Talk:Linear form

HOM_k(V,k)

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

Hom_k(U,V) means the set of all linear mappings from U to V, while k specifies the field under which the mappings operate. So Hom_k(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)

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)

Edit-summarizing with "disambiguation", you added, as a HatNote on Linear functional

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.

and it now reads

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.

   Thanks for your attention.
--Jerzy•t 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 (talk • contribs) 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)

"Continuous linear functional" listed at Redirects for discussion

A discussion is taking place to address the redirect Continuous linear functional. The discussion will occur at Wikipedia:Redirects for discussion/Log/2020 June 19#Continuous linear functional until a consensus is reached, and readers of this page are welcome to contribute to the discussion. 1234qwer1234qwer4 (talk) 10:04, 19 June 2020 (UTC)

Recent edits so that content appears removed

MgKrupa and I appear to be coming close to an WP:Edit war, so I would like to explain the rational behind my recent edits here. Some of the content MgKrupa placed here is inapposite for this page. For example,

Functional analysis is a field of mathematics dedicated to studying vector spaces over ℝ or ℂ when they are endowed with a topology making addition and scalar multiplication continuous.

Such objects are called topological vector spaces (TVSs).

Prominent examples of TVSs include Euclidean space, normed spaces, Banach spaces, and Hilbert spaces.

These are facts about functional analysis, not linear forms. They belong here only insofar as they are necessary to describe linear forms. In the example cited above, they do not; more to the point, they are little better than a "See Also" list, and so should appear there instead.

Other cases are less clear-cut. For example,

If X is a vector space over a field 𝕂 then 𝕂 is called X 's underlying (scalar) field and any element of 𝕂 is called a scalar.

Do we need this to describe linear forms? Well, we are almost certainly going to use the word "scalar" on this page. But in order to get to this quote, a reader has to already make it through the introduction, which contains

In linear algebra, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars….

In general, if V is a vector space over a field k, then a linear functional f is a function from V to k that is linear….

The set of all linear functionals from V to k, denoted by Hom_k(V,k), forms a vector space over k with the operations of addition and scalar multiplication defined pointwise. This space is called the dual space of V, or sometimes the algebraic dual space, to distinguish it from the continuous dual space.  It is often written V^∗, V′, V^# or V^∨ when the field k is understood.

A reader can't make it through all of that and still not understand what a scalar is.

Finally, some of the content that MgKrupa reinstated was already expressed more concisely elsewhere on the page. For example, his discussion of the relationship between real and complex functionals was already discussed under Linear form § Change of field.

Bernanke's Crossbow (talk) 22:28, 11 November 2020 (UTC)

---

"Do we need this to describe linear forms?"

According to Wikipedia:Manual of Style/Mathematics and Wikipedia:Make technical articles understandable and Wikipedia:Manual of Style, this article should follow the following guidelines (as well as others not listed here). It should be be written "one level down", which means:

• "An article about a mathematical object should provide an exact definition of the object, perhaps in a "Definition" section after section(s) of motivation."
• "When in doubt, articles should define the notation they use."
• "consider the typical level where the topic is studied (for example, secondary, undergraduate, or postgraduate) and write the article for readers who are at the previous level."
• "Articles should be as accessible as possible to readers not already familiar with the subject matter."
• "articles on undergraduate topics can be aimed at a reader with a secondary school background, and articles on postgraduate topics can be aimed at readers with some undergraduate background."
• "If an article requires extensive notation, consider introducing the notation as a bulleted list or separating it into a "Notation" section."
• "Writing one level down also supports our goal to provide a tertiary source on the topic, which readers can use before they begin to read other sources about it."

So the answer to your question is yes, we do need to describe basics such as "scalars". Besides Wikipedia policy, one reason for including the definition of a scalar is that a reader who only partially familiar with linear forms might erroneously think that "scalar" always means real number.

In addition, whenever it is reasonable and appropriate to do so, then important terminology that a reader may not be completely familiar with (either entirely unfamiliar with it or just vaguely familiar with it) should be briefly defined/described within this article, instead of just having a link to the article about the term (this is because ideally, a "typical" reader should not have to go down a rabbit hole of Wikipedia links and search through various articles in order to understand something stated in this article; this may, for instance, be helpful with terms that have multiple meanings, e.g. "normal" or terms whose definition is found deep inside some article's body and not in the lead/introduction). Mgkrupa 23:34, 11 November 2020 (UTC)

"Someme of the content MgKrupa placed here is inapposite for this page"

So remove that particular content. If it's sourced then explain why it should be removed. In general, if multiple authors who are experts in the article's subject matter thought that fact was important enough to include in their books/articles, then (assuming that it was added to the article) some justification should be given for its removal. In an ideal world, it should be discussed first on the talk page with notifications being given to major editors. Mgkrupa 23:34, 11 November 2020 (UTC)

"For example, his discussion of the relationship between real and complex functionals was already discussed under Linear form § Change of field."

