Definition talk:Test Function Space

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)