Affiliations: Department of Mathematics, Iowa State University, Ames, Iowa 50011, USA. firstname.lastname@example.org
Note:  Section 5.3 previously appeared in the conference proceedings of Computability in Europe 2015 .
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 and that ℓp is linearly isometric to a computable Banach space if and only if p is computable. These results hold in both the real and complex case.
Keywords: Computable analysis, computable model theory, ℓp spaces
Journal: Computability, vol. Preprint, no. Preprint, pp. 1-18, 2016