Skip to main content
SpringerLink
Log in

A Space with No Unconditional Basis that Satisfies the Johnson–Lindenstrauss Lemma

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

We give the first example of a nontrivial twisted Hilbert space that satisfies the Johnson–Lindenstrauss lemma. This space has no unconditional basis. We also show that such a space gives a partial answer to a question of Mascioni.

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, vol. 233. Springer, New York (2006)

  2. Benyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis, vol. 48. American Mathematical Society Colloquium Publications, Providence (1998)

    MATH  Google Scholar 

  3. Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Grundlehren der Mathematischen Wissenschaften, vol. 223. Springer, Berlin (1976)

    Book  Google Scholar 

  4. Bourgain, J., Tzafriri, L., Kalton, N.J.: Geometry of finite dimensional subspaces and quotients of Lp. GAFA, Springer Lecture Notes 1376, 138–175 (1989)

    MATH  Google Scholar 

  5. Cabello Sánchez, F., Castillo, J.M.F., Suárez, J.: On strictly singular nonlinear centralizers. Nonlinear Anal. 75(7), 3313–3321 (2012)

    Article  MathSciNet  Google Scholar 

  6. Cabello Sánchez, F.: Nonlinear centralizers in homology. Math. Ann. 358(3–4), 779–798 (2014)

    Article  MathSciNet  Google Scholar 

  7. Carro, M.J., Cerdà, J., Soria, J.: Commutators and interpolation methods. Ark. Mat. 33(2), 199–216 (1995)

    Article  MathSciNet  Google Scholar 

  8. Casazza, P.G., Nielsen, N.J.: A Banach space with a symmetric basis which is of weak cotype 2 but not of cotype 2. Stud. Math. 157(1), 1–16 (2003)

    Article  MathSciNet  Google Scholar 

  9. Casazza, P., Shura, T.J.: Tsirelson’s Space. With an Appendix by J. Baker, O. Slotterbeck and R. Aron. Lecture Notes in Mathematics, vol. 1363. Springer, Berlin (1989)

    Google Scholar 

  10. Castillo, J.M.F., González, M.: Three-Space Problems in Banach Space Theory. Lecture Notes in Mathematics, vol. 1667. Springer, Berlin (1997)

    Book  Google Scholar 

  11. Castillo, J.M.F., Ferenczi, V., González, M.: Singular twisted sums generated by complex interpolation. Trans. Am. Math. Soc. 369, 4671–4708 (2017)

    Article  MathSciNet  Google Scholar 

  12. John, F.: Extremum Problems with Inequalities as Subsidiary Conditions. Courant Anniversary Volume, pp. 187–204. Interscience, New York (1948)

  13. Johnson, W.B., Lindenstrauss, J.: Extensions of Lipschitz mappings into a Hilbert space. In: Conference in Modern Analysis and Probability (New Haven, Conn., 1982). Contemporary Mathematics, vol. 26, pp. 189–206. Amer. Math. Soc., Providence (1984)

  14. Johnson, W.B., Naor, A.: The Johnson-Lindenstrauss lemma almost characterizes Hilbert space, but not quite. Discrete Comput. Geom. 43(3), 542–553 (2010)

    Article  MathSciNet  Google Scholar 

  15. Kalton, N.J.: Nonlinear commutators in interpolation theory. Mem. Am. Math. Soc. 385, 1–85 (1988)

    MathSciNet  MATH  Google Scholar 

  16. Kalton, N.J.: Differentials of complex interpolation processes for Köthe function spaces. Trans. Am. Math. Soc. 333(2), 479–529 (1992)

    MathSciNet  MATH  Google Scholar 

  17. Kalton, N.J.: Twisted Hilbert spaces and unconditional structure. J. Inst. Math. Jussieu 2(3), 401–408 (2003)

    Article  MathSciNet  Google Scholar 

  18. Kalton, N.J., Montgomery-Smith, S.: Interpolation of Banach spaces. In: Johnson, W.B., Lindenstrauss, J. (eds.) Handbook of Geometry of Banach Spaces, vol. 2, pp. 1131–1175. Elsevier, Amsterdam (2003)

    Chapter  Google Scholar 

  19. Kalton, N.J., Peck, N.T.: Twisted sums of sequence spaces and the three space problem. Trans. Am. Math. Soc. 255, 1–30 (1979)

    Article  MathSciNet  Google Scholar 

  20. Mascioni, V.: On weak cotype and weak type in Banach spaces. Note Mat. 8(1), 67–110 (1988)

    MathSciNet  MATH  Google Scholar 

  21. Mascioni, V.: On Banach spaces isomorphic to their duals. Houst. J. Math. 19(1), 27–38 (1993)

    MathSciNet  MATH  Google Scholar 

  22. Pisier, G.: Volume of Convex Bodies and Banach Space Geometry. Cambridge Tracts in Mathematics, vol. 94. Cambridge University Press, Cambridge (1989)

  23. Suárez de la Fuente, J.: The Kalton centralizer on \(L_p[0,1]\) is not strictly singular. Proc. Am. Math. Soc. 141(10), 3447–3451 (2013)

  24. Suárez de la Fuente, J.: A weak Hilbert space that is a twisted Hilbert space. J. Inst. Math. Jussieu (2018). https://doi.org/10.1017/S1474748018000221. (to appear)

Download references

Acknowledgements

We are grateful to the referee for careful reading of the manuscript and valuable suggestions that lead us to improve considerably the quality and presentation of the paper.

Author information

Authors and Affiliations

  1. Instituto de Matemáticas Imuex, Universidad de Extremadura, Avenida de Elvas s/n, 06011, Badajoz, Spain

    Jesús Suárez de la Fuente

Authors
  1. Jesús Suárez de la Fuente
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jesús Suárez de la Fuente.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The author was supported in part by Project MTM2016-76958-C2-1-P and Project IB16056.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Suárez de la Fuente, J. A Space with No Unconditional Basis that Satisfies the Johnson–Lindenstrauss Lemma. Results Math 74, 126 (2019). https://doi.org/10.1007/s00025-019-1047-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-019-1047-2

Keywords

Mathematics Subject Classification

Search

Navigation