Talk:Examples of vector spaces

Unbounded sequences

in section "infinite coordinate space" there is a slight confusion about "unbounded sequences" which seem here to refer to infinite sequences while it usually means that the image of the sequence is unbounded (i.e. the set of all elements { x_i ; i\in\N } is unbound in F (whatever its topology may be), e.g. for C that (lim)sup|x_i|=∞). — MFH: Talk 19:27, 27 May 2005 (UTC)

Yes, unbounded is probably not the right word. I not sure what the proper language is for distinguishing between

1. a finite sequence
2. an infinite sequence with only finitely many nonzero terms
3. an infinite sequence with infinitely many nonzero terms.

I suppose one could gloss over the distinction between the first two items (which strictly speaking have different function domains) and call both finite sequences although that doesn't quite seem right to me. -- Fropuff 20:14, 2005 May 27 (UTC)

Fields

Why does this article talk so much about fields? the set of integers over the integers is a vector space, but not a field.. maybe this should be made clear (even though they aren't that common in usage) --yoshi 23:06, 11 January 2006 (UTC)

Fields are included in the definition of a vector space. Generalizations to commutative rings are called modules. The integers are indeed a Z-module, but that is not what this page is discussing. -- Fropuff 23:24, 11 January 2006 (UTC)

In that case I'm gonna add a see also section --yoshi 02:25, 12 January 2006 (UTC)

The vector space article already discusses modules in the section on generalizations. The link is somewhat out of place here. If there were an article on examples of modules, it might be appropriate to link it from here (I'm not claiming such an article should exist). -- Fropuff 02:41, 12 January 2006 (UTC)

A vector space is a module over a division ring -- not over a field. This article errs because the set of quaternions is a division ring but not a field. To speak of a vector space over the quaternions you can't define vector spaces only over fields. Jalongi (talk) 07:11, 19 July 2009 (UTC)

Maybe I'm misunderstanding what you're saying or I missed something in the article but it seems correct to me. The only time the article mentions quaternions is as a vector space over the reals which is a field. The sources I've checked all define a vector space to be over a field, do you have a reference defining it over a division ring? Off the top of my head I don't see that any of the theory would break if you defined it that way but maybe there just aren't a lot of applications.--RDBury (talk) 22:37, 19 July 2009 (UTC)

Hungerford's Algebra (which seems to be a popular reference and graduate text) defines a vector space to be a unitary module over a division ring. I misread the article's example involving the quaternions. The quaternions are vector space over the real numbers, but according to Hungerford's definition we could use quaternions as scalars and still have a vector space. One part of the theory that breaks down under the more general definition is that a matrix over a division ring does not have a well-defined rank. It has a left-rank and a right-rank, I think. We would just have to be careful about commutativity in proofs. Thanks for catching my error.Jalongi (talk) 15:59, 20 July 2009 (UTC)

You forgot to mention that Hungerford's Algebra has a yellow cover, pretty much all you need to say really. Somewhat ironically, Springer's website is one of the sources I mentioned above. It may be worthwhile to add Hungerford's definition to the Vector Space article in the Generalizations section. It's probably not a good idea to add it near the beginning since most of the article should be at a college freshman level of accessibility.--RDBury (talk) 19:33, 20 July 2009 (UTC)

Function spaces and generalized coordinate spaces

In this article, the notation F^X is used for the set of maps from X to the field F with finitely many nonzero terms (aka, the generalized coordinate space with basis isomorphic to X). I believe this is a bad idea: the notation Y^X is standard in set theory for the set of all maps from X to Y, and gives rise to the notion of an exponential object in category theory. In fact, in this article, the notation F^N is already mentioned as a notation for the set of all maps from the natural numbers to F.

I think the notation V^X should be given as a notation for the function space of all maps from X to V, then specialized to the case V=F. Then the generalized coordinate space should be defined as a subspace of F^X, with a different notation. I guess the correct notation would be to use the direct sum $\bigoplus _{X}\mathbf {F}$ or coproduct. Comments? Geometry guy 15:55, 12 February 2007 (UTC)

Really? I don't see where that notation is used. The only thing I see is F^∞ used to mean $\bigoplus _{i=1}^{\infty }\mathbf {F}$ rather than F^N. I believe this notation is somewhat standard (note that ∞ is not a set here, just a symbol). I agree that V^X should be used to mean the set of all maps from X to V. -- Fropuff 18:47, 12 February 2007 (UTC)

I must be hallucinating! You are right F^X does not seem to appear anywhere. I guess what disconcerted me was that I was expecting to find a discussion of Fⁿ and F^N as special cases of F^X, instead of which, I saw Fⁿ and F^∞ as special cases of the direct sum over X (what do you think of a notation like $\mathbf {F} ^{\oplus X}$ for this?). Anyway, there is nothing wrong with that approach - certainly I'm happy with the notation F^∞ - but I still think it would be nice to mention F^X explicitly and give some subexamples. Do you agree? Geometry guy 19:07, 12 February 2007 (UTC)

You're probably not the only one to hallucinate here. We should definitely clear things up and mention the notation F^X explicitly. The only notations I've seen used for generalized coordinate space are $\bigoplus _{X}\mathbf {F}$ or (F^X)₀. I'd be happy with either, although the first is more transparent. I'm okay with $\mathbf {F} ^{\oplus X}$ too, but I don't ever recall seeing it before. We should stick with standard notation (if it exists). -- Fropuff 19:32, 12 February 2007 (UTC)

Yeah, we should, and I know the subscript zero is standard in this field, but $\mathbf {F} ^{\oplus X}$ is surely as standard a translation of $\bigoplus _{X}\mathbf {F}$ as $E^{\otimes n}$ is a translation of $\bigotimes _{i=1}^{n}E$. Anyway, I'm sure one (or both) of us will edit this article in the near future. Geometry guy 22:29, 12 February 2007 (UTC)

(Re edit.) Nicely done! This adds something to the whole article. Geometry guy 09:28, 13 February 2007 (UTC)

I'm new to adding comments to Wikipedia, my apologies if I screw anything up. Here's my question...There seems to be an ambiguous statement in the Field vector space example. The article states that the basis of a field is the identity element, but it does not specify whether it is the multiplicitive identity element, or the additive identity element. Later in the article the basis is expressed as the set {0), but I don't see how a linear combination of zeros could result in any non-zero element of the field. Shouldn't the basis be {1}?

It's more or less implicit when you're talking about fields that identity element means multiplicative identity, but the article should be more specific. I couldn't find where {0} was given as a basis, but if it is then it's incorrect since no set containing 0 can be a basis. The basis of {0} is the empty set.--RDBury (talk) 19:43, 20 July 2009 (UTC)