Computable copies of ℓ p 1

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 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.

Analytic computable structure theory and $L^p$ spaces

“…Every separable atomic space has countably many atoms. So, the purely atomic case has already been resolved; namely ℓ p is computably categorical only when p = 2 and ℓ p n is computably categorical for all p, n [22], [23]. So, only the L p spaces of non-atomic but not purely atomic spaces remain to be examined, and a future paper will do so.…”

Degrees of and lowness for isometric isomorphism

“…Recently, a program to adapt the framework of computable structure theory to the continuous setting, that is, to metric structures, has emerged (see Melnikov [14], Melnikov and Ng [15], Melnikov and Nies [16], Nies and Solecki [17], McNicholl [12], Clanin, McNicholl and Stull [4] and Brown and McNicholl [2]). We contribute to this direction by introducing the study of lowness for isometric isomorphism of metric structures.…”

Computing the exponent of a Lebesgue space

“…In the case of countable structures, this is achieved by numbering the elements of the structure in a suitable way; namely so that the induced relations and operations on the natural numbers are computable. Recently, the field has expanded its purview by investigating metric structures such as metric spaces and Banach spaces (see eg Melnikov [19], Melnikov and Nies [21], Melnikov and Ng [20], McNicholl [17], Clanin, McNicholl and Stull [6], and Brown and McNicholl [3]). In the case of Banach spaces, a computable presentation is a numbering of a linearly dense sequence in such a way that the norm and the vector space operations can be computed.…”