Could someone please explain to me, why the embedding $\iota \colon C^0( \overline{\Omega}) \to L^2(\Omega)$ is not compact?
$\Omega=(0,1)$ and $ \overline{\Omega}$ denotes the closure of $\Omega$.
I already got the hint to use the sequence $f_k(x):=\sin(k\pi x) $ with $k \in \mathbb N$. Then I can prove, that $(f_k)$ converges weakly to $0$. But how to continue the proof?