Continuous Domains for Function Spaces Using Spectral Compactification

TL;DR

This paper develops a method to create continuous domains for function spaces over non-core-compact topological spaces using spectral compactification, enabling better computational analysis in functional analysis and PDEs.

Contribution

It introduces a spectral compactification approach to construct continuous domains for function spaces over non-core-compact spaces, bridging the gap with computable analysis.

Findings

Spectral compactification relates non-core-compact spaces to continuous domains.

Enables computations in native function spaces while performing analysis in the compactified domain.

Provides a Galois connection for transitioning between original and compactified function spaces.

Abstract

We introduce a continuous domain for function spaces over topological spaces which are not core-compact. Notable examples of such topological spaces include the real line with the upper limit topology, which is used in solution of initial value problems with temporal discretization, and various infinite dimensional Banach spaces which are ubiquitous in functional analysis and solution of partial differential equations. If a topological space $\mathbb{X}$ is not core-compact and $\mathbb{D}$ is a non-singleton bounded-complete domain, the function space $[\mathbb{X} \to \mathbb{D}]$ is not a continuous domain. To construct a continuous domain, we consider a spectral compactification $\mathbb{Y}$ of $\mathbb{X}$ and relate $[\mathbb{X} \to \mathbb{D}]$ with the continuous domain $[\mathbb{Y} \to \mathbb{D}]$ via a Galois connection. This allows us to perform computations in the native structure $[\mathbb{X} \to \mathbb{D}]$ while computable analysis is performed in the continuous domain $[\mathbb{Y} \to \mathbb{D}]$, with the left and right adjoints used for moving between the two function spaces.