Functional Analysis 1 | Metric Space - How to Measure Distances?

[@brightsideofmaths]

Please do the quiz to check if you have understood the topic in this video: thebrightsideofmathematics.com/courses/functional_analysis/overview/

[@rozil2763]

Great work man, I want to know where and why this functional analysis started and where it can be used please

[@leelapaturi8643]

We want more videos on other topic like real analysis

[@brightsideofmaths]

@@leelapaturi8643 Is coming.

[@stat5web]

@@rozil2763 One area I can talk about: it is extremely useul in convex optimization

[@keyyyla]

Pure Mathematics content on a didactical level like this is very rare on youtube. Keep up the great work.

[@brightsideofmaths]

Thank you very much. I am still producing new videos each week :)

[@TheSandkastenverbot]

Hoorray, functional analysis! Here's a wish list: - Arzela Ascoli - Banach Steinhaus - Spectrum of non-compact operators

[@weirdo-jw9kc]

Aw i love this channel. Excited for this new series. I loved measure theory.

[@zazinjozaza6193]

This is awesome, I just recently looked into functional analysis a little bit. If you could explain what exactly it means to complete a metric space and why it's an important concept that would be great. It is one of the first concepts I came across yet I had trouble understanding it. Great video can't wait for more.

[@ichkaodko7020]

we want too see as much as functional analysis there is. with your clear and concise explanation talent, we don't want to waste it but we want every ounce of it.

[@brightsideofmaths]

Thanks :)

[@nayjer2576]

You guys, producing such high quality content for pure math, are just incredibly helpfull. Truely an inspiration. If I make it through my math major, I hope I can produce someday some pure math videos too, about uncovered topics. :)

[@brightsideofmaths]

Thank you very much! I am producing even more videos now!

[@zhouryan6960]

The best channel ever for high school students like me who's aiming for a Math major in their undergrad

[@m2rahman]

Thanks for starting this new series. Looking forward to it.

[@李子熠]

it's incredible to have found this special math courses here

[@brightsideofmaths]

Thank you. I hope you enjoy it :)

[@vaxwiz]

Excellent, clear presentation....looking forward to the next video!

[@brightsideofmaths]

Thank you :) The interesting stuff is in production :)

[@FloriUchiha789]

You are amazing! I like how you explain every property with graphs :)

[@108titan]

loving this! could you maybe have a takeaway problem at the end with a solution as well? A big drawback when learning math independently is not being able to know if you're right or not

[@shilpasolanki2114]

Thank you so much for this video. I have my M.sc. semester exam next week and this video will help to clear my doubts. Looking forward to more videos.

[@PunmasterSTP]

How did your exam go?

[@robertschlesinger1342]

Worthwhile basic video. I look forward to the sequels.

[@MathIguess]

When you started with "hello and welcome", I knew I was in for some awesome mathematics lol Can't wait to watch this series :)

[@brightsideofmaths]

Nice :) And thanks for the support!

[@sejuprajapati2005]

Thankyou so much for making videos on function analaysis . You are a very good mathematics profacer

[@Xaarfai]

God bless. I have Functional Analysis the comming semster as a non mathematician and your videos are a great way to prepare for (and maybe even master) it :D

[@brightsideofmaths]

Thank you very much :)

[@ekaptsv]

Recently I have watched some of your measure theory videos. They are very clear! Happy to see a new course on your channel! Thank you and, please, keep up the good work! My wishlist: any standard course on Functional Analysis with focus on EXAMPLES would be OK.

[@PunmasterSTP]

Metric space? More like "The fact that more people don't know about this is a disgrace!" Thanks for putting up so much amazing content on KZbin.

[@tilkesh]

Best in this field. Thank you very much.

[@soumyaadabala]

at @3:10 why is the cartesian product of x and x

[@ivanleon4961]

I would love to see how hilbert spaces play a crucial role in Fourier Analysis. I also head that functional analysis is the bridge to Harmonic Analysis. I like to see that but not sure if that’s asking too much... Also unbounded operators lol

[@Ash-bc8vw]

There is a new video on that topic about the bridge I'll try to find the link and post it here

[@rgoswami]

These videos are fantastic and are a real highlight of my days. 💯💯

[@lukephillips7239]

Thank you bright side. You are making my math degree a lot easier!

