Math 131 120916 Ascoli Arzela and Stone Weierstrass (redone)

Theorem of Ascoli-Arzela.. Stone-Weierstrass Theorem (density of polynomials in the space of continuous functions on a closed interval with respect to the supremum norm metric). (Redid the lecture but perhaps didn't improve it)

[oosky]

Your lecture was a ray of light in the dark. I took almost all of the baby Rudin lectures you posted for a year, and without you, I would not have studied this course to the end. Thank you so much again. I will always be rooting for you across the sea -From a college student in Korea

[sarkaajsingh]

Dear Professor Ou, I would like to thank you very much from the bottom of my heart for doing this. I thoroughly enjoyed this series. You conveyed the material in an intuitive and beautifully pedagogical manner. This series is a crucial step towards one's education in higher Mathematics. There is no avenue for someone like me to study these things in my country as Pure Mathematics is wildly unpopular here, thus, only taught at the 3 top institutes in the country and nowhere else. So thank you for making education accessible for everyone all over the world. With this, I find myself humbled, though encouraged to pursue a further education in Mathematics. Thanks again and best wishes to you.

[Leon-qizx]

I just want to show my sincerely appreciation to you professor Ou! Thank you! It's been a wondeful journey for me and I've learned a lot! I have to say, for me this is the best Real Analysis resources on YouTube if someone wants to self learning analysis. Sometimes I compared your 2018 version and this one, cz for some of the videos 18 version has better quality, and some has better quality for this one. I will continue studying your rest of analysis courses. Again, thank you so much! For your reliability, trustworthiness and professionalism, hope you could upload more your math courses on YouTube!

[aa-ludu]

Far more enlightening than my professor's lectures on these two theorems!

[EulerGauss]

Thank you, professor Ou! I really appreciate it and it helped me a lot. This is really fun. I can't wait to watch Fourier analysis lectures.

[jushkunjuret]

Finally made it here, thank you Prof. Ou for such an amazing journey!

[kamrangupta]

Thank you very much for posting these lectures. This was extremely useful in helping me review advanced calc!

[thesunnydk]

Thank you Prof. Ou for the series! I really appreciate the geometrical explanations you try to give us to help us understand and also to allow us an intuition when a theorem will be useful in a subsequent proof. Please don't stop uploading your lectures! I am planning on continuing on with your complex analysis and fourier analysis series but I am wondering if it will still be useful to watch introduction to analysis after having watched this series.

[aleksandrb]

Thanks! Your lecture helped a lot, I failed to understand the part of Weierstrass theorem where we introduced Pn(x), but you did a good job at explaining that

[shihuazhang]

Thank you Professor Ou! I'll refer back to these videos again and again. Thank you!

[yearningcentipedes]

Thanks you a lot Professor Ou for these wonderful lectures about Real Analysis. I am in highschool and my dream is to be able to be able to do a PhD in pure mathematics or theoretical physics in the futre, and thanks to your video, I was able to come a bit closer to my goal. These lectures makes reading Rudin a lot easier, thanks again for this great and clear content, it was really entertaining to follow this course and I will, I hope, never forget about it.

[JaspreetSingh-zpnm]

thanks professor, though it started to hurt my brain but cool result, polynomials are dense in any space of continuous bounded functions

[APh_]

Just wanted to say thanks, these videos really helped me out :)

[manimaran]

Dear Prof. I like your lecture. I am really enjoying your videos. I beg to differ you. In Rudin book page number 165, Theorem number 7.33 is known as Stone-Weierstrass Theorem(1937). But what you mentioned as Stone-Weierstrass Theorem, which is not Stone-Weierstrass Theorem, which is Weierstrass Approximation Theorem (1885). If I am wrong, Please correct me.

[mingencho]

Hi Prof. Ou, thanks for the video! Just a quick question: given the existence of a sequence of polynomials that converges uniformly to a function f under the sup norm, does the same sequence converge uniformly to f under the Lp norms?

[sriramraghunathan]

Prof Ou, A suggestion - for the Weierstrass approximation theorem perhaps you might want to consider the proof using Bernstein polynomials which is constructive and far more easier to understand at the undergrad level. You could refer to the classic GF Simmons for the same.

[merlinpriyanga]

While proving uniform convergence of a subsequence in Ascole Arzela, why are you choosing different functions but on same point?

[NoBody]

Thank you, Professor Ou. Are there any videos about the other part of Rudin's book ?

[Kharukat]

Thank you, thank you Professor Ou! This video is very helpful. :)

[merlinpriyanga]

In your lectures on sequences and series of functions, which one is Ascoli Arzela theorem?