I did not notice that you created the new section named "Change of field" with some of content from the section named "Real and complex linear functionals" (that existed before your major changes). Had I noticed then I would have attempted to merge into your new section some of the information that I was restoring. Mgkrupa 23:34, 11 November 2020 (UTC)

"MgKrupa and I appear to be coming close to an WP:Edit war"

You deleted lots of appropriately sourced information without first justifying it in either the edit summary or the talk page. You undid my revision. In contrast, I kept most of your edits/reorganization, moved a couple of things around, and added back the information that you deleted (without justification). I definitely did not want to get into an edit war which is why I chose not to undo your edits by restoring the article to an earlier version. Mgkrupa 23:34, 11 November 2020 (UTC)

I forgot to mention that according to Wikipedia:Manual of Style/Lead section and Wikipedia:Manual of Style/Mathematics#Article introduction:

• "The lead sentence should state that the article is about a topic in mathematics"
• "The lead sentence should informally define or describe the subject."
• "The lead should, as much as possible, be accessible to a general reader, so specialized terminology and symbols should be avoided."
• "The first sentence should tell the nonspecialist reader what, or who, the subject is. It should be in plain English."
• "The lead section should include, when appropriate:" "An informal introduction to the topic, without rigor, suitable for a general audience. The appropriate audience for the overview will vary by article, but it should be as basic as reasonable. The informal introduction should clearly state that it is informal"
• "Editors should avoid lengthy paragraphs and overly specific descriptions – greater detail is saved for the body of the article."
• "In general, introduce useful abbreviations, but avoid difficult-to-understand terminology and symbols. Mathematical equations and formulas should be avoided when they conflict with the goal of making the lead section accessible to as broad an audience as possible. Where uncommon terms are essential, they should be placed in context, linked and briefly defined."
• "The lead should stand on its own as a concise overview of the article's topic."

So the introduction to this article must be reworded to comply with the above requirements. Mgkrupa 00:07, 12 November 2020 (UTC)

Also, according to the above Wikipedia policy, the technical definition of a linear form, as well as related definitions such as "kernel", "dual space", etc. (which can be, and had been, briefly defined), belong in the body of the article. This is one reason why I added the section "Definitions and preliminaries" to your version of the article. With your recent blanket revert to an earlier version of this article, you have since removed that section (along with all of my other most recent edits). Mgkrupa 00:54, 12 November 2020 (UTC)

@Mgkrupa:There's a lot of material here to address. Let me start off by saying I seem to have inadvertently attacked you by mentioning edit wars. I didn't mean to do so; in fact, I am very impressed by the work you have put into Wikipedia this past year (if not more!). I wasn't quite sure where to start my comments and noting that we had reverted each others' edits seemed as good a place as any. You took the time to rearrange the content you re-added to fit within the section headings I had set up and I am grateful for that. I think you'll find, if you reread it, that less material was removed than you think; it was just rephrased instead.

I disagree with you that "multiple authors who are experts in the article's subject matter thought [these] facts were important enough to include in their books/articles." Looking through the sections that had most of their content removed, I find citations to Wilansky, Modern Methods in Topological Vector Spaces and Narici & Beckenstein, Topological Vector Spaces. These aren't general linear algebra books for undergraduates; they're niche books within the functional analysis literature that happen to mention some results about linear forms in their introductory sections. They are certainly trustworthy sources for any results they claim, but I don't think we should draw any conclusions about what are appropriate topics on this page from them. (And, for the record, I don't think Narici & Beckenstein is well-written either.) For example, Friedberg, Insel & Spence's Linear Algebra doesn't mention hyperplanes at all (OTOH, it barely touches on linear forms in the optional section 2.6); Halmos' Finite Dimensional Vector Spaces spends a lot of time on linear forms, but doesn't talk about the real-complex correspondence either. This article isn't primarily for functional analysts; it's for general linear algebra, and just relying on functional-analytic sources can skew one's sense of what's important.

You have convinced me that the introduction is unnecessarily technical, but I strongly disagree that the "definitions" section from previous versions is the correct solution. The current introduction already defines the object ("An article about a mathematical object should provide an exact definition of the object, perhaps in a "Definition" section after section(s) of motivation.") so perhaps it should be moved to a definition section. Note too, that the guidelines recommend a "Definition [singular]" section — it's for defining the subject of the article, not every term mentioned in the article.

The problem with your definitions section is that either the terms were used uncommonly in the article itself or they were fairly standard. Having a gigantic "Definitions" section for terms uncommon in the article is disruptive, because it breaks topic locality. The terms are defined at the beginning and used far below; the introduction is displaced from the elaboration and examples by the (out-of-place) definitions. You'll notice I left the definitions of maximal vector subspace in place, even though that, strictly speaking, isn't a definition about linear forms — those definitions were near where they were used.

