Banach Spaces - Lec02 - Frederic Schuller

[@sardanapale2302]

As a mathematics professor I must admit I love seeing my colleague wrestling to prove that a Cauchy sequence is necessarily bounded (or getting totally confused on how to prove that the bounded operators form a Banach space). It happens to the best of us hahaha! Feels good to see that. These are not easy subjects to teach. That being said, I am loving these lectures and the ones he made on General Relativity. He is a very good teacher.

[@mousedorff453]

I guess that at some point, when you've completely assimilated the material and you've developed a strong intuition for it that you use primarily in research, you kind of forget about the nitty gritty stuff that went into the constructions of the tools you constantly use. Maybe he has just gotten to the point where all of this stuff is very second-nature to him.

[@paulhowrang]

I am glad a mathematics teacher is not teaching the subject, most of them spend time doing calligraphy on the board while knowing little what's all that math is actually worth doing

[@sardanapale2302]

@@paulhowrang : Mr Frederic Schuller is a mathematician. Sorry to break that news for you. I am also a mathematician (Pure Maths, ergodic theory) and my background is Engineer, Master in theoretical physics. Have a nice day :P

[@paulhowrang]

@@sardanapale2302 His training is indeed in maths, but his area has been theoretical physics. And he is a 'Dr' not 'Mr', and moreover your background is irrelevant here, my comment was not about you. Not everyone is an academician par excellence as Dr Frederic, at least not in teaching.

[@sardanapale2302]

@@paulhowrang I am a Dr too, I dont give a damn about titles. You can stick your rectification about me calling him a Mr where the sun does not shine. That being said, your initial comment was under my comment, so I assumed, apparently wrongly, that it was directed to me. Mr Schuller had, at the time of my comment, as a description of his area 'mathematical physics', NOT theoretical physics. Not sure you are able to make the difference between the two. Chill, go annoy someone else. Thanks.

[@ahmedrefaat7563]

I was struggling with mathematical physics 8 years ago and I wish I wish I have seen these lectures, it is so great. Thank you.

[@ark8949]

This is the quantum mechanics class I wish I took.

[@shinuito]

For anyone interested, when he's showing A is bounded, as in the limit of a cauchy sequence of bounded operators is bounded, he should've used completes of the real numbers. ||A_n|| is a cauchy sequence of real numbers by the reverse triangle inequality, and that allows you to replace lim ||A_n|| by a finite real number. You can also find this as a theorem in Reed and Simon volume 1 ~Thm III.2 depending on the edition.

[@MrAkashvj96]

Fascinating. Thank you !

[@SD19951]

Someone knows how it's possible to obtain the problem sheet?

[@dogukanozlu8021]

Thank you very much, professor!

[@michaellewis7861]

Use a sequence to show the operator is unbounded 31:00

[@millerfour2071]

8:50, 35:00, 39:15, 42:50, 47:30, 49:40, 54:53, 1:05:45, 1:18:27, 1:38:37, 1:44:13, 1:47:54

[@neelmodi5791]

At 1:19:00 on the blackboard, I don't see why the boxed expression being less than epsilon for every f implies that the supremum is less than epsilon. Surely it is possible that epsilon could be equal to the supremum? This is probably nitpicky since we could just start by taking the epsilon to be eps/2 or something and then that is less than epsilon, but just want to confirm that to be absolutely concise one should do this.

[@SD19951]

Totally right. I think there's another "problem" like this when he lets m tends to infinity at 1:13:40. When you use a limit in a disequality you have to add the equal sign to the disequality symbol. Anyway, as you pointed out, it's not a problem since epsilon is totally arbitrary.

[@themenace4716]

You are totally right. Has anyone found the correct way to do it?

[@00TheVman]

@@themenace4716 It is not too hard to fix these mistakes as it is enough to show that the terms get arbitrarily small. Having a strict inequality or not, or using 2*epsilon instead of epsilon won't make a difference in these circumstances. But if you are interested here is a brief summary of the proof drive.google.com/file/d/1gI1UYvUCpiTJasmrZg2YeY8IWWa6iNEU/view?usp=sharing

[@sanketthakkar4496]

how can we find problem sheet for this lecture?

[@junfan02]

1:22:45 if the {fn} converges to f in V, isn't V a Banach space instead of just being a normed space?

[@jonathansharir-smith6683]

No -- he is using that D_A is dense in V to make this conclusion (so V is the closure of D_A, so every f in V is a limit point of D_A). That is different than saying that every Cauchy sequence in D_A converges to some f in D_A (which is the completeness requirement).

