Computable structures and operations on the space of continuous functions

摘要

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.