Definability in Function Spaces

Abstract

We study $C (X) \cap L^p (X)$, the set of all continuous functions in $L^p (X)$, as a subspace of $L^p (X)$, and show that it is $\Pi^0_3$-complete when $X$ is Polish locally compact, while it is $\pi^1_1$-complete when $X$ is Polish not $\sigma$-compact. We also show that the subspace of Riemann integrable functions and, for every $k =1,2, \dots, \infty$, $C^k (\mathbb{R})$ are $\Pi^0_3$-complete in $L^p (\mathbb{R})$. In contrast the subspace of all everywhere differentiable functions is $\pi^1_1$-complete in $L^p (\mathbb{R})$. If $X$ is locally compact, we consider $C (X)$ endowed with the compact-open topology and establish the complexity of some of its subspaces, including $C_0(X)$, $C_{00}(X)$ and $UC (X,d)$.