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! π 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/ β² Timestamps : 00:00 - Introduction 00:23 - Pointwise convergence 02:35 - Uniform convergence 08:29 - Conclusion Other videos on this channel that might interest you : β What is an area under the curve ? https://youtu.be/E0Ryo199-Gs β Everything you DREAM to know about The LEBESGUE Integral : https://youtu.be/Oigh-j52CqE β The Myth of the Horizontal LEBESGUE Integral ? | Γpisode 3 : https://youtu.be/zUg_KVBd7e4 β Where Is The Circle ? | Gauss Integral : https://youtu.be/U2xmox321_k β How Many Squares on the ChessBoard ? | Bijection & Counting : https://youtu.be/2adERtKWCek β The Weierstrass Function is Everywhere Continous, Nowhere Differentiable : https://www.youtube.com/watch?v=37tG_qvBb3M β The BATMAN Equation : https://youtu.be/5SVRLxCBu9E β The Time Table IS Inside your Cup : https://youtu.be/tBoE568elCE β Can you solve the 6 enigma ? https://youtu.be/ikDqP5b9lgM