The completion of $C_c(\mathbb{R}^k)$ with $\|f - g\|_p$ as a metric

Question

In Rudin RCA, 3.15, he remarks that because $C_c(\mathbb{R}^k)$ is dense in $L^p(\mathbb{R}^k)$ (i.e. every point of $L^p(\mathbb{R}^k)$ is either in or a limit point of $C_c(\mathbb{R}^k)$) and because $L^p(\mathbb{R}^k)$ is a complete metric space, $L^p(\mathbb{R}^k)$ is the completion of $C_c(\mathbb{R}^k)$. My question is, is this completion of a metric space unique, because he uses the word "the". In other words is the completion of $C_c(\mathbb{R}^k)$ necessarily $L^p(\mathbb{R}^k)$?