Example of an infinite-dimensional space, and why its dimension is infinity Check out my Matrix Algebra playlist: • Matrix Algebra Subscribe to my channel: / @drpeyam
Does the thm below is true for infinite-dim vector space? "Every vector in a vector space can be written in a unique way as a finite linear combination of the elements in this basis."
It’s a bit more complicated than that, for example x^2 cannot be written as a finite linear combination of e^(inx), but as an infinite series. There’s something called a Hamel basis, look it up!
An example from field theory would be the field extension ℝ/ℚ with [ℝ:ℚ]=∞ the same order as |ℝ|. Therfore ℝ can be interpreted as an uncountable infinite dimensional ℚ vectorspace :)
Well the best you can do is prove that any basis could not be listed out because it would lead to a contradiction. You can do this easily with the space of all functions, but continuous functions, differentiable functions, etc are harder as mentioned in the video because you need axiom of choice. Funny enough, the space of analytic functions is countably infinite, which is one of the coolest distinctions between analytic and infinitely differentiable in my opinion.
Does it really make sense to represent a sequence as an infinite dimensional vector? A sequence has an order built into it where the components of a vector do not. No idea if i will get a response on this 4 year old video lol.
Let R be the set of reals and N of positive integers. I don't know if this is a widespread notation but for me R^N is the set of real sequences and R^∞ the set of real sequences that are 0 in a finite number of steps. That means that the family F=(δ_n)_n∈N is free in both vectorial spaces made from R^N and R^∞ yet it is a basis of R^∞ but not of R^N. Furthermore I don't think R[X] and (whatever is the notation for the space of power series) are isomorphic, as R^N and R^∞ are not. I'm saying this because even if you can prove there are bases of R^N and (not twice) I don't think - though I could be very wrong on this - that there is an isomorphism between them and bases of R^∞ and R[X].
By your definition of ℝ^∞, it is not isomorphic to ℝ[[x]] (This would ne the notation for power series over ℝ) but instead ℝ[[x]] would be isomorphic to ℝ^ℕ. You have to remember that we aren't necessarily talking about convergence when we are dealing with the space of power series. In ℝ[[x]] you also have non-convergent series like 1+1x+4x²+27x³ +(...)+ nⁿxⁿ +(...). If you would wan't to talk about converent series you'ld first have to make sure that your set even is a space! For example the set of all convergent power series with a fixed radius of convergence r∈ℝ form a space! But finding a isomorphism to one of those would be more tricky :P
The sia sequence is my new favorite sequence.
You killed me with the SIA sequence professor 😂😂😂
Hahaha
Sears' sequence: 😒
Sia's sequence: 🤩
Lemme throw some Infinity-sided die for ya
*throws*
It seems to be rolling forever :D
Compactness in my linear algebra? It's more likely than you think!
As an example for infinite dimension I used to mention the set of all (real or complex) one variable polynomials.
Infinite dimension ( all math )
More illustrations on the basis of continuous functions PLEASE!!
First!
Thank you for making such interesting videos
😊
Thanks a gain D peyam its also a nice video
R vector space over Q as the field of rationals is also an infinite dimensional vector space. Nice video.
Great example!
This guy is way too underated
1:50 yeah, dot dot dot. So let's continue this sequence logically
Does the thm below is true for infinite-dim vector space?
"Every vector in a vector space can be written in a unique way as a finite linear combination of the elements in this basis."
It’s a bit more complicated than that, for example x^2 cannot be written as a finite linear combination of e^(inx), but as an infinite series. There’s something called a Hamel basis, look it up!
Can dimension be uncountable?
Mmmmmh, depends on how you define a basis, check out Hamel basis
An example from field theory would be the field extension ℝ/ℚ with [ℝ:ℚ]=∞ the same order as |ℝ|. Therfore ℝ can be interpreted as an uncountable infinite dimensional ℚ vectorspace :)
Well the best you can do is prove that any basis could not be listed out because it would lead to a contradiction. You can do this easily with the space of all functions, but continuous functions, differentiable functions, etc are harder as mentioned in the video because you need axiom of choice. Funny enough, the space of analytic functions is countably infinite, which is one of the coolest distinctions between analytic and infinitely differentiable in my opinion.
Does it really make sense to represent a sequence as an infinite dimensional vector? A sequence has an order built into it where the components of a vector do not. No idea if i will get a response on this 4 year old video lol.
Let R be the set of reals and N of positive integers.
I don't know if this is a widespread notation but for me R^N is the set of real sequences and R^∞ the set of real sequences that are 0 in a finite number of steps. That means that the family F=(δ_n)_n∈N is free in both vectorial spaces made from R^N and R^∞ yet it is a basis of R^∞ but not of R^N.
Furthermore I don't think R[X] and (whatever is the notation for the space of power series) are isomorphic, as R^N and R^∞ are not. I'm saying this because even if you can prove there are bases of R^N and (not twice) I don't think - though I could be very wrong on this - that there is an isomorphism between them and bases of R^∞ and R[X].
By your definition of ℝ^∞, it is not isomorphic to ℝ[[x]] (This would ne the notation for power series over ℝ) but instead ℝ[[x]] would be isomorphic to ℝ^ℕ. You have to remember that we aren't necessarily talking about convergence when we are dealing with the space of power series. In ℝ[[x]] you also have non-convergent series like 1+1x+4x²+27x³ +(...)+ nⁿxⁿ +(...). If you would wan't to talk about converent series you'ld first have to make sure that your set even is a space! For example the set of all convergent power series with a fixed radius of convergence r∈ℝ form a space! But finding a isomorphism to one of those would be more tricky :P
This is notation that I have seen as well
Dr. Peyam. Could i be smart like you?
😌
I’m not that smart, haha
πm sir is the smartest
where is part when we talk about minecraft
well, i understand nothing because i'm Vietnamese.
but anyway how the hell that this video has lesser views than Baby Shark???
Love you as a human being and love from India ❤️🇮🇳
Minecraft April Fools?