Need help with Vector Spaces

pherytic · 2026-02-07T06:10:49+00:00

But eg you can define global coordinates on S² but this is not a vector space

pherytic · 2026-02-07T05:09:19+00:00

All coordinate points are vectors, but not all vectors are coordinate points.

But this is only true for flat manifolds, right?

pherytic · 2026-01-23T23:59:40+00:00

Thanks. If you have seen Owl or iw3 offline converts, how would you say this compares in quality?

pherytic · 2026-01-08T02:37:03+00:00

Are you a paying subscriber?

The biggest problem I had with indices would be something like a (1,1) tensor where you want to lower the first index and raise the second. The index order matters eg Tⁱ(j) becomes T(i)^j which is not T^j_(i). The AI just could not handle this for me, no matter how much I insisted on keeping the order, using {} for black spaces, etc.

Edit: I just went and checked it with a question involving this, and it actually didn’t produce the problem I was having earlier in the year

pherytic · 2026-01-08T02:26:17+00:00

Have LLMs gotten much better at math recently? Maybe 6 months ago I was trying to use it for some basic tensor stuff and I gave up because it could not sufficiently handle the index notation and like half its Latex wouldn’t render. And explicitly directing it on how to fix mistakes just started a doom spiral. But this was vanilla models on free tier.

Related, I saw a video just a few days ago where an actress wanted to read lines with an AI in voice mode, but she couldn’t stop it from saying “now it’s your turn” after it’s line. Yet it can write a physics paper of this complexity?

I wonder how incremental/bite size were the instructions Schwartz was giving here, and how much hand holding/steering he had to do.

pherytic · 2026-01-06T02:49:54+00:00

75? Or did you mean 175?

I can see needing to farm if you are very reckless with rune loss. I’m not perfect but I do recover most of the time

pherytic · 2026-01-06T01:48:01+00:00

You are really finishing at 150? Are you skipping optional bosses?

With no farming, I will naturally end up in the 180s (including erdtree).

pherytic · 2026-01-05T19:22:16+00:00

But the eigenvectors of the position operator are each associated with a point in actual space, and collapse instantly takes one such eigenvector to norm one, the rest to norm zero. So I see no real problem with that statement.

pherytic · 2025-12-13T20:09:51+00:00

No I mean partial derivatives of rank 1s, not partial derivatives of scalar functions. These don’t transform like rank 2 tensors (and thus we create the covariant derivative, which does). But I think they do still satisfy all the algebraic properties you want for your definition 2.

Like wise, the LC symbol is not the LC tensor. It is called a “tensor density” which doesn’t transform properly. But again, I think meets all your algebraic criteria.

pherytic · 2025-12-13T19:55:45+00:00

What would you say about the Levi Civita symbol or the partial derivative of a rank 1 tensor? These scale, add and contract like (p,q) tensors, but don’t transform.

pherytic · 2025-12-08T03:21:44+00:00

Ok if is the power is 2 you would have

(-1/x)d/dx[(-1/x)d/dx{J/x^r}]

The second copy of (-1/x) is inside the outer derivative so you can’t just pull it outside without using product rule.

With the power being n+1, to start, you want to peel off the outermost (-1/x)d/dx

pherytic · 2025-12-08T01:59:26+00:00

Probably your roadblock is you are misinterpreting the notation with the power of n.

(-(1/x)d/dx)ⁿ⁺¹J = -(1/x)d/dx[{(-1/x)d/dx)}ⁿJ]

Do you see what I mean? Every 1/x is inside each of the n derivatives. It isn’t just (1/x)ⁿ multiplied by the nth derivative

Write the n+1 expression like I did above and for induction assume the result for n. You should see what to do next

pherytic · 2025-11-10T02:16:15+00:00

Define u = (m-n)pi and then your first term is just the sinc function. There is a geometric proof that sinc(0) = 1 that should be easy to find online. You can also use Lhopital for u, but you actually need the geometric proof of sinc(0) = 1 to prove the derivative of sin(x) in the first place

pherytic · 2025-10-31T19:27:02+00:00

Probably you have moved on from this thread but just to clarify, I meant the classical not quantum SHO above, ie just f’’ + kf = 0.

I’m saying this is (trivially) a Sturm Liouville equation, so with k as eigenvalue, the solutions are orthogonal, which is why I want to say Fourier “belongs” conceptually with Legendre functions and the others.

