Computable structure theory on Banach spaces

Abstract

In this dissertation we investigate computability notions on several different Banach spaces, namely the separable $L^p$-spaces and $C[0,1]$. It was demonstrated by McNicholl \cite{TM} that the halting problem is a necessary and sufficient condition for the existence of computable isometric isomorphisms between any two computable representations of the purely atomic $L^p$-spaces (e.g. $\ell^p$) where the underlying measure space is generated by finitely many atoms. In the case where the underlying measure space is generated by finitely many atoms (such as in $\ell^p_n$), McNicholl also proved that it is always possible to find an algorithm that computes isometric isomorphisms between any two computable representations. Clanin, McNicholl, and Stull \cite{CMS} proved a similar result. Namely they proved that for any two computable representations of a non-atomic $L^p$-space (e.g. $L^p[0,1]$) there is always a computable isometric isomorphism between them. We both continue and complete the classification of the separable $L^p$-spaces up to degree of categoricity by investigating the hybrid $L^p$-spaces, whose underlying measure spaces consist of both atomic and non-atomic parts, and determine how much computational power is necessary and sufficient to compute isometric isomorphisms between any two copies of these spaces. Secondly, we continue a line of inquiry initialized by Melnikov and Ng in 2014, who proved that for $C[0,1]$ (i.e. the Banach space of all continuous functions on the closed unit interval) there is a pair of computable representations between which there is no computable isometric isomorphism. They achieved this by constructing one of the representations in such a manner that the constant unit function \textbf{1} is not computable, contrasting with the other representation in which \textbf{1} is computable. We show in Chapter 5 that given any computable representation of $C[0,1]$ as a Banach space the halting set always computes \textbf{1}. We also determine how much extra computational power beyond that of the halting set is sufficient to compute the modulus operator $|\cdot|$ within any computable representation. Lastly, we use these two results to determine how much power is sufficient to compute an isometric isomorphism between any two computable representations of a restricted class of representations of $C[0,1]$