Computably Isometric Spaces

Melnikov, Alexander G.

doi:10.2178/jsl.7804030

Cited by 52 publications

(75 citation statements)

References 30 publications

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“…In particular, Melnikov and Nies showed that computably presentable compact metric spaces are ∆ 0 3 -categorical and that there is a computably presentable Polish space that is not ∆ 0 2 -categorical [26]. At the same time, Melnikov showed that the Cantor space, Urysohn space, and all separable Hilbert spaces are computably categorical (as metric spaces), but that (as a metric space) C[0, 1] is not [24]. Recently, Greenberg, Knight, Melnikov, and Turetsky announced an analog of Goncharov's syntactic characterization of relative computable categoricity for metric spaces [13].…”

Section: 23mentioning

confidence: 99%

Analytic computable structure theory and $L^p$ spaces

2019

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We continue the investigation of analytic spaces from the perspective of computable structure theory. We show that if p ≥ 1 is a computable real, and if Ω is a nonzero, non-atomic, and separable measure space, then every computable presentation of L p (Ω) is computably linearly isometric to the standard computable presentation of L p [0, 1]; in particular, L p [0, 1] is computably categorical. We also show that there is a measure space Ω that does not have a computable presentation even though L p (Ω) does for every computable real p ≥ 1.Laboratoire lorrain de recherche en informatique et ses applications,

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Section: 23mentioning

confidence: 99%

Analytic computable structure theory and $L^p$ spaces

2019

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“…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.…”

Section: 2mentioning

confidence: 99%

Computable copies of ℓ p 1

McNicholl

2017

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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 ∆ 0 2 -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.

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“…We continue here the program, recently initiated by Melnikov and Nies (see [9], [8]), of utilizing the tools of computable analysis to investigate the effective structure theory of metric structures, in particular L p spaces where p ≥ 1 is computable. Specifically, we seek to classify the L p spaces that are computably categorical in that they have exactly one computable presentation up to computable isometric isomorphism.…”

Section: Introductionmentioning

confidence: 99%