Springer-Verlag, Berlin, 1983. MR 0724434 | Zbl 0557.46020
[17] B. B. Panda O. P. Kapoor: A generalization of local uniform convexity of the norm. J. Math. Anal. Appl. 52 (1975), 300-308. DOI 10.1016/0022-247X(75)90098-0 | MR 0380365
[18] B. Prus: On Bynum's fixed point theorem. Atti Sem. Mat. Fis. Univ. Modena 38 (1990), 535-545. MR 1076471 | Zbl 0724.46020
[19] M. M. Rao Z. D. Ren: Theory of Orlicz Spaces. Marcel Dekker Inc., New York, 1991. MR 1113700
[20] A. E. Taylor D. C. Lay: Introduction to Functional Analysis. John Wiley & Sons, New York (second edition), 1980. MR 0564653
[21] G. L. Zhang: Weakly convergent sequence coefficient of product space. Proc. Amer. Math. Soc. 117 (1993), no. 3, 637-643. DOI 10.1090/S0002-9939-1993-1152993-1 | MR 1152993 | Zbl 0787.46021
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