[@brightsideofmaths]

Thanks a lot! And thanks for the support :)

[@AliAhmed-jw8gy]

thanks so much .your explaination is very clear and easy to understand.

[@AliAhmed-jw8gy]

please make a video on distribution and sobolev sapce .

[@nzuckman]

Woohoo! I'm so glad you're covering FA! 😁

[@qiaohuizhou6960]

0:30 what it is about functional analysis 1:58 metric space

[@maths4fun103]

This is a wonderful presentation, I need this on controlled metric space

[@brightsideofmaths]

Very good :)

[@remlatzargonix1329]

Thank you for doing these videos!

[@ZanyRocket]

Hmm, how do metric spaces in functional analysis relate to the notion of a metric in differential geometry? I know they must be different, since in DG you can have negative distances, and the metric is a local property rather than a function on pairs of points.

[@brightsideofmaths]

It's related but first of all very different. When we will talk about inner products in this course, you might see the connections. In differential geometry, you put an inner product to each point on the manifold and also call this a "metric".

[@putin_navsegda6487]

It's my summer holidays and i begin this playlist. Next semester we have a course of Functional Analysis and it's said to be VERY difficult, one of the hardest courses.

[@brightsideofmaths]

You can do it :)

[@putin_navsegda6487]

@@brightsideofmaths Thank you ! I hope i will !

[@ojomuolanshile5166]

@@putin_navsegda6487How was the course,did you pass it?

[@MrWater2]

Could be that we only need the 2 properties to define a metric instead of 3? I mean that the transitive propertie could be demostrasted using (1: d(x,y) = 0) and (3, triangle inequality)? Also the quiz ask about that

[@brightsideofmaths]

Transitive property? You mean the symmetry? How would you prove it?

[@MrWater2]

@@brightsideofmaths Yes, that d: X\times X\to R satisfy that 1)d(x,y)=0 iff x=y and 2)d(x,y)=0 and so d is a metric. We can take z=x in the second condition, and combine with the first on so d(x,y)

[@brightsideofmaths]

@@MrWater2 Your triangle inequality looks strange.

[@thiloherold7087]

Great video and as usual good explanation! :) But a question: what is the difference between a norm and a metric? Because as I understood it it has the same properties. Thanks!

[@brightsideofmaths]

Yeah, we will talk about this later on. A norm is defined on a vector space and measures lengths. A metric measures distances and always need to points for this.

[@nnofficial2414]

Great job! 👏

[@Amanpreetkaur-cp4fs]

Writing is great 🎉 And fresh 😅 Nice 👍

[@user-du3ow9er1y]

we appreciate your efforts to make ugly math understandable hhhh pleas keep them coming and we will support your content through paypal

[@abhijitmahata4149]

Can you refer me a book on functional analysis, I want to start reading

[@brightsideofmaths]

For example "Rudin" is very good :)

[@DanielJanzon]

Kreyzig is a good one too, the proofs are worked out in detail.

[@vaashis3397]

Beginning Functional Analysis by Karen Saxe.

[@mendelabramzon]

Kolmogorov, Fomin

[@keyyyla]

@@DanielJanzon I used Kreyszig in my B.Sc, the best book I read into.

[@abhijitmahata4149]

I am from India. Waiting for next part. Thank you

[@survivor1620]

hahn banach theorem along with some examples (how to extend function using this result?)

[@brightsideofmaths]

You may find that at the end of the video series: kzbin.info/aero/PLBh2i93oe2qsGKDOsuVVw-OCAfprrnGfr

[@survivor1620]

Thank you@@brightsideofmaths

[@arthurlbn]

Do you have something about convex optimization, convex analysis?

[@brightsideofmaths]

Not yet :)

[@dvolpatto]

Simply outstanding didactics, aiming to understand and not just a heavy proof-driven functional analysis course! Really good to understand the subject and get (geometrical) intuition. Thanks for sharing your videos! By the way (and out of curiosity), do you like the Kreyszig Introductory Functional Analysis book? I think the way you explain is similar to this book.

[@brightsideofmaths]

Much appreciated! I don't know this book but I will definitely have a look.

