Banach Fixed-Point Theorem

[@nadav7679]

I have a functional analysis exam coming up, so it was great to see the full details of the proof taken with care!

[@brightsideofmaths]

Thank you very much! Good luck and thanks for the support!

[@ivansidorov1384]

Thank you. Clear and consequent. I always enjoy your math videos.

[@brightsideofmaths]

You are very welcome :)

[@t.b.4923]

I really enjoyed your concise explanation. Keep up the work and your channel will grow!

[@DingHang04]

Very interesting to think this happens in real life

[@DavidLessure]

This is a great video, we just learned about it in class, and this explanation makes it make a lot more sense. As always thank you TheBrightSideOfMaths ☀️😎

[@brightsideofmaths]

Nice :) Thank you! And thanks for the support!

[@StratosFair]

A beautiful proof for a beautiful theorem

[@BreezeTalk]

This is high quality mathematics in my eyes

[@brightsideofmaths]

It is :)

[@DOROnoDORO]

Recently had to prove this in an analysis test :) turns out it's quite important for dynamic systems, my university's specialty

[@bestpixels596]

Which reference books are used to prove this theorems?

[@brightsideofmaths]

None. This proof is already mathematical folklore and can also be found in Wikipedia, for example.

[@MGoebel-c8e]

Nice to listen to someone speaking English in my own accent;) Good video, especially appreciate the constant reminders that this is no rocket science. One question and a couple of observations: On 4:27 why does it have to be an inequality? The argument would hold as well if there was an equal sign, no? The definition of the map was a little quick for me - had to pause and go back in order to realize that we were hopping from one point to the next. Why this map? Would have helped if you had talked more about what this implies, i.e. what insight this delivers that is helpful for all the use cases you mention at the beginning. That would be more insightful than the uniqueness proof at the end (only professional mathematicians would even demand a proof of that, for the rest of us that is obvious enough:))

[@mulletronuk]

4:27 using an inequality here is more general than an equality. Insight: Take any real number, and take the cosine of it in your calculator. Now take cos(Answer) repeatedly and watch it converge rapidly to a fixed point :)

[@brightsideofmaths]

Thanks! Now try to prove this cosine procedure by using the Banach fixed-point theorem :)

[@eduardoGentile720]

Here in Naples everybody calls this the Banach Caccippoli theorem hahaha

[@brightsideofmaths]

True :) I also know this name!

[@Risu0chan]

In France it's called the Picard (or Banach-Picard) fixed-point theorem, after Émile Picard. I didn't know Renato Caccioppoli's name. Interesting character, he was a pianist, an antifascist during Mussolini's era, playing La Marseillaise (French anthem) when il Duce was visiting… There is even a film about him.

[@brightsideofmaths]

@@Risu0chan Thanks! I did not know that :)

[@eduardoGentile720]

@@Risu0chan He is considered so important here in Naples that the math department of the Federico II (the most important university in the south of Italy) is called "department of Math and applications Renato Caccippoli"

[@debmallyachanda5384]

Absolute gem!

[@brightsideofmaths]

Thanks :)

[@awindwaker4130]

Beautiful proof

[@oldcowbb]

seems like it will be very useful in nonlinear control

[@brightsideofmaths]

Yes, definitely

[@tens0r884]

does the contraction have to be from X to X ? Does this not apply to X -> a different metric space as well ?

[@brightsideofmaths]

No, it has to be the same space in domain and codomain. Otherwise, the notion "fixed point" would not make much sense.

[@tens0r884]

@@brightsideofmaths You're correct but I should have mentioned that the two spaces X and Y have non trivial intersection, for example, a contraction that also shifts the points a bit. I'll give a concrete example, f: [0, 1] -> [0.75, 1.25] given the canonical metric

[@brightsideofmaths]

@@tens0r884 Then the Banach fixed-point theorem is not applicable :D

[@A_doe_wasting_her_life]

i was just wondering why is the idea of a cauchy sequence useful lol. NIce vid

[@brightsideofmaths]

Thanks!

[@satiremuch2643]

What if the distance is 0.9999... + 0.0000...1. How far away are they then? 1:54

[@brightsideofmaths]

What is your metric space here?

[@tens0r884]

0.0000...1 is not a real number (its not well defined)

[@satiremuch2643]

@@tens0r884 Thank you for the answer. Would you like to expound on that?

[@tens0r884]

​@@satiremuch2643 I mean your decimal representation doesnt make sense. A real number less than zero always has the representation \sum_{i = 1} a_i * 10^(-i)

[@satiremuch2643]

@@tens0r884 Ah ha.... my intention was to show (0 followed by infinitely many nines) + (0,0 followed by infinitely many zeros and a 1 at the end). 0.(9)n + 1/10n =1 Not any negative number. Like this en.wikipedia.org/wiki/0.999...#Rigorous_proof

[@Algebraictivist]

Cool

[@Basedgwad]

Vsauce

[@brightsideofmaths]

He?