Talk:One-form (differential geometry)

Merge to "linear functional"

This subject of this article is precisely the same as that of "linear functional"; the only difference is in the notation and terminology, which results from the fact that the respective authors of the two articles have arrived at the concept from the points of view of different areas of mathematics. Hence, the two articles should be merged. At the very least, they should be cross-referenced so that readers are made aware of the connection.

My preference is that "linear functional" be the foundation of the new single article, since that article treats the concept more "in its own right", whereas the present article seems to have arisen in the context of a particular application.--Komponisto 19:12, 22 July 2006 (UTC)

Merging would require a lot of very careful work, if anybody is willing to work on it. Oleg Alexandrov (talk) 02:49, 24 July 2006 (UTC)

1-form is also used to mean differential 1-form, so it is not completely clear that a merger is the best way forward: I guess most of the article can be merged into "linear functional" and cross-referenced, and the rest can be rewritten into a short note to reflect the application of the notion of a linear functional to differential 1-forms, with the alternative notion and terminology that goes along with this. Geometry guy 13:04, 13 February 2007 (UTC)

I agree. The concepts may coincide in Rⁿ, but the term one-form is normally used in the context of manifolds. I do not believe they should be merged. Greg Woodhouse 03:25, 23 March 2007 (UTC)

But the definition of "one-form" given at the beginning of this article is: " a linear function which maps each vector in a vector space to a real number, such that the mapping is invariant with respect to coordinate transformations of the vector space." If this is the case, then a one-form is by definition a linear functional. The fact that the term "one-form" is traditional in the context of manifolds is not relevant unless it actually means something different in that context. At the moment, there is no indication in either article that it does (with the possible exception of differential 1-forms, in which case I would readily go along with the suggestion of Geometry guy above).

Reading the two articles, one has the impression that they were written by two groups of mathematicians who are simply unaware of each other, and in particular of the fact that they are both using the exact same concept. This is exactly the sort of annoying situation that an encyclopedia should endeavor to correct. Komponisto 06:56, 23 March 2007 (UTC)

But, they are not the same thing. One concept is local, the other global. The difference is similiar to that between a vector (and element of a vector space) and a vector field (a section of the tangent bundle). In fact, that's what 1-forms are: sections of the cotangent bundle of a manifold. Even when the bundle is trivial, they're not the same thing. Greg Woodhouse 12:03, 23 March 2007 (UTC)

Again: if the term "one-form" is really only used in this sense (i.e. a map from points on a manifold to linear functionals on appropriate spaces), the article should point this out, with an explicit cross-reference to the linear functional article. The first sentence would be particularly misleading as it stands. Komponisto 21:50, 23 March 2007 (UTC)

There is another reason not to merge the articles: In mathematics, the term linear functional is normally used on the context of function spaces, which are Banach spaces, not finite dimensional vector spaces. The concepts are not equivalent, either, because if you drop the requirement that the space be finite dimensional, linear maps are no longer automaticaly continuous. That's where boundedness comes in. Greg Woodhouse 12:26, 23 March 2007 (UTC)

The term "linear functional" means precisely "linear map from a vector space to its scalars"; no more, no less. Continuity is not invloved (hence the term "continuous" or "bounded" linear functional in cases where it is). It is perfectly legitimate (and common) to use the term in the context of finite-dimensional vector spaces (which are in particular Banach spaces). Reference: Paul Halmos, Finite Dimensional Vector Spaces, 2nd ed. Springer. (Also see the italic text at the top of the linear functional article.) Komponisto 21:50, 23 March 2007 (UTC)

I can play the reference game, too. After an initial chapter that does not make any continuity assumptions, DeVito writes (in Functional Analysis, Academic Press, 1978, p. 17): "...it is clear that we have been exploring a blind alley. If we are to proceed any further, we must find another approach..." True, the assumption of continuity is usually made explicit in some form or another, but there is a point at which constant repetition turns into pedantry. It's also worth noting that Halmos had a pedagogic goal in Finite Dimensional Vector Spaces, namely using a setting familiar from introductory courses in linear algebra to set the stage for introducing more advanced concepts from analysis. Greg Woodhouse 22:26, 23 March 2007 (UTC)

I fail to see how the fact that "linear functional" is often used as a abbreviation of "continuous linear functional" (in contexts where the latter are of primary interest) makes any difference here. Abuse of language is common in all areas of mathematics, as you surely realize. Nor do I see the relevance of Halmos's "pedagogic agenda"; every author has one. (If you don't accept the "authority" of Halmos, try Lang's Algebra [p. 142 of the revised 3rd edition].)

But matters of terminology are a secondary issue anyway. The main point is that there are two different articles that purport to be about linear transformations from a vector space to the set of scalars. Each of them reads as if the other did not exist. I regard this as a problem.

Even if, as you say, the present article should really be about a related but distinct concept (sections of the cotangent bundle of a manifold), it doesn't appear that way at the moment. And even once this is corrected, the relation between the two concepts should still be discussed.

