Why care about infinite dimensions?

[@jacksyd]

I don’t usually write comments but I just wanted to say that I really enjoy your videos!!! I’m a final year undergrad math student and everything you talk about is so interesting and well articulated!!

[@greghmn]

"If you don't succeed, try again" is an ironic lesson since the function looks like someone tried to draw a wave and gave up toward the end.

[@zeleniyslon8]

One thing I found really interesting is the Moebius strip. I remember being shown in school what happens when you cut it through the middle and having my mind blown. It would be cool to see an explanation of why it should behave that way

[@ASackVideo]

This is a great suggestion! Thanks!

[@b1bbscraz3y]

@@ASackVideo does infinite dimensions prove simulation theory?

[@mitchkovacs1396]

Can you do a deeper dive into distances in infinite-dimensional spaces? At first glance in R^∞, it seems like the Euclidean metric is nonsense since the distance from the origin to (ε,ε,ε,...) is infinite even for arbitrarily small ε. Not only can you find points that are a finite distance away, but you can then define an equivalence relation on R^∞ by saying that P1 ~ P2 if dist(P1,P2) < ∞. I think it's interesting that we can do that with R^∞, but not R^n (fails the transitive property)

[@fejfo6559]

Yeah you have to define distance differently. You could define the distance between (x1,x2,...,xi,...) and (y1,y2,...,yi,...) to be the maximum of |xi-yi| (like in the video) Or as (x1-y1)²/2+(x1-y1)²/4 + ... + (xi-yi)/2^i + ... Or in many other ways.

[@mitchkovacs1396]

@@fejfo6559 The Euclidean metric does work, but for a relatively restricted neighborhood of points. The point (1, 1/2, 1/3, 1/4, ...) is a Euclidean distance of pi/sqrt(6) away from the origin.

[@ASackVideo]

You can do this in R^n, just everything is equivalent! As for R^∞, we call this the "ℓ^2" norm. The standard thing to do is restrict only to "square-summable" sequences, that is, sequences whose sum of squares converges. If you don't restrict, the topology you get isn't too much more interesting, it's a bunch of disconnected disjoint copies of the restricted version. There's a lot to say about the ℓ^2 space of sequences, but some notable bits are that it's complete and it's a Hilbert space.

[@danielyuan9862]

A quick "definition" of a complete metric space. A metric space is complete where, if for any sequence of points, the points get infinitely close to each other as you go further into the sequence, there exists a limiting point that the sequence that the point converges to. For example, the real number line excluding 0 isn't complete because the sequence 1, 1/2, 1/4, 1/8... are all real non-zero numbers that get closer to each other as the sequence continues, but there is no number (because 0 doesn't count) that the sequence converges to. But the real number line itself is complete.

[@dialgos7574]

Great Video as I am currently retaking multidimensional calculus and this is some nice motivation to revise the lectures. As a video Idea: I am really a fan of combinatorial game theory and I believe in the world of modern math it is highly underappreciated. Maybe a short video about surreal numbers would be cool! :D

[@jcsahnwaldt]

Great video! Minor gripe: The use of "x" in "

[@carlhopkinson]

Very clear and beautiful presentation. You have a gift.

[@jeremy.N]

Amazing video, i usually really dont like operator or functional analysis, but this was great

[@leehayes4019]

Excellent description, thanks for teaching me!

[@Kram1032]

You kinda moved from finite dimensions straight to a particular version of a continuum of dimensions, but there also are discretely-infinitely dimensional spaces, and in these infinite settings you can also have different ways the dimensions are topologically arranged (say, a regular fourier spectrum vs. spherical harmonics or such)

[@paradoxica424]

don’t forget the Hilbert Cube; a compact set homeomorphic to the standard infinite dimensional cube but with zero hypervolume

[@Kram1032]

@@paradoxica424 that just goes from 1D to 3D. No infinite dimensions here. Though you can certainly generalize it to higher dimensions

[@paradoxica424]

You are confusing Hilbert Curve with Hilbert Cube. The Hilbert Cube is the space given by the cartesian product of the intervals [0, 1/n] over all positive integers n.

[@Kram1032]

@@paradoxica424 ah, sorry, fair enough Sure, there are many ways to construct infinite-dimensional spaces

[@elisecastor5534]

Great video man, these ideas are always so interesting to think about. Also, they're much more fun than sitting in the back of the lecture hall.

[@ahasdasetodu6304]

Great video, however I had trouble understanding the jump from an infinite list of numbers to a function on the interval [0,1]. From the way it's presented in the video it seems like they are supposed to be equivalent but O don't think they are since a point as a list of numbers has countably many parameters whereas a function has unvountably many. I would appreciate it if someone could explain how is it supposed to work because I'm pretty sure I'm not understanding it correctly

[@ASackVideo]

You're correct. The space of functions [0,1] -> R has uncountable dimension. Interestingly, C[0,1] has only countable dimension!

[@SilverXenolupus]

These math videos are awesome!

[@smileyp4535]

This is actually amazing because this is how I explain the concept of "intersectionality" to people, everyone is an infinite set of voluntary and involuntary identity groups that they belong to (woman, man, both, neither, tall, short, pilot, movie fan, etc, etc) and these can change as we go through life. Intersectionality is how we understand that we are all unique but all fit in different groups and communities and how you can find egalitarian and equitable solutions to democracy and mutual aid political issues without just falling into "oppression olympics" or identity politics, for example just because someone might be a black woman doesn't mean they aren't *also* a facist (like Candice Owens) it's about the ideas and solutions they advocate for, while also keeping identity and such in mind but not letting it eclipse the real issue (or slip into the cracks)

[@h4ck314]

Very nice !

