Advertisement

Analytic computable structure theory and \(L^p\)-spaces part 2

  • Tyler Brown
  • Timothy H. McNichollEmail author
Article
  • 1 Downloads

Abstract

Suppose \(p \ge 1\) is a computable real. We extend previous work of Clanin, Stull, and McNicholl by determining the degrees of categoricity of the separable \(L^p\) spaces whose underlying measure spaces are atomic but not purely atomic. In addition, we ascertain the complexity of associated projection maps.

Keywords

Computable analysis Computable structure theory Functional analysis 

Mathematics Subject Classification

03D45 03D78 46B04 

Notes

Acknowledgements

We thank Diego Rojas for proofreading and several valuable suggestions. We also thank the referee for suggesting many significant improvements.

References

  1. 1.
    Anderson, B., Csima, B.: Degrees that are not degrees of categoricity. Notre Dame J. Form. Log. (2016).  https://doi.org/10.1215/00294527-3496154 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cembranos, P., Mendoza, J.: Banach Spaces of Vector-Valued Functions, Lecture Notes in Mathematics, vol. 1676. Springer, Berlin (1997)CrossRefGoogle Scholar
  3. 3.
    Clanin, J., McNicholl, T.H., Stull, D.M.: Analytic computable structure theory and \(L^p\) spaces. Fund. Math. 244(3), 255–285 (2019)Google Scholar
  4. 4.
    Csima, B.F., Franklin, J.N.Y., Shore, R.A.: Degrees of categoricity and the hyperarithmetic hierarchy. Notre Dame J. Form. Log. 54(2), 215–231 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Fokina, E.B., Kalimullin, I., Miller, R.: Degrees of categoricity of computable structures. Arch. Math. Logic 49(1), 51–67 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    McNicholl, T.H.: A note on the computable categoricity of \(\ell ^p\) spaces. In: Evolving Computability, Lecture Notes in Computer Science, vol. 9136, pp. 268–275. Springer, Cham (2015)Google Scholar
  7. 7.
    McNicholl, T.H.: Computable copies of \(\ell ^p\). Computability 6(4), 391–408 (2017)Google Scholar
  8. 8.
    Melnikov, A.G.: Computably isometric spaces. J. Symb. Log. 78(4), 1055–1085 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Melnikov, A.G., Nies, A.: The classification problem for compact computable metric spaces. In: The Nature of Computation, Lecture Notes in Computer Science, vol. 7921, pp. 320–328. Springer, Heidelberg (2013)Google Scholar
  10. 10.
    Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Perspectives in Mathematical Logic. Springer, Berlin (1989)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsIowa State UniversityAmesUSA
  2. 2.Columbia CollegeColumbiaUSA

Personalised recommendations