The Fundamental Theorem of Functional Analysis
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- čas přidán 29. 08. 2024
- Here is the most important theorem in functional analysis: A linear transformation T is bounded if and only if it is continuous. This allows us to easily check whether an operator is continuous, and is the quintessential fact that is the genesis to the whole field. Enjoy!
Functional Analysis Overview: • Functional Analysis Ov...
Is the Derivative Continuous? • is the derivative cont...
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I'm watching him for 3 years, I learned a lot, thank you so much 🥰❤❤❤
More functional analysis and Banach/Hilbert spaces please
I've started watching your channel this month. I had heard your name before but never watched a video before than. What I absolutely love about this channel is that your contents are lit and you approach things as a PURE mathematician would (as much as possible on a CZcams video) without using stuff like notation abusing or things that really hit me hard. I'm a physics student, by the way.
Thanks so much!!! 😁
I'm also a physics student! What I like about Dr. Peyam is that he's always in a good mood. He really enjoys math => and that makes me now also enjoy math :)
I imagine all the people expecting another video about a fun arithmetic fact, and now it "doesn't even have to be a Banach space" :P. Cool video though!
YOU ARE AMAZING THANK YOU SO MUCH
You are also very funny by the way XD
You are doing the world a great service just by being yourself and expressing yourself this way!
Thank you!!!
Very nice presentation! As a suggestion for a possible future video, please talk about the Arzela-Ascoli theorem!
Another fantastic video. Thank you Dr. Peyam!
Can you Explain brich-dyre cojecture and L-function Doctor ?!!
Ok. Thanks.
Awesome to learn!
What would be an example of an unbounded linear transformation? As far as I know T can be represented by a matrix, so couldn't we bound every transformation by some notion of maximum scaling? maybe the max eigenvalue? Hence every linear transformation would be bounded? I've got no notion functional analysis but the title made curious
Check out the description. Differentiation is unbounded
For x in the unitary ball.
awesome to learn thanks
yay, linear transformations and vector spaces!
Is there an analogy of this for metrics in general?And if so,is the statement " 2 metrics d1 and d2 on V are equivilent iff there is c>0 such that d1(x,y)
No idea
I would not say Fundamental, but it is used everywhere ^^
then, what would be the fundamental theorem of functional analysis?
@@6754bettkitty I would not consider anything here a fundamental thing currently, functional analysis is more about a basic for modern analysis so we are doing kind of everything from weak derivatives to weak limit oder compact operators ^^
Hahn--Banach Theorem is definitely more fundamental!
Simple and nice ;)
Hi, thanks for the video and I have one question: Would it be correct if we didnt set epsilon to be 1 and we chose our constant C to be epsilon/delta?
The constant cannot depend on delta
You probably mean epsilon, dont you??
I think C
@@drpeyam Thank you for your reply
Isn’t the definition you wrote down for continuity in the beginning real uniform continuity and not general continuity?
It’s linear, so continuity is equivalent to uniform continuity
@@drpeyam oh didn’t know this, thanks!
During the step where you estimate ||T(x delta/||x||)|| you say that ||x*delta/||x|| ||< delta which is obviously wrong as it equals delta. This is precisely why we need the ||v|| < delta => ||T(v)|| < 1 in the first place. Otherwise nice video!
You should open and close your pen like they do with katanas while writing appears in the board.
With all due respect, Dr. Peyam. The beauty of Functional Analysis lies within its independence of dimensionality; either in finite-dimensional or infinite-dimensional..
To infer, under linear transformation that 《T(v-w)》 = 《T(v)》- 《T(w)》 is fun on finite-dimensional space..
We use Zorn's lemma upon idntification of prescribed Hamel basis and voila! We get the result..
For, infinite-dimensional, it aint that straightforward nor fun.. as we need to resort things using Schauder Bases..
And I think this is the beauty of functional analysis being the inter-twined saga of various branches in maths..
Nice vid, Dr. Peyam!
The definition of "bounded" given here seems to be the definition of Lipschitz continuity?
Well all Lipschitz continuous functions are uniformly continuous which are also continuous for linear functions.
So, only for linear functions including the zero vector?
It can't be a linear transformation on a vector space if it doesn't have the zero vector, since the zero vector has to be in a vector space.
Linear functions on a banach space. The vector space could be polynomials, Lp functions, R^n, complex numbers, etc.
Is oreo healthy now? Not seen it since many videos
She’s alright, she has arthritis in her legs, which isn’t very good :(
@@drpeyam oh
Can you suggest some research papers which undergraduate can read @Dr Peyam
@@drpeyam .
😄😁nice
Hi DrP, I want you to buy a fairly large canvas (about the size if the board in this vid), and paint it with a coat or two of gesso. Then record an episode on canvas, instead of a board (make it a cool episode). Then sign the canvas, with a small date, and sell it to me.