![The Arzela-Ascoli Theorem](https://writelatex.s3.amazonaws.com/published_ver/1550.jpeg?X-Amz-Expires=14400&X-Amz-Date=20240727T015400Z&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAWJBOALPNFPV7PVH5/20240727/us-east-1/s3/aws4_request&X-Amz-SignedHeaders=host&X-Amz-Signature=89b9fc495eae6abcfda3325dfa1b65e685784ddf47939f21c30da2f452b5efdd)
The Arzela-Ascoli Theorem
Author
Oliver Williamson
Last Updated
9 years ago
License
Creative Commons CC BY 4.0
Abstract
We will form a proof of the Arzela-Ascoli Theorem through use of the Heine-Borel theorem. We will also be considering some notions of compactness on metric spaces. The Arzela-Ascoli Theorem then allows us to show compactness, letting us state and prove Peano's existence theorem, pertaining to the existence of the solutions of a type of ODE. Then we will state the Kolmogorov-Riesz compactness theorem, allowing us to show compactness in $L^p$ spaces, building from the Arzela-Ascoli Theorem.
![The Arzela-Ascoli Theorem](https://writelatex.s3.amazonaws.com/published_ver/1550.jpeg?X-Amz-Expires=14400&X-Amz-Date=20240727T015400Z&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAWJBOALPNFPV7PVH5/20240727/us-east-1/s3/aws4_request&X-Amz-SignedHeaders=host&X-Amz-Signature=89b9fc495eae6abcfda3325dfa1b65e685784ddf47939f21c30da2f452b5efdd)