[@MathematicsMadeSimple1]

This was fantastic, I like the way the topic was introduced.

[@ActionPhysics]

The dual vector space exists in linear algebra without any restriction to being bounded. I don't see why the restriction to bounded linear map also named as dual. Are these 2 notions different ?

[@alejandropacheco8666]

The two spaces are different. One is the algebraic dual that contains the other one: the topological dual. The topogical dual is usefull to formulate the riesz theorem for hilbert spaces.

[@chasebender7473]

Also, in typical linear algebra courses you only deal with finite dimensional vector spaces, in which has all linear maps are bounded (equiv. continuous).

[@reinerwilhelms-tricarico344]

What does he mean with ADDU and CADI ? I forgot but can't find it anywhere defined.

[@elosant2061]

He explains it here: kzbin.info/www/bejne/o5PZZIdnatOKp5I

[@stipepavic843]

thxxx alot, finaly found your channel , great lecturer with deep math, subbed!!

[@themenace4716]

@Frederic Schuller, could you provide the problem sheet, please?

[@miqaelgali3161]

and the text books too

[@giannisniper96]

to show that the operator is not bounded, can you say for all M in R you can find a function in C1 such that sup(...)>M?

[@johnbee6160]

yes. that is the same statement as the definition for convergence to infinity; hence unbounded.

[@ricardocesargomes7274]

One lecture beauty!(IFCE-Brazil)

[@shadabahamed9362]

Can someone give me the problem sets?

[@andrea.dibiagio]

Oh man I wish I found these when I was studying for my masters degree.

[@jackdaniel8763]

Same remark Andrea at university which have you studied?

[@GaryPansey]

A Banach Space is a complete normed space (every Cauchy sequence converges). A Norm has 4 properties: >=0; =0 only when the function being normed is the 0 vector; scalar multiplication holdes; and; the norm of a sum = the sum of the norms.

[@goblin5003]

If you want to be even cleaner, you can omit the >=0 property for the norm definition since you can deduce it from the 3 others

[@Blank-here2234]

is hahn banach theorem covered here...sorry I haven't gone through the video...if yes i will go through it

[@zfengjoe]

Not the way Hahn-Banach extends bounded function from a linear subspace (not necessarily dense). Here only extends from a dense subset. It's a standard approximate argument.

[@donoyonkai7225]

Very good videos...I'll be aiming an A+ this time...Arigato Sensei!!!

[@farahali5754]

الحرق في أيدي من الجنب ولكن اعرف يميني ويساري فقط لكن الغير الأشخاص لا ما اعرف يمينهم من يسارهم

[@SphereofTime]

1:00

[@anirbanmukherjee5240]

A Banach space is not really a generalization of a Hilbert space. The norm in the Hilbert space follows from the inner product. So yes, the Banach space is more general in that it admits norms that don’t correspond to an inner product. But it also entirely lacks the structure of the inner product.

[@AmitSharma-qe5ej]

suggest book

[@RamSingh-fe6ju]

Kya baat hai

[@OhadAsor]

awesome

[@ernestomamedaliev4253]

1:04:45 "everybody happy?" hahaha

[@gaulindidier5995]

My only problem is that he set a sequence A which is normaly a_n the same as A for his linear map and that he set f as an element. f should be strictly reserved to functions.

[@mousedorff453]

It's just notation. Moreover, a lot of the spaces on which these linear maps are defined on are function spaces so using f makes a lot of sense (it's also what some books tend to do).

[@SkyFoxTale]

Great lecture, thanks for uploading! It’s a bit too pedantic for me. Haven’t these students had analysis and topology already?

[@berserker8884]

Yeah sometimes it goes into way too much detail, like showing that P in not bounded. A student who had several analysis courses would take seconds to know why the example works. Still, I love this professor for his vision, skill and love for what he teaches.

[@jelmar35]

These seem like physics students to me, not mathematics

[@konstantin7596]

@@jelmar35 so?

[@chasebender7473]

With recent political issues, BLM might be able to replace BLT in terms of recognizability, and bring forth the modern mathematical terminology via modern social strife.

[@michaellewis7861]

58;00

[@maciej12345678]

so much chalk go off

[@kparag01]

Teach without books ...my god

[@howmathematicianscreatemat9226]

Yes, that’s what an authentic mathematician is like 😊😅