Skip to main content
SpringerLink
Log in

Strict convexity of Orlicz sequence spaces equipped with p-Amemiya norms

  • Original Research
  • Published:
Indian Journal of Pure and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this paper, criteria for strict convexity of Orlicz sequence spaces equipped with the p-Amemiya norms \((1\le p\le \infty )\) are presented. The obtained results widen the characterization of strict convexity of Orlicz sequence spaces. From our results it follows that there are Orlicz sequence spaces which are strictly convex for p-Amemiya norms with \(1<p<\infty \) only, that is, they are neither strictly convex for Luxemburg norm corresponding to the case \(p=\infty \) nor for Orlicz norm corresponding to the case \(p=1\).

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.

Similar content being viewed by others

References

  1. Cao, L., Wang, T.: Some notes on K(x) in Musielak-Orlicz spaces. Natur. Sci. J. Harbin Normal Univ (4), 1–4 (2000)

    MathSciNet  MATH  Google Scholar 

  2. Chen, L., Cui, Y.: Complex extreme points and complex rotundity in Orlicz function spaces equipped with the p-Amemiya norm. Nonlinear Analysis 73(5), 1389–1393 (2010)

    Article  MathSciNet  Google Scholar 

  3. Chen, S.: Geometry of Orlicz spaces, vol. 356. Dissertationes Mathematicae (1996)

  4. Cui, Y., Duan, L., Hudzik, H., Wisla, M.: Basic theory of p-Amemiya norm in Orlicz spaces: Extreme points and rotundity in Orlicz spaces endowed with these norms. Nonlinear Analysis 69(5-6), 1796–1816 (2008)

    Article  MathSciNet  Google Scholar 

  5. Cui, Y., Hudzik, H., Li, j., Wisla, M.: Strongly extreme points in Orlicz spaces equipped with the p-Amemiya norm. Nonlinear Analysis 71, 6343–6364 (2009)

  6. Cui, Y., Hudzik, H., Meng, C.: On some local geometry of Orlicz sequence spaces equipped with the Luxemburg norm. Acta Mathematica Hungarica 80(1), 143–154 (1998)

    Article  MathSciNet  Google Scholar 

  7. Cui, Y., Hudzik, H., Nowak, M., Puciennik, R.: Some geometric properties in Orlicz sequence spaces equipped with Orlicz norm. Journal of Convex Analysis 6(1), 91–113 (1999)

    MathSciNet  MATH  Google Scholar 

  8. Cui, Y., Hudzik, H., Wisla, M.: Monotonicity properties and dominated best approximation problems in Orlicz spaces equipped with the p-Amemiya norm. Journal of Mathematical Analysis and Applications 432(2), 1095–1105 (2015)

    Article  MathSciNet  Google Scholar 

  9. Cui, Y., Hudzik, H., Wisla, M., Wlazlak, K.: Non-squareness properties of Orlicz spaces equipped with the p-Amemiya norm. Nonlinear Analysis Theory Methods and Applications 75(10), 3973–3993 (2012)

    Article  MathSciNet  Google Scholar 

  10. Duan, L., Xu, J., Cui, Y.: Extreme points and rotundity in Orlicz sequence spaces equipped with p-Amemiya norm. Journal of Jilin University (Science Edition) 50(5), 902–906 (2012)

    MathSciNet  Google Scholar 

  11. Foralewski, P., Hudzik, H., Szymaszkiewicz, A.: Local rotundity structure of Cesaro Orlicz sequence spaces. Journal of Mathematical Analysis and Applications 345(1), 410–419 (2008)

    Article  MathSciNet  Google Scholar 

  12. Hudzik, H., Maligranda, L.: Amemiya norm equals Orlicz norm in general. Indagationes Mathematicae 11(4), 573–585 (2000)

    Article  MathSciNet  Google Scholar 

  13. Kaczmarek, Radoslaw: Uniform rotundity of Orlicz function spaces equipped with the p-Amemiya norm. Mathematische Nachrichten 291(10), 1514–1532 (2018)

    Article  MathSciNet  Google Scholar 

  14. Kaminska, A.: Rotundity of Orlicz-Musielak sequence spaces. Bull. Acad. Polon. Sci. Math 29(3-4), 137–144 (1981)

    MathSciNet  MATH  Google Scholar 

  15. Kaminska, A.: Strict convexity of sequence Orlicz-Musielak spaces with Orlicz norm. Journal of Functional Analysis 50(3), 285–305 (1983)

    Article  MathSciNet  Google Scholar 

  16. Krasnoselskii, M.A., Rutickii, Y.B.: Convex Functions and Orlicz Spaces. P. Noordhoff Ltd (1961)

  17. Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces. I Sequence Spaces 350(2), 86–157 (1973)

    Google Scholar 

  18. Luxemburg, W.A.J., Zaanen, A.C.: Conjugate spaces of Orlicz spaces. Indagationes Mathematicae 59, 217–228 (1956)

    Article  MathSciNet  Google Scholar 

  19. Milnes, H.W.: Convexity of Orlicz spaces. Pacific Journal of Mathematics 7(3), 1451–1486 (1957)

    Article  MathSciNet  Google Scholar 

  20. Musielak, J.: Orlicz spaces and modular spaces, vol. 1034. Springer (1983)

  21. Nakano, Hidegore: Topology and linear topological spaces. Maruzen Co.Ltd (1951)

  22. Orlicz, W.: A note on modular spaces. Bull. Acad. Polon. Sci. Math. Astronom. Phys 9, 157–162 (1961)

  23. Rao, M.M., Ren, Z.: Theory of Orlicz Spaces. Marcek Dekker Inc (1991)

  24. Wisla, M.: Geometric properties of Orlicz spaces equipped with the p-Amemiya norms-results and open questions. Comment. Math 55, 183–209 (2015)

    MathSciNet  MATH  Google Scholar 

  25. Wisla, M.: Orlicz spaces equipped with the s-norms. Journal of Mathematical Analysis and Applications 483(2), 1–30 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Nature Science Foundation of China (11871181).

Author information

Authors and Affiliations

  1. Department of Mathematics, Harbin University of Science and Technology, Harbin, 150080, China

    Xiaoyan Li & Yunan Cui

  2. Department of Mathematics, Harbin Normal University, Harbin, 150025, China

    Xiaoyan Li

Authors
  1. Xiaoyan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yunan Cui
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yunan Cui.

Additional information

Communicated by T S S R K Rao.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, X., Cui, Y. Strict convexity of Orlicz sequence spaces equipped with p-Amemiya norms. Indian J Pure Appl Math 53, 660–671 (2022). https://doi.org/10.1007/s13226-021-00157-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13226-021-00157-x

Keywords

Search

Navigation