Degree Type

Dissertation

Date of Award

2019

Degree Name

Doctor of Philosophy

Department

Mathematics

Major

Mathematics

First Advisor

Timothy H. McNicholl

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 $C[0,1]$.

Copyright Owner

Tyler Anthony Brown

Language

en

File Format

application/pdf

File Size

76 pages

Share

COinS