Lecture 1: Basic Banach Space Theory
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- Опубліковано 16 лис 2022
- MIT 18.102 Introduction to Functional Analysis, Spring 2021
Instructor: Dr. Casey Rodriguez
View the complete course: ocw.mit.edu/courses/18-102-in...
UA-cam Playlist: • Lecture 1: Basic Banac...
An introduction to Banach space theory, including vector spaces, norms and important examples of normed spaces.
License: Creative Commons BY-NC-SA
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MIT 18.102 Introduction to Functional Analysis, Spring 2021
Instructor: Dr. Casey Rodriguez
View the complete course: ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/
UA-cam Playlist: ua-cam.com/video/uoL4lQxfgwg/v-deo.html
You should pin it
Good day MIT OCW, do you have an Electric power system video lectures?
where can I find the answers of the assignments of this course
On behave of BUK, sir did it all, thanks for the helpful resources
Timestamps and Summary of Lecture 1: Basic Banach Space Theory
0:00 - Motivation and Introduction
Functional analysis works with vector spaces that are sometimes infinite-dimensional, using the techniques of analysis to study their structure & functions defined on them. Besides existing as a pure mathematical discipline, it finds applications in partial differential equations and physics.
5:27 - Review of Vector Spaces
Vector spaces are defined over a field (here, R or C) and are closed under addition and scalar multiplication. Every vector space has a basis, or a maximal spanning set; its dimension is the cardinality of any basis (this is well-defined). e.g. Finite-dimensional vector spaces over R: R^n (n-fold Cartesian product)
13:46 - An Infinite-Dimensional Vector Space
The set of continuous functions from [0, 1] to the complex numbers forms a vector space (sum, multiple of any elements are still in the VS). But the countably infinite set {1, x, x^2,...} is linearly independent in C([0, 1]).
16:26 - Norms
To perform analysis, we require a concrete notion of "proximity" or "size" on our spaces. A norm assigns to each vector in a vector space some nonnegative real number; as a map it is positive semidefinite, homogeneous, and satisfies the triangle inequality.
A seminorm is similar to (but weaker than) a norm; it satisfies the latter two conditions but is not necessarily definite. A vector space endowed with a norm is a normed vector space: these are central objects of study in functional analysis.
21:15 - Metric Spaces
A metric on a set provides a notion of distance between its points. It is an identity-indiscernible, symmetric, and triangle inequality-satisfying function; a space with such a distance function is called a metric space (has the metric topology)
23:08 - Norms Induce Metrics
Given a norm on a vector space, it induces a metric in the natural way. This allows us to talk about ideas like convergence, completeness, etc. with regards to normed vector spaces.
26:11 - Examples of Norms (and NVS)
Euclidean norm on R^n, C^n provide us with the most familiar notion of distance. We can consider more broadly the family of norms called p-norms, of which the Euclidean norm is a special case (taking p = 2). When p = ∞ we consider the norm that picks out the maximal component of a vector in n-space.
32:45 - The Space C_∞{X}
From a metric space X we consider the space C_∞{X} of all continuous, bounded functions from X to the complex numbers. This indeed forms a vector space - on it, we can introduce the supremum norm that measures the maximal value a function takes. This turns C_∞(X) into a normed vector space. If we take X to be some compact interval like [0, 1], the boundedness of continuous functions are automatic.
39:24 - The Supremum Norm and Uniform Convergence
When asking what convergence in this metric means for functions in C_∞(X), we find it translates to uniform convergence in the familiar sense from real analysis.
42:20 - l^p-Spaces
The l^p spaces consist of p-summable sequences. When p = 2, these are square-summable, etc. and these are all defined to be the sequences on which the respective p-norms resolve finitely.
46:38 - Banach Spaces
We are interested in special cases of normed vector spaces that mimic the situation/structure in Euclidean spaces, namely, their completeness. We have seen that norms (on vector spaces) give rise to metrics, with respect to which we make sense of convergence of sequences. Cauchy sequences are those whose terms tend arbitrarily close; every convergent sequence is Cauchy, but the converse doesn't necessarily hold (the rationals have many "holes": one can construct a sequence of rationals that close in on sqrt(2) but can't converge to it in Q). Spaces for which the converse does hold, i.e. Cauchy sequences converge, are complete with respect to that given metric. Banach spaces are normed vector spaces that are complete with respect to the metric induced by the norm.
49:52 - Examples of Banach Spaces
R^n, C^n are Banach spaces with respect to any of the aforementioned p-norms. These provide relatively trivial examples, but foundational ones.
