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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm36-12-2015,
author = {Alexander G. Melnikov and Keng Meng Ng},
title = {Computable structures and operations on the space of continuous functions},
journal = {Fundamenta Mathematicae},
volume = {233},
year = {2016},
pages = {101-141},
zbl = {06575007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm36-12-2015}
}
Alexander G. Melnikov; Keng Meng Ng. Computable structures and operations on the space of continuous functions. Fundamenta Mathematicae, Tome 233 (2016) pp. 101-141. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm36-12-2015/