An Introduction to Compact Sets

Compact sets are the foundation that modern mathematics is built on, and here we explore their definition and properties. Optimization problems are justified because continuous functions take maximum values on compact sets, our first spaces of functions get magnitudes and distance because of compact sets. And all of this is because Maurice Frechet wanted to generalize the work of Weierstrass. //Books Walter Rudin - Principles of Mathematical Analysis 3rd Edition - https://amzn.to/3MDHUis Gerald Folland - Real Analysis: Modern Techniques and Their Applications - https://amzn.to/3JmK6Np R. Shankar - Principles of Quantum Mechanics - https://amzn.to/40825xj Maxwell Rosenlicht - Introduction to Analysis - https://amzn.to/3NFYiAC Tom Apostol - Mathematical Analysis 2nd Edition - https://amzn.to/3mwxFlC //Exercises - Prove the product to sum identities for exponentials first for rational numbers then for real numbers. //Watch Next The Real Analysis Survival Guide https://youtu.be/v5rD0B-zfXw Mastering the Art of Reading a Proof: By Example https://youtu.be/WPWf_s6Q1QQ Introduction to Control Theory https://youtu.be/0v4WFmOm764 //Music Provided by Epidemic Sound Use this referral link to get a 30 day free trial with Epidemic Sound for your YouTube channel: https://www.epidemicsound.com/referral/644nao //Recording Equipment Canon SL3: https://amzn.to/3nZ11KU Canon T6i: https://amzn.to/3FUpkQh Rode VideoMic: https://amzn.to/3lhldGa Blue Yeti Microphone: https://amzn.to/3I1y88N Yeti Nano Microphone: https://amzn.to/3I1mriA SanDisc 256GB SD Card: https://amzn.to/3E3LOOr Neewer 5600K USB LED Lights: https://amzn.to/3xvB9cN Neewer 18 inch Ring Light: https://amzn.to/2ZvgCsc Camera Power Adapter: https://amzn.to/3D3upUu DISCLAIMER: The links above in this description may be affiliate links. If you make a purchase with the links provided I may receive a small commission, but with no additional charge to you :) Thank you for supporting my channel so that I can continue to produce mathematics content for you! 0:00 Introduction 1:52 ChatGPT still can't math... 4:36 What is a compact set? 6:20 Compact sets are closed 8:32 Prisms are closed 9:20 The Heine Borel Theorem 10:17 Frechet's Definition 10:58 Wrap up