50:39 - C_∞(X) is Banach
A useful nontrivial example - the normed vector space of continuous, bounded functions X -> C actually forms a Banach space with respect to the supremum metric. The process of showing that it is a Banach space amounts to exhibiting completeness, and is instructive in demonstrating the general procedure for showing that a space is Banach: first take a Cauchy sequence & come up with a candidate for its limit, then show that this proposed limit lies in the space, and finally show that the convergence does occur.
you're great! this comment need to be pinned, oh maybe it can be implemented into youtube sections
7:58
the sound of hard chalk on thick slate is marvelous
I had no idea Anthony Fantano was a functional analysis professor at MIT
I’ve needed this course for a couple of years now. Looking for good open resources regarding such an advanced topic was hard. The search is over. Thanks OCW! You’re da GOATs of open learning.
Thank you, MIT OCW team! Thank you, Dr. Rodriguez! These videos will change a lot of people's lives. It's a great contribution to the education!
Awesome to see a functional analysis series uploaded.
Thank you so much OCW & Dr. Rodriguez for putting video lectures to these notes! Since last year's 18.102 notes were published to the website, they have been the most valuable and easily-approachable resource for me when learning functional analysis.
glad to be of help
Wow!! Thanks very much MIT!! Really appreciate the amazing sharing of resources and hopefully one day we will get open courses on the key undergraduate and graduate math subjects!
There is a reason that the ground goes down. If anyone else saw the electric outlets on the wall, the thumb doesn't contact any of the bare connectors but the finger below the thumb does and many people get electrocuted unless the ground is down.
have been waiting for his functional analysis for a long time!!! 😃Thank you!
Dr Rodriguez is an absolute Top G. My man dropped the Real Analysis course (which I'm still devouring) a few months ago.
This was long awaited.
Thank Prof!
why tate is also top g is because he has an audience that watch functional analysis lol
@@Random-sm5gi A series of Tate will uniformly converge toward Dr. Rodriguez? Lol, I have no idea what I am talking about.
Did one of your students build the motiontracking camera?
Thank you MIT OCW and Dr. Rodriguez. I enjoy the course very much!
More functional analysis courses please!! more advanced functional analysis courses! perhaps even non-linear functional analysis :D
Ah, the functional analysis. Together with topology, my favorite subjects in uni.
I hope your teacher knew how functional analysis was used. Mine did not. The MIT professor mentioned PDEs. I wish I knew it.
Do you learn many things about unbounded operators?
What was your major?
Hey self-learners! At 45:11, it should be p = 1, not p = ∞.
"You've taken linear algebra. You've taken calculus." Buddy, not even close. But please, continue! >.
Smiles along in internet
Счас возьму линейную алгебру, аналитическую геометрию, отшлифую все это математическим анализом и вычислю оптимальное количество пива, водки и их соотношение которое залью в себя сегодня вечером
Then what tf are you doing here lmao, go play with trig or something
Did u even understand anything then?
@@jongxina3595 only that you're smarter
Thank you for this opportunity to learn such a valuable material
Why is it valuable?
@@sonjak8265 May he meant to be accessible, since It is very difficult to find exotic abstract academic knowledge through a quickly and direct explanations - this channel has provided them since then. And personally, for me, this lecture is valuable because gives you a deeper notion on some of the problems you can solve with these material.
Thank you professor for this excellent lecture.
A lecture on the unit disk would be appreciated!
Thank you Sir for these 2 series: Real Analysis and Functional Analysis. Can we be expectant of Complex Analysis?
Beautiful course
I will follow him
Follow him wherever he may go
There isn't an ocean too deep
A mountain so high it can keep me away
Millions of thanks to Dr. Casey Rodriguez who makes maths courses accessible, ocean not too deep and mountain not too high any more.
Thank you, MIT. I hope this can help out clarify some parts, i.e., the norm of magnitude space and the linear independent space, that I sometimes confuse at. As long as I can clearly understand the context about definitions well, I can correct what I mistake at. God bless you always.
I am freestyle developer in Brazil and since started my career did never have contact with computer science academics lectures, but after some years of experience I am loving all the free lectures MIT is providing in youtube. I have watched almost the whole channel since then, and I loved this one too. I would love now to set up a goal to study in MIT personally, but don't even know how much is the financial and general requirements to get into CS grad program.
