Definition talk:Zero-Limit Sequence in Schwartz Space

I repeat my plea: you can source your "corrections", can you?

I understand that the original page was sourced directly from the Sasane work. Unless Sasane doesn't know what he's talking about, and is actually wrong, it is suboptimal to just change it because you know better.

So please don't just change stuff like this, even though everybody knows you're by far the cleverest person here, without first reviewing the source from which the "incorrect" version is printed, because it might just be the case that what is there is a simplified entry-level version of the definition which is used for lesser mortals who are struggling with this material at first sight. --prime mover (talk) 20:42, 24 October 2022 (UTC)

Looking at it more closely, it seems that the original was not "incorrect", but a special case of the more general form that you expanded it to. However, such is not how the Sasane work presented it, and this has resulted in the source citation no longer being accurate. I suggest and recommend that the more general version be presented separately, while the specific version where the limit is $\bszero$ should subpage it. --prime mover (talk) 20:58, 24 October 2022 (UTC)

I thought I just corrected an obvious mistake, since:

$\paren 1$: The page is named Definition:Convergent Sequence/Schwartz Space but the convergent sequence was not defined, instead only zero sequence.

$\paren 2$: The assumption "Let $\phi \in \map \SS \R$ be a Schwartz test function" was there, but $\phi$ was nowhere used.

If you want to revert my correction, then pleas rename this pate to Definition:Zero Sequence/Schwartz Space and delete the assumption of $\phi$.--Usagiop (talk) 21:33, 24 October 2022 (UTC)

But I will ask for checking the source beforehand.--Usagiop (talk) 21:43, 24 October 2022 (UTC)

I agree that the title should be changed. For Schwartz spaces general convergence is not discussed - only the convergence to $\mathbf 0$ is provided. This is done because distributions are linear, so one can incorporate the limit into the sequence itself and solve a modified problem without loss of generality. Indeed $\phi$ here is not being used here and can be removed. Fixing the rest of the content of the page is suboptimal - there are pages linked to this one resting on convergence to $\mathbf 0$. Please revert to the previous version. You are free to rename the original page according to your taste.--Julius (talk) 23:11, 24 October 2022 (UTC)

I've reverted it, Julius -- please feel to determine what the name of this page now needs to be. --prime mover (talk) 05:09, 25 October 2022 (UTC)

Once the dust has settled, a page explaining and proving the above paragraph would then be essential to remove the confusion that this talk page exemplifies. --prime mover (talk) 05:09, 25 October 2022 (UTC)

Let me note that convergence to $\mathbf 0$ is just a special case of the general convergence. Not a real issue. But feel free to revert it as you like. --Usagiop (talk) 00:52, 25 October 2022 (UTC)

But if you revert and rename this page, you still need a page for the definition of $\phi_n \stackrel \SS {\longrightarrow} \phi$ which looks exactly like this. --Usagiop (talk) 01:01, 25 October 2022 (UTC)

You're the one who wants it, so feel free to create it. --prime mover (talk) 05:07, 25 October 2022 (UTC)

For now we should keep this page as it is unless two things are done. Firstly, we need a definition page for a general convergent sequence. Secondly, we need to prove that, thanks to linearity of tempered distributions, $\phi_n \stackrel \SS {\longrightarrow} \phi$ iff $\phi_n - \phi \stackrel \SS {\longrightarrow} 0$. If we had these two pages, then this one could be simplified considerably by defining it as a convergent sequence with limit zero. I think we should still keep this definition in this or a simplified form, because in certain contexts it reduces the notational load. As for the title, we already have Definition:Space of Zero-Limit Sequences, so we could use Zero-Limit Sequence/Schwartz Space.--Julius (talk) 07:48, 25 October 2022 (UTC)

Would it be a proof? We can write it this way and say that $\phi_n \stackrel \SS {\longrightarrow} \phi$ if and only if $\phi_n - \phi \stackrel \SS {\longrightarrow} 0$ but that would be a definition since we haven't defined any convergence aside from convergence to 0. It'd be like giving the definition of a null sequence and saying that $a_n \to L$ if $a_n - L$ is null, which is fine if a bit obfuscatory. But here it nicens the notation a little. [also sorry for just contributing on talk pages and not making any mainspace edits, I have been meaning to get re-stuck in to editing] Caliburn (talk) 16:08, 25 October 2022 (UTC)

I believe there is an analogical result on somewhere here for real sequences, i.e. if $\sequence {x_n}$ converges to $l$, then $\sequence {x_n -l }$ converges to $0$. If we could find that then the same logic would apply here. What I mean is that we should write another definition page for general convergence, then this one would be a special case, and then show the equivalence.--Julius (talk) 20:43, 25 October 2022 (UTC)

As I told you, this was an obvious mistake of the creator of this page. Or, can you show us your reference for this stupid definition of the convergence sequence, prime mover? --Usagiop (talk) 14:53, 25 October 2022 (UTC)

We only have your word for it that it's an "obvious mistake" unless you can find a solid reference that backs up what you say. Without that everything you say is just hot air. --prime mover (talk) 16:46, 25 October 2022 (UTC)

Please double check your source and correct the mistakes on this page and the other related pages. I have no access to your book but I know this definition is wrong. --Usagiop (talk) 14:57, 25 October 2022 (UTC)

If you have no references then you're SOL. Your bossy attitude stinks. --prime mover (talk) 16:00, 25 October 2022 (UTC)

The test function space counterpart taken from from the same book looks normal. A similar definition for Schwartz space must be found in the book. --Usagiop (talk) 16:34, 25 October 2022 (UTC)

Can we stop fighting over whether this is a mistake or not? The page has been renamed and now reflects the content better. It may happen that sometimes the titles are simplistic because spelling our all the details would make it impractically long. But as we add more stuff, eventually the degeneracy gets broken and we have to distinguish between all the variations.--Julius (talk) 20:32, 25 October 2022 (UTC)