This is a nice video, but I believe the statement of the theorem as given here is slightly wrong. Your quantifiers in the definition of a contraction mapping are the wrong way around. It should be ‘there exists a q such that for each x, y’ rather than ‘for each x, y there exists a q’. This may seem inconsequential, but it is very important to the theorem that there is one ‘uniform’ constant q that works for all points in X under the map T. If not, we can potentially have a sequence of points in X such that values of d(Tx, Ty) come arbitrarily close to d(x, y), and this violates what is necessary to prove the Banach Fixed Point Theorem.
@Diego Marra The problem is that such a ‘biggest q’ doesn’t necessarily exist. There exist subsets of [0,1) that have no greatest element in the standard ordering (indeed the set itself has this property).
Well explained, sir! Kindly explain what you mean on 2:30 by the statement "q in (0,1) implies that the RHS should be greater." Common sense suggests that it should be smaller since it q is a fraction of 1.
Well Explained. The Canada map example was really helpful.
I really appreciate the map example!
This is a nice video, but I believe the statement of the theorem as given here is slightly wrong. Your quantifiers in the definition of a contraction mapping are the wrong way around. It should be ‘there exists a q such that for each x, y’ rather than ‘for each x, y there exists a q’. This may seem inconsequential, but it is very important to the theorem that there is one ‘uniform’ constant q that works for all points in X under the map T. If not, we can potentially have a sequence of points in X such that values of d(Tx, Ty) come arbitrarily close to d(x, y), and this violates what is necessary to prove the Banach Fixed Point Theorem.
@Diego Marra The problem is that such a ‘biggest q’ doesn’t necessarily exist. There exist subsets of [0,1) that have no greatest element in the standard ordering (indeed the set itself has this property).
@Diego Marra Very good thing to question though!
good catch!
Wow. So clearly explained. Thank you!
Nice explanation Sir.Your this video is useful to me to understand the concept of Contraction mapping.
this is a great explanation, helped me with Picard-Lindelof proof
ruclips.net/video/EoImJHpsocQ/видео.html
love that, very easy to understand, thx!
Best explanation I have ever seen. Keep it up ❤
Fantastic explanation!
very helpful!
thx
great job boys!
Well explained, sir! Kindly explain what you mean on 2:30 by the statement "q in (0,1) implies that the RHS should be greater." Common sense suggests that it should be smaller since it q is a fraction of 1.
お、おう
大きい地図に小さい地図を重ねると？？？ってなっちゃうけど、
小さい地図をびろーんと伸ばすとどこか一点は不動っぽいのはわかる･･･かな🤔😀👍
really helpful
very helpful
To gret
T:X-R
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