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Local rotundity structure of Cesàro–Orlicz sequence spaces. (English) Zbl 1155.46007

Summary: Some criteria for extreme points and strong U-points in Cesàro–Orlicz spaces are given. In consequence, we find a Cesàro–Orlicz sequence space different from \(c_{0}\) which has no extreme points. Some examples show that in these spaces the notion of a strong U-point is essentially stronger than the notion of an extreme point. Various examples presented in this paper show that there are some differences between criteria for extreme points and strong U-points in Orlicz spaces and in Cesàro–Orlicz spaces. We also show that the uniqueness of the local best approximation needs the notion of an SU-point, that is, the notion of an extreme point is not strong enough here.

MSC:

46B20 Geometry and structure of normed linear spaces
46B45 Banach sequence spaces
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[1] Nieuw Arch. Wiskd., 16, 47-51 (1968), Programma van Jaarlijkse Prijsvragen (Annual Problem Section)
[2] Bandyopadhyay, P.; Huang, D.; Lin, B. L., Rotund points, nested sequence of balls and smoothness in Banach spaces, Comment. Math., 44, 2, 163-186 (2004) · Zbl 1097.46009
[3] Bennett, G., Factorizing the classical inequalities, Mem. Amer. Math. Soc., 120, 576 (1996) · Zbl 0857.26009
[4] Chen, S. T., Geometry of Orlicz Spaces, Dissertationes Math. (Rozprawy Mat.), vol. 356 (1996), Polish Acad. Sci.: Polish Acad. Sci. Warsaw · Zbl 0873.46018
[5] Chen, S. T.; Cui, Y. A.; Hudzik, H.; Sims, B., Geometric properties related to fixed point theory in some Banach function lattices, (Handbook of Metric Fixed Point Theory (2001), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht), 339-389 · Zbl 1013.46015
[6] Cui, Y. A.; Hudzik, H., Some geometric properties related to fixed point theory in Cesàro sequence spaces, Collect. Math., 50, 3, 277-288 (1999) · Zbl 0955.46007
[7] Cui, Y. A.; Hudzik, H., Packing constant for Cesàro sequence spaces, Nonlinear Anal., 47, 2695-2702 (2001) · Zbl 1042.46505
[8] Cui, Y. A.; Hudzik, H.; Meng, C., On some local geometry of Orlicz sequence spaces equipped with the Luxemburg norm, Acta Math. Hungar., 80, 1-2, 143-154 (1998) · Zbl 0914.46004
[9] Cui, Y. A.; Hudzik, H.; Petrot, N.; Suantai, S.; Szymaszkiewicz, A., Basic topological and geometric properties of Cesàro-Orlicz spaces, Proc. Indian Acad. Sci., 115, 4, 461-476 (2005) · Zbl 1093.46013
[10] Cui, Y. A.; Jie, L.; Płuciennik, R., Local uniform nonsquareness in Cesàro sequence spaces, Comment. Math., 37, 47-58 (1997) · Zbl 0898.46006
[11] Cui, Y. A.; Meng, C.; Płuciennik, R., Banach-Saks property and property \((β)\) in Cesàro sequence spaces, Southeast Asian Bull. Math., 24, 201-210 (2000) · Zbl 0956.46003
[12] Diestel, J., Sequences and Series in Banach Spaces, Grad. Texts in Math., vol. 92 (1984), Springer-Verlag: Springer-Verlag Berlin
[13] P. Foralewski, H. Hudzik, R. Płuciennik, Orlicz spaces without extreme points, submitted for publication; P. Foralewski, H. Hudzik, R. Płuciennik, Orlicz spaces without extreme points, submitted for publication · Zbl 1194.46011
[14] P. Foralewski, H. Hudzik, A. Szymaszkiewicz, Some remarks on Cesàro-Orlicz sequence spaces, submitted for publication; P. Foralewski, H. Hudzik, A. Szymaszkiewicz, Some remarks on Cesàro-Orlicz sequence spaces, submitted for publication · Zbl 1198.46017
[15] Grza̧ślewicz, R.; Hudzik, H.; Kurc, W., Extreme and exposed points in Orlicz spaces, Canad. J. Math., 44, 3, 505-515 (1992) · Zbl 0766.46007
[16] Jagers, A. A., A note on Cesàro sequence spaces, Nieuw Arch. Wiskd., 22, 113-124 (1974) · Zbl 0286.46017
[17] Kantorovich, L. V.; Akilov, G. P., Functional Analysis (1977), Nauka: Nauka Moscow, (in Russian) · Zbl 0555.46001
[18] Krasnoselskiıˇ, M. A.; Rutickiıˇ, Ya. B., Convex Functions and Orlicz Spaces (1961), P. Nordhoff Ltd.: P. Nordhoff Ltd. Groningen, (translation from Russian)
[19] Lee, P. Y., Cesàro sequence spaces, Math. Chronicle New Zealand, 13, 29-45 (1984) · Zbl 0568.46006
[20] Leibowitz, G. M., A note on the Cesàro sequence spaces, Tamkang J. Math., 2, 151-157 (1971) · Zbl 0236.46012
[21] Lindenstrauss, J.; Tzafriri, L., Classical Banach Spaces I. Sequence Spaces (1977), Springer-Verlag: Springer-Verlag Berlin · Zbl 0362.46013
[22] Lindenstrauss, J.; Tzafriri, L., Classical Banach Spaces II. Function Spaces (1979), Springer-Verlag: Springer-Verlag Berlin · Zbl 0403.46022
[23] W.A.J. Luxemburg, Banach function spaces, thesis, Delft, 1955; W.A.J. Luxemburg, Banach function spaces, thesis, Delft, 1955 · Zbl 0068.09204
[24] Maligranda, L., Orlicz Spaces and Interpolation, Semin. Math., vol. 5 (1989), Universidade Estadual de Campinas: Universidade Estadual de Campinas Campinas, SP, Brazil · Zbl 0874.46022
[25] Maligranda, L.; Petrot, N.; Suantai, S., On the James constant and \(B\)-convexity of Cesàro and Cesàro-Orlicz sequence spaces, J. Math. Anal. Appl., 326, 312-331 (2007) · Zbl 1109.46026
[26] Musielak, J., Orlicz Spaces and Modular Spaces, Lecture Notes in Math., vol. 1034 (1983), Springer-Verlag: Springer-Verlag Berlin · Zbl 0557.46020
[27] Rao, M. M.; Ren, Z. D., Theory of Orlicz Spaces (1991), Marcel Dekker Inc.: Marcel Dekker Inc. New York · Zbl 0724.46032
[28] Shiue, J. S., Cesàro sequence spaces, Tamkang J. Math., 1, 19-25 (1970) · Zbl 0215.19504
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