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.
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.
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!
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!
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! =)
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?
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.
@@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 RU-vid 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
