I really enjoy this series and look forward to more videos ! An idea of interesting topic to cover would be projection theorems in Hilbert spaces and Banach spaces (with proofs ?). In any case, thank you for your work !
At 3:24 you say that compact operators are very close to matrices. Aren't they exactly equivalent because they're both linear? Can't every compact operator be represented as a matrix? (even if it's infinite-dimensional?)
A matrix is usually a finite table. Of course, you could represent compact operators by infinite-dimensional tables but I would be careful with the name "matrix" then.
@@brightsideofmaths the only other exposure I’ve had to FA is Frederic Schuller’s quantum theory lectures, and I struggled to understand some of the stuff he discussed about self-adjointness, essential self-adjointness, Schwartz space, Sobolev space, and other related topics. I’m interested in the fact that the standard position and momentum operators in QM are unbounded and not self-adjoint unless we restrict them to Schwartz space-at least I think that’s what I learned from Schuller, but it’s been a while so I don’t know if that’s correct! Anyway, I’d love to learn more about those sorts of things. And it would also be good to see examples of residual spectra!
Functional analysis heckin slaps! Just like this video! Thank you so much for the video!
Glad you liked it!
I really enjoy this series and look forward to more videos ! An idea of interesting topic to cover would be projection theorems in Hilbert spaces and Banach spaces (with proofs ?). In any case, thank you for your work !
After all, new aww piece. This series is really cool.
Great video as always, looking forward to the proof (or a sketch of it) of the main result !
The proof is really a lot of work. I won't do it in the next video immediately.
@@brightsideofmaths I think you could break it up into multiple videos if necessary
Spectracus!
Awesome series.
At 3:24 you say that compact operators are very close to matrices. Aren't they exactly equivalent because they're both linear? Can't every compact operator be represented as a matrix? (even if it's infinite-dimensional?)
A matrix is usually a finite table. Of course, you could represent compact operators by infinite-dimensional tables but I would be careful with the name "matrix" then.
This is gold ❤
So excited to watch this!
Would you ever do a video series on scattering theory? I understand if not, but it would be pretty neat since it uses spectral theory.
Good idea! :)
The eigenvectors of the operator T in the example are simply (1,0,0,0,…), (0,1,0,0,…), (0,0,1,0,…), etc., is that right?
Yes, indeed!
When will we see a next video?! Love the series! Btw, can you work out the examples too?
Yeah, thanks :)
The next video will come soon!
Any plans to continue this FA series soon?
Yes! The plans are there. Any wishes?
@@brightsideofmaths the only other exposure I’ve had to FA is Frederic Schuller’s quantum theory lectures, and I struggled to understand some of the stuff he discussed about self-adjointness, essential self-adjointness, Schwartz space, Sobolev space, and other related topics. I’m interested in the fact that the standard position and momentum operators in QM are unbounded and not self-adjoint unless we restrict them to Schwartz space-at least I think that’s what I learned from Schuller, but it’s been a while so I don’t know if that’s correct! Anyway, I’d love to learn more about those sorts of things. And it would also be good to see examples of residual spectra!
@@synaestheziac All very good topics I want to cover :)