# Talk:Linear form

WikiProject Mathematics (Rated Start-class, Mid-importance)
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Field: Algebra

## HOMk(V,k)

> I'm not familiar with the notation HOMk(V,k). What does it mean? 15:46, 20 February 2009 (UTC)

Homk(U,V) means the set of all linear mappings from U to V, while k specifies the field under which the mappings operate. So Homk(V,k) is the set of all mappings from the vector space V to the scalar field k, i.e. a linear functional from U to k. Unknown (talk) 03:40, 12 May 2010 (UTC)

## Kernel

The page mentiones "kernel" but doesn't define it. —Ben FrantzDale 03:45, 1 December 2006 (UTC)

## One-Form

Not a good idea to merge in the article on one-form because the one-form has a special use in relativity —The preceding unsigned comment was added by 81.86.127.58 (talk) 23:19, 14 March 2007 (UTC).

I disagree. This issue is currently being addressed at Talk: One-form; see the discussion there. Komponisto 00:32, 26 March 2007 (UTC)

## Dual space

The notion of dual space is already discussed in great detail in the article Dual space. The purpose of this article is not to list the various properties of dual spaces. In particular, the assertion that the dual space has the same dimension as V is not true in general (only if V has finite dimension). In fact, the algebraic dual space of an infinite-dimensional vector space never has the same dimension as V. So I would prefer to leave the discussion of the dual space to the dedicated article on the subject. siℓℓy rabbit (talk) 19:03, 7 December 2008 (UTC)

I relied on http://mathworld.wolfram.com/DualVectorSpace.html. Whoever wrote the original dual space material probably meant n-vector, with reference to the case where V is Rn. I'll take your word for it re infinite-dimensional spaces being an exception, and that "the same dimension" would not be understood as both infinite.
I have no objection to saying less about dual-spaces on this page. My edit was to return the intro to the scope it had when I raised the issue of the dual space being a "k-vector space", but with improved clarity. You could remove "is a vector space over the field k" to be consistent.Paul V. Keller (talk) 21:46, 7 December 2008 (UTC)
Why would I remove the latter statement? The sentence is intended to convey that the set of all linear functionals on a vector space is <dramatic pause> a vector space and that this vector space is called the dual space. I am assuming that somehow this is not as clear as it should be? siℓℓy rabbit (talk) 22:18, 7 December 2008 (UTC)
"vetor space over the field k" is giving a property of the dual space, which your protested earlier was to no purpose given that there is a whole page on dual spaces. If you think it is inappropriate to mention the similarity of the dual space to the space V in the finite dimensional case to help give a sense of what a space of linear functions on Rn or any other space might look like, if you cannot see how mentioning something like that might be useful to someone seeing the dual space and linear functional definitions for the first time, you might as well just define the dual space and leave it at that, or not even define the dual space at all in the intro.Paul V. Keller (talk) 23:05, 7 December 2008 (UTC)
Strictly speaking, the article does just define dual spaces. The vector space structure they support is part of their definition. You seem to be insisting that we should think of the dual space not as a vector space but as a set? siℓℓy rabbit (talk) 23:28, 7 December 2008 (UTC)

## disambiguation "hatnote"

### "Linear functional" Dab HatNote

Copied from User_talk:Boud#.22Linear_functional.22_Dab_HatNote so that the discussion takes place in a more convenient place. Boud (talk) 19:39, 25 March 2012 (UTC)

This article deals with linear transformations from a vector space it its field of scalars. These transformations may be functionals in the traditional sense of functions of functions, but this is not necessarily true.

This article deals with linear maps from a vector space to its field of scalars.  These maps may be functionals in the traditional sense of functions of functions, but this is not necessarily the case.

(I haven't examined who contributed to the changes of wording.)
I have several interrelated concerns about it, and would appreciate any assistance you could offer in resolving them.

