Computable copies of ℓp¹

Article type: Research Article

Authors: McNicholl, Timothy H.

Affiliations: Department of Mathematics, Iowa State University, Ames, Iowa 50011, USA. mcnichol@iastate.edu

Note: [1] Section 5.3 previously appeared in the conference proceedings of Computability in Europe 2015 [8].

Abstract: Suppose p is a computable real so that p⩾1. It is shown that the halting set can compute a surjective linear isometry between any two computable copies of ℓp. It is also shown that this result is optimal in that when p≠2 there are two computable copies of ℓp with the property that any oracle that computes a linear isometry of one onto the other must also compute the halting set. Thus, ℓp is Δ20-categorical and is computably categorical if and only if p=2. It is also demonstrated that there is a computably categorical Banach space that is not a Hilbert space. These results hold in both the real and complex case.

Keywords: Computable analysis, computable model theory, ℓ^p spaces

DOI: 10.3233/COM-160065

Journal: Computability, vol. 6, no. 4, pp. 391-408, 2017