[@ishaangoud3180]

Functional Analysis, better version of Real/Complex Analysis 👏🏻

[@chanwoochun7694]

Hi, I recently finished your series on complex analysis which was fantastic. Do you plan to cover reproducing kernel Hilbert space sometime in the future for the functional analysis series? There is a growing interest in RKHS due to the popularity of machine learning, but quality online content on RKHS is currently lacking. I would appreciate it if you could consider it! Many thanks.

[@brightsideofmaths]

Very good idea! I might do that soon :)

[@thomascousins9150]

Can you comment on iff conditions for equality in property (3), ie d(x,y) = d(x,z) + d(z,x) iff ?

[@brightsideofmaths]

I general, there is not much to say about this. Do you have an explicit metric space in mind?

[@thomascousins9150]

@@brightsideofmaths I guess that the point z must be x or y or an internal point on the "line" segment XY

[@brightsideofmaths]

@@thomascousins9150 In an abstract general metric space, we just don't have "line" segments upfront. Therefore, I was asking if you think of some special metric spaces here?

[@thomascousins9150]

I guess I'm just thinking of ordinary Euclidean space R³ with the Pythagorean metric

[@cerbahsamir8617]

is functional analysis closely related to variational calculus or not ?

[@brightsideofmaths]

It is.

[@jackdavis6315]

Sweet! Would it be too much to cover positive-operator-valued measures? Coming from qm

[@Jeffrey561]

Sir please on Banach Spaces, Linear Transformation.And sir , Simmons is enough for reference on Functional Analysis? Or Any??

[@brightsideofmaths]

Which books do you mean exactly? Simmons's "Introduction To Topology And Modern Analysis" is a very good book. Of course, this is what you can use for functional analysis :)

[@Jeffrey561]

@@brightsideofmaths Oakay Sir Thank you

[@Domzies]

I picked this course for next semester, its the one im most excited for. Hopefully this stupid virus wont interfere again and force us to study from home.

[@brightsideofmaths]

I also don't like teaching from home exclusively. However, I hope that I can help you here with the videos :)

[@schatjepatatje767]

Nice video dawg, terry crews certified.

[@willkimoto1218]

Does anyone know if metric spaces have to be strictly convex? If not, how can you define distance between two points without having the line segment that connects them entirely contained in the space?

[@brightsideofmaths]

Okay maybe another question helps: How would do define a "line segment" in a metric space?

[@YorkiePP]

Hell yes man I'm so excited for FA :))

[@mikamikamusic7792]

Is the metric d the same as the metric tensor g

[@brightsideofmaths]

No :)

[@lcfrod]

Excellent. Thanks a lot .

[@brightsideofmaths]

Glad you liked it!

[@adithyagiri7933]

Great....keep bringing them all...

[@suwandi178]

Sound interesting, i like it.

[@gracewanjiru5967]

Hello.Would love you to do vector spaces in functional analysis

[@brightsideofmaths]

Great suggestion!

[@xeno5031]

Will this course cover all topics in functional analysis ?

[@brightsideofmaths]

No course can cover all topics in functional analysis, I am afraid. Sorry! But I give a good overview about the most important topics here :)

[@davidizquierdo8769]

Hello :) Thanks for all your videos, they are awesome without exception. Have you ever thought of doing a series on Perturbation Methods or Approximation Theory? Regards

[@Spacexioms]

Are you following a text book for this course, thank you!

[@brightsideofmaths]

No, not really. However, a lot of functional analysis textbooks do a similar approach.

[@daviidayala4987]

hey! sorry, a question: what program do u youse to draw and write?

[@brightsideofmaths]

Xournal :)

[@daviidayala4987]

@@brightsideofmaths thank you very much! I've just met your channel, I love it!

[@achillesmichael5705]

hallo gutes video hast du das thema auch auf deutsch zufällig?

[@brightsideofmaths]

Noch nicht, aber wird kommen :)

[@أحمدرمضانعلي-ث6ذ]

Where is the norm space ?

[@paominlienguite2397]

Subscribed today 🥰

[@sandorszabo2470]

I think that the notion of metric space belongs more to topology. People say functional analysis is an infinite dimensional linear algebra. However it is true that many structure has metric space property.

[@brightsideofmaths]

Yes, we definitely use some topology in this course!

