Man I really appreciate how that first slide broke down the differences in such a clear and simple way! Thanks again so much for making and sharing all of these wonderful videos.
Hilbert spaces! finally! i have listened to every single videos in order from "start learning mathematics" just to get there! what an amazing journey it was and what a great teacher you are! thank you professor! (of course trying to learn quantum mechanics is the cause for all this madness!)
@@brightsideofmaths Of course! I am following the very useful map of mathematics you have given in your website! And I have no choice but to go up to manifolds for general relativity! I don't think there is a single area of mathematics you have missed which is needed for modern physics! Was it a deliberate choice?
@@brightsideofmaths beautiful! mission accomplished! Then I think we can say that the dark side of mathematics is any kind of mathematics which has no use in physics! (just joking!)
hey thanks a lot for these videos. I am currently taking a functional analysis lecture and it is awesome to hear the fundamental definitions explained by someone else :) weiter so!!
So, this is Hilbert space. I remember back in university I saw a super thick book with 'Hilbert space' on its hardcover. It was at that moment Hilbert space casted huge shadow over my soul. Now it doesn't seem to be very terrifying... at the first look.
Hi, thank you very much. I have a question; at 6:58 you said that we can define a special norm based on inner product, and norm must satisfy three properties, I have no problem with first two properties, but about 3rd one, can't obtained it. ||x+y|| =< ||x|| + ||y|| for the left side, ||x+y|| = sqrt() = sqrt( + + + ). now the right side, sqrt() + sqrt(). I don't know how to continue!
A really brief recap of trigonometry would be useful when you defined the inner product using cos(\alpha). Some time ago I was trying to read Time Series Theory and Methods from Blackwell and Davis, they have a chapter (2) about Hilbert spaces. I wish I would have your videos in that moment! Probably I'm going to try to read again that book when I finish this series. Thanks again!
What's that about linearity in the second component only ? Shouldn't it be linear in both its arguments anyway ? And I'm also not 100% clear on why we need the conjugate when we're dealing with complex numbers, anyone got an example ?
@7:13 I am not sure on this but I think one cannot measure angles in a hilbert space. Angles can only be measured if the hilbert space is real. Complex hilbert spaces do not have the concept of angles...
Yes, standard angles only exist in real vector spaces, but the general concept still holds in complex vector space if one wants to generalize it. It's an abstract concept anyway :)
@@brightsideofmaths As far as I know the standard formula u.v=|u||v|cost will not work on complex spaces no matter how one generalizes the concept of angle. See www.people.vcu.edu/~rhammack/reprints/cmj210-217.pdf, the very last paragraph just before the references. I think any generalization of angles must depend on analytic continuation, but the conjugates in the metric destroys it.
I had a question, is norm and inner product defined only for vector spaces over real or complex field? Is it not definable for any vector space over an arbitrary field, for ex. over Q? What I'm guessing is that there are some problems that come in defining it for such cases, either losing consistency or usefulness. If yes, what are these problems?
Nice explaination, but you could've extended a little more on Hilbert Spaces and throw some examples!! I'm recomminding this video to lots of friends! =)
@@brightsideofmaths Oooooh I didn't even catch your accent, you have good diction (unlike me, even in my mother tongue). Even on 1.5x speed I can understand everything. But now that I think about it, the RUclips CC auto-generator detects the language as German in some of your videos, I should've realized lol
Do some definitions of the inner product over ℂ behave in a way that swaps the notation order like so, < k x, y> = k < x, k y > = conj(k) ? 5:57 Lol nvm 6:17
Man I really appreciate how that first slide broke down the differences in such a clear and simple way! Thanks again so much for making and sharing all of these wonderful videos.
Thanks :)
Fantastic ability to explain abstract ideas! Thank you so much.
Hilbert spaces! finally! i have listened to every single videos in order from "start learning mathematics" just to get there! what an amazing journey it was and what a great teacher you are! thank you professor! (of course trying to learn quantum mechanics is the cause for all this madness!)
Nice! I also have a series about Hilbert Spaces and Unbounded Operators. See description :)
@@brightsideofmaths Of course! I am following the very useful map of mathematics you have given in your website! And I have no choice but to go up to manifolds for general relativity! I don't think there is a single area of mathematics you have missed which is needed for modern physics! Was it a deliberate choice?
@@batmanrobin6711 Yes, this is my goal :)
@@brightsideofmaths beautiful! mission accomplished! Then I think we can say that the dark side of mathematics is any kind of mathematics which has no use in physics! (just joking!)
this is a very concise and perfect definition
hey thanks a lot for these videos. I am currently taking a functional analysis lecture and it is awesome to hear the fundamental definitions explained by someone else :) weiter so!!
