# Definition talk:Vector Subspace

Looks like there is a need for the page Definition:Linear Manifold to define the concept of a closed linear subspace? --prime mover 09:50, 17 December 2011 (CST)

- You may have misunderstood. A linear manifold is what is defined here as a linear subspace. Then, a linear subspace is meant to be inherently closed. I have adapted the statement for clarity. --Lord_Farin 09:55, 17 December 2011 (CST)
- Having done that, I want to point out that it is less ambiguous to write closed linear subspace every time Conway (my source for this terminology) writes linear subspace. --Lord_Farin 09:58, 17 December 2011 (CST)
- I wouldn't have a problem with renaming this page "linear manifold" (along with some associated rewording), then adding a new page "linear subspace" to define "closed linear manifold". The name "vector subspace" can still sit there with a redirect. My source work (Warner) is, I have been informed, unusual in its terminology and symbology, so I'm more than happy to defer to a more mainstream set of definitions etc. --prime mover 17:19, 17 December 2011 (CST)
- Mainstream terminology is better, indeed. However I haven't encountered 'linear manifold' outside Conway, and hence am quite reluctant to let it prevail. Other opinions/references on this? --Lord_Farin 02:25, 19 December 2011 (CST)

- I wouldn't have a problem with renaming this page "linear manifold" (along with some associated rewording), then adding a new page "linear subspace" to define "closed linear manifold". The name "vector subspace" can still sit there with a redirect. My source work (Warner) is, I have been informed, unusual in its terminology and symbology, so I'm more than happy to defer to a more mainstream set of definitions etc. --prime mover 17:19, 17 December 2011 (CST)

When $T$ is a closed subset of $S$, isn't it automatically a $K$-vector space? --Lord_Farin 03:49, 3 February 2012 (EST)

Pm, what is your idea for Definition:Linear Manifold? Something like Definition:Linear Subspace being a transclusion to this page and referenced one?

I think that could work when this page is given an 'about' tag, something like:

- 'This page is about a subspace of a general vector space. Some types of vector spaces have a narrower definition of a subspace. See Definition:Linear Subspace.'

or maybe simply applying the about template:

*This page is about Linear Subspace in the context of Vector Space. For other uses, see Linear Subspace.*

What do you say? --Lord_Farin 17:24, 3 February 2012 (EST)

- Surely it should just be as simple as "a linear manifold is a vector subspace of a Hilbert space that yadayada ... whatever the details. Also see ..." etc. but neatened up according to our HR. After all, from what I understand it's a vector subspace with extra conditions on it, same as a vector space is a module with extra conditions on it. Or is it more subtle than that? --prime mover 18:10, 3 February 2012 (EST)

- It is more subtle. As I said above, a linear manifold
*is*a linear subspace on PW; a linear subspace on a Hilbert space is a linear manifold that is closed. At least, that's what Conway says. So there is a problem with double nomenclature; hence my thoughts about a disambiguation (as nobody ever speaks about a vector subspace of a Hilbert space). --Lord_Farin 03:17, 4 February 2012 (EST)

- It is more subtle. As I said above, a linear manifold

- In that case, a) A page "Linear Subspace" which has 2 sections: 1: a link to Vector Subspace, explaining that it's the same thing, and 2: A statement that it is a "linear manifold" which is closed. Then b) A page "Linear Manifold" which contains a redirect to Vector Subspace and a category indicator to Hilbert Space. My point is: if there's a term that is used, we need an entity on ProofWiki so that a user who enters it will be directed to
*some*page that either defines it or tells the user that it means the same thing as something we*have*defined. - But as you point out, I'm not familiar with the details as I haven't studied any of this (I'm learning as I go, my formal education stops at an MMath). All I was originally doing was pointing out that "Linear Manifold" needed some sort of page (the nature of which I was guessing at) to achieve the above effect.
- Oh, and while we are about it, we would need to make a link to it from "Manifold", either as a disambiguation or (if it's the same thing) some words explaining what their conceptual connection is. --prime mover 03:56, 4 February 2012 (EST)
- And, apologies, we already discussed this in December, I completely forgot (early onset alzheimers). I hope I've been consistent at the very least. --prime mover 04:13, 4 February 2012 (EST)

