Computable copies of ℓp
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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 Rᵖ. It is also shown that this result is optimal in that when p /= 2 there are two computable copies of Rᵖ with the property that any oracle that computes a linear isometry of one onto the other must also compute the halting set. Thus, Rᵖ is ∆⁰-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.
Comments
This is a manuscript of an article published as McNicholl, Timothy H. "Computable copies of ℓp." Computability 6, no. 4 (2017): 391-408. The final publication is available at IOS Press through http://dx.doi.org/10.3233/COM-160065. Posted with permission.