[1] : Mas então tu tá assistindo isso pra quê? Análise funcional é um assunto SUPER avançado em matemática. Tipicamente é um curso de mestrado e doutorado (tem uma versão mais "leve" pra mestrado) e é bom ter visto equações diferenciais antes. Assistindo isso, você vai sentir que tá "entendendo" mas na real, não está. Se quiser, tente fazer os exercícios e exames desse curso que vai ficar claro. [2] Ir pro MIT é um nonsense total. O estudante médio do MIT é MUITO bem preparado e está estudando desde muito cedo com material muito bom, em exatas, tipicamente eles estudaram/competiram nas IMO (Olimpíada Internacional de Matemática). Tentar ir pra lá sem bom preparo... você vai ser destruído. Fora que um mês no MIT custa 6500 dólares (quase 40 mil reais por mês, um ano vai custar quase meio milhão de reais).
@@gustavoturm pra quê? para ter o que todo mundo que assiste esse video quer ter: conhecimento. só o que eu li é sua perspectiva se você fosse entrar la, a sua visão de dificuldade, que é diferente da minha, você enxerga isso como se fosse algo bizarramente difícil, mas vendo essas aulas todas eu acredito que não seja impossível. E sim, eu percebo que precisa ter sido desde o inicio extremamente privilegiado, coisa que eu nunca fui, eu sempre conquistei as coisas sozinho sem precisar dos meus país (porque eles morreram) e vou continuar conquistando. Mas eu não ligo para o quão difícil seja, ou que demore mais um tempo, ou que eu seria destruído, eu apenas continuarei estudando ( o que já faço todo dia ) para refutar a dificuldade do MIT. E mesmo se for caro, até la eu ganho experiencia o suficiente como Dev para ganhar o necessário para se manter.
@@snk-js[1] : Tu não entendeu a pergunta: Não é "pra que tu tá fazendo isso?", é "pra que tu tá fazendo isso sem nem 1/10 do conhecimento que precisa pra entender isso?".
"Minha visão de dificuldade?" Vamo fazer o seguinte, isso é uma prova desse curso de análise funcional: ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/7720fe590b773188648d100c22969cd0_MIT18_102s21_midterm.pdf Responde e me mostra as resposta pra ver como é que é essa "tua visão de dificuldade".
[2] : Tu entendeu completamente errado o que foi dito. O ponto não é que é difícil, é que é difícil *e caro.* Tu tá fazendo 50 mil reais por mês como dev? Isso - sendo bem otimista - é O MÍNIMO que tu precisa ter pra pagar a mensalidade do MIT. Tá aqui a página com os valores: sfs.mit.edu/undergraduate-students/the-cost-of-attendance/annual-student-budget/
That's so great! I feel like if getting into grad school at MIT is something you really want to do, then just go ahead and do it and don't let anyone else tell you otherwise! Grad school admissions are *extremely* competitive and you need to have a well rounded profile in order to have a chance to get into any of the programs. Some of these requirements include (disclaimer: this list is by no means comprehensive) good grades, relevant research experience and/or publications and strong recommendation letters from supervisors that can vouch for the quality of research work you have done. Some universities also require you to take a GRE and an English language test. The latter only really applies to you if you come from a country that does not speak English as a first language or if your bachelors/masters degree was not in English. To my knowledge, most PhD positions are fully/partially funded and may require you to also do some TA work in addition to your research. Depending on the school, you may have to pay a fee to send in your application. Good luck and I really hope you find the courage and motivation to go through with this!
@@anonymousperson9757 beautiful still precise words. Grateful for your message. I entirely studied humanities since starting to really like numbers, so its still really far away from me to achieve excellence in natural sciences, yet, I am naturally at researching and being curious itself which make it easier to become a scientist in the long run. But as you mentioned I really must start to focus on academic works and quality research. I will figure out how to solve these requirements along the time. Thanks
Oh I cannot thank you enough!!! Thank you!!
Im from a Machine Learning background(masters at the moment) and you are really good and smoothly going from concept to concept, in my opinion. I hope to continue this series during the holiday season.
Did you manage to go through it? :)
@@user-wl2rb3rh5c איזה גבר, אז אני לא היחיד פה אה?
@@heyman620
חחח יאפ, נראה כמו מרצה סבבה האמת, סקרן לראות איך יהיה, מה לומד?
@@user-wl2rb3rh5c אני כבר לא ממש עושה קורסים, אני מאסטרנט והמחקר קשור ללמידת מכונה. אבל התואר הראשון היה במדעי המחשב (לכן לא למדתי אנליזה פונקציונלית)
@@user-wl2rb3rh5c תנסה גם לקרוא את element of theory of functions and functional analysis, אני אישית מרגיש שהוא בונה דברים ממש יפה.
Great job
Did we show that u(x) is continuous? I think this can be proved by convergence of infinity norm implies uniform convergence of the sequence, then the pointwise limit is continuous.
Can we get an ocw on relativistic quantum field theory by mit on UA-cam? Hope for getting reply.
