Completion Theorem

Definition

For a Metric Space $(X,d)$ there exists complete Metric Space $(Y,d^{\prime })$ and an isometric embedding $e:X\to Y$ for which $e(X)$ is a dense subset of $Y$. The space $(Y,d^{\prime })$ is unique up to metric equivalence. The space $(Y,d^{\prime })$ is called the completion of $(X,d)$.