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    • The Real Analysis Survival Guide  
Mastering the Art of Reading a Proof: By Example    • Mastering the Art of Reading Proofs: ...  
Introduction to Control Theory    • Introduction to Control Theory  

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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