Having moved through the preceding compactness ideas, we now study Arzela-Ascoli compactness criterion. The focus keyword for this lesson is Arzela-Ascoli compactness criterion. This page records the definitions, theorems, proofs, examples, and practice items needed for undergraduate work in compactness and its consequences in metric spaces. :::theorem[Arzela-Ascoli Theorem] Let $K$ be a closed subset of $C[0,1]$ with the uniform metric. Then $K$ is compact if and only if $K$ is uniformly bounded and equicontinuous. ::: :::proof Given that $K$ is closed in $C[0,1]$. To prove the criterion. Compactness gives uniform boundedness and, by a finite-cover argument in the uniform metric, equicontinuity. Conversely, uniform boundedness and equicontinuity allow a diagonal subsequence on a countable dense set; equicontinuity upgrades convergence to uniform convergence. Closedness keeps the limit in $K$. $\square$ ::: :::solved-problem The family of all continuous $f:[0,1]\to[0,1]$ satisfying $|f(s)-f(t)|\leq |s-t|$ is compact. ::: :::solution It is uniformly bounded, equicontinuous with $\delta=\varepsilon$, and closed under uniform limits. Arzela-Ascoli gives compactness. $\square$ ::: :::solved-problem The closed unit ball in $C[0,1]$ is not compact because it is not equicontinuous. ::: :::exercise [1] State the main definition or theorem from this lesson. [2] Give one example where the hypothesis is used. [3] Explain one common mistake related to this topic. ::: :::answer [1] See the first formal block of the lesson. [2] The examples in the lesson provide standard choices. [3] A common mistake is to apply a Euclidean compactness rule in an arbitrary metric space. ::: :::faq[Frequently Asked Questions] Q: Why is the metric important? A: Compactness, boundedness, convergence, and openness are all interpreted using the chosen metric. Q: What should students check first? A: Identify the space, the metric, and the exact compactness criterion being used. ::: :::call-to-action[Continue to Compactness Criteria in Sequence Spaces] subtitle: Review the definitions, proofs, and examples before moving to the next compactness idea. button: Next Lesson | /functional-analysis/metric-spaces/compactness-and-its-consequences-in-metric-spaces/compactness-criteria-in-sequence-spaces button-ghost: Previous Lesson | /functional-analysis/metric-spaces/compactness-and-its-consequences-in-metric-spaces/uniform-boundedness-in-function-spaces :::