Abstract

A criterion for λ-points of the unit ball in Orlicz spaces generated by arbitrary Orlicz functions (that is Orlicz functions which vanish outside zero and which attain infinite values to the right of some point u>0 are not excluded) and equipped with the Orlicz norm is given. Moreover, Orlicz spaces with the λ-property are characterized. In contrast to results of H. Y. Sun and C. X. Wu [λ-property of Orlicz space LM, Comment. Math. Univ. Carolin. 31 (1990) 731-741], Orlicz spaces considered by us need not always have the λ-property.

Author information

Contact details are reproduced from the original publication and may be historical.

Adam Bohonos

School of Mathematics, West Pomeranian University of Technology, Al. Piastów 48/49, 70-310 Szczecin, Poland

abohonos@zut.edu.pl

Keywords

  • Orlicz space
  • lambda-point
  • lambda-property
  • extreme point
  • Orlicz norm

Mathematics Subject Classification

46E30, 46E40, 46B20

References

  1. R. M. Aron, R. H. Lohman: A geometric function determined by extreme points of the unit ball of a normed space, Pacific J. Math. 127 (1987) 209–231.https://doi.org/10.2140/pjm.1987.127.209Google Scholar
  2. R. M. Aron, R. H. Lohman, A. Suárez Granero: Rotundity, the C.S.R.P., and the λ-property in Banach spaces, Proc. Amer. Math. Soc. 111 (1991) 151–155.https://doi.org/10.1090/S0002-9939-1991-1030732-0Google Scholar
  3. C. Benett, R. Sharpley: Interpolation of Operators, Academic Press, London (1988).Google Scholar
  4. S. T. Chen: Geometry of Orlicz Spaces, Diss. Math. 356 (1996) 204 p.Google Scholar
  5. S. T. Chen, Y. A. Cui, H. Hudzik: Isometric copies of l1 and l∞ in Orlicz spaces equipped with the Orlicz norm, Proc. Amer. Math. Soc. 132(2) (2004) 473–480.https://doi.org/10.1090/S0002-9939-03-07099-0Google Scholar
  6. S. T. Chen, H. Y. Sun, C. X. Wu: λ-property of Orlicz spaces, Bull. Pol. Acad. Sci., Math. 39 (1–2) (1991) 63–69.Google Scholar
  7. Y. Cui, L. Duan, H. Hudzik, M. Wisła: Basic theory of p-Amemiya norm in Orlicz spaces (1 ≤ p ≤ ∞): Extreme points and rotundity in Orlicz spaces endowed with these norms, Nonlinear Anal. 69 (2008) 1796–1816.https://doi.org/10.1016/j.na.2007.07.026Google Scholar
  8. Y. A. Cui, H. Hudzik, M. Nowak, R. Płuciennik: Some geometric properties in Orlicz sequence spaces equipped with Orlicz norm, J. Convex Analysis 6(1) (1999) 91–113.https://doi.org/10.68381/jca06007Google Scholar
  9. Y. A. Cui, H. Hudzik, R. Płuciennik: Extreme points and strongly extreme points in Orlicz spaces equipped with the Orlicz norm, Z. Anal. Anwend. 22(4) (2003) 789–817.https://doi.org/10.4171/zaa/1174Google Scholar
  10. P. Foralewski, H. Hudzik, R. Płuciennik: Orlicz spaces without extreme points, J. Math. Anal. Appl. 361 (2010) 506–519.https://doi.org/10.1016/j.jmaa.2009.07.015Google Scholar
  11. A. S. Granero: λ-property in Orlicz spaces, Bull. Pol. Acad. Sci., Math. 37 (7–12) (1989) 421–431.Google Scholar
  12. H. Hudzik: On the distance from the subspace of order continuous elements in L1 + L∞, Funct. Approximatio, Comment. Math. 25 (1997) 157–163.Google Scholar
  13. H. Hudzik, A. Kamińska, M. Mastyło: Local geometry of L1 ∩ L∞ and L1 + L∞, Arch. Math. 68 (1997) 159–168.https://doi.org/10.1007/s000130050045Google Scholar
  14. H. Hudzik, L. Maligranda: Amemiya norm equals Orlicz norm in general, Indag. Math., New Ser. 11(4) (2000) 573–585.https://doi.org/10.1016/S0019-3577(00)80026-9Google Scholar
  15. S. G. Krein, Ju. I. Petunin, E. M. Semenov: Interpolation of Linear Operators, AMS Trans. of Math. Monographs 54, Amer. Math. Soc., Providence (1982).Google Scholar
  16. R. H. Lohman: The λ-function in Banach spaces, in: Banach Space Theory (Iowa City, 1987), B.-L. Lin (ed.), Contemp. Math. 85, Amer. Math. Soc., Providence (1989) 345–351.https://doi.org/10.1090/conm/085/983393Google Scholar
  17. J. Musielak: Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer, Berlin (1983).https://doi.org/10.1007/BFb0072210Google Scholar
  18. M. M. Rao, Z. D. Ren: Theory of Orlicz Spaces, Marcel Dekker, New York (1991).Google Scholar
  19. H. Y. Sun, C. X. Wu: λ-property of Orlicz space LM, Commentat. Math. Univ. Carol. 31 (1990) 731–741.Google Scholar

Suggested citation

A. Bohonos, R. Płuciennik. “λ-Points in Orlicz Spaces.” Journal of Convex Analysis 21 (2014), No. 1, 147–166. https://doi.org/10.68381/jca21007

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Published by Heldermann Verlag, 2014. Rights now held by Banach Press LLC.