Talk:Direct product

Huh? Direct product of vector spaces

I cut out the following from the section Vector space direct product because it is insane:

Note that a direct product for a finite index $\prod _{i=1}^{n}X_{i}$ is identical to the direct sum $\oplus _{i=1}^{n}X_{i}$. The direct sum and direct product only differs for infinite indices, where the elements of a direct sum are zero for all but for a finite number of entries.

It has been added by User:Piil on 22 sept 2004. The correct definition is in the article Tensor product. Was there some deeper meaning behind this seemingly insane statement? linas 17:12, 5 Feb 2005 (UTC)

Insane? As far as I know, that is correct. What does it say in the tensor product article? err... I'll go check it out. Lethe | Talk 18:51, Feb 5, 2005 (UTC)

The only thing I see on the tensor product article about direct products is this statement: "This construction is unrelated to the direct product of M and N." So ... I don't understand why you want to replace the explanation on direct products with a link to tensor products. I'm reverting. -lethe ^talk 18:54, Feb 5, 2005 (UTC)

Lethe is right, the statement is correct as it stands. You are confusing the direct product with the tensor product — these aren't the same thing. Most courses on linear algebra don't talk about the direct product, only the direct sum. The reason being is because the two are identical on a finite number of factors (exactly as is stated). To see the difference you have to take an infinite number of factors (and so obtain an infinite dimensional vector space). -- Fropuff 04:14, 2005 Feb 6 (UTC)

radical suggestion

I'm going to make a radical suggestion that we just move everything on this page to other articles (existing or newly-created) and turn this one into a disambiguation page. Before there's an outcry, let me give some reasons:

1. The topological product really has no business being here. No one (that I know of) calls the product topology the "direct product topology".
2. The product in category theory is a common generalisation which doesn't belong in an article without lots of examples.
3. All that leaves is the direct product of groups and vector spaces. There's nothing that makes these more significant than other occurrences of categorical products, except that they are the ones called "direct product". That's a pretty flimsy reason to justify an exposition of them apart from their main articles, IMO.

All of this is related to my personal belief that all of the "product" articles collectively are in a confusing and sorry shape. Some things are misnamed, some articles have no apparent reason for their content organisation, other things aren't clarified enough, etc. At the heart of the matter seems to be a failure to organise, name, and clarify topics by keeping in mind their category theory meaning. This doesn't mean you have to know category theory to understand anything, but category theory does point a clear direction of how things should be organised, and it's not the direction we're going.

There are 4 major ideas going on in all these articles, based on 2 criteria with 2 options each: first, product or coproduct/sum; second, external or internal. That makes

1. (External) product
2. Internal product
3. (External) coproduct/sum
4. Internal coproduct/sum

A lot of things are named "sum" that are really products, and a few things that are "internal" aren't clearly identified that way (since "external" is a kind of default). For example, direct sum of groups is actually about the internal direct product of groups. Also, in many cases, you can form the product/sum like you do the sum/product, as objects, but it's not a universal object. For example, if you have an abelian group, you can take the product of them as groups, and as an object, this is still an abelian group, but no longer satisfies universal property. Similarly, you can take the "abelian" sum of arbitrary groups, but it's not universal. This is sometimes called the "weak direct product" or "restricted direct product". This distinction between what is an object and what actually is universal is missing in many places. You don't have to mention it directly, but it seems it should guide the presentation. Revolver 19:54, 31 August 2005 (UTC)

A lot of the above, my ideas have changed on. See the discussion at the wikiportal talk page. Although, I still think the "radical suggestion" is good. This should be a disambig page. Revolver 05:05, 1 September 2005 (UTC)

I would disagree with the split. All these types of direct products (OK, at least product of groups, modules and topological spaces) share the same idea of using the cartezian product. I think they are all best understood if in the same place. Oleg Alexandrov (talk) 23:37, 13 January 2007 (UTC)

I second Revolver's suggestion. I certainly agree that I do not know of reputable sources that use the phrase "direct product" for topological spaces. One would normally just say "product". Probably the section "Topological space direct product" should be merged into the product topology page. In the mathematical literature, the term "direct product" is used primarily for groups and modules (the latter includes vector spaces and representations). Ebony Jackson (talk) 05:49, 2 December 2020 (UTC)

confusion over direct product vs direct sum

This is a subsuggestion of the radical idea section that I feel very strongly needs to be called out explicitly:

I'm pretty sure where it says w.r.t the direct product: "For abelian groups which are written additively, it is also called the direct sum, denoted by <figure omitted>" is incorrect, the direct product is the possibly infinite product, where there can be possibly infinite number of non-identity coordinates. The direct sum is the subgroup of the direct product consisting of elements where only finitely many coordinates can be non-identity. - —Preceding unsigned comment added by 71.197.234.28 (talk • contribs)

I have changed the sentence to explicitly say what I believe was intended: "For abelian groups which are written additively, it may also be called the direct sum of two groups...". - grubber 14:53, 17 April 2007 (UTC)

Direct product of functions?

The direct product of two functions would make a simpler example than the direct product of two groups. However, perhaps the terminology "direct product of functions" is not in standard use. A post on mathoverflow related to this is: http://mathoverflow.net/questions/106496/functions-that-can-be-written-as-direct-products-of-other-functions-question-ab

Tashiro~enwiki (talk) 16:00, 27 June 2015 (UTC)

I agree that "direct product of functions" is not a standard term, so I would not add it to this article. Ebony Jackson (talk) 05:15, 2 December 2020 (UTC)

Product of infinitely many objects.

'We can even talk of products of infinitely many objects e.g. R x R x R...' Just out of interest shouldn't there be another line or so explaing what this product is like describing its cardinality for instance.

Poshida (talk) 14:37, 7 October 2015 (UTC)

Removal of metric and norm section

I propose the 'metric and norm' section be removed because it does not seem particularly relevant to the article. At the very least it should be incorporated into the rest of the article so as not to occupy a whole section. - 25 May 2019‎ Joel Brennan

I wonder whether the sentence is appropriate at all.

As far as I remember, I saw a direct product defined only on equationally-defined algebraic structures with one underlying set A and one or more operations f: A ×...×A → A. In these cases, the operations on the direct product can be defined pointwise.

This wouldn't be applicable to metric spaces, since a metric maps to the nonnegative real numbers; pointwise lifting doesn't make sense. The same for a norm.

Of course, a metric can be defined on the Cartesian product of metric spaces, in various ways, but this isn't what the article speaks about.

If I'm right (I'm far from being sure about it), the sentence could be turned in a counter-example to distinguish a direct product from a Cartesian product. - Jochen Burghardt (talk) 14:32, 26 May 2019 (UTC)

I too agree to the removal of the "Metric and norm" section. Ebony Jackson (talk) 05:49, 2 December 2020 (UTC)