Define $L_2(\Omega)$ as the completion of $C_c(\Omega)$ in scalar product norm

Question

I am reading Aubin's Applied Functional Analysis. In Chapter $6$, he defines $L_2(\Omega)$ as the completion of $C_c(\Omega)$ in the norm topology induced by the scalar product \begin{align*} \|f\|^2 = \int_{\Omega} \langle f(x), f(x) \rangle dx, \end{align*} where $\Omega \subset \mathbb R^n$ is an open set and $C_c(\Omega)$ denotes functions with compact support. He claims the space acquired by completion is indeed isomorphic to the $L_2$ space (with Lebesgue integral) without proof. I am wondering how this isomorphism is defined?