Nice lecture... Sir which source book you are using?
Finally thank you
Quite interesting.
Will OCW publish abstract algebra videos please ?
🎉🎉🎉🎉🎉🎉🎉been waiting for this for a long time
This is great.❤
Can it be applied in economics
My dreams come true. I just want to watch some lectures about functional analysis.
Did he say R2 in conjunction with space?
Around 1:02:00 , the instructor mentions that C is complete , meaning that for all x belongs to X, u(x) = lim u_n(x). What i got confused is that he mentioned that it has a point wise limit(space of continuous bounded functions), however, this would mean given an x, and epsilon, we can find an N, such that u_n(x) converges to u(x). But by completeness of C, it says for every x , un(x) has a limit( isnt this uniform convergence in C)? and if so, doesnt that show uniform convergence in set of bounded continuous functions? Im confused
I‘m not sure I fully understood your question, but add my thoughts here:
- complete space here means that a sequence of u_n belonging to X has a limit function which also belongs to X
- uniform convergence is defined as there exists an N such that for all x: |u_n(x)-u(x)| < epsilon. So you choose an N for all x, where with pointwise you can choose an N for all x seperately. Therefore uniform is stronger.
Okay, I’m also confused. Since u(x) is bounded and has complete domain, its compact. Now the functions are continuous(?), therefore uniform continuous?😅
the domain D is not compact, for example, D=complex numbers, so D is complete but not compact. Also, u(x) is bounded not means continuous, such as Heavisde function, hence we need to show u is continuous under those conditions@@oreo-sy2rc
This automated motion-tracking is very bothering. Why not a wide-angle lens instead ?
At 45:11, why is the sequence {1/j}_j=1^infinity not in l^infinity? The supremum of the sequence is simply 1, is it not?
Yup had the same doubt
I can say our indian professor have much better teaching skill than anyone else
it would be great if we have video lecture for commutative algebra as well
ua-cam.com/play/PLq-Gm0yRYwTjBziGqSW9kFF9o2l5ECDvY.html
@@leandrocarg thanks for your help, but I mean MIT's course in this case.
Man where were these lectures when I was taking my functional analysis class 😭
Anyone knows why {fn=x^n } is linearly independent ?
Recall that linear independence can be formulated as "no nontrivial relations between a set of vectors" in the sense that any linear combination, say, c_{a_1}*v_{a_1} + ... + c_{a_n}*v{a_n} + ... = 0 implies that each field scalar c_{a_i} = 0. Here we consider the set {1, x, x^2,...}: a countably infinite set consisting of all elements of the form x^n.
Suppose we have some nontrivial relation, i.e. c_0(1) + c_1(x) + c_2(x^2) + ... = 0. This forces all of the c_i to be zero, because intuitively, terms of different degree "cannot cancel each other out". As a slightly more tractable example, you can consider finitely many of these terms in a polynomial: if we have ax^2 + bx + c = 0, it's not possible to configure the purely scalar coefficients in a way that allows for this to hold unless a = b = c = 0.
Because we've shown that the set {x^n}_{n >= 0}, considered as continuous & bounded functions [0, 1] -> C are linearly independent, it follows that C([0, 1]) cannot have any basis of finite cardinality; hence it is an infinite-dimensional vector space over R or C.
anyone tried to prove the statement at 28:55?
spectral theorem
Thanks mit
I need data analytics full course please help me
In the last theorem that he proved, where did he use the fact that X is a metric space??????????
Hello, I have just graduated from Makerere University with BSc in Education (Mathematics and Economics)
I want to enroll for Masters program next year in Applied mathematics. So am preparing myself by undertaking this course.
I have finished homework one, how can I submit it and you look at it please?
Thanks so much DOC.
Hi @Believer-zj2cz, OCW is not a distance-learning program, has no registration or enrollment option, so unfortunately, we are not able to provide interaction or direct contact with MIT faculty, staff, or students. It's best to think of OCW as a free online library of course materials that you can study at your own pace.
For interactive study from MITx, you can browse options here: openlearning.mit.edu/courses-programs/mitx-courses/. Good luck with your studies!
Is this his first time?
"Automated motion tracking" bro who u tracking here???
Abstract
A growing lattice with a label system of growth in every direction from the centre to infinity
whole numbers for measuring size in connection to the size of the lattice label to size and size
of the knot so that you can solve a measure from the knot size and size from the label in three
dimensions to help solve Euler Brick problem?
remember the difference between a label and a size.
I will show each three slices of the lattice.
slice one centre is 0L0 this is the middle of space x,y,z
0L0 is a label and it has size as well its size is the knot size.
