@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 -
