Functional Analysis 12 | Continuity

[@Independent_Man3]

Thank you for another video. I would like to see future videos on implicit function theorem and inverse function theorem.

[@pontifexmaximus_e]

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|.

[@tianchengxue5377]

I like your approach of using the squeeze theorem to explain sequential limit convergence.

[@pleasetakemyadvice]

He has not used the Squeeze Theorem, but just the fact that a

[@timwestlund3072]

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

[@brightsideofmaths]

Yeah, I like that :)

[@PunmasterSTP]

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.

[@cindywu9623]

Thank you so much! These videos are absolutely top notch.

[@brightsideofmaths]

Thank you very much for your support :)

[@manzoorhussainshigri7610]

thank you so much sir! God bless you

[@sergiohuaman6084]

fantastic video! thanks for sharing

[@birdeye700]

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

[@tim-701cca]

There is no need to do this because he used the continuity of inner product above to prove.

[@bilalghermoul3634]

Good explanation

[@pan19682]

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

[@ahmedamr5265]

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

[@brightsideofmaths]

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

[@ahmedamr5265]

Ah yes, thanks a lot :)@@brightsideofmaths

[@osvaldonava5827]

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.

[@PunmasterSTP]

I've seen him mention that he uses Xournal.

[@AlexisPapic]

This is very good! Can anyone recommend a book to follows this lectures?

[@andyl.5998]

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.

[@callumgilfedder9097]

Great video

[@brightsideofmaths]

Thanks!

[@oskaradolfson7450]

Thanks for the great videos! What topics are you exactly intending to cover in this functional analysis series?

[@brightsideofmaths]

I want to cover a lot of topics :) This will be my biggest series ever :D

[@shibshankardey5934]

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

[@simonepirrera3855]

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!

[@brightsideofmaths]

Thank you for your question. Please note here that we already choose a sequence (x_n) that has a limit in X.

[@simonepirrera3855]

​@@brightsideofmaths Thanks again! Your videos are helping me a lot with my PhD research. I will start supporting you on Steady today :-)

[@PunmasterSTP]

@@simonepirrera3855 I'm just curious; what are you researching and how is it going?

[@simonepirrera3855]

​@@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)

[@PunmasterSTP]

@@simonepirrera3855 That sounds really cool, and exciting. I'm glad you're getting some results!

[@xwyl]

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.

[@TheAaronDrew]

Great videos. Good job! Btw, I'm interested what software you might use for the vids. Making vids is good way to learn.

[@brightsideofmaths]

I use Xournal :)

[@TheAaronDrew]

@@brightsideofmaths Thanks much:)

[@scollyer.tuition]

I like the [] notation for the pre-image. Is that your own invention? I don't think that I've seen it before.

[@brightsideofmaths]

I also like the notation. It is not my invention but not used very often, sadly.

[@scollyer.tuition]

@@brightsideofmaths I'm stealing it. As Picasso said: great artists steal; good artists borrow.

[@brightsideofmaths]

@@scollyer.tuition Please do it :)

[@tsifj]

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?

[@brightsideofmaths]

There is no flipping. It's the same idea applied again :)

[@arturo3511]

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 ?

[@arturo3511]

Also standard metric means d(x,y) = |x-y| for Reals and ||x-y|| for complex number? Thank you

[@rohityparasnis]

Shouldn't you have limit superior instead of limit (lim-sup instead of lim) on the LHS at 6:46?

[@pneujai]

agreed, indeed we should have limsup≤f(x~)≤liminf now limsup≤liminf implies lim exists

[@zoedesvl4131]

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.

[@brightsideofmaths]

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.