- In that case, a) A page "Linear Subspace" which has 2 sections: 1: a link to Vector Subspace, explaining that it's the same thing, and 2: A statement that it is a "linear manifold" which is closed. Then b) A page "Linear Manifold" which contains a redirect to Vector Subspace and a category indicator to Hilbert Space. My point is: if there's a term that is used, we need an entity on ProofWiki so that a user who enters it will be directed to

I think it's done. Many pages need to be adapted to link directly to the appropriate material, though. --Lord_Farin 05:58, 4 February 2012 (EST)

Linking done. --Lord_Farin 06:14, 4 February 2012 (EST)

- Works for me. Thx for insights in the work on Zorn's Lemma by the way. --prime mover 07:05, 4 February 2012 (EST)

## Non-Empty?

Which of the criteria for the definition here makes sure that the subspace is not empty? I learned that by definition a linear subspace has to have an element, is that implied here? --GFauxPas 14:53, 11 March 2012 (EDT)

- Yes. It is stated that a subspace is a vspace itself. Then, a vspace is a group under addition, hence nonempty. --Lord_Farin 19:00, 11 March 2012 (EDT)

Re: whether the definition I added for $\R^n$ is a result or whether it's a definition: Khan and Fraleigh both have it as a definition. I don't have enough understanding of the general definition at the top of the page to determine how the $\R^n$ definition plays into the grand scheme of things. If One-Step Vector Subspace Test is both necessary and sufficient, then it looks like the definitions would be equivalent, but that page only has it as a sufficient condition. --GFauxPas 19:01, 12 March 2012 (EDT)

- We've had a similar conversation before. If there are several different ways of defining an entity, and all are equivalent, then we can do one of two things:

- a) Write a large page (complete with transclusions, if need be) and cite all the definitions, then prove piecemeal that they are all equivalent
- b) Pick one definition (the most intuitively straightforward, IMO), prove that all the other statements are equivalent to it, and then mention that "some sources use this as the definition".

- b) is the way a lot of the pages have been structured. It makes it easier to manage.

- Incidentally, as your new definition only covers the specific instance of $\R^n$ and not the general vector space, I'd be prepared to discount it as a definition. Fraleigh and Khan only treat the conventional real-number space (it's all that's needed in elementary applied maths and physics, and that's where those texts are directed) so that's what they define.

- So, IMO, this statement is safest as a result not a defn. --prime mover 19:09, 12 March 2012 (EDT)

- Agreed, but then it's years ago I learned about 'the grand scheme'... Nonetheless, firmly agreed. --Lord_Farin 19:12, 12 March 2012 (EDT)

- I'm having a circularity problem. $\mathbb{W}$ is a subspace, so it's non empty, and that's how I know it contains the zero vector, because it's closed under multiplication by zero scalar. $\mathbb{W}$ is a subspace because it passes the One-Step Vector Subspace Test. One of the conditions for the Two-Step Vector Subspace Test is that $\mathbb{W}$ is non-empty. I know it's non empty, because it contains the zero vector. he;p --GFauxPas 19:31, 12 March 2012 (EDT)

- It's circular because of the inference '$W$ is a subspace as it passes One-Step Vector Subspace Test'. You chose an involved way of saying: 'it passes the subspace test as it passes the subspace test'. This is bound to be interpreted as circularity, because it
*is*. --Lord_Farin 19:38, 12 March 2012 (EDT)

- It's circular because of the inference '$W$ is a subspace as it passes One-Step Vector Subspace Test'. You chose an involved way of saying: 'it passes the subspace test as it passes the subspace test'. This is bound to be interpreted as circularity, because it

- Fair enough. I gave the $\R^n$ stuff its own page. --GFauxPas 19:44, 12 March 2012 (EDT)

- Good stuff. I renamed it to match the definition we already have for this item (refactoring in progress). --prime mover 03:10, 13 March 2012 (EDT)