Computable structures and operations on the space of continuous functions
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.