Judging from your explanation in your videos, you have a very deep understanding of the topic and you helped me alot to understand functional analysis. I would really love to see some videos about the applications on elliptic PDE's and also about the Sobolev space (maybe also about the Hölder space), since this was the most difficult part for me to understand. Thank you for the great videos anyway!
Wow, what a nice proof ! Btw, I have a question which might be outside the scope of this course : judging by your drawings, it seems the spectrum has infinite (uncountable) cardinality. Is it always the case ?
Thanks for your reply ! Also correct me if I'm wrong but it seems you haven't proved the middle part of claim c) with the expression of the spectral radius. Any hints on how I could prove it myself ?
@@brightsideofmaths Thanks for your reply bro. I am a researcher working on convex optimization, if you wanna promote your videos in China or make video about optimization theory, please contact me if you want.🤣
It appears when you write T^-2, you mean the linear operator formed by multiplying T inverse with T inverse. I am starting to understand you can work with exponents of functions the same way as with numbers. In particular in the Taylor series you are applying other exponents to change the ^-1 exponent which is normally used as inverse notation. Do you have videos that delve into this? Does this only work with linear operators?
What about the last bit of proving that the spectral radius is the limit of a sequence of operator norms ? This doesn't seem to follow directly from the two other results 🤔 Any hints or reference ?
@@brightsideofmaths oh no worries, I know a proof of this result ! But I just thought maybe it's a direct consequence of the two other statements and I was a bit puzzled haha
This is an excellent video and the proof involves a lot of interesting knowledge. Would be even better if elaborate why the composition of linear functionals and analytic functions is still analytic :)
![Functional Analysis 32 | Normal and Self-Adjoint Operators [dark version]](http://i.ytimg.com/vi/9k6qkzIPfBg/mqdefault.jpg)
![Functional Analysis 32 | Normal and Self-Adjoint Operators [dark version]](/img/tr.png)
Judging from your explanation in your videos, you have a very deep understanding of the topic and you helped me alot to understand functional analysis. I would really love to see some videos about the applications on elliptic PDE's and also about the Sobolev space (maybe also about the Hölder space), since this was the most difficult part for me to understand. Thank you for the great videos anyway!
You are such a knowledgeable mathematician, truly outstanding.
Thank you so much. I will need a long time to unpack your videos.
You are welcome! Of course, this was not easy stuff here but getting some ideas and remembering the important facts is possible.
Wow, what a nice proof ! Btw, I have a question which might be outside the scope of this course : judging by your drawings, it seems the spectrum has infinite (uncountable) cardinality. Is it always the case ?
Of course, this is possible. However, also finite spectrum is possible. Just take the identity operator.
Thanks for your reply !
Also correct me if I'm wrong but it seems you haven't proved the middle part of claim c) with the expression of the spectral radius. Any hints on how I could prove it myself ?
Really really thank you ! could you please recommend any textbook i can refer to about functional analysis?
I always would go with Rudin :)
@@brightsideofmaths Thanks for your reply bro. I am a researcher working on convex optimization, if you wanna promote your videos in China or make video about optimization theory, please contact me if you want.🤣
It appears when you write T^-2, you mean the linear operator formed by multiplying T inverse with T inverse. I am starting to understand you can work with exponents of functions the same way as with numbers. In particular in the Taylor series you are applying other exponents to change the ^-1 exponent which is normally used as inverse notation. Do you have videos that delve into this? Does this only work with linear operators?
It's a very good idea for a video! Powers of functions are usually understood with the composition in mind.
What about the last bit of proving that the spectral radius is the limit of a sequence of operator norms ? This doesn't seem to follow directly from the two other results 🤔
Any hints or reference ?
Oh, this is some work with sequences, so more basic than expected. I guess you find it in wikipedia.
@@brightsideofmaths oh no worries, I know a proof of this result ! But I just thought maybe it's a direct consequence of the two other statements and I was a bit puzzled haha
Oh, I see. Sorry for the confusion.@@StratosFair
At the end of the vedio, I cannot see how Hahn Banach theorem applys, could you give some hints? Thanks!
I’d like to understand why the function f_{l} is holomorphic. I supoose you can use the continuity of l and the series expansion of the resolvent(?)
Yes, you use the series expansion of the resolvent. Just try to write it done.
This is an excellent video and the proof involves a lot of interesting knowledge. Would be even better if elaborate why the composition of linear functionals and analytic functions is still analytic :)
You need more sample runs than palatable to construe any result with this method.
Thank you
When will this series continue? :)
After I have finished the Real Analysis series, which is very soon :)
@@brightsideofmaths makes sense :) keep up your good work!
care to explain why lambda is not in the spectrum?
Yes, we chose it like that.