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!
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.
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.
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.
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?
Set Theory Some notions from Analysis Linear Algebra Basic notions from Abstract Algebra & Topology are very helpful and most importantly Mathematical Maturity.
![Functional Analysis 7 | Examples of Banach Spaces [dark version]](http://i.ytimg.com/vi/qjrgiZOnqdE/mqdefault.jpg)
![Functional Analysis 7 | Examples of Banach Spaces [dark version]](/img/tr.png)
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!
ALL OF YOUR VIDEOS ARE AWESOME!
can you open the function analysis No. 5 (Cauchy Seq and Complete Space ) to let everyone can see? thanks
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.
Why should we care about this space? Euclidian or hilbert space seem to bo the same job.
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)
Hello,
You are doing a very good job! Thank you for strengthening my intuiton while remaining rigorous :)
this is so well explained and easy to understand that I am afraid I am kidding myself and in reality understand nothing about this!
Great intro video for Functional Analysis, easy to digest.
Glad you like it. We will do harder stuff later :)
@@brightsideofmaths Looking forward to it! XD
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.
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.
Amazing content, very easy to grasp. Thank you! Subscribed!
Danke Ihnen für die tolle Erklärung!
thanks for ur efforts,,,, and ur term,,, very useful
Does a Banach Spaced form an Abelian Group(+) under addition . Like a vector space?
Yes!
@@brightsideofmaths Thank you!
Merci beaucoup compadre 👌 sehr gute videos brudi
Thanks for uploading..very much appreciated..😊😊
In spacetime the norm is c²t² - x² - y² - z², which clearly doesn't satisfy (c) at 3:25. So spacetime is not a Banach space, right?
It's not a standard norm if it can be negative. So yes, in that regard, it's not a Banach space. The theory about Krein spaces helps there.
@@brightsideofmaths You are awesome, thank you.
Is it also equipped with open set topology?
What do you mean exactly?
Hello, what software do you use to make your videos?
See my website in the description. There is an FAQ :)
First video thazcan explain this to me
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?
Yes, both things are true. The dimension does not play a role in this definition, a priori.
What do you need to understand this course
Set Theory
Some notions from Analysis
Linear Algebra
Basic notions from Abstract Algebra & Topology are very helpful
and most importantly Mathematical Maturity.
Sir please explain metric completion theorem and its uniqueness please do explain sir I'll be waiting sir
Coming soon :)
@@brightsideofmaths I will be waiting sir thank you for replying sir
Nice!
Sir please make the videos of linear space
Linear algebra course is in the making :)
@@brightsideofmaths where???
@@umarhabiblmsonly2206 Here on my desk.
Great
아주 좋은 비디오. 한국에서 감사요!
Mmmm😊
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