An incomplete metric space can be placed inside a complete metric space without changing its original distances. This process is called completion. Isometries and isometric embeddings express exact preservation of distance. The construction by equivalence classes of Cauchy sequences is the standard abstract method for completing a metric space. :::definition[Completion of a Metric Space] Let $(X,d)$ be a metric space. A complete metric space $(X^*,d^*)$ is called a $completion$ of $(X,d)$ if: (i) $X$ is a subspace of $X^*$. (ii) Every point of $X^*$ is the limit of a sequence in $X$. ::: :::example The real line $\mathbb{R}$ is the completion of $\mathbb{Q}$ with the usual metric. ::: :::example The closed interval $[0,1]$ is the completion of $(0,1)$, $[0,1)$, $(0,1]$, and $[0,1]$ with the usual metric. ::: :::example Every complete metric space is its own completion. ::: :::definition[Isometry] Let $(X,d)$ and $(X',d')$ be metric spaces. A mapping $f:X\to X'$ is called an $isometry$ if $$\begin{equation*} d'(f(x),f(y))=d(x,y),\qquad x,y\in X. \end{equation*}$$ ::: :::definition[Isometric Embedding] An isometry $f:X\to X'$ is called an $isometric embedding$ of $X$ into $X'$. ::: :::definition[Isometric Metric Spaces] Two metric spaces are called $isometric$ if there exists an onto isometry between them. ::: :::remark Every isometry is one-to-one. If $f(x)=f(y)$, then $d(x,y)=d'(f(x),f(y))=0$, so $x=y$. ::: :::theorem[Existence of Completion] Every metric space has a completion. ::: :::proof Given that $(X,d)$ is a metric space. To prove that $X$ has a completion. Let $\widehat X$ be the set of all Cauchy sequences in $X$. Define $\{x_n\}\sim\{y_n\}$ if $\lim_n d(x_n,y_n)=0$. This is an equivalence relation. Let $\widetilde X$ be the set of equivalence classes. Define $$\begin{equation*} \rho(\widetilde x,\widetilde y)=\lim_{n\to\infty}d(x_n,y_n), \end{equation*}$$ where $\{x_n\}\in\widetilde x$ and $\{y_n\}\in\widetilde y$. This is well-defined and is a metric. Map $x\in X$ to the class of the constant sequence $(x,x,x,\ldots)$. This map preserves distances exactly. The resulting metric space is complete, and every point is the limit of a sequence from the embedded copy of $X$. Hence every metric space has a completion. $\square$ ::: :::theorem[Uniqueness of Completion up to Isometry] Any two completions of a metric space are isometric. ::: :::proof Given that $(X_1,d_1)$ and $(X_2,d_2)$ are completions of $(X,d)$. To prove that $X_1$ and $X_2$ are isometric. For $u\in X_1$, choose a sequence $\{x_n\}$ in $X$ with $x_n\to u$ in $X_1$. Define $T(u)=\lim_{X_2}x_n$. Equivalent Cauchy sequences give the same limit, so $T$ is well-defined. If $u,v\in X_1$, then $$\begin{equation*} d_2(Tu,Tv)=\lim_{n\to\infty}d(x_n,y_n)=d_1(u,v). \end{equation*}$$ Thus $T$ is an isometry. Density of $X$ in both completions makes $T$ onto. Hence the completions are isometric. $\square$ ::: :::exercise [1] Define completion. [2] Define isometry. [3] Why is every isometry one-to-one? [4] What is the completion of $\mathbb{Q}$? ::: :::answer [1] A complete metric space containing the original space densely as a subspace. [2] A distance-preserving map. [3] Equal images have zero distance, so the original points have zero distance. [4] $\mathbb{R}$. ::: :::call-to-action[Continue to Constructed Metrics] subtitle: Finish the section with practice problems on constructing and verifying metrics. button: Next Lesson | /functional-analysis/metric-spaces/introduction-to-metric-spaces/constructed-metrics-and-practice-problems button-ghost: Previous Lesson | /functional-analysis/metric-spaces/introduction-to-metric-spaces/complete-function-spaces-and-sequence-problems :::