You're the man hahaha this is the most comprehensible real and functional analysis course video set ever. It helps a ton to demystify the more technical and rigorous definitions from the course itself. Thank you!!!!
Go back to the previous video where Hilbert spaces are introduced. If we are on the complex field, we always have to conjugate when multiplying one of the arguments by a scalar. It is to ensure the positive definiteness of the inner product
I thought it was quite intuitive the way you built metric > norm > inner product Are there any good reasons to go the opposite direction? That what my linear algebra book does
The other way starts with a well-known structure and goes to extract some essential parts. My way was building a house from the ground with each floor. The other way is looking at a finished house and what the components are. Both ways are fine, I guess :)
@@jared805 Take a look at Linear Algebra done right (ch. on inner product spaces); there you'll find a very good presentation of the opposite direction.
Very high quality videos all of these, much impressed! Doing a course on Functional Analysis right now and needed to refresh my basic knowledge and this is helping a lot
Why is that the complex conjugation of scaler product at 6:40 ??? I thought that complex conjugation is needed in the second part of the product. aha in this they do it in the first part not in the second, ok.
First of all, and as always, thanks for the nice video :-) 7:09 To go from 1. to 2. line you use linearity in the second argument and from 2. to 3. you use linearity in the first argument. Isn't there a complex conjugate missing at some point?
Very nice video, a quicker way to show the CS inequality is to consider the function that maps any real number t to norm(x + ty)^2 Using the properties of the scalar product, we can see that this function is a second order nonnegative polynomial in t Non-negativity implies then that its discriminant is lesser or equal to 0, which directly yields the desired inequality
I am not the author of this wikipedia article ;) You can check my video about the definition of the inner product. The thing is that two different definitions are in common use.
@@brightsideofmaths This proof looks like a byproduct of calculating the norm of the orthogonal part. And in this process, Pythagoras theorem is also proved. But I want an intuitive method. The most important thing to use is linearity, we can safely assume ||x||=1 and transform the original inequality to ^2=p^2=^2 (because o is orthogonal to x). This proof also relies on linearity and positivity. In my proof I didn't consider complex numbers.
![Functional Analysis 12 | Continuity [dark version]](/img/n.gif)
You're the man hahaha this is the most comprehensible real and functional analysis course video set ever. It helps a ton to demystify the more technical and rigorous definitions from the course itself. Thank you!!!!
Wouldn´t it be a great idea to start lectures on Differential Geometry? Humble request!
Yes please
Captions really would help people like me, who can't understand English very well. Anyway, love your videos, they are really helpful. Thanks a lot.
Thanks! Some nice subscribers always provide subtitles but they are not available for all videos yet. I am sure that they will come :)
Cauchy-Schwarz? More like “cool and smart!” I’m loving your videos; keep up the amazing work.
cringe
@@afaale1 Then my work here is done 😎
can you explain why we conjugate the after we pull out from the 2*Re(...)?
Go back to the previous video where Hilbert spaces are introduced. If we are on the complex field, we always have to conjugate when multiplying one of the arguments by a scalar. It is to ensure the positive definiteness of the inner product
I thought it was quite intuitive the way you built metric > norm > inner product
Are there any good reasons to go the opposite direction? That what my linear algebra book does
The other way starts with a well-known structure and goes to extract some essential parts. My way was building a house from the ground with each floor. The other way is looking at a finished house and what the components are. Both ways are fine, I guess :)
@@brightsideofmaths for the record i much prefer yours
@@jared805 Take a look at Linear Algebra done right (ch. on inner product spaces); there you'll find a very good presentation of the opposite direction.
Very high quality videos all of these, much impressed! Doing a course on Functional Analysis right now and needed to refresh my basic knowledge and this is helping a lot
How did the rest of your course go?
@@PunmasterSTP Not that great because I found I was still a bit underprepared, but I thoroughly enjoyed it anyway. Thanks for asking!
@@ChristopherZAR I'm glad it was enjoyable, and you are most welcome for my asking!
Thanks to you
Thank you this is very nice. Your measure theory course is very nice as well.
Why is that the complex conjugation of scaler product at 6:40 ??? I thought that complex conjugation is needed in the second part of the product. aha in this they do it in the first part not in the second, ok.
First of all, and as always, thanks for the nice video :-)
7:09 To go from 1. to 2. line you use linearity in the second argument and from 2. to 3. you use linearity in the first argument. Isn't there a complex conjugate missing at some point?
The complex conjugation is there. Note that in the absolute value and in the real part it does not make a difference.
it was much easier to prove it by 0 . and using the solution to the polynomial b ^2 - 4ac
Sure, this is also possible :)
amazing job. though i still have some difficulties understanding it. how do i get the PDF for better study? thank you
Thanks! The PDF is the description with a link on Steady :)
What prerequisites do I need to fully understand this proof?
See here: thebrightsideofmathematics.com/startpage/
Just some minor feedback. The red, green and orange line colors could be changed. There may be viewers with color perception issues,
Very good point! Thanks :)
These videos are so cool!
Very nice video, a quicker way to show the CS inequality is to consider the function that maps any real number t to norm(x + ty)^2
Using the properties of the scalar product, we can see that this function is a second order nonnegative polynomial in t
Non-negativity implies then that its discriminant is lesser or equal to 0, which directly yields the desired inequality
Yes, that is correct :)
6:25 why conjugate comes out?
This is in the definition of the inner product: it's antilinear in the first argument.
@@brightsideofmaths in wikipedia conjugate comes out if it is in the second place not in the first place in inner product could you check it sir?
I am not the author of this wikipedia article ;)
You can check my video about the definition of the inner product. The thing is that two different definitions are in common use.
@@brightsideofmaths oh ill see ill check it later and ill subscribe now good day
Maybe starting from the norm of y parallel is less or equal to the norm of y, then square the both side, would be much easier to complete the proof.
Why do you know that the norm of y parallel is less or equal to the norm of y?
The proof looks highly constructed and counterintuitive. Is there a natural way to prove it?
I tried to make it intuitive. In my opinion, this is the most natural proof for the inequality.
@@brightsideofmaths This proof looks like a byproduct of calculating the norm of the orthogonal part. And in this process, Pythagoras theorem is also proved.
But I want an intuitive method. The most important thing to use is linearity, we can safely assume ||x||=1 and transform the original inequality to ^2=p^2=^2 (because o is orthogonal to x). This proof also relies on linearity and positivity.
In my proof I didn't consider complex numbers.
Hi, Thanks again for the clear explanations. By the way, are you planning to talk about Hölder's inequality also?
Yeah, of course. This is coming after the inner products :)
Wow, two videos in a row. How did we deserve that? :D
Great Video!
Hello sir,
I love your content and way of teaching so please sir start sessions on discrete mathematics! Humble request
So good, what program you using to make the videos?
OBS :)
Your videos are outstanding; Please start lecture on Differential equations.
Thanks :) I will do that!
Keep it lit mate
AHHH MATHE
is that a german accent? sry cant overhear it^^
Yeah, it is! Glad you notice it. I don't try to cover it :D
@@brightsideofmaths yeah no need to cover it anyways^^ thank you for the video
А БУНЯКОВСКИЙ ГДЕ БЛЯТЬ???
Absorbed in "Cauchy" :)
@@brightsideofmaths :(((
@@ДаниилЗайченко-в1ф That is what happens if your a student of a famous mathematician. It's not a fair life.
I think a more elegant proof for Cauchy-Schwartz is to use the quadratic formula: ghomi.math.gatech.edu/LectureNotes/LectureNotes0U.pdf
Yes, it's a matter of taste :)