Talk:Projective tensor product

Close paraphrasing

Sections "Seminormed spaces" and "Examples" paraphrase too closely their respective sources. Added the relevant notice for now. I might try to shorten them down into a couple sentences and put them elsewhere later but if someone else wants to then go ahead. ByVarying | talk 16:39, 10 August 2023 (UTC)

@Mgkrupa:: Thanks again for agreeing to discuss these concerns. Below is my response about WP:PARAPHRASE and WP:LIMITED regarding section Seminormed spaces, followed by my suggested change for how to remedy this issue and reduce the overall level of technical detail on the page. Later, I'll have a suggested change for the examples section.

It looks as though this section is based on the first full paragraph on p. 435 and the first full paragraph of p. 438, of Trèves. The problem comes from the fact that these two paragraphs, in their entirety, are closely paraphrased, not summarized. So, although you asked for specific sentences that are problematic, the problem doesn't come from the way the paraphrasing is done in any particular sentences independently; instead, it comes from the volume of text that is closely paraphrased, as a whole. Now, I see how, as you say, WP:LIMITED might be argued, and I know that it is especially difficult to summarize mathematical statements rather than paraphrasing them. But I do think that there are opportunities to summarize more here (rather than paraphrase), which means that WP:LIMITED doesn't apply. Namely, Prop. 43.1 is an elaboration of what precedes it in that paragraph in the book. Restating it is not necessary to summarize the essential content of the paragraph (namely, that a seminorm on $X\otimes Y$ can be built out of the seminorms on $X$ and $Y$, and that that seminorm induces the projective topology). Just stating that essential summarized content, together with the definition of the "tensor product" of the seminorms, would be an adequate summary; WP:LIMITED would now apply to our restatement of the definition.

To summarize the content on p. 438, we could say something like, "when the topologies on $X$ and $Y$ are induced by families of seminorms, the projective topology on $X\otimes Y$ is induced by the family of their tensor products." I will do something similar when I give the suggested new text.

Here is my suggestion:

Remove sections "Normed spaces" and "Seminormed spaces." These will be incorporated in a new section preceding "Properties."

Add a section called "Definitions" before "Properties." This will consist of: (1) the universal property currently in "Properties," characterizing the projective tensor product for all locally convex topological vector spaces; (2) a statement of the construction of the tensor product of seminorms and the fact that it generates the projective topology just characterized in the universal property; (3) a sentence saying that the tensor product of seminorms which are also norms is a norm, and that this construction therefore carries over to normed spaces. I'll split the post here so I can sign and then give my suggested text. ByVarying | talk 19:31, 10 August 2023 (UTC)

Here is suggested text for the new "Definitions" section:

Let $X$ and $Y$ be locally convex topological vector spaces. Their projective tensor product $X\otimes _{\pi }Y$ is the unique locally convex topological vector space with underlying vector space $X\otimes Y$ having the following universal property:

For any locally convex topological vector space $Z$, if $\Phi _{Z}$ is the canonical map from the vector space of bilinear maps $X\times Y\to Z$ to the vector space of linear maps $X\otimes Y\to Z$; then the image of the restriction of $\Phi _{Z}$ to the continuous bilinear maps is the space of continuous linear maps $X\otimes _{\pi }Y\to Z$.

When $X$ and $Y$ are seminormed spaces, the topology of $X\otimes _{\pi }Y$ is induced by seminorms constructed from those on $X$ and $Y$ as follows. If $p$ is a seminorm on $X$, and $q$ is a seminorm on $Y$, define their tensor product $p\otimes q$ to be the seminorm on $X\otimes Y$ given by

$$(p\otimes q)(b)=\inf _{r>0,\,b\in rW}r$$

for all $b$ in $X\otimes Y$, where $W$ is the balanced convex hull of the set $\left\{x\otimes y:p(x)\leq 1,q(y)\leq 1\right\}$. The projective topology on $X\otimes Y$ is generated by the collection of such tensor products of the seminorms on $X$ and $Y$.

When $X$ and $Y$ are normed spaces, this definition applied to the norms on $X$ and $Y$ gives a norm on $X\otimes Y$ which generates the projective topology.

What do you think? I think this reduces paraphrasing to a minimum, collects together the definitions, and gets rid of some less important detail. And, as this reflects, having looked at more sources, I am open to reintroducing the $\pi$ subscripts. (I originally removed them because the source I was familiar with didn't use them.) Thanks for reading. ByVarying | talk 20:16, 10 August 2023 (UTC)

Thank you for your work and my apologies for the lateness of my reply (I've been busy). You removed a lot information about projective norms, which I think is important to understanding projective tensor products. I suggest restoring some of this information. Your thoughts? Mgkrupa 17:52, 4 September 2023 (UTC)

Thanks for replying. I think that the extra detail in the section Projective norms in the old revision here is textbook-like elaboration that does not constitute a summary. The sentence starting with "Given..." and ending with a sum of simple tensors is a "trivial" derivation from the definition of the norm. The next one is exactly what you would expect for the completion. The next one (with $\lambda _{i}$) is a "trivial" consequence of what precedes it. The sentence about unit balls of Banach spaces might be added to the current Properties section.

Respectfully, my opinion is that this is mostly unencyclopedic restatement of facts that appear as propositions and problems in the textbooks. If a way could be found to present some of it in a more natural and encyclopedic way, I would be glad to have it in the article. ByVarying | talk 15:37, 14 September 2023 (UTC)

Since you said to go ahead and make my changes, I will do that. ByVarying | talk 20:49, 10 August 2023 (UTC)

References

• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.

@ByVarying: I want to personally thank you for this major rewrite. Functional analysis is not exaclty my area of expertise, but I want to say that the result is a *lot* more readable and accessible than the previous version. As has been discussed previously a few times with the previous main editor of this article, the purpose of wikipedia is not to "teach" by explicitly writing down the most minute details of a subject in a fully deductive fashion (to the point of making it nearly inaccessible to anyone except the most motivated persons with already a background in the subject). Instead, it is to present a coherent and accessible overview of the topic, and refer to referenced sources for anyone interested in exploring further. Again, thank you! PatrickR2 (talk) 19:25, 14 August 2023 (UTC)

Hi PatrickR2, thank you for the compliment about some of my contributions here. I'd rather not involve myself in any past content disagreement you seem to be describing, but hope anyone who would have otherwise disagreed with recent changes to the page will find my reasoning above persuasive. ByVarying | talk 22:53, 14 August 2023 (UTC)

Bibliography

I am splitting the "Bibliography" section into a "References" section (for sources cited in the article) and "Further reading" section, and removing the references that, based on their tables of contents and indexes, don't talk about this topic. ByVarying | talk 04:09, 11 August 2023 (UTC)

Spaces of absolutely summable families

I am going to remove the subsection of "Examples" with this title, and am starting a discussion in case there is an objection to this. There are at least two possible arguments to remove this subsection. First among these is that this appears to be more of a result about nuclear spaces than about the projective tensor product itself—indeed, that is the way that the only source for the example, pp. 179–184 of Schaefer's Topological Vector Spaces, frames it. Second, looking at what's in this subsection now and what's in the source, it would be impossible to adequately summarize without giving excessive weight and excessive detail to an example that is, again, only in one source, and, again, apparently related somewhat coincidentally to the topic of this article. ByVarying | talk 18:33, 16 August 2023 (UTC)