Banach Fixed Point Theorem

[@rafaelac.1643]

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.

[@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.

[@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!

[@lowerbound4803]

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

[@stefanocarini8117]

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

[@benmorris8824]

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

[@Stochine]

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

[@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.

[@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! ❤️❤️❤️

[@aisains983]

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

[@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.

[@paslogique5983]

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

[@hilario-H]

You're amazing, Dr!

[@thellvll]

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

[@bandamkaromi]

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

[@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

[@twoglassesofwhiskey.324]

Thank You, This is really great explanation.

[@bansishah2091]

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

[@drpeyam]

I already have one :)

[@iamreallybadatphysics]

Amazing proof, you made it very clear 😃

[@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?

[@iamreallybadatphysics]

Underrated time-stamp 8:54 🤣

[@許家溢]

This video is very helpful . Thank you Dr!

[@drpeyam]

You’re welcome!!!

[@vinh-ducle8529]

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

[@leonoravandeurs9899]

Amazing video, Thanks!!

[@princeardalan]

Good video. Thank you for making these.

[@fahimullah8490]

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

[@drpeyam]

Brezis Functional Analysis

[@fahimullah8490]

@@drpeyam thanks a lot.

[@ITz_Moren0]

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

[@salvatoregiordano6816]

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

[@TheWhyNotSeries]

yes

[@sugarfrosted2005]

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

[@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

[@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!

[@justindab0bm]

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

[@mtaur4113]

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

[@drpeyam]

Great remark

[@sushrutdeshpande5599]

Actually incredible!!!

[@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 😂

[@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

[@faizanurrehman6220]

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

[@109hardcore]

That's one of the definitions of continuity.

[@riyajain5202]

Great effort sir! Tysm sir !

[@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 :)

[@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

[@estuardodiaz2720]

Great video!!

[@arifmehmood4627]

absolutely good

[@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

[@ruby4046]

Thank you thank you thank you ❤️❤️😭

[@rizkyagungshahputra215]

does x* is a periodical point?

[@drpeyam]

Yes, a point of period 1

[@jazminsutcliff4106]

Such a great teacher!

[@sosolola9012]

Thank you

[@tekaaable]

Dr Peyam sent me here!

[@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.

[@sugarfrosted2005]

Oh, I know where I heard this before now.

[@remlatzargonix1329]

Nice!

[@Leo-io4bq]

please use lower case letters o.O

[@drpeyam]

No one can read them

[@ninakuup21]

d(a,b) indeed my friend

[@outdoorjoe1021]

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

[@rajbirvirdi4571]

*dabs*

[@siekensou77]

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

[@wladkhayra5345]

Mrbk

[@quantumchill5237]

ecks dee