inner product with one argument fixed is a linear functional, linear functional are continuous, continuous function valued at a limit point equals to the limit of function values as input goes to infinity => lim[n] = => = 0 //f continuous at a limit point : if lim[t→x]f(t) = f(x) : ∀ε>0 in Y (dY(f(x),f(t)) < ε → ∃δ>0 in X (dX(x,t)) //fx(y)=⟨y,x⟩ denote the inner product function. Note that this is a linear functional -- that is, it is linear in y, and maps vectors to scalars. //It is a well-known theorem that linear functionals are continuous (on the entire space) if and only if they are bounded. Here, "bounded" means that there exists a constant M such that |f(y)|≤M|y| for all y in the space. //That the inner product functional is bounded now follows from the Cauchy-Schwarz Inequality: |f(y)|≤|x||y|.
An alternative proof that the orthogonal complement is closed (it relies on the fact that f^-1[M] is closed for all closed sets M iff f is continuous. This was not proven in the video, but it is easy to prove). Let U be an arbitrary subset of an inner product space, then we note that the orthogonal complement of U is the intersection of the orthogonal complements of all one point sets {u} in U. However, as the function f(x)= is continuous, all orthogonal complements of the one point sets are closed as they are the preimage of the closed set {0}. Now we get that the orthogonal complement of U is an intersection of closed set and thus it is closed. QED
Continuity? More like "I can't wait to see" what's up next in this course! Thanks again for making and sharing all of these very high-quality lectures.
The last step toward the end is not trial, and it needs extra steps. : =. = . + = . + 0 = .. As limit (Xn-X^) =0, for any fixed U , . ----> 0, the rest follows
Thanks so much for the video! you're a hero! One doubt: In example (c), I guess the map f:X->[0,inf) and not f:X->R. Otherwise if we test the first definition of continuity we can pick an open interval e.g. (-3,-1) in Y which doesn't have an inverse
Dr. Großmann (The Bright Side Of Mathematics) offers his lecture notes in PDF to his Steady members. But if you want a book in the traditional sense, I'd recommend Introductory Functional Analysis with Applications by Erwin Kreyszig.
It would be agood idea to present us many more applications concerning the theory in maths you are dealing with. i think there is a version of this kind in german language many thanks for your efforts
Example (c), 5th minute: do we actually need X to be a Banach space so that the limit point of each sequence is still in X? PS: thank you for sharing your knowledge!
@@PunmasterSTP I work on control theory. Currently I am focusing on the identification of continuous time dynamical systems. In this context concepts from functional analisys (e.g. function norms) are often used in the literature. It is going quite well I guess... we are about to submit a first result 🙂 (for me it's the first, I started my PhD in November)
I love your videos and your style. How do you make your videos? Do you have a digital pen or an iPad? Which app do you use? I would like to make this kind of videos.
Could you plz upload a video on the uniform and absolute continuity and their difference along with continuity? I am struggling to see any video on them. It will be helpful to many I believe
In the proof of sequential continuity of normed spaces, how applying the limit on the similar expression with reversed x_n, x tilde flips the inequality?
Epsilon-delta definition for continuity would be easier. And using the definition for sequence limit in (c) would be more fundamental thus better than the double inequality trick.
So if I were under Germany's mathematic education system, when would I usually take courses like this one? Senior undergrad or master? I think your series of fuctional analysis can be considered as a (student-friendly) classic functional analysis course like many other math education systems.
This course is for everyone after they learnt a typical Analysis I and II course. So usually, you would take such a Functional Analysis course in the second or third year of your studies. Yeah, I try to do a more or less classical course in this topic because it is so important in so many fields that some groundwork just has to be taught here.
![Functional Analysis 13 | Bounded Operators [dark version]](http://i.ytimg.com/vi/ENdE6esHdZk/mqdefault.jpg)
Thank you for another video. I would like to see future videos on implicit function theorem and inverse function theorem.
inner product with one argument fixed is a linear functional, linear functional are continuous, continuous function valued at a limit point equals to the limit of function values as input goes to infinity => lim[n] = => = 0
//f continuous at a limit point : if lim[t→x]f(t) = f(x)
: ∀ε>0 in Y (dY(f(x),f(t)) < ε → ∃δ>0 in X (dX(x,t))
//fx(y)=⟨y,x⟩ denote the inner product function. Note that this is a linear functional -- that is, it is linear in y, and maps vectors to scalars.
//It is a well-known theorem that linear functionals are continuous (on the entire space) if and only if they are bounded. Here, "bounded" means that there exists a constant M such that |f(y)|≤M|y| for all y in the space.
//That the inner product functional is bounded now follows from the Cauchy-Schwarz Inequality: |f(y)|≤|x||y|.
An alternative proof that the orthogonal complement is closed (it relies on the fact that f^-1[M] is closed for all closed sets M iff f is continuous. This was not proven in the video, but it is easy to prove).
