As an example for infinite dimension I used to mention the set of all (real or complex) one variable polynomials.
@omniverse681 Жыл бұрын
Infinite dimension ( all math )
@cobalius4 жыл бұрын
Lemme throw some Infinity-sided die for ya *throws* It seems to be rolling forever :D
@sugarfrosted20055 жыл бұрын
Compactness in my linear algebra? It's more likely than you think!
@Eden-mn6rt3 жыл бұрын
This guy is way too underated
@Mircor555 жыл бұрын
R vector space over Q as the field of rationals is also an infinite dimensional vector space. Nice video.
@drpeyam5 жыл бұрын
Great example!
@snowflake82352 жыл бұрын
Love you as a human being and love from India ❤️🇮🇳
@rohunse55555 жыл бұрын
First! Thank you for making such interesting videos
@drpeyam5 жыл бұрын
😊
@Vampianist35 жыл бұрын
More illustrations on the basis of continuous functions PLEASE!!
@snuffybox Жыл бұрын
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.
@doria_bolognese5 жыл бұрын
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."
@drpeyam5 жыл бұрын
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!
@etienneparcollet7275 жыл бұрын
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].
@stydras33805 жыл бұрын
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
@willnewman97835 жыл бұрын
This is notation that I have seen as well
@cubicardi80115 жыл бұрын
1:50 yeah, dot dot dot. So let's continue this sequence logically
@112BALAGE1125 жыл бұрын
Can dimension be uncountable?
@drpeyam5 жыл бұрын
Mmmmmh, depends on how you define a basis, check out Hamel basis
@stydras33805 жыл бұрын
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 :)
@orangeguy54635 жыл бұрын
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.
@newtonnewtonnewton15875 жыл бұрын
Thanks a gain D peyam its also a nice video
@shandyverdyo76885 жыл бұрын
Dr. Peyam. Could i be smart like you? 😌
@drpeyam5 жыл бұрын
I’m not that smart, haha
@yrcmurthy83235 жыл бұрын
πm sir is the smartest
@miloglin82874 жыл бұрын
where is part when we talk about minecraft
@baongocnguyenhong56745 жыл бұрын
well, i understand nothing because i'm Vietnamese. but anyway how the hell that this video has lesser views than Baby Shark???