L is not just a label knot it has size as well.
rule for this lattice
A=B
pick size of gaps in lattice 20mm
pick size of knots L in my lattice is a squares diaginal = 10mm
example
-----------------------------------------------------------------------------------
example of lattice
slice 0 centre 0L0 and centre of lattice
2L0 2L0 2L0 2L0 2L0
1L0 1L0 1L0 1L0 1L0
0L0 0L0 (0L0) 0L0 0L0
-1L0 -1L0 -1L0 -1L0 -1L0
-2L0 -2L0 -2L0 -2L0 -2L0
slice 1 centre 0L1 but not centre of whole lattice
2L1 2L1 2L1 2L1 2L1
1L1 1L1 1L1 1L1 1L1
0L1 0L1 (0L1) 0L1 0L1
-1L1 -1L1 -1L1 -1L1 -1L1
-2L1 -2L1 -2L1 -2L1 -2L1
slice -1 centre 0L-1 but not centre of whole lattice
2L-1 2L-1 2L-1 2L-1 2L-1
1L-1 1L-1 1L-1 1L-1 1L-1
0L-1 0L-1 (0L-1) 0L-1 0L-1
-1L-1 -1L-1 -1L-1 -1L-1 -1L-1
-2L-1 -2L-1 -2L-1 -2L-1 -2L-1
------------------------------------------------------------------------------------
so a point on the lattice
example = 0L0
so the lattice is infinite in every way so a line will look like this.
example y = 0L0 + 0L0 + 0L0 = 20mm*3 = y this line on the lattice
so a 60mm line.
example x = 0L0 + 1L0 + -1L0 = 20mm*3 = x this on the lattice so a 60mm line.
the hard point is a diagonal line
example = 0L0 + -1L0 + 1L0 + (Knot=1 whole and 2 halves of a knot)
so 60mm because A=B so (0L0 + -1L0 + 1L0) = 60mm + knots
now a square
example = 1L0 1L0
0L0 0L0
example = 1L0 1L0 1L0
0L0 0L0 0L0
-1L0 -1LO -1L0
now a cube
1L1 1L1 1L1 1L0 1L0 1L0 1L-1 1L-1 1L-1
0L1 0L1 0L1 0L0 0L0 0L0 0L-1 0L-1 0L-1
-1L1 -1L1 -1L1 -1L0 -1LO -1L0 -1L-1 -1L-1 -1L-1
By Aaron Cattell
Please don't move the camera. Adjust/fix it at some appropriate position.
Good lecture. ( Would be even better with a camera operator who doesn't keep moving it all the time. )
What's the text book of this course?
From the course syllabus, "There is no assigned textbook for this course. Instead, we will follow lecture notes written by Professor Richard Melrose when he taught the course in 2020, as well as lecture notes taken by an MIT student who took the class with Dr. Rodriguez in 2021. These can be found in the Lecture Notes and Readings section." ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/pages/lecture-notes-and-readings/
Best wishes on your studies!
@@mitocw
Thanks😍♥️
Too much goog vedio
1:00
hello and velcome to functional analyziz
anthony fantano's side hustle
I like how you actually put it in the title...
"Theory"
Nevermind he writes with his left hand... enough shown.
He kinda lost it at the end
the movement of the camera is so annoying
Yes! Yes! I understood some of these words!
Play video MIT self study
ty ty meow
I have no doubt about his knowledge but he looks less interested in teaching.
Probably has to do with the empty Covid era classroom.
Seriously, this guy is why reading the edited, vetted course book is superior to the lecture. If you were truly a neophyte you'd be lost by so many of the skipped steps and failed contextual references. This lecture is targeted at grad students who want too feel nostalgic.
ez
promosm
First to comment lol
big achievement
A child of five years cannot understand this, the wonderful of language is that you can explain anything even child can sense, I think this the old system of teaching mathematics
It's interesting that some professors still write a bunch of definitions and equations on a chalk board the way it has been done for well over 120 years. A lot of that could be written out and provided to the students before class; this would permit the professor to spend more time answering questions and analyzing the information during class.
Yeah nah. A lot of students still prefer being walked through the contents and that's what Professor Rodriguez is doing. Math is more about developing intuition rather than rote memorisation and it is important to see how equations and formulae come into existence from intuition. Paper hand outs before class work well for other courses.
I’d argue that this lecture style gives sort of a storyline for the student to follow as it unfolds. I think it’s very effective for following proofs
The real problem for Physicists is Truth in Labelling. Putting individual's possessive names over the top of self-defining informational identification instead of behind it in the Bibliography, ..no one knows what the reality of Actuality is.
Totally irrelevant
wtf
indeed