Open Access
2000/2001 Definability in Function Spaces
Alessandro Andretta, Alberto Marcone
Real Anal. Exchange 26(1): 285-310 (2000/2001).


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 $UC (X,d)$.


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Alessandro Andretta. Alberto Marcone. "Definability in Function Spaces." Real Anal. Exchange 26 (1) 285 - 310, 2000/2001.


Published: 2000/2001
First available in Project Euclid: 2 January 2009

zbMATH: 1015.28003
MathSciNet: MR1825510

Primary: 03E15 , 28A05

Keywords: borel hierarchy , projective hierarchy

Rights: Copyright © 2000 Michigan State University Press

Vol.26 • No. 1 • 2000/2001
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