Effective Banach spaces

TL;DR

This thesis explores the relationship between higher order recursion theory and computability in analysis, demonstrating an uncountably infinite computability structure on a Banach space of operators on almost periodic functions.

Contribution

It introduces an uncountable computability structure on a Banach space, addressing a key question in computability in analysis and physics.

Findings

Existence of an uncountable computability structure

Application to Banach space of bounded linear operators

Insight into higher order recursion theory and analysis

Abstract

This thesis addresses Pour-El and Richards' fourth question from their book "Computability in analysis and physics", concerning the relation between higher order recursion theory and computability in analysis. Among other things it is shown that there is a computability structure that is uncountable. The example given is a structure on the Banach space of bounded linear operators on the set of almost periodic functions.