Functional Analysis 7 | Examples of Banach Spaces

[@ROni_ROmio]

Please...we hope to get one course about type of convergence ...proba..a.s....Lp ..distribution ..and relation between them😍😍😅😅🤗🤗

[@이효건-o4o]

This is a masterpiece, piercing most important things in a video around 10 minutes

[@stevilimfax8296]

I think that you lectures perfectly and always hit the center of a topic! i have just one objection that your cursor is really small and when you are showing something on a board it is really hard to find it :D, never the less keep on going!

[@brightsideofmaths]

Thanks! I guess, I already corrected the cursor problem in the newer videos :)

[@scollyer.tuition]

There's a tiny omission in this video: when you talk about l^p spaces, you are considering the general case of the field F being either R or C, and you say that you mentioned that Cauchy => complete with the usual norm in the first example for these fields. In fact, the first example only states this fact for R; it is of course true for C too, but it may have been clearer to have treated C as a separate example. Of course, I'm being a little picky here. Apart from that, nice video. I think it was very helpful to omit the proof of the l^p norm at this stage.

[@brightsideofmaths]

Yeah, you are right. I also should have talked about an example about complex numbers. However, it didn't want to lose too much time before showing the interesting examples :)

[@arturo3511]

By assumptions all individual x^(k)'s themselves belong to l^p, so the horizontal sequence converge (is that right). Now I'm a bit confused about the vertical sequences, why do we assume the vertical sequences x^(k) are a cauchy sequence ? Furthermore, we want to show that all Cauchy sequence in X converge, so we assume that we can put all these sequence vertically and call it a cauchy sequence. Wouldn't we have to prove this? How do we know this set of sequences (that converge in lp by definition) form themselves together a Cauchy sequence and how do we know these form all the Cauchy sequence ? In other words how do we know that all Cauchy sequences will appear when we iterate through x^k(s) ? Thanks a lot

[@evolindica244]

Very helpful and concise. Thank you very much for making the series of functional analysis!

[@brightsideofmaths]

Glad you enjoyed it!

[@mathieumaticien]

I understood up to the point where you said x_n^(k) -> x*_n for all k. But I don't understand why you did the whole thing with epsilon' and the less than or equal to part. You have that ||x^(k) - x*||^p = lim_N lim_l sum(|X_n^(k) - x_n^(l)|^p), where each absolute values are strictly < epsilon (no prime)? I'm just lost on where you got ||x^(k) - x*||

[@ВероникаВоробьева-и6к]

Thank you so much for your videos, they are very clear and pleasant! Looking forward to the next episode

[@sparrowpower5466]

A suggestion: Try to write in the middle of the screen. You usually write in the lower part of the screen which causes difficulty to watch on youtube because most of the functions of youtube are also in the lower part of the screen.

[@brightsideofmaths]

Thanks and good suggestion! I try to have this in mind in my next videos.

[@MusicKnowte]

Everything got difficult at this video i am now scared

[@brightsideofmaths]

That is no problem. Some things are just technical at some point. Don't worry! Other things might be easier again :)

[@zazinjozaza6193]

Great, can't wait for more!

[@ahippiething]

Thank you so much for the amazing videos! I just started taking functional analysis and I can only understand the subject if I watch your videos

[@brightsideofmaths]

You are very welcome :)

[@PunmasterSTP]

How did the rest of the class go?

[@rakshithasp1279]

Diameter of a set and distance between two set regarding this topic plz do vedio sir

[@brightsideofmaths]

:)

[@glorialam9059]

You didn't show that l^p(N,F) is a linear space first - what does it mean to some infinite Cauchy sequence? Do you just put it of summing infinite things element-wise? - you gonna end up in fundamental paradoxes without strict axiomatic justification you can do it.

[@brightsideofmaths]

We add and scale elementwise. This gives the linear space.

[@farzanahdarwish8818]

Amazing! These videos are really useful and comprehensive. Thank you.

[@ferry7185]

can I ask something about F, is F can be defined for scalar in vector space that is F is field in general? not just real and complex number

[@brightsideofmaths]

Here, F is either R or C. That is common for functional analysis.

[@xwyl]

I watched this at least six times to understand the proof. There are 2 obstacles: 1. The English is hard to understand sometimes. A subtitle would help a lot. 2. Crucial logical steps should be as clear and eye-catching as possible. These steps should better have at least pop-up dialogues with crucial contents in them. When some steps use previous definitions, more stress and clarity should be added (maybe pop-ups plus screenshots of the definitions with hilights). Wish you make the Bright Side much brighter, and rescue more from the dark side.