Finally, let me point out that finite-dimensionality is no more intrinsic to the concept of a differential manifold (and its tangent and cotangent bundles) than it is to that of a Banach space. I trust I don't need to cite a reference for this statement. Komponisto 23:11, 23 March 2007 (UTC)

Returning to the previous discussion for a moment, I think this whole presentation is heavily biased towards the coordinate point of view, where it is a common shorthand to blur the distinction between tensors and tensor fields. That's fine so long as the reader appreciates that there is a distinction. I favor a more balanced approach, giving equal wait to the coordinate based and coordinate free approaches. Greg Woodhouse 12:33, 23 March 2007 (UTC)

I agree with you here. In fact, a greater emphasis on the coordinate-free approach would probably help to clarify the relationship between "one-forms" and linear functionals. My desire is simply that this relationship, whatever it is, be explicitly acknowledged. Komponisto 21:50, 23 March 2007 (UTC)

Here, we certainly agree. But how to draw the connection? Vector bundles globalize vector spaces, and (differential) 1-forms globalize the concept of the dual concept. If I am a little over-emphatic on this point, it's because this concept is an important one. I still remember picking up a book on general relativity and reading that contravariant vectors are quantities that transform in such and such a way, and covariant vectors are quantities that transform in such and such a way. My reaction was essentially, Huh? That presentation provided (me, at least) with essentially no insight. Of course, given a pair of charts, the coordinates have to transform in some way. Maybe if formulas seemed more intuitive to me, I would have studied physics. But I see that, in one sense at least, I'm really arguing your point. It's important to try and indicate why a vector space V is not the same as its dual V^*., and all this talk of 1-forms vs. dual vectors just seems to cloud the issue. I see no good reason to talk about 1-forms in the conttext of a single vector space unless you are trying to motivate the concept of 1-forms on manifolds. From an algebraic point of view, both articles are essentially about the same thing, but the language employed points in very different directions. Maybe a single article, say on vector spaces, could make the two different lines of investigation explicit. I should apologize for being so cranky. I guess it's Friday, and I felt a bit like I was being talked down to. BTW, this is neither here nor there, but if you want a reference on infinite dimensional manifolds, you might consider Lang's Differentiable Manifolds. He goes to some not inconsiderable pains to state and prove fresults for Banach spaces and manifolds wherever possible. Greg Woodhouse 01:00, 24 March 2007 (UTC)

(Indeed--the Lang book is the one I had in mind). Rereading the present article, I have noticed that the word "manifold" does not appear even once! So I think one of the first steps should be to correct this.

Tell me if you would agree with the following: "A 1-form on a vector space is a linear functional. A 1-form on a differential manifold is a section of the cotangent bundle--in other words, a map that sends each point of the manifold to an element of the cotangent space (i.e. a continuous linear functional on the tangent space) at the point in question." (Actually, as you hinted at before, it seems that the latter usage is obtained from the former by means of the same abuse of language that permits some people to say "tensor" when they mean "tensor field".) If this were used as the first two sentences of the article, it might be followed by: "A 1-form on a manifold may thus be viewed as a linear functional that varies from point to point, in the same way that a vector field (resp. tensor field) may be thought of as a vector (resp. tensor) that varies from point to point."

What I suggest is creating some material on sections of cotangent bundles for this article, and moving most or all of the current material over to linear functional. We can then put a note at the top saying: "This article deals with 1-forms on manifolds. For 1-forms on vector spaces, see linear functional." Komponisto 05:40, 25 March 2007 (UTC)

It seems to me like there's a lot of discussion here, and it's just the two of us. I don't want to make a mountain out of a mole hill. Hmm...maybe a new article discussing "globalization" in general would be helpful. I just don't see merging this article, but I'm sympathetic with your desire to be clear about the relationship between the global concept (differential 1-forms) and the algebraic concept (the dual space/covecors). Greg Woodhouse 12:20, 26 March 2007 (UTC)

Firstly, I don't think the number of people taking part in this discussion has the slightest bearing on the substantive issues involved here. Anyone is free to participate. It is in the nature of things (and justly, I believe) that the future of an article will be decided by those who care the most.

Secondly, regarding the course of action to take, this seems to me like a no-brainer. The situation as it stands is unacceptable (and that is a point on which I will insist). All the material in the present article, save the last stub-of-a-section ("Differential one-forms") belongs at linear functional, if it belongs anywhere at all. The subject of that last section on differential forms should be the subject of this entire article--otherwise this article is 100% redundant. So the obvious solution seems to be to write an article on differential 1-forms and put it here, while transferring the rest to linear functional where it belongs. Note that this would no longer technically be a merger, but merely a rewrite of the present article (and possibly linear functional too).

I made some concrete proposals for carrying out this project above, which you unfortunately did not specifically address. I will now make another; unless you or someone else has a specific objection (which will necessarily lead to a discussion), I will proceed with this plan and begin editing the article(s). My next proposal is a section on the construction of the canonical 1-form on the cotangent bundle, and its relation to the canonical 2-form (reference: Lang, Differential and Riemannian Manifolds, pp 146-147). Komponisto 02:13, 28 March 2007 (UTC)