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!
Note: The thumbnail was taken from en.wikipedia.o...

[@rafaelac.1643]

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

[@tofu8676]

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.

[@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.

[@knightfanta3548]

Your smiles are as nice as your videos. Thanks.

[@hassanalihusseini1717]

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!

[@tofu8676]

well it was shown that xn converges to some x, you still need to show that this x is fixed point of T tho.

[@hassanalihusseini1717]

@@tofu8676 Thank you for your answer! Now I can see it better.

[@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

[@iamreallybadatphysicsbutda8198]

Underrated time-stamp 8:54 🤣

[@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 =)

[@stevekaszycki8629]

Thanks for the video! Any chance of more fixed point videos? Brouwer, Kakutani, and Tarski would all make for fixating videos. Hopefully that pun is convincing enough.

[@drpeyam]

Hahahaha

[@justindab0bm]

Subscribed because of that 10/10 d(a,b) joke

[@aisains983]

0:07 - Thanks for watching! Oh... no problem, CYA!

[@rajbirvirdi4571]

*dabs*

[@twoglassesofwhiskey.324]

Thank You, This is really great explanation.

[@stefanocarini8117]

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

[@mtaur4113]

The proof does not work if d(T(a), T(b)) < d(a,b) without a specific constant c

[@drpeyam]

Great remark

[@outdoorjoe1021]

I didn't catch the joke because I am not a native speaker.

[@jackstacks3989]

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.

[@leonoravandeurs9899]

Amazing video, Thanks!!

[@paslogique5983]

I was going to see the proof of this theorem tomorrow in topology class, guess I just spoiled myself

[@Stochine]

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

[@maggomor2151]

As long as M is not infinity, you are omitting a factor in the analytical short-form solution to the geometric series, no? Did you leave it out because in the later limit considerations it falls out anywas as N \to \infty and M > N?

[@phscience797]

Great proof, but there is something I obviously don‘t quite get: Consider the function f(x) = 1 - x ^ 2 like in the thumbnail. For the metric space we want (J, |•|) for some interval J and the normal real metric d(a, b)=| a - b |. Now, one can derive* that, for f to be a contraction, it must hold that for all a, b from J: 1 > |a+b|, which would imply that J = (-0.5, -0.5), otherwise there could be two numbers from J whose sum is greater than or equal to 1. That contradicts the theorem because there is no fixed point there, it’s at 1/sqrt(2)! *Here’s my derivation: d( f(a), f(b) ) = | (1 - a^2) - (1 - b^2) | = | b^2 - a^2 | Wlog, let b = a + h => b^2 = a^2 + 2ah + h^2 q | b - a | = q | h | >= | 2ah + h^2 | = | h | | 2a + h| q >= 2a + h = a + b Can you spot the mistake? I am pretty confused!

[@lowerbound4803]

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

[@wladkhayra5345]

Mrbk

[@evangelosnikitopoulos9623]

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!

[@drpeyam]

That theorem inspired me too! ❤️❤️❤️

[@iamreallybadatphysicsbutda8198]

Amazing proof, you made it very clear 😃

[@sosolola9012]

Thank you

[@vinh-ducle8529]

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

[@princeardalan]

Good video. Thank you for making these.

[@keshavgodani6649]

Sir, why have you not explained about the applications of Banach fixed pt th. , the have taught this so amazing that now I want to see you teaching all the applications and there proofs of BFPT

[@drpeyam]

The Mean Value Theorem and Fixed Points kzbin.info/www/bejne/sHbIZn1pjZt8iac

[@hillarioushilariokasaquerm684]

You're amazing, Dr!

[@thellvll]

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

[@julesleb]

Dr Peyam you are superb and i’m very grateful i found you your videos are really helpful. But why do you write in capital letters?

[@drpeyam]

Because I can only write in cursive and people can’t read my handwriting

[@Leo-io4bq]

please use lower case letters o.O

[@drpeyam]

No one can read them

[@morgengabe1]

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

[@drpeyam]

Welcome back!

[@benmorris8824]

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

[@sugarfrosted2005]

This reminds me a bit of the bourbaki-witt fpt, though the proof is totally different. That theorem has several cute proofs.

[@timurpryadilin8830]

Wonderful video! Peyam, can you please make a proof why does the l'Hôpital rule work. It is realy not obvious to me

[@drpeyam]

Coming in the next month or so :)

[@bansishah2091]

loved how simply you explained this. can you make a playlist on teaching real analysis pleaseeeee ?

[@drpeyam]

I already have one :)

[@bandamkaromi]

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

[@faizanurrehman6220]

Please briefly describe how continuity allows to move limit inside (t(x))?

[@109hardcore]

That's one of the definitions of continuity.

[@許家溢]

This video is very helpful . Thank you Dr!

[@drpeyam]

You’re welcome!!!

[@fahimullah8490]

Superb explanation. Please recommend the functional analysis book you follow.

[@drpeyam]

Brezis Functional Analysis

[@fahimullah8490]

@@drpeyam thanks a lot.

[@riyajain5202]

Great effort sir! Tysm sir !

[@ninakuup21]

d(a,b) indeed my friend

[@salvatoregiordano6816]

You explained it so well! Thank you! Is this the same of the contraction mapping theorem?

[@TheWhyNotSeries]

yes

[@ruby4046]

Thank you thank you thank you ❤️❤️😭

[@sugarfrosted2005]

Oh, I know where I heard this before now.

[@himanshumallick2269]

Which "areas" do you mainly deal with in general? From your previous videos, I suspect that you are interested especially in analysis and differentiable geometry(?)

[@drpeyam]

No differential geometry! I’m interested in Partial Differential Equations!

[@himanshumallick2269]

@@drpeyam So are you looking forward to the Navier Stokes equation problem?

[@drpeyam]

Hahahahaha, good one 😂

[@sushrutdeshpande5599]

Actually incredible!!!

[@arifmehmood4627]

absolutely good

[@remlatzargonix1329]

Nice!

[@quantumchill5237]

ecks dee

[@Albkiller22]

Hey Dr peyam what's up? why don't you do some videos on statistics?

[@drpeyam]

I’m a mathematician, I don’t know any statistics

[@sofianechalal3433]

Booom !!!!!!

[@tofu8676]

brutal

[@estuardodiaz2720]

Great video!!

[@tekaaable]

Dr Peyam sent me here!

[@rizkyagungshahputra215]

does x* is a periodical point?

[@drpeyam]

Yes, a point of period 1

[@michaelempeigne3519]

Solve. 615 + x^2 = 2^y where "x, y" are elements of Z

[@drpeyam]

The solutions are y = log2 (615 + x^2) for x in R

[@nicholasr79]

@@drpeyam y= (615+x^2)/[log(2)], no?

[@drpeyam]

Nicholas Sway By log2 I mean log base 2, so it’s log base 2 of 615 + x^2

[@nicholasr79]

@@drpeyam 👍

[@michaelempeigne3519]

@Eŭkalipto Muziko thx; I corrected it.

[@siekensou77]

Bit difficult to watch this video cuz of the angle...

[@TheRedfire21]

Never seen this way of proving it, so cool especially the series part! the way i learned it was way less elegant for sure lol

[@ITz_Moren0]

Great video, now that I’m coursing analysis I can finally understand all of this demonstrations and theorems, cheers.

[@jazminsutcliff4106]

Such a great teacher!