Definition talk:Distribution

So we are not just talking via a template – the notation $\innerprod T \phi$ creates a useful analogy with the inner product, since when $T$ is "actually a locally integrable function", (when $T$ is the distribution associated with a locally integrable function) it is simply the $L^2$ inner product, (just from how the distribution associated with a locally integrable function is defined) which also uses the notation $\innerprod \cdot \cdot$. As usagiop also mentions, this notation is often used in general dual spaces, sometimes to appeal to the same analogy with the $L^2$ inner product. Caliburn (talk) 11:57, 26 May 2022 (UTC)
 * Yes, the dual pair a generalization of the inter product. An inner product is a symmetric bi-linear map like $L^2\times L^2\to\C$ but a dual pair may be asymmetirc like $L^p\times L^q\to\C$, where $\frac{1}{p}+\frac{1}{q}=1$.--Usagiop


 * What I am trying to say is that the way the sentence is worded is ambiguious.


 * "$a$ can sometimes be written as $b$" literally means "There are times when it is appropriate to write $a$ using the notation $b$ instead."


 * I'm asking the question as to whether it was supposed to say: "$a$ can sometimes be seen as $b$", that is, the notation depends on what sources you use, in other words "some sources use the notation $b$ where uses $a$."


 * If it is sometimes preferable to use the $\innerprod T \phi$ notation, because the way it is being used particularly lends itself towards that notation, then we need to expand on that sentence and explain what those circumstances are. It would probably make sense to write a subpage and transclude it. --prime mover (talk) 12:08, 26 May 2022 (UTC)


 * I would think here the author meant "may also be written as", as in alternative notation. But indeed this analogy is worth discussing. We don't even have a page for the $L^2$ inner product (it's part of what I'm writing to set up $L^p$ spaces properly) and I'm yet to look at Julius's distribution work, this is all on my to do list for after my last exam. Caliburn (talk) 12:13, 26 May 2022 (UTC)


 * Best of all good wishes to you on that final point --prime mover (talk) 15:17, 26 May 2022 (UTC)


 * I agree with Caliburn's interpretation. Sometimes is redundant. Even further, one often uses both $\map T \phi$ and $\innerprod T \phi$, equivalently, during a discussion.--Usagiop

I simply mean these are two ways to denote exactly the same thing. No strings attached. On the other hand, in more involved studies one can notice that dual-space notation is very natural and, one could say, carries more information. However, in my material duality was not really needed anywhere, so I mostly used a simpler mapping-like notation. Reasoning behind dual spaces will require a bit more reading. --Julius (talk) 15:34, 26 May 2022 (UTC)


 * Thanks, that explains it. --prime mover (talk) 16:06, 26 May 2022 (UTC)