Let $\Omega\subset\mathbb{R}^{N}$ be a bounded domain and $f_{1}:\Omega\to\mathbb{R}$ be an element of $C_{0}(\Omega)$. Define $C_{0}(\Omega) := \{f\in C(\overline{\Omega})\,|\, f|_{\partial\Omega}=0 \}$. Then, I can immediately see that \begin{align*} ||f_{1}||_{L^{2}(\Omega)}&=\bigg(\int_{\Omega}|f_{1}(x)|^{2}dx\bigg)^{\frac{1}{2}}\\ &\leq \bigg(\int_{\Omega}(\max_{\overline{\Omega}}|f_{1}|)^{2}dx\bigg)^{\frac{1}{2}}\\ &= |\Omega|^{\frac{1}{2}}\max_{\overline{\Omega}}|f_{1}|\\ &= C(\Omega)||f_{1}||_{C_{0}(\Omega)}<\infty \end{align*}
In fact, I can see that for $1\leq p \leq\infty$, I have $||f_{1}||_{L^{p}(\Omega)} \leq C(\Omega)||f_{1}||_{C_{0}(\Omega)}$. So, I can at least say that is embedded $C_{0}(\Omega)$ to $L^{p}(\Omega)$.
Now, my problems are the followings :
1. How to ensure that the embedding is compact at least for $1<p<\infty$?
2. We know that $L^{p}$-spaces are spaces for equivalent classes of functions up to zero Lebesgue Measure. So, how to give a precise meaning of the embedding to $L^{p}$ since we have two different objects here.
Any clear explanation is pretty much appreciated! Thank you!