What Is UNIFORM Convergence ? Why Do We Care ? | Episode 1 - Sequences Of Functions
We know how to define without any ambiguity the convergence of a sequence of reals, and more generally in finite dimension, the convergence of a sequence of vectors, thanks to the equivalence of norms. On the other hand, it is much more difficult to define the convergence of a sequence of functions. Several natural, useful but not equivalent definitions are possible, among which we distinguish simple/pointwise convergence and uniform convergence. Those are the two modes of convergence that we we will discover intuitively and VISUALIZE together. The emphasis is on visualizations, as well as the transmission of the mathematical "feeling" that is hidden behind the sometimes off-putting formalism. For those who would like to support me: https://fr.tipeee.com/kobipy/ A comment, a like or a share are just as appreciated! 😉 The animations were made in Python, with the Manim module : ttps://www.manim.community/ The background music is : Piano Music by: Bensound.com/free-music-for-videos License code: NUU8ZZSW39K0YG4M Some of the images were taken on : https://www.vecteezy.com/ ⏲ Timestamps : 00:00 - Introduction 00:23 - Pointwise convergence 02:35 - Uniform convergence 08:29 - Conclusion Other videos on this channel that might interest you : ◆ What is an area under the curve ? https://youtu.be/E0Ryo199-Gs ◆ Everything you DREAM to know about The LEBESGUE Integral : https://youtu.be/Oigh-j52CqE ◆ The Myth of the Horizontal LEBESGUE Integral ? | Épisode 3 : https://youtu.be/zUg_KVBd7e4 ◆ Where Is The Circle ? | Gauss Integral : https://youtu.be/U2xmox321_k ◆ How Many Squares on the ChessBoard ? | Bijection & Counting : https://youtu.be/2adERtKWCek ◆ The Weierstrass Function is Everywhere Continous, Nowhere Differentiable : https://www.youtube.com/watch?v=37tG_qvBb3M ◆ The BATMAN Equation : https://youtu.be/5SVRLxCBu9E ◆ The Time Table IS Inside your Cup : https://youtu.be/tBoE568elCE ◆ Can you solve the 6 enigma ? https://youtu.be/ikDqP5b9lgM
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Comments
The math is very clearly explained and visualized. My only complaint is that the background music is loud enough to distract from that math.
7I feel the pace of the video is a bit fast if you were trying to mimic the 3b1b style. I guess there must be a little more space for viewers to thing and let more advanced audience to speed up the video for themselves. And I think more real examples of two different types of convergence could make the video more fun. I guess the kids and sports example didn't did the job well because first there isn't a metric space defined on sports and second the function is descreate. But anyway I enjoyed! Well done!
5.5I think the labels are switched on the inner boxes just past 7 minutes into the video.
6.5Can be useful for pupils/students (You should correct the inversion at 7'10")
5.9A beautiful art piece
2.5I really liked it, because I remember struggling with this concept during my lectures in university. It was a bit hard to follow though, due to the fact that it's in french and I had to constantly follow the subtitles. The content is presented in a pretty strict mathematical way which is great for people who studied mathematics, but not so great for people who are new to this stuff and are trying to figure out what any of this means. Overall good video, better than most but not the best.
7.2I think your animations were superb! For instance, the fact that you paused and played the animations a couple of times were a very good way to encourage people to pause and ponder on those concepts. Also, the examples of the kids/sports were very well fit to the explanations of the more tricky definitions. Good luck with your channel and for the contest!!!
7I think that this is a high quality video. The language (French instead of English) makes it difficult for me to accurately appraise it.
6.5is french allowed (personnally i do not care). Classical definition slow and a little boring
4.1Well done. Good example and clarification between simple and uniform convergence. Good follow up questions at the end.
7.8