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 $p\ge 1$ is a computable real, and if $\Omega$ is a nonzero, non-atomic, and separable measure space, then every computable presentation of $L^p(\Omega )$ 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 $\Omega$ that does not have a computable presentation even though $L^p(\Omega )$ does for every computable real $p\ge 1$.

References in corpus (1)

• Complexity theory for spaces of integrable functions