Uploaded October 2023 | Updated September 2026, 20 hours ago
Here we talk about continuity and its connections with compactness. Compactness was first defined around the same time as metric spaces, by Maurice Frechet, and they are more or less a generalization of finite sets. Their interactions with continuous functions have opened up the entire field of functional analysis over the past century, so let's look at the interactions between these important objects.
To try everything Brilliant has to offer—free—for a full 30 days, visit brilliant.org/ThatMathThing . The first 200 of you will get 20% off Brilliant’s annual premium subscription.
(This video was sponsored by Brilliant)
//Books
Rudin - amzn.to/3KeL0IC
Dummit and Foote Abstract Algebra amzn.to/3QxGvwN
Folland Real Analysis amzn.to/3dCuCao
Hungerford Algebra amzn.to/3pIJyGV
Pedersen Analysis NOW amzn.to/3c27MZo
//Watch Next
Real Analysis Survival Guide youtu.be/v5rD0B-zfXw
This is why you have imposter syndrome youtu.be/5CN_KovhHbY
SoME2 Entry youtu.be/3ARBkAM2hRg
Real Analysis:
How to think of Real Numbers youtu.be/fD9VggL7RYU
The Banach Fixed Point Theorem youtu.be/_I790SqNDjs
The Real Analysis Survival Guide youtu.be/v5rD0B-zfXw
Control Theory:
The Analyticity of the Laplace transform youtu.be/FIMkbFQL6XM
Introduction to Control Theory youtu.be/0v4WFmOm764
//Music
Most music provided by Epidemic Sound with the exception of one song by Tom Fox at Johnny Harris' Patreon
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 Start
1:14 Maurice Frechet
1:58 Motivation through Finite Sets
3:36 Brilliant.org Sponsorship
5:37 Continuity via Open Sets
11:01 Continuous Images of Compact Sets
12:51 Uniform Continuity and Heine's Theorem
18:37 Frechet Not Appreciated
19:38 Maximum value on Compact Sets
24:03 Wrap Up
Here we talk about continuity and its connections with compactness. Compactness was first defined around the same time as metric spaces, by Maurice Frechet, and they are more or less a generalization of finite sets. Their interactions with continuous functions have opened up the entire field of functional analysis over the past century, so let's look at the interactions between these important objects.
To try everything Brilliant has to offer—free—for a full 30 days, visit brilliant.org/ThatMathThing . The first 200 of you will get 20% off Brilliant’s annual premium subscription.
(This video was sponsored by Brilliant)
//Books
Rudin - amzn.to/3KeL0IC
Dummit and Foote Abstract Algebra amzn.to/3QxGvwN
Folland Real Analysis amzn.to/3dCuCao
Hungerford Algebra amzn.to/3pIJyGV
Pedersen Analysis NOW amzn.to/3c27MZo
//Watch Next
Real Analysis Survival Guide youtu.be/v5rD0B-zfXw
This is why you have imposter syndrome youtu.be/5CN_KovhHbY
SoME2 Entry youtu.be/3ARBkAM2hRg
Real Analysis:
How to think of Real Numbers youtu.be/fD9VggL7RYU
The Banach Fixed Point Theorem youtu.be/_I790SqNDjs
The Real Analysis Survival Guide youtu.be/v5rD0B-zfXw
Control Theory:
The Analyticity of the Laplace transform youtu.be/FIMkbFQL6XM
Introduction to Control Theory youtu.be/0v4WFmOm764
//Music
Most music provided by Epidemic Sound with the exception of one song by Tom Fox at Johnny Harris' Patreon
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 Start
1:14 Maurice Frechet
1:58 Motivation through Finite Sets
3:36 Brilliant.org Sponsorship
5:37 Continuity via Open Sets
11:01 Continuous Images of Compact Sets
12:51 Uniform Continuity and Heine's Theorem
18:37 Frechet Not Appreciated
19:38 Maximum value on Compact Sets
24:03 Wrap Up