[@jimmy_colombia]

excellent content, please add transcript, to be able to understand in other languages

[@apollo261]

Great video, but I have a doubt. When you explained infinite dimensions, you said it takes infinitely many numbers to define a point, but it's a countably infinite of numbers. And then you jump straight into the function case, when a real function in [0,1] needs uncountable infinitely many numbers to be defined. The only way I see this could make sense is if you interpret the continuity requirement as only needing to know the values of the function for all rational numbers between 0 and 1 (they are countable infinitely many) and then extending the function to the irrational values by using the fact that every irrational number can be expressed as a limit of a sequence of rational numbers and using the continuous nature of the function to define it for them. Is this the reason?

[@ASackVideo]

Yeah, I was a little loose with defining infinite dimensional spaces. The space of functions R->R is uncountable dimension (but still infinite dimensional of course!) C[0,1] it turns out has only countable dimension for exactly the reason you state. As a vector space, we can prove this rigorously with the injective map C[0,1] -> R^Q given by sending f -> {f(q)} for each rational q, and R^Q is isomorphic to R^N by picking any enumeration of the rationals.

[@apollo261]

Ok, thank you so much! This is such an interesting topic, and you explained it clearly. Your videos are amazing in general, keep the good work 👍

[@maximilianosotomayorga4977]

did not undesrtand much but was very beautiful

[@ryanmarck7901]

Mathematically this is correct!

[@shawnxihaowu7638]

Wow! Amazing!

[@ncb4_69]

Happy late new year sir,

[@nif4345]

5:45 you missed a ) on the third iteration of T

[@pielover267]

Hey this is great!

[@adamrummer5342]

why are we able to map [0,1] (an interval in R) -> R^{inf} when the first is uncountably infinite and countably infinite?

[@lysikasaito]

He is talking about an uncountably infinite vector space, R^{inf}. Not a countably infinite vector space.

[@adamrummer5342]

@@lysikasaito right ok thanks. I suppose I was confused by his introduction using the limit of a sequence of finite dimensions I didn't study infinite vector spaces, do both the countable an uncountable spaces exist as separate ojects?

[@ASackVideo]

My presentation may have been a bit sloppy here: These are two different infinite dimensional spaces, the space of sequences and the space of functions. Their dimensions are indeed different, the first being countable dimension and the second being uncountable dimension. Interestingly, the space of continuous functions over R has countable dimension because a continuous function is determined by its values at the rationals.

[@lysikasaito]

@SackVideo There are at least as many continuous functions as real numbers. There exists an f(x) = r, a constant function, for each r in R. So there are certainly not countably many continuous functions. Am I misunderstanding? That said, the sequence space is also uncountable, which is what I think you are meaning to prove with the rational points argument. Usually the rational points argument is used to prove that the sequence space is also uncountable. You have your argument backwards I believe.

[@ASackVideo]

As an R-vector space, C[0,1] has countable dimension. There's a difference between number of elements and dimension. R^2 is uncountable but is 2 dimensional. The space of all functions (not just continuous) from R to R is uncountable dimension, and the cardinality of the set is actually bigger than just R.

[@jake967]

+

[@BaptistPiano]

I’m confused as to how a point in infinite dimensional space represents a function on the reals from 0,1. The number of reals between 0,1 is not a countable infinity whereas the number of dimensions in an infinite dimensional point is a countable infinity right?

[@ASackVideo]

The space of all functions from [0,1] → ℝ has uncountable dimension!

[@HyperCubist]

You said you've never seen a hypercube but have a gif of one at the beginning of the video...

[@ASackVideo]

It’s only a projection, not a real one.

[@HyperCubist]

@@ASackVideo Well ... what would a real one look like?

[@HyperCubist]

And what if I told you, you've never seen a real cube in your life? Only projections of cubes?

[@HyperCubist]

And what's the difference between seeing a 2-D projection of a 4-D object (the gif), and seeing a 2-D projection of a 3-D objects(cubes on a screen, or in real life)?

[@iquemedia]

is dmt an infinite dimension?

[@ib_concept]

Fibanoci sequences

[@HeWhoProclaims]

Isn't finite just as theoretical as infinite? Just as you can't go far enough out to determine whether or not the universe is infinite or finite, It's also the same with going in. We don't know if matter is infinitely deep or eventually has a finite end. So pretty much all dimensions, all planes, and all matter can either be infinite or finite. So everything can be one of these options 1. If everything's infinite, then nothing's finite. 2. If everything's finite, then nothing is infinite. 3. Then the third option, duality of the two. A. (Infinitely finite or/and a finite of infinities) mixed with B. (Infinite infinities or/and finite finites.) I guess the important question for humanity to answer would be exactly this. Until this is answered. We really have no clue what our dimensional planes have in store for us. But that's just a theory.. A game theor.. wait.. Just a regular Theory!

[@floppy8568]

so this video was about... ...continuous functions. Not infinite-dimensional points.

[@ib_concept]

Stokes theorem

[@JohnDlugosz]

I lost you at the beginning. You have f(x) = some stuff that involves x; looks like a normal definition of a function. So I don't understand what function you are trying to find.

[@ASackVideo]

The "definition" of f involves f itself, so it's not clear if there actually is a function that satisfies the equation. For example, there's no function f such that f(x) = f(x) + 1.

[@mrosskne]

this video would be awesome if I could hear it

[@ib_concept]

P-adics

[@sebastianmanterfield3132]

🤬