[@brightsideofmaths]

Yes, you are right. I definitely should put up subtitles here! This is not an easy proof, of course, but maybe you can tell me the points where I lost you.

[@xwyl]

A minor place e.g. 5:25 'and now we have to show...' this is one of our objectives, and is better hilighted with something eye-catching to let readers know this is what you're gonna solve next. 8:50 'Or in other words, for an arbitrary epsilon...' here uses the definition of limit of convergent sequence, and a screenshot of the definition is better shown here in a pop-up. 9:45 uses the definition of Cauchy sequence, here also needs to show the definition clearly. And at the end of a proof, there better be a very short summary of the main steps of the proof.

[@zassSRK]

@@brightsideofmaths I'm lost at 6:30. I understand up to this point, but I don't understand when you say that taking this to the power of p is less than or equal to summing up the other possibilities. I'm not following here what you mean or why it is true.

[@MrWater2]

@@zassSRK Because you are summing the forth row and other numbers that are nonnegative so it must be that is at least equal if all the other numbers in the summand are zero but if at least one them is not zero is must be less. I guess the best way is to write the summation and you are going to notice that you have the same as in the left hand side plus other numbers

[@achillesmichael5705]

how many videos you reckon until you reach operators?

[@brightsideofmaths]

With operators we will start very soon. Maybe Video 10 or so :)

[@OffuAnthony-qd1hw]

Pls I just started watching your videos, and in this video you said you'd prove if that is a norm, but you didn't later prove it. Also help me show that lp(R) (1≤p≤infinity) is a vector space over R

[@chianchen776]

8:14 wasn't the point to prove L^p is complete, and so it's a Banach space? I'm kind of knotted by "because it's a Banach space", because if it is a Banach space, then it is complete. (Aha, it was meant to be so that it's Banach space so it's in C, so it's complete).

[@ekeebobs7520]

Thank you very much sir! Your video's have been helpful but I've got a suggestion. Please make use of a visible poster of maybe color the items you are talking about so we won't be misled. The pointer being used is small and I got confused sometimes when watching the video. Thanks again for the video's.

[@williamslima9181]

Your videos are great. Can you tell us what equipment and software do you use? Best regards.

[@brightsideofmaths]

Thank you very much :) I use the nice and free program Xournal.

[@williamslima9181]

@@brightsideofmaths Very interesting. It is difficult to see good quality handwritten text. Do you use a digitizing board!? Doesn't seem that you use a mouse to write.

[@brightsideofmaths]

@@williamslima9181 Yeah, of course, I use a graphic tablet to write the mathematics.

[@ferry7185]

in 10:47, why we get the norm is less than or equal? are we already get norm is less than?

[@xinghuashuying]

We know that the expression inside the limit works, assuming that it converges (it has to, since the norm is a function, and the inside limit forms a Cauchy Sequence on F) it converges to the boundary (supremum or infimum in the case of the reals), which is defined with

[@PunmasterSTP]

Examples of Banach spaces? More like "Astounding mathematics teaching-places!" These videos are amazing; thanks so much for making them.

[@ahmedamr5265]

Amazing video as usual, thanks! I think the answer to the 2nd question in the quiz on Steady is incorrect. It doesn't satisfy the positive definiteness of a norm.

[@brightsideofmaths]

Thank you very much :) I will check the question!

[@brightsideofmaths]

I corrected it! Thanks again!

[@qiaohuizhou6960]

2:13 Lp space 4:37 lp is Banach space

[@Studiacapta-qo2bo]

Ich jetzt versuche Deutsch zu lernen und es würde mir sehr sehr hilfreich sein wenn du diese Videos auch auf Deutsch übersetzen könntest. Danke schön immer für diese tollen Videos!

[@brightsideofmaths]

Yes, at some point you find these on the second channel.

[@bikenzhumagazy965]

why did you put (l) to the degree of x tilda while proving convergence, it was a bit suspicous! ?Cuz x tilda should have been constant sequence when x(k) was approuching to it...

[@brightsideofmaths]

What do you mean?

[@MrWater2]

Have this example a visual or geometric interpretation? For example base on what I've learn from you a sequence is just a map from N to R (or C and maybe more but now I don't know yet) so visually it's just a curve on a 2-d plane where it's pretty easy to see what does it mean to converge, the same applied to a generic metric space where we see the points (whatever they are) are closer and closer (because we have a notion of distances with the metric). So in this example, how can I see the convergence of sequences of sequences or the p-norm? it is related to the uniform convergence? I mean I imagine a sequence that's a curve, and take a subsequence from it and that's another curve, and so on. Thanks again by the way.

