Skip to main content
SpringerLink
Log in

Non-\(l_n^{(1)}\) point and uniformly non-\(l_n^{(1)}\) point of Orlicz–Bochner function spaces

  • Original Paper
  • Published:
Banach Journal of Mathematical Analysis Aims and scope Submit manuscript

Abstract

Let \(L_M\) be an Orlicz function space endowed with the Orlicz norm or the Luxemburg norm, and let X be a Banach space. In this paper we characterize the non-\(l_n^{(1)}\) point and the uniformly non-\(l_{n}^{(1)}\) point of Orlicz–Bochner function space \(L_M(\mu ,X)\). As the immediate consequences some criteria for non-square point and uniformly non-square point of \(L_M(\mu ,X)\) are obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aksoy, A.G., Khamsi, M.A.: Nonstandard Methods in Fixed Point Theory. Springer, Berlin (1990)

    Book  Google Scholar 

  2. Alherk, G., Hudzik, H.: Uniformly non\(-l_n^{(1)}\) Musielak–Orlicz spaces of Bochner type. Forum Math. 1, 403–410 (1989)

    Article  MathSciNet  Google Scholar 

  3. Beck, A.: A convexity condition in Banach spaces and the strong law of large numbers. Proc. Am. Math. Soc. 13, 329–334 (1962)

    Article  MathSciNet  Google Scholar 

  4. Chen, S., Hudzik, H.: On some convexities of Orlicz and Orlicz–Bochner spaces. Comment. Math. Univ. Carolin. 29, 13–29 (1988)

    MathSciNet  MATH  Google Scholar 

  5. Chen, S.: Geometry of Orlicz spaces. Dissertationes, Mathematicae Warszawa, Warszawa (1996)

  6. Diestel, J., Uhl Jr., J.J.: Vector Measures. Mathematical Surveys and Monographs, vol. 15. American Mathematical Society, Providence (1977)

    Google Scholar 

  7. Garcia-Falset, J., Liorens-Fuster, E., Mazcunan-Navarro, E.M.: The fixed point property and normal structure for some B-convex Banach spaces. Bull. Aust. Math. Soc. 63, 75–81 (2001)

    Article  MathSciNet  Google Scholar 

  8. Garcia-Falset, J., Liorens-Fuster, E., Mazcunan-Navarro, E.M.: Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings. J. Funct. Anal. 233, 494–514 (2006)

    Article  MathSciNet  Google Scholar 

  9. Gong, W., Zhou, C., Dong, X.: Uniformly non-\(l_n^{(1)}\), locally uniformly non-\(l_n^{(1)}\) and non-\(l_n^{(1)}\) properties in Orlicz–Bochner function spaces endowed with the Orlicz norm. J. Math. Anal. Appl. 462, 1283–1297 (2018)

    Article  MathSciNet  Google Scholar 

  10. Giesy, D.: On a convexity condition in normed linear spaces. Trans. Am. Math. Soc. 125, 114–146 (1966)

    Article  MathSciNet  Google Scholar 

  11. Giesy, D., James, R.: Uniformly non-\(l_n^{(1)}\) and B-convex spaces. Studia. Math. 48, 61–69 (1973)

    Article  MathSciNet  Google Scholar 

  12. Grzaślewicz, R., Hudzik, H., Orlicz, W.: Uniform non-\(l_n^{(1)}\) property in some normed spaces. Bull. Pol. Acad. Sci. Math. 34, 161–171 (1986)

    MATH  Google Scholar 

  13. Hu, Z., Lin, B.: Extremal structure of the unit ball of \(L_p(\mu, X)^*\). J. Math. Anal. Appl. 200, 567–590 (1996)

    Article  MathSciNet  Google Scholar 

  14. Huang, S.Z., Neerven, J.: B-Convexity, the analytic Radon–Nikodym property, and individual stability of \(C_0\)-semigroups. J. Math. Anal. Appl. 231, 1–20 (1999)

    Article  MathSciNet  Google Scholar 

  15. Hudzik, H.: Uniformly non-\(l_n^{(1)}\) Orlicz space with Luxemburg norm. Studia. Math. 8, 271–284 (1985)

    Article  Google Scholar 

  16. Hudzik, H.: Locally uniformly non-\(l_n^{(1)}\) Orlicz space. In: Proceedings of the 13th Winter School on Abstract Analysis, Srni 1985, Rend. Circ. Mat. Palermo., 10, 49–56 (1985)

  17. Hudzik, H., Kamińska, A., Kurc, W.: Uniformly non-\(l_n^{(1)}\) Musielak–Orlicz spaces. Bull. Pol. Acad. Sci. Math 35, 441–448 (1987)

    MATH  Google Scholar 

  18. James, R.: Uniformly nonsquare Banach spaces. Ann. Math. 280, 542–550 (1964)

    Article  Google Scholar 

  19. Kamińska, A., Turett, B.: Uniformly non-\(l_n^{(1)}\) Orlicz–Bochner space. Bull. Pol. Acad. Sci. Math. 35, 211–218 (1987)

    MATH  Google Scholar 

  20. Lin, P.: Köthe–Bochner Function Spaces. Birkhauser, Boston (2003)

    MATH  Google Scholar 

  21. Shi, Z., Wang, Y.: The locally uniformly non-square points of Orlicz–Bochner sequence spaces. Math. Nachr. 290, 920–929 (2017)

    Article  MathSciNet  Google Scholar 

  22. Shi, Z., Wang, Y.: The non-square points of Orlicz–Bochner sequence spaces. Southeast Asian Bull. Math. 41, 249–258 (2017)

    MathSciNet  MATH  Google Scholar 

  23. Shi, Z., Wang, Y.: Uniformly non-square point and representation of functionals of Orlicz–Bochner sequence spaces. Rocky Mt. J. Math. 48, 639–660 (2018)

    Article  MathSciNet  Google Scholar 

  24. Wang, T., Shi, Z., Li, Y.: On uniformly nonsquare points and nonsquare points of Orlicz spaces. Comment. Math. Univ. Carolin. 33, 477–484 (1992)

    MathSciNet  MATH  Google Scholar 

  25. Zhang, D.: Non-squareness of Orlicz–Bochner Spaces. Archive of Shanghai University, Shanghai (2009)

    Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions that helped improve the quality of this manuscript. This work is supported by the National Science Research Project of Anhui Educational Department (KJ2019A0487) and National Natural Science Foundation of China (11771273).

Author information

Authors and Affiliations

  1. Department of Mathematics, Anhui Normal University, Wuhu, 241000, People’s Republic of China

    Wanzhong Gong & Kangji Wang

Authors
  1. Wanzhong Gong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kangji Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wanzhong Gong.

Additional information

Communicated by Mieczyslaw Mastylo.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gong, W., Wang, K. Non-\(l_n^{(1)}\) point and uniformly non-\(l_n^{(1)}\) point of Orlicz–Bochner function spaces. Banach J. Math. Anal. 14, 1177–1200 (2020). https://doi.org/10.1007/s43037-020-00057-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s43037-020-00057-y

Keywords

Mathematics Subject Classification

Search

Navigation