Distributions 5 | Regular Distributions

[@ifyhu92]

Your videos are amazing! I love you, please keep creating content! You change my life in such a good way!

[@brightsideofmaths]

Wow, thank you! :)

[@swatiyadav1631]

Nice series. Befor starting a book on distribution theory, I watched this which is very helpful. I request you to provide some further applications of distributions.

[@davidsewell4999]

Which book did you read? I am starting an applied functional analysis book and this has been quite helpful

[@swatiyadav1631]

@@davidsewell4999 The book I was supposed to go with is " S. Kesavan, Functional Analysis, TRIM 52, Hindustan Book Agency, New Delhi", but due to circumstances, I didn't carry this forward. However, please suggest to me the book you prefer. This might help me in future.

[@davidsewell4999]

@@swatiyadav1631 I see. I will give that book a look. I can’t say that it’s better but I am reading “Applied Functional Analysis” by DH Griffel. The first chapter starts with distributions and test functions which is why I was watching this lecture series. I was most interested in understanding Hilbert Spaces better but distributions has been an interesting first topic

[@swatiyadav1631]

@@davidsewell4999 Thank you.

[@em70171]

I love you for these!!!

[@zachchairez4568]

You’re the absolute greatest!

[@mathjitsuteacher]

Very nice video series. I had never studied distributions before and I wanted to have an idea of what it was about. Your videos are very nice and precise. Congratulations!

[@cameronspalding9792]

Where’s the next video

[@harryhirsch2024]

If f is (locally) integrable, it can be discontinuous, and therefore is not differentiable in the strong sense, so there is no D^alpha(f) in the strong sense.

[@brightsideofmaths]

But it can also be differentiable, where D^alpha(f) in the strong sense exists.

[@raneena5079]

Hello, I have a question. If the distribution is only defined on test functions, how come I regularly see it used on functions without compact support (like sin) in differential equations and physics contexts?

[@brightsideofmaths]

One can extend the definitions in some sense. Which example do you have in mind?

[@raneena5079]

@@brightsideofmaths The only definitions I have seen are the one you presented and the informal one (function that is 0 or ∞ with area 1) used in differential equations

[@AliJoohy]

Is there any other lectures?

[@brightsideofmaths]

I will upload the next lectures in future. However, it will need some time.

[@AliJoohy]

@@brightsideofmaths It was very nice lectures.

[@AliJoohy]

Can I have your contact?

[@lasha-georgemath8961]

Hello! I have Question: I don't understand the resultant idea. why should we border the distribution function with that thing on the right of the inequality. also I assume that either side of inequality lives in different spaces, is it right? And also, do we know the properties of test function space? Thanks a lot in advance

[@lasha-georgemath8961]

And also 5:50 why does Convergence of T(\psi_k) to 1 means that linear map T is not continuous?

[@brightsideofmaths]

Which inequality?

[@brightsideofmaths]

For continuous maps T(\psi_k) should go to 0 :)

[@carl3260]

@@lasha-georgemath8961 For any linear map: T(0) = T(0+0) = T(0) + T(0) => [by subtracting T(0)s] T(0) = 0 (where 0 in brackets is the "zero function", on the right just the number zero). So the fact that T(phi_k) --> 1 as phi_k --> phi=0 => T(phi_k) -/-> T(phi) as phi_k --> phi so not continuous.

[@carl3260]

Re: "why ...", it's just to present an alternative definition of distributions (that will presumably be useful later, similar to in real analysis where different definitions of the same thing are useful at different times). In fact it's just an alternative definition of the continuity aspect. Loosely speaking, this is saying that if the value of a linear map T [LHS] doesn't get too big (i.e. is bounded) relative to the gradients of any (K-support) test function [RHS], then T is continuous and so is a distribution. Note: both sides of the inequality are just numbers: T maps to R, norms map to R

[@sniperking4901]

when is the next video coming?

[@brightsideofmaths]

Sorry, I am not so fast in producing the videos as I like to be. Next month, it should be ready.

[@rodas4yt137]

@@brightsideofmaths Thanks for these videos! They are the only introduction to distributions available on YT! Please keep the series on until the end!

[@8feetxiao]

You are SOOOOO helpful, thanks!!

[@ecamusle]

Thank you so much for this!!

[@ayatnassar873]

Thank you so much

[@mattetor6726]

Thank you!

[@qrubmeeaz]

Wonderful!! Any chance you would continue this series to applications of Sobolev spaces to PDEs?

[@brightsideofmaths]

Thank you very much! Yes, that is the plan. However, I am not so fast as I want to be. So you need to be patient for while. Sorry!

[@evionlast]

Super!

[@shohamsen8986]

Good job producing videos on these topics. Ever consider making the thing more informal, more like a classroom lecture. You can make bigger videos that way.

[@Hold_it]

Yay :D

[@darkstudios001]

Great!