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  1. What is the Brezis-Kato theorem?Where can I find the details?

  2. $\Omega$ 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 $K(\Omega)$?

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For the second question, $C_0((\Omega)$ is not equal to $K(\Omega)$. Take a sequence of nonzero functions $\phi_i\in K(\Omega)$ that have pairwise disjoint supports, satisfying $\phi_i\to0$ uniformly, and the support of $\phi_i$ tends to the boundary of $\Omega$ as $i\to\infty$. Then the sequence of functions $u_i=\phi_1+\ldots+\phi_i$ converges uniformly but the limit is not compactly supported in $\Omega$.

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