Computable copies of ℓ^p

2015/04/18 by Timothy H. McNicholl, McNicholl, Timothy H.
Computer Science · Mathematics · #03D45 #03D78 #46B25 #Benford’s Law and Fraud Detection #Cellular Automata and Applications #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.1504.04664

openalex publication_date 2015/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

\beginabstract 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 ℓ^p. It is also shown that this result is optimal in that when p ≠ 2 there are two computable copies of ℓ^p with the property that any oracle that computes a linear isometry of one onto the other must also compute the halting set. Thus, ℓ^p is Δ₂⁰-categorical and is computably categorical if and only if p = 2. It is also shown that there is a computably categorical Banach space that is not a Hilbert space and that ℓ^p 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.

Related

• The isometry degree of a computable copy of ℓ^p
• Analytic computable structure theory and L^p spaces
• A note on the computable categoricity of l^p spaces
• Analytic computable structure theory and L^p-spaces part 2
• On the complexity of classifying Lebesgue spaces