Many standard spaces used in analysis are complete. The proofs follow a common pattern: begin with a Cauchy sequence in the metric, extract coordinatewise or pointwise candidate limits, prove that the candidate belongs to the same space, and finally prove convergence in the original metric. This lesson records the main completeness results needed later. :::theorem[Completeness of Finite-Dimensional p-Metric Spaces] Let $X=\mathbb{R}^n$ and let $d_p(x,y)=(\sum_{i=1}^{n}|x_i-y_i|^p)^{1/p}$ for $p\geq 1$. Then $(X,d_p)$ is complete. ::: :::proof Given that $X=\mathbb{R}^n$ with metric $d_p$. To prove that $(X,d_p)$ is complete. Let $\{x^{(m)}\}$ be Cauchy and write $x^{(m)}=(x_1^{(m)},\ldots,x_n^{(m)})$. Since $$\begin{equation*} |x_i^{(m)}-x_i^{(r)}|\leq d_p(x^{(m)},x^{(r)}), \end{equation*}$$ each coordinate sequence is Cauchy in $\mathbb{R}$. Let $x_i=\lim_m x_i^{(m)}$ and $x=(x_1,\ldots,x_n)$. Finite coordinate convergence gives $d_p(x^{(m)},x)\to 0$. Hence $(X,d_p)$ is complete. $\square$ ::: :::corollary The space $\mathbb{R}^n$ with the maximum metric $d_\infty(x,y)=\max_{1\leq i\leq n}|x_i-y_i|$ is complete. ::: :::proof Given that $\{x^{(m)}\}$ is Cauchy in $d_\infty$. To prove that it converges. Each coordinate sequence is Cauchy in $\mathbb{R}$ and hence converges. Since there are finitely many coordinates, convergence is in the maximum metric. Hence the maximum metric space is complete. $\square$ ::: :::theorem[Completeness of Summable Sequence Spaces] Let $p\geq 1$ and let $X=\ell_p$ with metric $d_p$. Then $(X,d_p)$ is complete. ::: :::proof Given that $X=\ell_p$. To prove that $X$ is complete. Let $\{x^{(n)}\}$ be a Cauchy sequence, where $x^{(n)}=\{x_k^{(n)}\}$. Each fixed coordinate sequence is Cauchy, so define $x_k=\lim_n x_k^{(n)}$. The Cauchy condition gives uniform control of tails, which shows $\sum |x_k|^p<\infty$ and $d_p(x^{(n)},x)\to 0$. Hence $\ell_p$ is complete. $\square$ ::: :::theorem[Completeness of Bounded Sequence Spaces] The space of all bounded scalar sequences with metric $d(x,y)=\sup_k |x_k-y_k|$ is complete. ::: :::proof Given a Cauchy sequence of bounded sequences. To prove that it converges in the supremum metric. Coordinatewise limits define a sequence $x$. A fixed Cauchy comparison term shows that $x$ is bounded. The Cauchy condition then passes to the coordinatewise limit and gives $\sup_k |x_k^{(n)}-x_k|\to 0$. Hence the bounded sequence space is complete. $\square$ ::: :::theorem[Completeness of Bounded Function Spaces] Let $B(S)$ be the set of bounded scalar-valued functions on a nonempty set $S$ with the uniform metric. Then $B(S)$ is complete. ::: :::proof Given a Cauchy sequence $\{f_n\}$ in $B(S)$. To prove that it converges in $B(S)$. For each $x\in S$, the scalar sequence $\{f_n(x)\}$ is Cauchy. Define $f(x)=\lim_n f_n(x)$. A fixed comparison function shows that $f$ is bounded. Passing the Cauchy estimate to the pointwise limit gives uniform convergence $f_n\to f$. Hence $B(S)$ is complete. $\square$ ::: :::theorem[Completeness of Continuous Functions with Uniform Metric] Let $X=C[a,b]$ with the uniform metric. Then $X$ is complete. ::: :::proof Given a Cauchy sequence in $C[a,b]$. To prove that it converges in $C[a,b]$. It converges uniformly to a bounded function by completeness of bounded functions. Since the uniform limit of continuous functions is continuous, the limit belongs to $C[a,b]$. Hence $C[a,b]$ is complete. $\square$ ::: :::exercise [1] Why do Cauchy sequences in $\mathbb{R}^n$ give Cauchy coordinate sequences? [2] State the completeness theorem for $B(S)$. [3] Why is $C[a,b]$ complete under the uniform metric? ::: :::answer [1] Each coordinate difference is bounded above by the metric distance. [2] $B(S)$ is complete under the uniform metric. [3] Uniform limits of continuous functions are continuous. ::: :::call-to-action[Continue to Coordinatewise Convergence] subtitle: Next, compare coordinatewise, pointwise, and uniform convergence. button: Next Lesson | /functional-analysis/metric-spaces/introduction-to-metric-spaces/coordinatewise-and-uniform-convergence button-ghost: Previous Lesson | /functional-analysis/metric-spaces/introduction-to-metric-spaces/complete-and-incomplete-metric-spaces :::