Talk:Distribution (mathematics)

Completely agree about moving discussion of TVS topologies to another article

No one looking for a straightforward explanation of distributions will be able to get through the current article.

As far as the continuity of distributions is concerned, all one needs to say in this article is that a linear functional $T:{\mathcal {D}}(U)\to \mathbf {R}$ (or $\mathbf {C}$) is continuous if and only if it is sequentially continuous i.e., if $\varphi _{n}\to \varphi$ in ${\mathcal {D}}(U)$ (supports of all $\varphi _{n}$ are contained in a compact subset $K$ of $U$ and the sequence and all of its derivatives converge uniformly on $K$), then $T(\varphi _{n})\to T(\varphi )$ in $\mathbf {R}$. Explanations of sequential continuity vs. continuity in TVS can be given elsewhere.

Alternatively, you could say that $T$ is continuous if and only if for every compact subset $K$ of $U$ there exist $C>0$, $n\in \mathbf {N}$ such that $|T(\varphi )|\leq C\sum _{|\alpha |\leq n}\|\partial ^{\alpha }\varphi \|_{\infty }$ for every $\varphi$ supported in $K$. But I think the sequentially definition is more accessible.

However, these changes would require a lot of rewriting of the first part of this article.Reader634 (talk) 07:45, 12 December 2021 (UTC)

Comment

I think the recent changes are absolutely in the right direction. I made a few minor edits, mostly to remove "scare" quotes about the difficulties of defining the topology on the space of distributions (it's enough to refer to the relevant article). Also, I think more of the discussion on topologies can be moved elsewhere, and all (or nearly all?) of the material on $C^{k}$ spaces can be removed from this article, since $C^{\infty }$ test functions are the fundamental ones for distributions. As a minor point, I'd denote the regular distribution associated with multiplication by a locally integrable function $f$ by $T_{f}$ rather than $D_{f}$ (which could be confused with differentiation) but I don't think there's a standard notation here. Reader634 (talk) 09:08, 31 December 2021 (UTC)

Some notations not explained

Same as in Spaces of test functions and distributions: notation $L_{c}^{p}$ first appearing in 6 (Spaces of distributions) seems not to be explained anywhere on this page. Mamuka Jibladze (talk) 17:23, 31 January 2022 (UTC)\]

KINDLY DO NOT ERASE OTHER PEOPLE'S POSTS WHEN YOU POST YOUR OWN. 2601:200:C000:1A0:7097:9C72:4BFB:717D (talk) 00:11, 18 May 2022 (UTC)

Tempered distributions

"Tempered distributions" is used but not defined in this article. I've heard they are "distributions that are upper bounded by polynomials", which I'm not quite sure how to make precise. I've also heard the more and difficult definition in terms of (the extension of?) linear functions on Schwartz function. Needs to be made clear in this article. Jess_Riedel (talk) Jess_Riedel (talk) 19:07, 16 March 2024 (UTC)

Tautological sentence

“This leads to the space of (all) distributions on U, usually denoted by D’(U) (note the prime), which by definition is the space of all distributions on U”

What does this even mean? It sounds like a joke. The space of all distribution on U is by definition the space of all distributions on U? This is simply stating “X is by definition X” Siupa (talk) 09:18, 14 May 2025 (UTC)