Very interesting to think this happens in real life

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

@brightsideofmaths

Ай бұрын

You are very welcome :)

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

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

@brightsideofmaths

Ай бұрын

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

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

Ай бұрын

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

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

This is high quality mathematics in my eyes

@brightsideofmaths

Ай бұрын

It is :)

Beautiful proof

Absolute gem!

@brightsideofmaths

23 күн бұрын

Thanks :)

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

Ай бұрын

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

Ай бұрын

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

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

@brightsideofmaths

Ай бұрын

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

@tens0r884

Ай бұрын

@@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

Ай бұрын

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

seems like it will be very useful in nonlinear control

@brightsideofmaths

Ай бұрын

Yes, definitely

Here in Naples everybody calls this the Banach Caccippoli theorem hahaha

@brightsideofmaths

Ай бұрын

True :) I also know this name!

@Risu0chan

Ай бұрын

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

Ай бұрын

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

@eduardoGentile720

Ай бұрын

@@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"

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

@brightsideofmaths

Ай бұрын

Thanks!

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

@brightsideofmaths

Ай бұрын

What is your metric space here?

@tens0r884

Ай бұрын

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

@satiremuch2643

Ай бұрын

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

@tens0r884

Ай бұрын

​@@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

Ай бұрын

@@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

Vsauce

@brightsideofmaths

Ай бұрын

He?