Computable structures and operations on the space of continuous functions

Alexander G. Melnikov ¹   ; Keng Meng Ng ²

¹ Institute of Natural and Mathematical Sciences Massey University Auckland, New Zealand
² Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University Singapore

Fundamenta Mathematicae, Tome 233 (2016) no. 2, pp. 101-141

Résumé

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.

Détail

DOI : 10.4064/fm36-12-2015

Keywords: ideas machinery effective algebra investigate computable structures space continuous functions unit interval sup has infinitely many computable structures non equivalent computable isometry investigate usual operations necessarily computable every computable structure among other results there computable structure which computes scalar multiplication does compute operation pointwise multiplication functions another unexpected result there exists computable structure making computable banach algebra results have implications study number computable structures various commonly signatures