[@letslearnmathstudents4174]

Excellent way

[@qi1436]

thanks sir

[@eric3813]

Das wird interessant 😄👍

[@mayankvashishtha5714]

Please discuss some applications to quantum mechanics,bro i love your videos🤩

[@brightsideofmaths]

Thank you! I will do that for sure :)

[@trantrunghieu2332]

I love it.

[@funnykira1330]

Super interessant, ich freue mich bereits auf diese Serie. Ich habe dieses Semester Analysis III für Ingenieure (komplexe Analysis) abgeschlossen und damit das letzte Mathemodul meines Studiums leider beendet. Ich möchte mich nun neben meines Studiums weiter in Bereichen der Mathematik bilden und deine Videos sind eine gute Grundlage dafür. Danke dir! Übrigens, hast du vielleicht mathematische Literaturempfehlungen für einen Ingenieuren mit folgenden Kenntnissen: Kenntnisse in der linearen Algebra Kenntnisse in mehrdimensionaler und komplexer Analysis Kenntnisse in Integraltransformationen und partiellen DGLs. Denn leider ist die fortgeschrittenere interessante Literatur meist für Mathematikstudenten ausgelegt :/.

[@brightsideofmaths]

Das freut mich sehr. Es gibt sehr viele fortgeschrittene Mathematik-Bücher, die auch für Ingenieure oder Physiker verständlich geschrieben sind. Ich würde dir einfach mal deine Uni-Bibliothek empfehlen und dass du dort mal ein paar Bücher durchschauen. Was dich anspricht, thematisch wie auch von der Präsentation, kannst du einfach mal lesen :) Ich kann dir gerne ein paar Empfehlungen geben, aber dann müsstest du sagen, welche Themen dich so interessieren :)

[@funnykira1330]

@@brightsideofmaths Komplexe Analysis ist eigentlich das Thema was mich am meisten interessiert und leider ist die Literatur (zumindest für mich) meist zu schwer geschrieben.

[@tlli5066]

@@funnykira1330 Eigentlich ab diesem Stage lohnt sich mal zu überlegen, ob man sich schnell den Stoff von Rudin mal gönnt. Damit man gewisse Grundlagen hat, mathematische Sprache sprechen zu können. Beispielsweise wenn du diese Kenntnisse hättest, dann kannst du dich eigentlich problem los mit weiteren Themen und Maßtheorie und Integration, sowie Differentialgleichung und Topologie anschließen.

[@mattetor6726]

Great videos!

[@Tareq--ri6qe]

Thanks ❣️❣️❣️❣️

[@yBazo82]

so good

[@bakytzhansapy7369]

very good

[@brightsideofmaths]

Thank you! Cheers!

[@juliogodel]

Greaaat! Thanks a ton!!!

[@keishamselex1689]

really helpful;

[@丁哲元]

functional analysis + linear algebra + statistics = machine learning

[@strai5150]

Oops!! Got little bit late to hop in here for the new series 😁

[@LuanaBarbosadaSilva-p1r]

Muito Bom!

[@changhanchao9654]

yes yes yes!

[@MindeyI]

So, you just skip a bunch of interesting things, and jump right into metric spaces? Ah... Was expecting more of topology indeed.

[@brightsideofmaths]

Yeah, I needed to find my priorities :)

[@sarkarsubhadipofficial]

❤

[@duckymomo7935]

tfw he does distribution theory before functional analysis hehe

[@RaeTheActuary]

❤🎉🎉🎉🎉🎉🎉❤

[@enochtetteycommey7907]

shidaa

[@MasayoshiTakanaka-w5k6c]

You are a german guy

[@stsowi]

This accent is thicc!

[@brightsideofmaths]

And you will find nice functional analysis behind it :)

[@shabnamghalichi8534]

Those lines are not clear and i can not watch exactly

[@brightsideofmaths]

Which lines?

[@philippfarag]

deutsch-englischer Dialekt xD

[@brightsideofmaths]

Of course!

[@philippfarag]

@@brightsideofmaths i have problems to understand the german notations, is there a cool german version of this very nice course? XD or a nice dictionary for this? :D .... ......

[@brightsideofmaths]

@@philippfarag When I created this course, I also wanted to do German videos about it. However, then I decided that this is probably not needed. I would suggest that you use Wikipedia for translation of the notation. However, maybe I can also create a nice dictionary there. It's a good idea.

[@philippfarag]

@@brightsideofmaths thank you very much for your work and support :-)