Distributions 4 | Space of Distributions

[@TheSandkastenverbot]

Deine Videos sind richtig klasse! Du erklärst selbst Angst-Themen für Studenten wie Maßtheorie richtig verständlich und so motivierend, dass man sich weiter darin vertiefen will!

[@LL-wu5ui]

Thank you so much for your videos. I currently have a subject called "Partial differential equations 2" where we study Distributions, Fourier and Soboljev spaces. Sometimes it's really hard to understand such advanced mathematics, but you made it a lot easier for me. I also have "Measure theory" and plan on watching your videos (and take notes) as well. I really appreciate you and your effort, there aren't many videos explaining such advanced things, and you do it so well!

[@SergeyB1995]

Great video! I wanted to understand the topic from 2016 (out of curiousity mostly) but couldn't find such a simple and clear explanation. This series made my day!

[@uttaranchoudhurry]

The Explanations are crystal clear .....Great Work... I am in love with these videos. Hope to see more.

[@sabrinariah1195]

thank you for your videos we need more exercices

[@brightsideofmaths]

Thanks a lot! They will come :)

[@mattetor6726]

I like that you keep it real! Pun intended

[@whatitmeans]

Could you add to the next video the geometrical/visual/draw view of the proyection being made with the definition of the Tf distribution as the integral Tf(phi) = int{f*phi}.... is a proyection like a dot product? What are the arguement variables of the functions "f" and "phi"? Are the same? int_R^n{f(vec{x})*phi(vec{x})*dvec{x} ?

[@billbulgari]

Thank you very much for these videos about Distributions! Are you going to release another related video? Do you recommend any books?

[@brightsideofmaths]

You are welcome and thank you. There are more videos planned. I just need more time for them than I thought :)

[@billbulgari]

@@brightsideofmaths Take your time. Do you suggest any books? I study from Szekeres - Modern Mathematical Physics.

[@kayebennett7867]

Hi, nice video, and thank you for this amazing series. In physics is quite common to use "operator-valued distributions", this made me ask two questions. First of all, how general can one formulate the theory of distributions? From the definition given here, I guess that you should be able define distributions whose values lie in an arbitrary normed space, correct? Can this be generalised even further? What is the minimum structure needed to define distributions (even if you might loose some nice properties)? On the other hand, my other question is, how specific are all the concepts you are showing in this series? Do they still hold true for more general definitions of distributions or you really need the condition that T must be real valued?

[@brightsideofmaths]

Very good questions. I cannot answer these in a short KZbin comment. One can generalize a lot of notions for operator-valued distributions. I would do this a weak sense but this is definitely something for a whole video.

[@kayebennett7867]

@@brightsideofmaths thanks for your response, would be nice to have at least one video commenting about generalisations, even if it's an appendix once the series is finished.

[@apostoloskountouris5144]

A quick question concerning the statement at 9'29" about the reconstruction of a function from the knowledge of the 'numbers' that represent the distribution... I am not sure to understand what this statement implies. Is some analytical reconstruction implied or is there an algorithm for some kind of numerical approximation of the function so to speak ?

[@Aakash-pt5ux]

Fantastic!

[@apostoloskountouris5144]

I find the notes at web2.ph.utexas.edu/~mwguthrie/t.theory_of_distributions.pdf particularly useful and complementary to these excellent videos. These notes closely follow (in almost-everywhere sense ) Halperin/Schwartz small book www.worldcat.org/title/introduction-to-the-theory-of-distributions/oclc/56910581

[@bokistotel]

Great videos. Will there be a next video?

[@brightsideofmaths]

Of course. I am working on it :)

[@scollyer.tuition]

This may be a rather dim question as I'm watching this series at high speed, but at 8:05, you define the distribution via an integral over R^n; however, since the test function has compact support, should it not make more sense to define the integral only over its support? Or am I talking nonsense? Also, (having checked wikipedia to make sure I'm right), your definition of a distribution makes clear that it is a linear functional on C\infty - is there any reason that you don't mention this? For me, knowing this demystifies the subject to a large extent. In fact, with your nice series, I'm wondering why I was so scared of distributions for so long; they're far less tricky than I imagined.

[@brightsideofmaths]

There is no difference between the integral over the whole R^n and the integral over the support :)

[@japedr]

Is D' ("D prime") the same thing as the dual space?

[@brightsideofmaths]

With the correct topology in mind, this is exactly correct.

[@juliogodel]

When the next one is available? :)

[@brightsideofmaths]

Don't worry. It is on my mind. Maybe I finish it and the end the month :)

[@MrOvipare]

I am used to see Dirac's delta as a test function, so I'm a bit confused of the notation delta(phi)... Are we using Dirac's delta as a test function to measure some other test function phi?

[@brightsideofmaths]

The delta is not a test function but a linear map defined on the test function. That is the crucial point here.

[@MrOvipare]

@@brightsideofmaths ah! Thank you :)

[@arsenelupin123]

I'm not sure I quite understand the necessity for the continuity condition on the test functions. Why do we have to impose that to define a distribution?

[@brightsideofmaths]

Yes, the part is very essential as we will see later!

[@arsenelupin123]

@@brightsideofmaths Can't wait. As a physicist, just saying that it's a linear map tells me pretty much everything I need to know in practice. But good to know the formal details.

[@mrhilinisrine8540]

could you prove the linearity in first example (a) , you said it's obvious but i didn't get it!!!!!!!

[@cameronspalding9792]

Is there a part 5

[@dacianbonta2840]

Woudn't be more straightforward to use T for T-est functions (and T-rond for there space) and D for D-istributions (and D-rond for there space)

[@brightsideofmaths]

Maybe. But I can't change common notations :D