Uploaded February 2023 | Updated September 2026, 9 hours ago
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 - amzn.to/3MDHUis
Gerald Folland - Real Analysis: Modern Techniques and Their Applications - amzn.to/3JmK6Np
R. Shankar - Principles of Quantum Mechanics - amzn.to/40825xj
Maxwell Rosenlicht - Introduction to Analysis - amzn.to/3NFYiAC
Tom Apostol - Mathematical Analysis 2nd Edition - 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 youtu.be/v5rD0B-zfXw
Mastering the Art of Reading a Proof: By Example youtu.be/WPWf_s6Q1QQ
Introduction to Control Theory youtu.be/0v4WFmOm764
//Music Provided by Epidemic Sound
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//Recording Equipment
Canon SL3: amzn.to/3nZ11KU
Canon T6i: amzn.to/3FUpkQh
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Neewer 5600K USB LED Lights: amzn.to/3xvB9cN
Neewer 18 inch Ring Light: amzn.to/2ZvgCsc
Camera Power Adapter: 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
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 - amzn.to/3MDHUis
Gerald Folland - Real Analysis: Modern Techniques and Their Applications - amzn.to/3JmK6Np
R. Shankar - Principles of Quantum Mechanics - amzn.to/40825xj
Maxwell Rosenlicht - Introduction to Analysis - amzn.to/3NFYiAC
Tom Apostol - Mathematical Analysis 2nd Edition - 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 youtu.be/v5rD0B-zfXw
Mastering the Art of Reading a Proof: By Example youtu.be/WPWf_s6Q1QQ
Introduction to Control Theory 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:
epidemicsound.com/referral/644nao
//Recording Equipment
Canon SL3: amzn.to/3nZ11KU
Canon T6i: amzn.to/3FUpkQh
Rode VideoMic: amzn.to/3lhldGa
Blue Yeti Microphone: amzn.to/3I1y88N
Yeti Nano Microphone: amzn.to/3I1mriA
SanDisc 256GB SD Card: amzn.to/3E3LOOr
Neewer 5600K USB LED Lights: amzn.to/3xvB9cN
Neewer 18 inch Ring Light: amzn.to/2ZvgCsc
Camera Power Adapter: 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
