Extremely clear explanations of a very beautiful result ! It was just not instantly clear for me that S^n was converging to zero, but it is actually easy to show by induction and using the definition of operator norm that norm(S^k)≤norm(S)^k for all k, so the result follows :)
I didn't know a matrix could be analytical. Calculus proves that functions can be expanded into a taylor series, but can it be shown that a matrix can be expanded into a taylor series and what would it even mean? It would approximate the matrix ? :0
Quick question about this "Taylor expansion" : can we identify and say that (T-\lambda)^{-k-1} is the "k-th derivative" of the map \lambda \mapsto (T - \lambda)^{-1} ?
Looking at f(lambda) = inverse of T-lambda, can we say that knowing the value of this function at only one point (lambda zero) is enough to know all its values in C minus the spectrum?
Hey! big fan of your page. I have request in functional analysis. Please do unbounded operators and adjoints also can you explain duality relationship also. Thank you
![Functional Analysis 31 | Spectral Radius [dark version]](http://i.ytimg.com/vi/7iNcVjLUINw/mqdefault.jpg)
![Functional Analysis 31 | Spectral Radius [dark version]](/img/tr.png)
Extremely clear explanations of a very beautiful result !
It was just not instantly clear for me that S^n was converging to zero, but it is actually easy to show by induction and using the definition of operator norm that norm(S^k)≤norm(S)^k for all k, so the result follows :)
Can you prove this by induction?
Very clear explanation- thanks!
I didn't know a matrix could be analytical. Calculus proves that functions can be expanded into a taylor series, but can it be shown that a matrix can be expanded into a taylor series and what would it even mean? It would approximate the matrix ? :0
Matrices have addition and multiplication as numbers. It works the same :)
Hi! Thanks for your beautiful videos! What tool do you use to write? It seems to work well, I'd like to buy one!
This the free program Xournal :)
@@brightsideofmaths thanks! I was also curious about the drawing tablet/pen
@@rodas4yt137 Just a graphics tablet. Everyone will do :)
@@brightsideofmaths ok thanks!
Wonderful video.
Thanks for visiting :)
Quick question about this "Taylor expansion" : can we identify and say that (T-\lambda)^{-k-1} is the "k-th derivative" of the map \lambda \mapsto (T - \lambda)^{-1} ?
Yeah, you just have to know how to define derivatives where the values of the function are in a Banach space.
@@brightsideofmaths this is a related question: Why is there no factorial in the denominator?
At the end how did you use sup=1/inf? This is not true generally. Right?
Looking at f(lambda) = inverse of T-lambda, can we say that knowing the value of this function at only one point (lambda zero) is enough to know all its values in C minus the spectrum?
Not exactly. The analytical property works only locally.
Hey! big fan of your page. I have request in functional analysis. Please do unbounded operators and adjoints also can you explain duality relationship also. Thank you
Will come eventually :)
When will this series continue? :)
I have already planned the content. So I guess in a few weeks, there will be need videos :)