Definition talk:Test Function Space

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Lack of consensus

I see that that the original article is constantly being reverted back and forth. If the original article looks ugly, please present your point here. I understand that sometimes the material here gets repetitive (e.g. same statement presented semanticaly and mathematically in the same line), but a short reminder is not always bad thing. It often helps students to finally understand the meaning of math symbolism.--Julius (talk) 11:39, 26 May 2022 (UTC)

I've reverted it back to the original version so we can discuss it. Personally, I only create new pages for spaces of things when there is additional structure added, (for example, a topology or some vector space structure) and would usually do this as a subpage of the "thing". Otherwise I would just add "we denote the space of all such $\phi$ by $\map {\mathcal D} {\R^d}$" (just using this example). I think we should use this page to describe the test function space with its canonical topology, so that we are not just saying "the space of test functions is the set of all test functions". Thoughts? Admittedly I have yet to look through the distributions stuff and I'm not sure where the topology is outlined, only making this comment in passing. Caliburn (talk) 11:48, 26 May 2022 (UTC)

I just tried to improve the following misleading exposition:

Let $\phi : \R^d \to \C$ be a test function.
Then the set of all $\phi$ is called the test function space and is denoted by $\map \DD {\R^d}$.

In the first line, a specific test function is assigned to the symbol $\phi$. But in the second line, the symbol $\phi$ runs over all test functions. Is this a preferred ProofWiki style to define a set? --Usagiop

I agree, fwiw, I just reverted since there's obviously contention. Caliburn (talk) 12:05, 26 May 2022 (UTC)
While Usagiop is technically correct, I prefer the more explicit nature of the page as Julius has presented it.
I can't conceive of any circumstances where the presentation as given here is genuinely going to be misleading, so unless it causes serious pain in the accuracy-perception glands, I'd recommend it stays as it is. --prime mover (talk) 12:16, 26 May 2022 (UTC)
Besides, starting the page with "Then" makes us look illiterate. --prime mover (talk) 12:18, 26 May 2022 (UTC)
I agree not misleading, but I think it's slightly redundant without highlighting any other structure the space can be given over it being a set of test functions as above. I'd field the suggestion of mentioning the topology here. Caliburn (talk) 12:27, 26 May 2022 (UTC)
Well whatever we do, leaving the page just as "Then the set of all test functions is called the test function space and is denoted by $\map \DD {\R^d}$" is obviously wrong.
I'll give it a proper workover. --prime mover (talk) 12:48, 26 May 2022 (UTC)
OK, I also made a mistake. Then was wrong. On the other hand, I said misleading because it can be understood as $\set \phi$. Having fixed $\phi$ to be a test function, there is no other choice. I also agree to Caliburn's suggestion. --Usagiop

The first sentence implies that $\phi$ is specific? To me it is exactly the opposite. If there is no "such that" then I personally treat "a test function" as "any test function". Anyway, this page was meant to have a page for the standard symbol denoting "space of test functions". Would it be better to write $\map \DD {\R^n} = \set {\phi : \phi \textrm{ is a test function}}$? As for topology, I can see a reason transcluding it here as a related property, but I do not recall the topology actually being a part of the definition. Otherwise, we also need to introduce topologies for once-differentiable, twice-differentiable function spaces etc. I will have to check the books for that. Maybe we could coin a different space, i.e. a test function space with naturally induced topology or smth?--Julius (talk) 15:20, 26 May 2022 (UTC)

Topology is defined here Definition:Convergent Sequence in Test Function Space. OK, specific is confusing. Just a any function, but the any function is then fixed. You cannot vary it below. --Usagiop
I'm missing something, what's the relevance of topology here? --prime mover (talk) 16:05, 26 May 2022 (UTC)
I assume Caliburn means that the space of the test function space $\DD$ is related to topological vector space. It is also essential to introduce distribution. It is helpful at least to refer to its topology. --Usagiop
To me the point in creating a separate definition for "Space of X" is to introduce some additional structure to it, beyond just saying "The Space of X is the set of all X", otherwise I'd just include it as a note at the end of the definition of X. ("We denote the set of all such $X$ by $\map S X$" or similar) In this case it'd be that of a TVS as usagiop says. Just a personal suggestion, this is how I have been setting up $L^p$ spaces and the like. Caliburn (talk) 17:13, 26 May 2022 (UTC)
$\map \DD {\R^n} = \set {\phi : \phi \textrm{ is a test function}}$ is actually quite a good compromise, except in that format we have lost our link to test function.
I'm on it. --prime mover (talk) 15:23, 26 May 2022 (UTC)
I've taken the liberty to attempt to improve the notation. Hopefully it is indeed considered an improvement.
I also feel compelled to respond to "The first sentence implies that $\phi$ is specific?" The way it was written, it reads as, "Let $\phi$ be any one test function." It is then problematic to use the same variable in an implicitly quantified manner, since they do not refer to the same thing. This is why typically such definitions are structured by defining the thing, and then saying "where <notation X> denotes <thing X>". — Lord_Farin (talk) 18:06, 26 May 2022 (UTC)