Let U be an arbitrary subset of an inner product space, then we note that the orthogonal complement of U is the intersection of the orthogonal complements of all one point sets {u} in U. However, as the function f(x)= is continuous, all orthogonal complements of the one point sets are closed as they are the preimage of the closed set {0}. Now we get that the orthogonal complement of U is an intersection of closed set and thus it is closed. QED
Yeah, I like that :)
I like your approach of using the squeeze theorem to explain sequential limit convergence.
He has not used the Squeeze Theorem, but just the fact that a
Continuity? More like "I can't wait to see" what's up next in this course! Thanks again for making and sharing all of these very high-quality lectures.
Thank you so much! These videos are absolutely top notch.
Thank you very much for your support :)
thank you so much sir!
God bless you
fantastic video! thanks for sharing
The last step toward the end is not trial, and it needs extra steps. : =. = . + = . + 0 = .. As limit (Xn-X^) =0, for any fixed U , . ----> 0, the rest follows
There is no need to do this because he used the continuity of inner product above to prove.
Thanks so much for the video! you're a hero!
One doubt: In example (c), I guess the map f:X->[0,inf) and not f:X->R. Otherwise if we test the first definition of continuity we can pick an open interval e.g. (-3,-1) in Y which doesn't have an inverse
Thanks a lot! Good question but remember: we don't need to have an inverse to calculate preimages :)
You can check here: tbsom.de/s/sls
Ah yes, thanks a lot :)@@brightsideofmaths
Good explanation
This is very good! Can anyone recommend a book to follows this lectures?
Dr. Großmann (The Bright Side Of Mathematics) offers his lecture notes in PDF to his Steady members.
But if you want a book in the traditional sense, I'd recommend Introductory Functional Analysis with Applications by Erwin Kreyszig.
It would be agood idea to present us many more applications concerning the theory in maths you are dealing with. i think there is a version of this kind in german language many thanks for your efforts
Great video
Thanks!
Example (c), 5th minute: do we actually need X to be a Banach space so that the limit point of each sequence is still in X?
PS: thank you for sharing your knowledge!
Thank you for your question. Please note here that we already choose a sequence (x_n) that has a limit in X.
@@brightsideofmaths Thanks again! Your videos are helping me a lot with my PhD research.
I will start supporting you on Steady today :-)
@@simonepirrera3855 I'm just curious; what are you researching and how is it going?
@@PunmasterSTP I work on control theory. Currently I am focusing on the identification of continuous time dynamical systems.
In this context concepts from functional analisys (e.g. function norms) are often used in the literature.
It is going quite well I guess... we are about to submit a first result 🙂 (for me it's the first, I started my PhD in November)
@@simonepirrera3855 That sounds really cool, and exciting. I'm glad you're getting some results!
Thanks for the great videos! What topics are you exactly intending to cover in this functional analysis series?
I want to cover a lot of topics :) This will be my biggest series ever :D
I love your videos and your style. How do you make your videos? Do you have a digital pen or an iPad? Which app do you use? I would like to make this kind of videos.
I've seen him mention that he uses Xournal.
Could you plz upload a video on the uniform and absolute continuity and their difference along with continuity? I am struggling to see any video on them. It will be helpful to many I believe
In the proof of sequential continuity of normed spaces, how applying the limit on the similar expression with reversed x_n, x tilde flips the inequality?
There is no flipping. It's the same idea applied again :)
I like the [] notation for the pre-image. Is that your own invention? I don't think that I've seen it before.
I also like the notation. It is not my invention but not used very often, sadly.
@@brightsideofmaths I'm stealing it. As Picasso said: great artists steal; good artists borrow.
@@scollyer.tuition Please do it :)
Great videos. Good job! Btw, I'm interested what software you might use for the vids. Making vids is good way to learn.
I use Xournal :)
@@brightsideofmaths Thanks much:)
Shouldn't you have limit superior instead of limit (lim-sup instead of lim) on the LHS at 6:46?
agreed, indeed we should have limsup≤f(x~)≤liminf
now limsup≤liminf implies lim exists
Epsilon-delta definition for continuity would be easier. And using the definition for sequence limit in (c) would be more fundamental thus better than the double inequality trick.
What does it mean all subsets in a discrete metric space are opposites (4:15) and which definition does it use to imply that it is continuous ?
Also standard metric means d(x,y) = |x-y| for Reals and ||x-y|| for complex number? Thank you
So if I were under Germany's mathematic education system, when would I usually take courses like this one? Senior undergrad or master? I think your series of fuctional analysis can be considered as a (student-friendly) classic functional analysis course like many other math education systems.
This course is for everyone after they learnt a typical Analysis I and II course. So usually, you would take such a Functional Analysis course in the second or third year of your studies.
Yeah, I try to do a more or less classical course in this topic because it is so important in so many fields that some groundwork just has to be taught here.