Talk:Tensor product

This page is poorly constructed

Well, maybe that's an overstatement, but it appears as though mathematicians have clogged this page up with soggy, pointless math jargon and made it irritatingly difficult to use as a reference for, you know, actually calculating things with tensor products. If all else fails, can somebody actually push the examples to the top as a quick reference and push the frivalous tedium to the bottom, or even better, package the frivalous tedium into a completely separate page? Or give me permission to do so.

THIS PAGE IS THE BEST I′VE FOUND ANYWHERE ON THE INTERNET FOR NON-MATHEMATICIANS. PLEASE KEEP IT AVAILABLE AS IS, UNDER ANOTHER NAME IF YOU MUST. I′m a physicist who has been studying and using tensors and tensor products on and off over several years. I′ve used practically every major site and pdf on the Inet, as well as many texts, to try to break thru mathematicians' jargon and abstraction, and the more limited understanding of others, to get to a clearer understanding of the finer points, and EXAMPLES. After discovering this page it′s replaced Gower′s as the one I go to whenever I need a refresher. IMHO it starts simply, at the best point, with just 2 vector spaces, and builds logically from there, and has a splendid set of links to further details such as Hilbert spaces TP and topological spaces TP etc. It includes excellent examples, and does include enough of the mathematical concerns in later sections to make you aware of them and where to go for more detail. If it′s incorrect anywhere then certainly those places should be corrected, but for me it′s far superior pedagogically to anything else available on this subject. Relabel it ″Tensor Products for Non-mathematicians″ if necessary, but please, please keep it available. JHE-37 (talk) 17:58, 10 October 2021 (UTC)

There is need to define the tensor product of vector spaces $V\otimes W$ as there are two interpretations:

1. $V\otimes W=\{v\otimes w\mid v\in V,w\in W\}$ which is not a vector space but a set of product states
2. $V\otimes W={\text{span}}(\{v\otimes w\mid v\in V,w\in W\})$ which is a vector space by taking the span (all linear combinations) of product states — Preceding unsigned comment added by 193.171.241.212 (talk) 10:26, 5 February 2022 (UTC)

Only the second interpretation is correct. A definition of the tensor product is given somewhere in the article, but it is rather difficult to find it. From the content of this talk page and the tags at the top of the article, it seems that everybody agree thay the article is awfully written. The problem is to find an expert that is willing to find the time for rewriting the article. D.Lazard (talk) 11:23, 5 February 2022 (UTC)

The first three sections seem to be too much

In my opinion, the first three sections of this article are a nice attempt to explain tensor products, but they need to be revised or removed (specifically, "Intuitive motivation and the concrete tensor product", "Baby step towards the abstract tensor product: the free vector space", "Using the free vector space to "forget" about the basis".

The main problem with these sections is that they're too wordy and overcomplicated. In general, writing lengthy, huge explanations in an introductory setting deceives, misguides, and confuses readers into thinking something is more complicated than it actually is.

I think when a reader visits they page, they will want a quick but rigorous, no-nonsense explanation of a tensor product, and they probably don't want to have to parse through dozens of paragraphs to understand a tensor product. There are already many online PDFs a reader can turn to for more in depth explanations. So, does anyone care if I try to revise these paragraphs to make them less wordy, or if I reorganize them (maybe place them somewhere near the bottom so it doesn't scare off readers)? Since it seems like the real discussion begins at "The definition of the abstract tensor product" (which it should in my opinion), I'm also inclined to remove them all together, but anyone can let me know if they have any opposition. Lltrujello (talk) 05:12, 23 November 2020 (UTC)

I agree the first three sections are lengthy or too lengthy. I would not altogether remove them though: if the article starts out with

$V\otimes W:=F(V\times W)/{\sim }$

then (either before or immediately after that) we need to include at least a brief explanation of the free vector space.

As far as these first aspects are concerned, I suggest the following ordering:

• a brief recollection on the free vector space
• the definition of the tensor product.
• Explain (here I suggest that we do spend a bit of ink; currently this fact is just mentioned in §5.2, but not explained) how the choice of a basis for V and W yields a basis for V \otimes W. This explanation should be comprehensible from reading the preceding part about the free vector space. The goal is not to require readers (who know vector spaces) to jump to other articles for understanding this bit of information.

Jakob.scholbach (talk) 13:16, 23 November 2020 (UTC)

inconsistent font usage

I made the fonts consistent in this edit http://en.wiki.x.io/w/index.php?title=Tensor_product&type=revision&diff=1111854170&oldid=1111816282 now undone by an editor of the article.

My reason was that the inconsistent font uses that have been restored by that undoing editor of the page are confusing. The weak point is where the Latin lower case letter vee in the undoing editor's inconsistently preferred math markup {{math|·}} looks like the Greek lower case letter nu in fonts that are clearer, such as that of <math>·</math>. This is evident in this snip from the article: "An element of the form $v\otimes w$ is called the tensor product of v and w." The problem is that the two different math markup formats, <math>·</math> and {{math|·}}, use different fonts. It is true that, often enough, the font usage that is inconsistently preferred by the undoing editor is slightly quicker to type. My experience is that the LaTeX format <math>·</math> is to be preferred as it often is in the article. Chjoaygame (talk) 11:22, 23 September 2022 (UTC)

Definition

In the definition from bases, it says:

The tensor product $V\otimes W$ of V and W is a vector space which has as a basis the set of all $v\otimes w$ with $v\in B_{V}$ and $w\in B_{W}.$

Out of the blue appears the tensorproduct $v\otimes w$, just the thing that should be defined. Madyno (talk) 12:33, 25 November 2022 (UTC)

One usage of the symbol $\otimes$ is for vector spaces; the other is for vectors. For basis vectors $v\in B_{V}$ and $w\in B_{W}$ the expression $v\otimes w$, which is a basis vector for $V\otimes W$, should be thought of as a primitive symbol that doesn't reduce to anything simpler. Elements of $V\otimes W$ are sums of such symbols with numerical coefficients. You can of course take the tensor product of non-basis vectors—the result will be written in terms of tensor products of basis vectors, as described in the paragraphs following the one containing the passage you quoted. I think it would be helpful to many readers if some elementary examples could be added to the article showing how the notation works and what it is useful for. Will Orrick (talk) 03:48, 26 November 2022 (UTC)

Tensor product of n vector spaces for n>2

The article defines tensor products of a finite nonnegative integer $n$ vector spaces for only the case $n=2$. However, the "General tensors" section uses tensor products for general $n$, so we need to define somewhere tensor products in general (finite nonnegative integer). Thatsme314 (talk) 19:31, 17 October 2023 (UTC)

The case n > 2 is defined in section § Associativity. D.Lazard (talk) 20:48, 17 October 2023 (UTC)

Good catch! Now, we just need to define it for $n=0$. Also, it might be nice to re-organize the article to emphasize the general definition. Thatsme314 (talk) 01:00, 18 October 2023 (UTC)