Functional Analysis 6 | Norms and Banach Spaces

[@wdacademia2329]

A quick remark: it may seem rather ad-hoc to restrict ourselves to R and C as the only two options for the field. Or in the definition of a metric space, having the metric be a function into R (why not some other ordered topological field). It turns out though, that when we consider all desired properties for our analysis we get nice categoricity results. For example, up to isomorphism, R is the unique complete ordered field as well as the unique complete Archimedean field (order completeness in the former and metric completeness in the latter). Another result is that R and C are the only connected, locally compact, topological fields. So these restrictions are not so ad-hoc after all.

[@samtux762]

Why should we care about this space? Euclidian or hilbert space seem to bo the same job.

[@PunmasterSTP]

Norms and Banach spaces? More like "Now these videos grace us" with tremendous amounts of knowledge and understanding. Thank you so much for making and uploading all of them!

[@rin-or3no]

Thanks. I know its hard to make videos so fast but I enjoy this. Waiting for the next one. (One video each day will be great)

[@jordanmatin8498]

Hello, You are doing a very good job! Thank you for strengthening my intuiton while remaining rigorous :)

[@parianhatami]

ALL OF YOUR VIDEOS ARE AWESOME!

[@zoedesvl4131]

I'd like to add some generalization here, many of which will be learned in the future I think. There is a hidden property of this metric d induced by norm: d is translate invariant. To be precise, we naturally have d(x+z,y+z)=||(x+z)-(y+z)||=||x-y||=d(x,y) for all z in X. But is norm always a thing? By Axiom of Choice, any vector space is normable (i.e., we are able to define a norm whatsoever), but some norm results in abnormal structure, in which case we don't want to admit the existence of that norm. (If you are interested, learn this theorem: A topological vector space X is normable if and only if its origin has a convex bounded neighborhood.) To solve this problem, mathematicians introduced a more generalized topological vector space: F-space. A topological vector space X is called a F-space if it has a complete translate invariant metric (i.e. d(x,y)=d(x+z,y+z) for all z in X). So the blue box at the end will be updated in the future: norm be 'upgraded' to translate invariant metric. By the way this channel is great! I always prefer recommend this channel to people learning math over many others. It's important to keep the seriousness of mathematics and this channel deals with it nicely.

[@brightsideofmaths]

Thank you for your generalisations and the recommendations :) I will cover Fréchet-spaces maybe in the end of this series. I really like them but I have the feeling that it is easier first to do a lot of functional analysis with Banach spaces before going into this direction.

[@batmanrobin6711]

this is so well explained and easy to understand that I am afraid I am kidding myself and in reality understand nothing about this!

[@raycopper9229]

Great intro video for Functional Analysis, easy to digest.

[@brightsideofmaths]

Glad you like it. We will do harder stuff later :)

[@raycopper9229]

@@brightsideofmaths Looking forward to it! XD

[@kristiantorres1080]

Amazing content, very easy to grasp. Thank you! Subscribed!

[@ishaangoud3180]

Does a Banach Spaced form an Abelian Group(+) under addition . Like a vector space?

[@brightsideofmaths]

Yes!

[@ishaangoud3180]

@@brightsideofmaths Thank you!

[@L23K]

Danke Ihnen für die tolle Erklärung!

[@ROni_ROmio]

thanks for ur efforts,,,, and ur term,,, very useful

[@Independent_Man3]

Difference between Hilbert space and Banach space ?? What if the norm is induced by an inner product and X is complete with respect to this norm? Is the following true? Banach space = complete normed vector space Hilbert space = complete inner product space Further, what if the vector space is infinite dimensional?

[@brightsideofmaths]

Yes, both things are true. The dimension does not play a role in this definition, a priori.

[@davidaugustyn9234]

What do you need to understand this course

[@vipilvijay7116]

Set Theory Some notions from Analysis Linear Algebra Basic notions from Abstract Algebra & Topology are very helpful and most importantly Mathematical Maturity.

[@pebotin]

Thanks for uploading..very much appreciated..😊😊

[@rakshithasp1279]

Sir please explain metric completion theorem and its uniqueness please do explain sir I'll be waiting sir

[@brightsideofmaths]

Coming soon :)

[@rakshithasp1279]

@@brightsideofmaths I will be waiting sir thank you for replying sir

[@nachomacho7027]

Merci beaucoup compadre 👌 sehr gute videos brudi

[@samtux762]

Is it also equipped with open set topology?

[@brightsideofmaths]

What do you mean exactly?

[@efamily2854]

Hello, what software do you use to make your videos?

[@brightsideofmaths]

See my website in the description. There is an FAQ :)

[@ski34able]

First video thazcan explain this to me

[@umarhabiblmsonly2206]

Sir please make the videos of linear space

[@brightsideofmaths]

Linear algebra course is in the making :)

[@umarhabiblmsonly2206]

@@brightsideofmaths where???

[@brightsideofmaths]

@@umarhabiblmsonly2206 Here on my desk.

[@zazinjozaza6193]

Nice!

[@adilaid4443]

Great

[@MrBorderlands123]

아주 좋은 비디오. 한국에서 감사요!

[@leewilliam3417]

Mmmm😊

[@MrArmas555]

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