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visualizing infinitely dimensional spaces
april 24th, 2023got really into hilbert spaces, fourier and all that this semester. recently, i thought of a way of representing infinitely dimensional spaces, like how we can process audio from 20 Hz to 20 kHz each have different dB amplitudes, and fourier is a really huge deal there but we have to kind of think of every Hz as its own coordinate in a way, hence giving an uncountably large basis in a way. but what's cool with fourier is it puts things in angles and shit, so it gave me the idea of representing each coordinate with a specific angle around a point, a bit like this, which resembles a spider plot, except there's no piecewise linear interpolation involved, and it's the axis segment lengths that indicate how big or small a coordinate is :
and yes, it can actually be an infinitely-dimensional thing since there are infinitely many angles. it's not really ℝ² as a vector space btw, the vectors i'm talking about just have some kind of subset β ⊆ B(0ℝ², 1) as its orthonormal basis, so its vectors are more like ℝ·β ≝ {λx : λ ∈ ℝ, x ∈ β}.
what's even cooler is that you can also plot functions from ℝ to ℝm or some E where (E, ⟨,⟨) is a hilbert space for which the basis is indexed over ℝ or some part of it, using the basis {(x,y) ∈ ℝ² : (x+yi)2n = (−1)n, 0 ≤ arg(x+yi) < π} in n dimensions, irrational rotations over discrete indices, or arctan otherwise to decide which angle we assign each coordinate to. i call it a mascara plot because it looks a bit like a mascara brush :
also you might think what i showed you above is continuous is some kinda way... and it's absolutely NOT. for it to be continuous, it has to have amplitudes continuously decreasing and increasing for the same angular coordinates. to test if a function is continuous, you can check if each plane passing through the x-axis intersected with the mascara graph around this very axis is comprised of a continuous curve ; that's even equivalent, i think, but i'm too lazy to actually check (idrk how that would break tho lol). notice this only works up to hilbert spaces for which the basis is indexed over ℝ or part of it or smth, if you're indexing over ℝℝ or some shit then that's your problem.
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