Skip to main content
SpringerLink
Log in

Stability of ε-Isometries on ℒ∞-Spaces

  • Published:
Acta Mathematica Scientia Aims and scope Submit manuscript

Abstract

In this article, we discuss the stability of ε-isometries for ℒ∞,λ-spaces. Indeed, we first study the relationship among separably injectivity, injectivity, cardinality injectivity and universally left stability of ℒ∞,λ-spaces as well as we show that the second duals of universally left-stable spaces are injective, which gives a partial answer to a question of Bao-Cheng-Cheng-Dai, and then we prove a sharpen quantitative and generalized Sobczyk theorem which gives examples of nonseparable ℒ∞-spaces X (but not injective) such that the couple (X, Y) is stable for every separable space Y. This gives a new positive answer to Qian’s problem.

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. Albiac F, Kalton N J. Topics in Banach Space Theory. Graduate Texts in Mathematics 233. New York: Springer, 2006

    MATH  Google Scholar 

  2. Argyros S A, Haydon R G. A hereditarily indecomposable ℒ ∞-space that solves the scalar-plus-compact problem. Acta Math, 2011, 206(1): 1–54

    Article  MathSciNet  Google Scholar 

  3. Avilés A, Sánchez F C, Castillo Jesús M F, Gonzalez M, Moreno Y. On separably injective Banach Spaces. Adv Math, 2013, 234: 192–216

    Article  MathSciNet  Google Scholar 

  4. Bao L, Cheng L, Cheng Q, Dai D. On universally left-stability of ε-isometry. Acta Math Sin, Engl Ser, 2013, 29(11): 2037–2046

    Article  MathSciNet  Google Scholar 

  5. Benyamini Y, Lindenstrauss J. Geometric Nonlinear Functional Analysis I. Amer Math Soc Colloquium Publications, Vol 48. Providence, RI: Amer Math Soc, 2000

    MATH  Google Scholar 

  6. Bourgain J, Delbaen F. A class of special ℒ ∞-space. Acta Math, 1980, 145: 155–176

    Article  MathSciNet  Google Scholar 

  7. Cheng L, Dong Y, Zhang W. On stability of nonlinear non-surjective ε-isometries of Banach spaces. J Funct Anal, 2013, 264: 713–734

    Article  MathSciNet  Google Scholar 

  8. Cheng L, Dai D, Dong Y, et al. Universal stability of Banach spaces for ε-isometries. Studia Math, 2014, 221: 141–149

    Article  MathSciNet  Google Scholar 

  9. Cheng L, Cheng Q, Tu K, Zhang J. A universal theorem for stability of ε-isometries of Banach spaces. J Funct Anal, 2015, 269: 199–214

    Article  MathSciNet  Google Scholar 

  10. Dai D, Dong Y. On stability of Banach spaces via nonlinear ε-isometries. J Math Anal Appl, 2014, 414: 996–1005

    Article  MathSciNet  Google Scholar 

  11. Dai D, Zheng B. Stability of a pair of Banach spaces for ε-isometries. Acta Math Sci, 2019, 39B(4): 1163–1172

    Google Scholar 

  12. Dilworth S J. Approximate isometries on finite-dimensional normed spaces. Bull London Math Soc, 1999, 31: 471–476

    Article  MathSciNet  Google Scholar 

  13. Dutrieux Y, Lancien G. Isometric embeddings of compact spaces into Banach spaces. J Funct Anal, 2008, 255: 494–501

    Article  MathSciNet  Google Scholar 

  14. Fabian M, Habala P, Hájek P, Montesinos V, Zizler V. Banach Space Theory. The Basis for Linear and Nonlinear Analysis. New York: Springer, 2011

    MATH  Google Scholar 

  15. Figiel T. On non linear isometric embeddings of normed linear spaces. Bull Acad Polon Sci Math Astro Phys, 1968, 16: 185–188

    MathSciNet  MATH  Google Scholar 

  16. Gevirtz J. Stability of isometries on Banach spaces. Proc Amer Math Soc, 1983, 89: 633–636

    Article  MathSciNet  Google Scholar 

  17. Godefroy G, Kalton N J. Lipschitz-free Banach spaces. Studia Math, 2003, 159: 121–141

    Article  MathSciNet  Google Scholar 

  18. Gruber P M. Stability of isometries. Trans Amer Math Soc, 1978, 245: 263–277

    Article  MathSciNet  Google Scholar 

  19. Hyers D H, Ulam S M. On approximate isometries. Bull Amer Math Soc, 1945, 51: 288–292

    Article  MathSciNet  Google Scholar 

  20. Johnson W B. Communication. mathoverflow, 2012. http://mathoverflow.net/questions/72750/density-character-and-cardinality

  21. Mazur S, Ulam S. Sur les transformations isométriques d’espaces vectoriels normés. C R Acad Sci Paris, 1932, 194: 946–948

    MATH  Google Scholar 

  22. Omladič M, Šemrl P. On non linear perturbations of isometries. Math Ann, 1995, 303: 617–628

    Article  MathSciNet  Google Scholar 

  23. Qian S. ε-Isometric embeddings. Proc Amer Math oc, 1995, 123: 1797–1803

    MathSciNet  MATH  Google Scholar 

  24. Šemrl P, Väisälä J. Nonsurjective nearisometries of Banach spaces. J Funct Anal, 2003, 198: 268–278

    Article  MathSciNet  Google Scholar 

  25. Sobczyk A. Projection of the space (m) on its subspace co. Bull Amer Math Soc, 1941, 47: 938–947

    Article  MathSciNet  Google Scholar 

  26. Tabor J. Stability of surjectivity. J Approx Theory, 2000, 105: 166–175

    Article  MathSciNet  Google Scholar 

  27. Wolfe J. Injective Banach spaces of type C(T). Israel J Math, 1974, 18: 133–140

    Article  MathSciNet  Google Scholar 

  28. Zippin M. The separable extension problem. Israel J Math, 1977, 26: 372–387

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou, 362000, China

    Duanxu Dai

Authors
  1. Duanxu Dai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Duanxu Dai.

Additional information

This work was supported by the Natural Science Foundation of China (11601264), and the Natural Science Foundation of Fujian Province of China (2019J05103), and the Outstanding Youth Scientific Research Personnel Training Program of Fujian Province and the High level Talents Innovation and Entrepreneurship Project of Quanzhou City (2017Z032).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dai, D. Stability of ε-Isometries on ℒ∞-Spaces. Acta Math Sci 39, 1733–1742 (2019). https://doi.org/10.1007/s10473-019-0619-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-019-0619-2

Key words

2010 MR Subject Classification

Search

Navigation