"When in doubt, articles should define the notation they use" refers to circumstances where there is substantial variation in notation. For example, $\|x\|_{w}$ can mean the $L^{w}$ norm if the letter p is already taken…but it could also mean a weak norm — one defined via the action of some dual space. So an article should disambiguate. And that's not even the disaster with T3 & T4 vs. regular & normal topological spaces, where there are two competing conventions in the literature!

"Kernel" is not an "uncommon term" that belongs in the body of the article. The two linear algebra textbooks I cite above mention it long before they hit linear forms (and Bretscher, Linear Algebra with Applications mentions kernels in chapter 3 (of 9) without ever getting to forms!).

Anyways, long story short, I'll try to rewrite the introduction more gently while keeping it flowing into the rest of the article.

Thanks for your critiques, Bernanke's Crossbow (talk) 04:47, 12 November 2020 (UTC)

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"I disagree with you that "multiple authors who are experts in the article's subject matter thought [these] facts were important enough to include in their books/articles."" - I take this particular sentence back. I am no longer arguing this (although I'm not say that I think it's wrong, I'm just not going to argue for it because in terms of my overall response, it's not important).

""When in doubt, articles should define the notation they use" refers to circumstances where there is substantial variation in notation." Can you please provide a linear that for this? In particular, your statement that there must be "substantial variation in notation"?

"These aren't general linear algebra books for undergraduates; they're niche books within the functional analysis literature that happen to mention some results about linear forms in their introductory sections." - If you don't like my functional analysis references then you can add your own references.

"those definitions were near where they were used."- Thank you for improving the article with this particular change. Unless such a definition is used repeatedly in the article, I currently see no reason why I would move such a definition.

""Kernel" is not an "uncommon term" that belongs in the body of the article." - Please be careful to remember that most people who read this article are not specialists in Mathematics. While a term like "kernel" is common in mathematics, for many (maybe even most) readers, it may only be just vaguely familiar (for instances, I'd wager than many aspiring physicists also read this article). As I posted above, the policy is: "Articles should be as accessible as possible to readers not already familiar with the subject matter." The definition of "kernel" is short and easily added to the article so I do not see what harm is done by include this term's definition (at least where it is first used).

"I strongly disagree that the "definitions" section from previous versions is the correct solution." I want to make it clear that I'm not attached to any particular wording or organization of the article's material. This includes the definition section that I added back. However, as I argue below, this article should have a definition section.

"perhaps in a "Definition" section" so perhaps it should be moved to a definition section."  and  "Note too, that the guidelines recommend a "Definition [singular]" section — it's for defining the subject of the article, not every term mentioned in the article." - (1) I do not disagree with what you wrote.

(2) But to make it clear, the Definition section is not necessarily only for the definition of the article's subject. Other thing such as certain notation or certain closely related terminology should be included there under certain conditions (which is the case with many other mathematics article).

(3) I argue that there should be a Definition section. Any single one of the following reasons is by itself enough to justify the inclusion of a "Definition" section: (3a) The formal definition of "linear form" is currently in the lead of the article and it should not be there. It should be in the body of the article. To stay consist with the majority of other mathematics article on Wikipedia, I argue that this formal definition should be in a section titled "Definition". (3b) Let's not forget that these articles should be useful for people of all backgrounds. Linear forms are generalized to modules. In particular, the term "linear form" may also refer to a linear transformation from a module over a commutative ring R into the R-module R. I argue that this definition of "linear form" should also appear somewhere in the article "Linear form". This generalization is not currently included in the article and unless it is given its own section at the end of the article (or placed into some appropriate section), then I think that it should be included in the definition section (at the end of it, and it should be clearly stated that this generalization is used in abstract algebra and not regular linear algebra). (3c) The rigorous definition of the sum, difference, and scalar multiples of linear forms should be defined somewhere in this article and unless it is given its own section or placed into some appropriate section, then it should be given in the definition section (because these operations are clearly pertinent to linear forms, their definitions should be included in this article; readers should not be redirected to another article for this definition). Mgkrupa 23:45, 12 November 2020 (UTC)

"I'll try to rewrite the introduction more gently while keeping it flowing into the rest of the article." I wish you luck. There is no reason why you should do this alone (it's not the case that only you, or only I, can edit this article). I want to help and I'm sure that we can work together to improve this article (along with any other editors that may appear). Although (as is Wikipedia's policy) we should not commit single edits that very substantially change the article's content without discussing it on the talk page (unless it's clear that there would be little reason to oppose such a large edit). If you suspect that I may oppose some particular edit then please give details in the edit summary for me to consider; I will do the same. Best of luck. Mgkrupa 23:45, 12 November 2020 (UTC)

Invisible math notation

With white text on black, there are some multiline voids where math. expressions are defined in the editable text. Specifics are the Wikipedia app, running on Android 10 Q in a Google Pixel XL cell. phone.

A conceivable cause could be use of .svg image format, which can be invisible, without viewers for it, in common display apps.

Regards, nikevich Nikevich 03:07, 1 January 2022 (UTC)