How did the rest of your class go?
So, this is Hilbert space. I remember back in university I saw a super thick book with 'Hilbert space' on its hardcover. It was at that moment Hilbert space casted huge shadow over my soul. Now it doesn't seem to be very terrifying... at the first look.
Thanks! :)
Hi, thank you very much.
I have a question; at 6:58 you said that we can define a special norm based on inner product, and norm must satisfy three properties, I have no problem with first two properties, but about 3rd one, can't obtained it.
||x+y|| =< ||x|| + ||y||
for the left side,
||x+y|| = sqrt() =
sqrt( + + + ).
now the right side,
sqrt() + sqrt().
I don't know how to continue!
4:42 I feel confused about why we need a complex conjugate, could anyone give me a further explanation?
Is there any situation that is not a real number when x,y is in complex space?
Take the vector x=(i,0), if you don't conjugate you have =i^2=-1 which contradicts positivity
To get real value
Llo😅ol😅p 😅lol😅😅😅😅😅😅😅l😅😅o😅😅😅😅lo
I always give thumbs up to alllll of your videos love you so much 😂❤❤❤
Best video on inner space I've seen till now 🔥🔥
Thanks :)
I loved it. Thank you so much.
Nice :)
A really brief recap of trigonometry would be useful when you defined the inner product using cos(\alpha). Some time ago I was trying to read Time Series Theory and Methods from Blackwell and Davis, they have a chapter (2) about Hilbert spaces. I wish I would have your videos in that moment! Probably I'm going to try to read again that book when I finish this series. Thanks again!
Could you recommend a book for functional analysis?
Rudin :)
Perfect!
What's that about linearity in the second component only ? Shouldn't it be linear in both its arguments anyway ? And I'm also not 100% clear on why we need the conjugate when we're dealing with complex numbers, anyone got an example ?
Ah, I got it ! It's to ensure positivity alright
Exactly :)
@7:13 I am not sure on this but I think one cannot measure angles in a hilbert space. Angles can only be measured if the hilbert space is real. Complex hilbert spaces do not have the concept of angles...
Yes, standard angles only exist in real vector spaces, but the general concept still holds in complex vector space if one wants to generalize it. It's an abstract concept anyway :)
@@brightsideofmaths As far as I know the standard formula u.v=|u||v|cost will not work on complex spaces no matter how one generalizes the concept of angle. See www.people.vcu.edu/~rhammack/reprints/cmj210-217.pdf, the very last paragraph just before the references.
I think any generalization of angles must depend on analytic continuation, but the conjugates in the metric destroys it.
There is a video also for topological spaces?
Not yet :)
Is linearity defined in second argument is common? because I try reading from some materials all says it defined linearity on first argument
Both definitions are common :)
fantastic
I had a question, is norm and inner product defined only for vector spaces over real or complex field? Is it not definable for any vector space over an arbitrary field, for ex. over Q?
What I'm guessing is that there are some problems that come in defining it for such cases, either losing consistency or usefulness. If yes, what are these problems?
You could generalise that concept but for us it makes sense to look at the R and C because they are complete.
Perfect
Hello sir which software u will use for class
I've seen him mention elsewhere that he uses Xournal.
Is that a double bass playing subtly in the background?
What? :D At which timestamp? :)
@@brightsideofmaths Maybe it's just the background noise :D Thanks for the great videos!
@@danielyoo828 There was a construction site outside at the time. Maybe that's it :D
Nice explaination, but you could've extended a little more on Hilbert Spaces and throw some examples!! I'm recomminding this video to lots of friends! =)
Thank you! Example you find in the next video :) ruclips.net/video/eiD6OueArHE/видео.html
Please I would like you to make video on pseudo differential operators. I am now understanding the mathematics I did in my undergrad
Bist du mittlerweile Univ.Prof. für Mathe? Immerhin hast du damals mein Studium gerettet 😂
"Damals" heißt? :D Ich bin ja schon alt ;)
Nice!
semi-inner product having all those properties except =0 =>X=0
Yeah!
oh yeah
The 100th thumb up
everything is more complicated when a particle becomes an ocean isnt it xD
lol
😊
Mmmm😊
Delicious mathematics
Terrible German english
That is the world we live in :)
@@brightsideofmaths Oooooh I didn't even catch your accent, you have good diction (unlike me, even in my mother tongue). Even on 1.5x speed I can understand everything.
But now that I think about it, the RUclips CC auto-generator detects the language as German in some of your videos, I should've realized lol
Do some definitions of the inner product over ℂ behave in a way that swaps the notation order like so,
< k x, y> = k
< x, k y > = conj(k)
? 5:57
Lol nvm 6:17
I define the inner product in this way :)