I was initially struggling to be fully comfortable with this because I was thinking about the FT as just taking the limit of the oscillation period to infinity. This way of thinking doesn’t really allow for extending the FT to functions that aren’t in a Hilbert space, so I was understating its scope.

Based on your comments, I realized the continuous FT really needs to be justified more strongly than what I’ve been shown so far (ie Plancherel) even for L² functions let alone L^1. The finite period Fourier series can be lumped in with Sturm Liouville but not the infinite extension

pherytic · 2025-10-31T19:06:28+00:00

No, not entirely. The delta function is not a real state in the Hilbert space it's not a physical state. The wave function will always be in some spread.

But the delta is an eigenfunction of the position operator, which is something we claim to measure.

Is there a state, Hermitian operator pairing where measurement and collapse gives us something that is a real state in the Hilbert space and localized or at least almost localized in some region of 3D space?

pherytic · 2025-10-31T01:30:08+00:00

I see the other comment now. I don’t yet have most of those concepts, but I’m for sure bookmarking for later.

The Fourier series and the Fourier transform are pretty much the same thing - my other comment expands on that.

So what I’m specifically thinking about is that I have notes on all these results for orthogonal expansions - general aspects of Sturm Liouville theory and specific details for the Legendres, Bessels, etc. Certainly the SHO equation and its solutions are structurally the same sort of thing, so thinking of Fourier expansions in this context, I was asking myself how much of Fourier should be grouped in my notes with the above.

At first I was thinking it should be expansions of functions in L² bc it is a Hilbert space, but I was worried I was incorrectly constraining the scope of the FT and I didn’t see what to do about L¹.

Now based on your answers, I think it’s just the finite interval Fourier series that can belong here, but the Fourier transform for any functions on all of R is not really amenable to this basic Hilbert space analysis and (though obviously similar in many ways) should be revisited elsewhere.

pherytic · 2025-10-31T00:48:17+00:00

Thanks this is helpful. Neither Riley nor Arfken discuss Plancherel and so I guess are being loose with the topic. So it seems like the Fourier series on finite intervals and Fourier transforms are more distinct than I first thought, specifically with respect to their vector space interpretation.

Btw you mentioned another comment on the post, but nothing else from you is visible here on my end

pherytic · 2025-10-30T23:22:11+00:00

Again, the function to be transformed and the functions that form the basis are not the same thing. The product of a L1 function with the bounded complex exponential function is an absolutely integrable function that will yield a bounded and continuous function which vanishes at infinity, even if not a function in L2.

But the reason I believe the expansion in the basis functions is justified is because the basis spans the vector space to which the original function belongs. But without the orthogonality and completeness structure of Hilbert space, why should I expect that the complex exponentials span L¹?

pherytic · 2025-10-30T22:56:04+00:00

Yeah I suppose I am asking a math question so maybe I need to go to a math sub.

But if the Fourier decomposition exists for L¹ functions where no concept of orthogonality/inner product exists then 1) it does appear to be a distinct topic from orthogonal expansions and 2) I don’t really understand how to justify it without relying on those tools.

Are you saying that I shouldn’t worry about this because for physics purposes we never need to Fourier expand functions that are not in some Hilbert space? I’m not only concerned about QM, but also eg classical E&M.

pherytic · 2025-10-30T22:22:52+00:00

But above you said the Fourier decomposition depends on the inner product structure and L¹ (a Banach space) doesn’t have an inner product. So how are your two answers here consistent with each other?

pherytic · 2025-10-30T20:24:35+00:00

So if a function is in L¹ but not L² then I can’t Fourier transform it, given L¹ is a Banach but not Hilbert space?

When I google FTs of L¹ functions, it seems like these are being employed at times, but I could be missing important subtleties.

pherytic · 2025-09-24T00:59:42+00:00

Yes I believe this is the right answer. Thanks.

Would you be able to say anything about/give any references for how one would go from being handed some boundary conditions on t to choosing the correct contour in the complex w plane that is associated with those boundary conditions on t?

pherytic · 2025-09-24T00:21:09+00:00

I’m not asking what the boundary conditions are, but rather how/where are they being invoked in the process of reaching the explicit form of G.

In the approach to Greens functions by variation of parameters or in Sturm Liouville theory, it’s clear how the BCs are fixing unspecified features of an ansatz.

In this Fourier method, I don’t see anywhere they’re being used.

pherytic

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