A Note on the Computable Categoricity of \(\ell ^p\) Spaces

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9136)

Abstract

Suppose that \(p\) is a computable real and that \(p \ge 1\). We show that in both the real and complex case, \(\ell ^p\) is computably categorical if and only if \(p = 2\). The proof uses Lamperti’s characterization of the isometries of Lebesgue spaces of \(\sigma \)-finite measure spaces.

Notes

Acknowledgement

The author thanks the anonymous referees who made helpful comments. The author’s participation in CiE 2015 was funded by a Simons Foundation Collaboration Grant for Mathematicians.

References

  1. 1.
    Ash, C.J., Knight, J.: Computable Structures and the Hyperarithmetical Hierarchy. Studies in Logic and the Foundations of Mathematics, vol. 144. North-Holland Publishing Co., Amsterdam (2000)MATHGoogle Scholar
  2. 2.
    Banach, S.: Theory of Linear Operations. North-Holland Mathematical Library, vol. 38. North-Holland Publishing Co., Amsterdam (1987). Translated from the French by F. Jellett, With comments by A. Pełczyński and Cz. BessagaMATHCrossRefGoogle Scholar
  3. 3.
    Cooper, S.B.: Computability Theory. Chapman & Hall/CRC, Boca Raton (2004)MATHGoogle Scholar
  4. 4.
    Fleming, R.J., Jamison, J.E.: Isometries on Banach Spaces: Function Spaces. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 129. Chapman & Hall/CRC, Boca Raton (2003)Google Scholar
  5. 5.
    Goncharov, S.: Autostability and computable families of constructivizations. Algebr. Log. 17, 392–408 (1978). English translationMathSciNetCrossRefGoogle Scholar
  6. 6.
    Harizanov, V.S.: Pure computable model theory. Handbook of Recursive Mathematics. Volume 1, Studies in Logic and the Foundations of Mathematics, vol. 138, pp. 3–114. North-Holland, Amsterdam (1998)Google Scholar
  7. 7.
    Lamperti, J.: On the isometries of certain function-spaces. Pac. J. Math. 8, 459–466 (1958)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Melnikov, A.G.: Computably isometric spaces. J. Symb. Log. 78(4), 1055–1085 (2013)MATHCrossRefGoogle Scholar
  9. 9.
    Melnikov, A.G., Ng, K.M.: Computable structures and operations on the space of continuous functions. Available at https://dl.dropboxusercontent.com/u/4752353/ Homepage/C[0,1]_final.pdf
  10. 10.
    Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Springer, Berlin (1989)MATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Iowa State UniversityAmesUSA

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