Analytic computable structure theory and $L^p$ spaces

Abstract

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,

On the Complexity of Classifying Lebesgue Spaces

“…Our proof implies that Lebesgue spaces are characterized among all Banach spaces by the success of an algorithm which attempts to build a formal disintegration of the given space. The notion of a formal disintegration is a development of the earlier notion of disintegration [30], [8]. The associated independence property vaguely resembles S-independence [9,11] in discrete completely decomposable groups, as well some other notions in the literature on discrete countable p-groups (c.f.…”

Analytic computable structure theory and $$L^p$$-spaces part 2

“…Together, these results determine the degrees of categoricity of separable spaces of the form L p (Ω) when Ω is purely atomic. In the paper preceding this, Clanin, McNicholl, and Stull showed that L p (Ω) is computably categorical when Ω is separable and nonatomic [3]. Here, we complete the picture by determining the degrees of categoricity of separable L p spaces whose underlying measure spaces are atomic but not purely atomic.…”

“…Here, we cover pertinent notions regarding external and internal direct sums of Banach spaces and the notion of complemented subspaces of an internal direct sum of Banach spaces. We also summarize additional background material from [3]. We will assume our field of scalars consists of the complex numbers, but all results hold for the field of real numbers as well.…”

“…With a pair (B # 0 , B # 1 ) of presentations of Banach spaces, there is an associated class of computable functions from B # 0 into B # 1 . Definitions of concepts such as these have become fairly well-known; we refer the reader to whom they are unfamiliar to Section 2.2.2 of [3].…”

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.…”