|WikiProject Mathematics||(Rated C-class, Mid-importance)|
Huh? Direct product of vector spaces
I cut out the following from the section Vector space direct product because its insane:
- Note that a direct product for a finite index is identical to the direct sum . 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)
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:
- The topological product really has no business being here. No one (that I know of) calls the product topology the "direct product topology".
- The product in category theory is a common generalisation which doesn't belong in an article without lots of examples.
- 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
- (External) product
- Internal product
- (External) coproduct/sum
- 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)
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 126.96.36.199 (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
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.