Computable structures and operations on the space of continuous functions

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.

Analytic computable structure theory and $L^p$ spaces

“…New results on Banach spaces began to appear in 2014. First, Melnikov and Ng showed that C[0, 1] is not computably categorical [25]. Then, McNicholl extended the work of Pour-El and Richards by showing that ℓ p is computably categorical only when p = 2 and that ℓ p is ∆ 0 2 -categorical when p is a computable real.…”

Uniform Procedures in Uncountable Structures

“…Nonetheless, it is much more natural to ask whether such results can be extended to definability of operations rather than of open subsets (e.g., + on a normed space that makes it a Banach space). See [16,17] for various natural examples of this sort.…”

“…This paper contributes to the general program (e.g. [17,18,14,16,20,19]) which aims to extend the key ideas and techniques of countable effective algebra to the study of uncountable structures. We follow the standard terminology of effective algebra [1,5] and say that a computable presentation (a computable copy, a constructivization) of a countably infinite algebraic structure A is an isomorphic copy C of A upon the domain ω in which all predicates and functions are (uniformly) Turing computable.…”

Computable copies of ℓ p 1

“…Ng have investigated computable categoricity questions with regards to the space C[0, 1] of continuous functions on the unit interval with the supremum norm. In particular, they have shown that C[0, 1] is not computably categorical as a metric space nor as a Banach space [9], [10]. The study of computable categoricity for countable structures goes back at least as far as the 1961 work of A.I.…”