Computable structures and operations on the space of continuous functions

Alexander G. Melnikov, and Keng Meng Ng. "Computable structures and operations on the space of continuous functions." Fundamenta Mathematicae 233.2 (2016): 101-141. <http://eudml.org/doc/283338>.

@article{AlexanderG2016,
abstract = {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.},
author = {Alexander G. Melnikov, Keng Meng Ng},
journal = {Fundamenta Mathematicae},
keywords = {continuous functions; effective categoricity; effective Banach space},
language = {eng},
number = {2},
pages = {101-141},
title = {Computable structures and operations on the space of continuous functions},
url = {http://eudml.org/doc/283338},
volume = {233},
year = {2016},
}

TY - JOUR
AU - Alexander G. Melnikov
AU - Keng Meng Ng
TI - Computable structures and operations on the space of continuous functions
JO - Fundamenta Mathematicae
PY - 2016
VL - 233
IS - 2
SP - 101
EP - 141
AB - 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.
LA - eng
KW - continuous functions; effective categoricity; effective Banach space
UR - http://eudml.org/doc/283338
ER -