Real Analysis 24 | Pointwise Convergence

[@raoufmedj6052]

I've been searching KZbin for a detailed well-presented explanation of this notion, and I must say, this is literally the best one out there. Great content!

[@brightsideofmaths]

Wow, thank you!

[@subhradeepghosh2523]

I was having some problem understanding pointwise convergence. This video cleared all my doubts. This is the best maths channel that I've found on KZbin till date.

[@brightsideofmaths]

Thank you very much :)

[@CrimsonSquaredX2]

Thatnk you for the illuminating examples and illustrations. I now understand that I had deeply undervalued the need for the notion of uniform convergence.

[@brightsideofmaths]

You are very welcome :)

[@malikialgeriankabyleswag4200]

You must have had great professors for your undergrad because the little detail about fixing the x's such that f_n(x)'s become a sequence in the real numbers as opposed to leaving x arbitrary so that f(x)'s are a sequence in the set of functions, this helped me to understand the real essence of the two kinds of convergence, better than anything else I've heard.. Very helpful! Thank you and God bless.

[@brightsideofmaths]

Nice to hear! You are welcome :)

[@sumittete2804]

4:19 how is it a parabola? Please explain. Here x is a variable. So it's supposed to be a linear function. Please, clarify my doubt.

[@brightsideofmaths]

@@sumittete2804 It''s a quadratic function x times x

[@sumittete2804]

@@brightsideofmaths Thank you very much sir...got it. Actually, I had mistaken that first "x" as multiplication.

[@travist1566]

Finally an explanation with visuals that makes sense!

[@brightsideofmaths]

Thank you for your support :)

[@sumittete2804]

4:19 how is it a parabola? Please explain. Here x is a variable. So it's supposed to be a linear function.

[@angelmendez-rivera351]

An interesting and simple example of pointwise convergence where the convergence appears weak is with the sequence f[n] of functions (-1, 1] -> R, (f[n])(x) = x^n. f : (-1, 1] -> R defined by f(x) := lim f[n] (n -> ♾) satisfies f(x) = 0 for every x in (-1, 1), but f(1) = 1. This is my favorite example, because it is quite an intuitive sequence to consider, and one which you expect to have nicer convergence properties.

[@hustle226]

I appreciate your efforts

[@IgorVladK]

great lecture, as always!

[@vijaysinghchauhan7079]

Spectacular Explanation.

[@brightsideofmaths]

Very nice! :)

[@sumittete2804]

4:19 how is it a parabola? Please explain. Here x is a variable. So it's supposed to be a linear function. Please, clarify my doubt.

[@Hold_it]

Great video! Can't wait for the one on uniform convergence :D

[@brightsideofmaths]

Already out as early access! :) I upload it on KZbin at the end of the week.

[@Hold_it]

@@brightsideofmaths Great, i will be waiting :D

[@parsatalaie9892]

tysm the example was amazing

[@x.s3429]

Hi Thanks a lot for this wonderful video. For example 2, i did get where is the case when x>0, fn(x) is not 0 for n

[@brightsideofmaths]

You are welcome! What exactly do you mean?

[@berke-ozgen]

Thank you for the content. I thought that 2nd example is not a good selection for this topic. Let me explain: when x goes to 0, fn(x) goes to infinity for increasing values of n. I am not sure that this function is even convergent. I guess it has 2 sub function sequences which convergences 0 and 1, respectively, which makes it non convergent.

[@brightsideofmaths]

Thank you very much. I don't get your problem with the second example because I show that the pointwise limit exist.

[@sumittete2804]

4:19 how is it a parabola? Please explain. Here x is a variable. So it's supposed to be a linear function. Please sir, clarify my doubt.

[@anirudhachakri4295]

Thank you so much!!

[@brightsideofmaths]

You are very welcome :)

[@chendian-jing6037]

For the second example "n^2x(1-nx)", I am a bit not sure. By definition, it requires for all epsilon > 0. Obviously, we cannot choose epsilon < 1/4 since in this case n = 1. If you choose epsilon = 1/8 for example, it implies n = 1/2. In this case 1/n = 2 > 1 which exceeds the domain [0,1]. Is it still pointwise convergent? I am confused.

[@brightsideofmaths]

Pointwise means that you fix the point x and then you check the sequences f_n(x) for convergence.

[@whendreamismathematics4709]

Awesome

[@predatoryanimal6397]

i have taken the integral of the function in the example (n^2 x (1- nx)) as two integrals n^2 x dx - n^3 x^2 dx from 0 to 1/n and it gave me a value of 1/6, does that mean that at the limit n-> infinity, there is an f(x) in the 0+ vicinity whose value is 1/6, or i have done something wrong? sorry for constantly bothering you with these questions but math is really tricky for me, thx for the answer in advance!

[@brightsideofmaths]

Great work calculating the integral here! It just not matter which n you choose, you always get the same value of the integral. You also see in sketch that the area under each curve has the same value. So you have found an example where lim ∫ f_n ≠ ∫ lim f_n

[@predatoryanimal6397]

@@brightsideofmaths thank you, your videos are very easy to understand! it makes much easier to study textbooks that use these concepts without properly explaining them at this level of simplicity and detail!

[@brightsideofmaths]

@@predatoryanimal6397 Thank you very much! I am glad that you can enjoy the videos :)

[@zyzhang1130]

I don’t understand why for the second example for x=0 the value is not undefined. Anyone can explain?

[@brightsideofmaths]

By definition, the value is always zero. Why should it be undefined in your opinion?

[@zyzhang1130]

@@brightsideofmaths because for x=0 we should use the first definition of f(x), and in the limit n is infinitely while x is 0 so u have a infinity times 0 situation

[@brightsideofmaths]

@@zyzhang1130 We first put in 0 and then send n to infinity. So we are at 0 the whole time.

[@zyzhang1130]

@@brightsideofmaths I see. But why it wouldn’t be reasonable to have the limit of the function first then give x a value?

[@brightsideofmaths]

That is not how *pointwise* convergence should work then :) (See the next video :)

[@pirzadaaakib4586]

Sir , could you please tell me the best book for convergence topic

[@brightsideofmaths]

You mean in the region of functional analysis?

[@hustle226]

You have to relate this topic to anything that makes this topic easier, I can't understand ,

[@joyzhao4069]

If you followed all the way from `start learning math’ and his previous lectures, you would have understood it better.

[@hustle226]

But you should do more efforts to make topic very easy

[@brightsideofmaths]

I really should do this, right!

[@hustle226]

What tha hell is going on

[@hustle226]

I don't understand nothing

[@hustle226]

Maths is a disgusting subject