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Computable structures and operations on the space of continuous functions

Tom 233 / 2016

Alexander G. Melnikov, Keng Meng Ng Fundamenta Mathematicae 233 (2016), 101-141 MSC: Primary 03F60; Secondary 03C57. DOI: 10.4064/fm36-12-2015 Opublikowany online: 7 December 2015

Streszczenie

We use ideas and machinery of effective algebra to investigate computable structures on the space $C[0,1]$ of continuous functions on the unit interval. We show that $(C[0,1], \sup)$ has infinitely many computable structures non-equivalent up to a computable isometry. We also investigate if the usual operations on $C[0,1]$ are necessarily computable in every computable structure on $C[0,1]$. Among other results, we show that there is a computable structure on $C[0,1]$ which computes $+$ and the scalar multiplication, but does not compute the operation of pointwise multiplication of functions. Another unexpected result is that there exists more than one computable structure making $C[0,1]$ a computable Banach algebra. All our results have implications for the study of the number of computable structures on $C[0,1]$ in various commonly used signatures.

Autorzy

  • Alexander G. MelnikovInstitute of Natural and Mathematical Sciences
    Massey University
    Auckland, New Zealand
    e-mail
  • Keng Meng NgDivision of Mathematical Sciences
    School of Physical and Mathematical Sciences
    Nanyang Technological University
    Singapore
    e-mail

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