[@brightsideofmaths]

A sequence is more like a discrete curve. Maybe my Real Analysis Series can help you there: tbsom.de/s/ra

[@nachomacho7027]

👌

[@awaismaths]

How did the second property of norm is not satisfied in proving lp subspaces? You were too quick in mentioning at 04:00

[@brightsideofmaths]

Without the pth-root, you would not be able to pull out the scalar with the absolute value like you want it for the second property.

[@saidsaterih3729]

Did you prove existence of this limit x_tild for Cauchy sequence in lp norm?

[@brightsideofmaths]

Yes, I did :) It's the last part of the video.

[@chianchen776]

7:43 well it might be my misunderstanding but, how do we be sure it forms a Cauchy sequence in 4th dimension? What if it behave periodically? As in the sequence formed by the 4th elements of all vector is periodic, such as a sinusoid.

[@brightsideofmaths]

Can that be if we assume that x^(k) is a Caucht sequence in l^p?

[@chianchen776]

Thank you for reminding me@@brightsideofmaths , I was stuck with intuition and now to look at the rigorous definition of Cauchy sequence (not just the intuitive idea) this made sense finally.

[@ahippiething]

I keep rewatching the video but there is one part that I didn't understand. so we know that for k,l > K any two sequences are at most epsilon prime distance from each other right? Then how can we be sure that x tilde is the limit? I know that the limit is one of the sequences with an index that is larger than K, but how are we so sure that it is x tilde?

[@brightsideofmaths]

Maybe, I don't understand the question correctly but we need to show that x tilde is the limit. Therefore, at 8:55, I write down the norm which should be arbitrarily small.

[@GODSONNWOSU-nb5or]

Straight to the point with no ambiguity, thanks for sharing ✌️

[@brightsideofmaths]

Thanks :)

[@kacper2028]

what the f**k is happening

[@brightsideofmaths]

Functional analysis :D

[@elhoplita69]

Hi I have one question: In minute 11:00 why did you set epsilon' := epsilon/2? Why is it neccesary?

[@brightsideofmaths]

Yeah, thanks for the question. We need something smaller than epsilon to get the strict inequality. Making epsilon half as big works :)

[@elhoplita69]

Thanks for your reply, I think I finally got it, great job not only in this video but also in the other ones :) Btw, are you planning to make videos about group theory someday??

[@brightsideofmaths]

@@elhoplita69 Yeah, I have some in my Start Learning mathematics series. However, I will also expand the topic in future.

[@qiaohuizhou6960]

Hi, thx for the video. I am wondering to show x-tilda is also an element in the Lp space, can I simply say that each component of x-tilde is a component of F hence, x-tilde must also be a sequence in F?

[@brightsideofmaths]

Of course, the x-tilde is a sequence in F. There you have the correct argument. However, you also have to show the second condition for being in l^p.

[@qiaohuizhou6960]

@@brightsideofmaths sorry I’m a bit confused. To show the Lp space is complete, we need to show every Cauchy sequence is convergent, I thought we have already shown in the previous step that x-tilde is indeed the limit, it seems the last step suggest we still need to show the limit is inside of the Lp space . If that’s the case, which step guarantee x^k-k-tilde is an element of Lp? Thank you so much for your patient!

[@brightsideofmaths]

@@qiaohuizhou6960 Yeah, you are quite right but we still have to be cautious here. We showed that the sequence x-tilde is the limit w.r.t. the p-norm. However, we haven't shown yet that x-tilde lies in the l^p-space itself.

[@walidbahjaoui8474]

lala 9wadt

[@safayassin2189]

those videos are helping me a lot, thanks!

[@brightsideofmaths]

You are very welcome and thanks for the support :)

[@rajshreejoshi9765]

Mathematics means headache for most of the people and MSc in mathematics is on another level. Teachers don't take your syllabus seriously they just need their payment on time. But tutors like you not just hope of students like us but also give us new vision to look at mathematics. Praise from India. 🙏❤️

[@mctab1]

Sir, can you give a definition of a dimension that you use when you tell that R is one-dimensional and {0} is zero-dimensional

[@brightsideofmaths]

That is the normal definition you find in linear algebra: the cardinality of a basis.

[@mctab1]

@@brightsideofmaths but in the case of {0} how can it be zero-dimensional? If it is zero-dimensional then there's no elements in the basis?

[@brightsideofmaths]

@@mctab1 Exactly! The empty set is a basis of the vector space {0}.

[@mctab1]

@@brightsideofmaths ok, thank you!

[@adilaid4443]

Nice work