What is the Brezis-Kato theorem?Where can I find the details?
$\Omega$ is an open subset of $R^N$ $K(\Omega):=\{u\in C(\Omega):supp\ u \text{ is a compact subset of }\Omega\},$ $BC(\Omega):=\{u\in C(\Omega):|u|_{\infty}=\sup_{x\in \Omega}|u(x)|<\infty\},$
The space $C_0(\Omega)$ is the closure of $K(\Omega)$ in $BC(\Omega)$ with respect to the uniform norm.
Does the space $C_0(\Omega)$ equal to $K(\Omega)$?