Banach Fixed Point Theorem

In this video, I prove the celebrated Banach fixed point theorem, which says that in a complete metric space, a contraction must have a fixed point. The proof is quite elegant and illustrates the beauty of analysis. This theorem is used for example to show that ODE have unique solutions under certain conditions. Enjoy!

[rafaelac.]

just seeing your happy mood made me be more interested in the theorem, thank you so much !!!!

[Davidamp]

I got an assignment that uses this Fixed Point theorem. Now I just remembered this video and all the ideas came back to my mind. Thank you for everything, Dr. Peyam

[KhaledAllen]

I love the way you teach this! It makes me really happy to see someone being playful while teaching math, instead of the dry boring stuff out there. Nice work.

[knightfanta]

Your smiles are as nice as your videos. Thanks.

[hassanalihusseini]

That was a really nice and understandable video about the Banch Fix point theorem. Honestly I thought after your Lemma 2 it was already proven, but I have to think a little bit about why Lemma 3 is needed.

Thank you so much for your effort!

[benmorris]

I hope every teacher in this world have same enthusiasm as this... So nice 👍

[evangelosnikitopoulos]

There is a special place in my heart for the Banach Fixed Point Theorem. I was not originally interested in math when I entered college, but I ended up taking an ODEs course in my sophomore year. One of the first things the course covered was the Banach Fixed Point Theorem and its application(s) to existence/uniqueness of solutions to certain classes of ODEs. It was first theorem whose full proof I saw and understood. It fascinated me and motivated me to start learning more math. Long story short, I am now a second year math PhD student, and I’m having a blast. I owe this partially to the Banach Fixed Point Theorem. Thanks for giving it the shoutout it deserves!

[tofu]

nice explanatory video, there is some interesting generalisation of this theorem which i came across in my exercise. the mapping T doesn't have to be contractive, it also suffices if T^n is contractive for some n in N and the theorem still holds.

edit: it's rather an extension than a generalisation of this theorem, because it uses the banach fix point theorem in its proof.

[lowerbound]

What a Lively lecture!! The giant tiger becomes a little kitty as the way you explain it. Thank you!! 🙂

[stefanocarini]

This is explained so brilliantly as always!!! :) I learned recently about this elegant theorem in the context of ODEs

[morgengabe]

I think it's been years since I saw one of your vids. Glad to remember you exist man! Great lecturer!

[jackstacks]

I want to thank you deeply for these videos. I studied a bit of math in college but always became very frustrated with the spoon feeding of information. It's wonderful to be able to just buy a dover book and search for a youtube video on a given subject that needs a bit more explanation or conceptualization.

[Stochine]

Thanks a ton for this video, honestly the best explanation/proof of this theorem I have seen!

[paslogique]

I was going to see the proof of this theorem tomorrow in topology class, guess I just spoiled myself <3. Thanks u !

[bandamkaromi]

other level math than before. Thank you, Dr. Peyam. Thank you.

[thellvll]

Amazing video! I really enjoyed learning about Banach fixed point theorem! Thanks.

[twoglassesofwhiskey.]

Thank You, This is really great explanation.

[kissoh]

Thank you for this, it was a very good video! "There can be only one" actually is from a movie (and series) called Highlander =)

[vinh-ducle]

thank you so much @Peyam, your video is very helpful and fun, love it

[iamreallybadatphysicsbutda]

Amazing proof, you made it very clear 😃