1. While "disambiguation" has senses to which Dab (and for that matter MoSDab) do not apply, i'm assuming you'd otherwise have said something in your summary to overcome the presumption that you meant "disambiguation of this WP article's title". Most obviously, that would IMO imply applying the discipline that the HatNote templates adhere to, where only the existing article(s) that would be contenders for the title being Dab'd are linked from within the HatNote (or Dab page). But the original and current versions both have four links, apparently without the expectation that the reader may have gotten to Linear functional even tho the topic they were seeking was linear map, vector space, scalar (mathematics), or functional (mathematics); i'm incredulous toward the implication that each of the four might be sought at "linear functional", and i await a reason why even "Linear map" might reasonably be sought there.
2. IMO, the wording "These transformations" (or "... maps") creates the presumption that the succeeding predicate of the sentence applies to the whole class specified, and thus makes "may" suggest that mathematicians take seriously a conjecture that every member of the class is a "functional in [etc.]", but the conjecture remains unproven -- rather than that one class is a proper subclass of the other, which (but for the choice of "these") i would have thot more likely.
3. Slightly compounding the ambiguity, "such" can be taken to mean either
1. "linear" or
2. "linear, and VS --> [associated-]scalar-field".
4. Bottom line, my best understanding is
1. that the HatNote is intended to quash the natural but false expectation that every linear functional is both a linear map and a functional,
2. that such confusion may be worth dispelling (but is not a matter of our having an article on linear maps that are functionals, and thus raises no need to assist users in navigating to any other actual WP article(s) on (a) topic(s) to which the title "linear functional" could apply), and
3. that the HatNote would better be removed, with that confusion-relief being provided by including -- probably immediately after the initial sentence of the existing lead 'graph -- something like:
(While some linear functionals are functionals -- i.e., functions of functions -- the terminology is not intended to imply that about every linear functional.)
5. Of course, if i am misunderstanding your intent, perhaps a different HatNote Dab (which would link only to additional actual WP articles whose topics could be called "linear functional") is needed to replace the confusing existing HatNote.

--Jerzyt 04:40, 23 March 2012 (UTC)

### proposed improved hatnote

To some degree i agree with your point - the disambiguation "hatnote" was written in Jan 2006, a long time ago in Wikipedia terms.

You refer to "such confusion" and "confusion-relief", so we seem to agree on that. There is confusion because there is ambiguity to someone who does not yet know what a linear functional means. Wikipedia:Dab describes one of the three important aspects of disambiguation as Ensuring that a reader who searches for a topic using a particular term can get to the information on that topic quickly and easily, whichever of the possible topics it might be.

So your objection to linking to vector space and scalar (mathematics) in the hatnote seems to be that the reader may be distracted into learning/checking what these things are instead of getting to the main article that s/he is really interested in, and that s/he should only branch off into prerequisite articles once s/he has got to that main article. IMHO this is a fair point. My motivation at the time was that Wikipedia is a wiki. Adding links generally makes it easier for people to understand words that they did not previously understand and that are prerequisites for understanding the main thing that they are interested in. But i do see the point of not replacing the task of the lead.

I also tend to agree that someone looking for linear map is not that likely to find this article first, and even if s/he does, the risk of ambiguity is low.

On the other hand, the point raised at Talk:Functional_(mathematics)#Serious_Problems in favour of the function of a function definition of functionals, which remains in the second sentence of the present version of the lead (erroneously presented as a special case). With this definition, it's not obvious to me that a functional that is linear is necessarily a linear functional (e.g. if the codomain is a tensor space, then the functional-that-is-linear is not a linear functional because it doesn't map to a field of scalrs). The terminology linear functional seems to be well-established in the sense defined in linear functional, but it doesn't seem to cover functionals that are linear.

I find it hard to believe that any typical search engine will go straight to the article functional (mathematics) when someone enters the expression linear functional and expects to find information about functionals (in general) that are linear. The search engine will give the linear_functional article as the most likely entry. I tried in a well-known search engine just now and did not get a link to functional. I tried in the mediawiki interface on en.Wikipedia and got sent straight to linear functional without getting a list of likely guesses to choose from.

Thus my proposal:

{{about|linear maps from a vector space to its field of scalars|functionals that are linear|functional}}

Do you (Jerzy) or anyone else object?

In parallel, the functional (mathematics) article needs some work as outlined at Talk:Functional_(mathematics)#Serious_Problems, but that should be discussed over there, not here. Boud (talk) 23:18, 25 March 2012 (UTC)

### improve

we should indexate

$\left(\begin{array}{c} x^1\\ \vdots\\ x^n \end{array}\right)$

to make more conceptually coherent with today's pragmatic usefulness. — Preceding unsigned comment added by Juan Marquez (talkcontribs) 04:04, 22 February 2013‎

The upper indexes is not a common today's usage in mathematics because possible confusion with exponents and orders of derivation. As far as I know upper indexes are of common use only in advanced tensor calculus, which is not the subject of this article. D.Lazard (talk) 08:26, 22 February 2013 (UTC)