Talk:Functional (mathematics)

Untitled

This article has been spawned from the article functional

Ae-a 20:05, 8 Jan 2005 (UTC)

Serious Problems

A function doesn't necessarily map a variable to a scalar. It can mape a variable to a vector (vector-valued function), tensor (tensor-valued function), function (function-valued function), funcitonal (functional-valued function). It's just a map whose argument is a variable.

Likewise, a funcitonal doesn't necessarily map a "vector space" to a "member of the field associated with that vector space" .... it could be a functional-valued functional, or it could map the functional G to "oranges" and all other functionals to "apples"

Also, saying that the argument of a functional is a vector space sounds too general, in Volterra's "Theory of Functionals" (1929) I think it's quite clear that the argument is a function (a function being a mathematical operation whose argument is a variable). Dr. Universe (talk) 14:54, 22 January 2010 (UTC)

This older version from 2006 put the function of a function definition as the main definition and linear functionals as a secondary definition. The present version seems to define functional as a one-criterion generalisation of linear functional, implying that any functional that is linear is necessarily a linear functional. It then defines a function of a function as a special case of the generalisation of linear functionals. However, it seems to me that linear functional is more restrictive than only requiring that a functional be linear. It also requires that the functional map to the reals. As Dr Universe states, a function in general maps to a set, not necessarily to the reals, so that set might not have any binary operations defined on it, let alone be a field.

The present lead is clearly incorrect regarding the relationship between a map from a vector space into its underlying scalar field and a function of a function. Any of the common definitions of function define it from a set X to a set Y. So a function of a function is from a set X to a set Y where the elements of X are functions x, each of which sends elements of some set Z_x to some set (maybe the same set) W_x. The oranges and apples example seems correct to me.

Speculation: i suspect that this may be a problem of applied mathematicians or theoretical physicists mostly being familiar with more "practical" - real-valued - examples of functions and functionals and assuming that pure (or applied) mathematicians have lost interest in the more general definition and the latter has become a museum piece only.

In any case, i agree that this article's lead appears to have a major problem. If standard usage is not clear enough, then i guess we will have to go by WP:RS for the historical and most common present-day usages. Boud (talk) 22:32, 25 March 2012 (UTC)

Restore the 2006 version of this article?

It looks like this article has regressed in readability. I noticed the comment below (from 2006): "I noted that this article is exceptionally clear, considering myself to the very core of the target audience. So please if you try to edit the article into a more professional, exact, encyclopedic form, do not destroy the understandability." I didn't understand what this article was talking about until I looked back at the August 2006 version. That version gives a very clear, understandable summary, explains the conflicting usage, and clearly points to Linear Functional -- whereas the current version essentially contains a stream of thoughts that don't really seem to help to define or explain the concept. AndrewBolt (talk) 17:49, 17 October 2016 (UTC)

I strongly agree with this. The sections that have been added are either unrelated (vector scalar product), or lack enough explanation to be helpful (locality) while still being too in-depth. Even the content that remains the same has been edited to lack any clarity. — Preceding unsigned comment added by 2607:B400:800:30:10CC:94BE:EEAA:2A85 (talk) 17:22, 1 February 2018 (UTC)

In total agreement with my predecessors above! MxBuck (talk) 20:40, 24 February 2022 (UTC)

Examples of integral as functional

The example fails to mention that the integral is a definite integral. <unsigned comment>

I have changed the section heading to "Definite integral", since the indefinite integral is clearly excluded. — Quondum^☏✎ 06:45, 16 March 2012 (UTC)

Comment

I always thought the "Functional" in "Functional Equation" was an adjective meaning having to do with functions, rather than the noun "functional" of this article. I am especially confused, since the Cauchy equation example looks at the function evaluated at different points, rather than an overall identity at all points (like y'+y^2=0, for instance).

--Baterista 08:09, 6 August 2005 (UTC)

...the integrand in Calculus of Variations is really a functional form, not a functional. The functional talked about here is the integral to be minimized/maximized.

The tone of this article

I noted that this article is exceptionally clear, considering myself to the very core of the target audience. So please if you try to edit the article into a more professional, exact, encyclopedic form, do not destroy the understandability.

Generally in Wikipedia I find many of the mathematics / physics articles very professional looking, but completely useless because they quite mercilessly use words that are even more incomprehensible than the article's main word.

Complex things may be made easier by adding examples. Wikipedia does not need the conciseness of a traditional papery encyclopedia.

193.110.109.30 13:51, 28 December 2006 (UTC)ETa

Yes, I agree with 193.110.109.30. In fact, I flipped over to the discussion on this page just to see if anyone had remarked on this very thing. Also agree with the "professional but useless" observation, in connection with the many dozens of math-related pages recently browsed.

68.7.44.49 10:04, 16 February 2007 (UTC)

As an applied mathematician I do not understand what is wrong with this entry. It is informative and goes straight to the point. Why does it need editing???

87.80.62.71 19:18, 23 March 2007 (UTC)

As an unapplied philosopher (ie layman) I also wondered why the tone-tag. I only came here to see if there was argument for the tag; there isn't. As a semi-experienced 'pedia editor, I'm simply removing said tag.Eaglizard 17:48, 19 April 2007 (UTC)

Examples?

I am not a professional mathematician, so I might be wrong about this. When I read this article I immediately thought of a functional as something like the divergence operator. If that is indeed an example, it might be a good idea to include it in the article. —Preceding unsigned comment added by Zoid62 (talk • contribs) 23:42, 9 August 2009 (UTC)

The difference between the operators and functionals is that the former transform functions into another functions, and the latter act on functions and produce a scalar value in the underlying field, e.g. a real number. Say, a certain norm of a function (a real non-negative value) is a functional, while the gradient or divergence are not. (Igny (talk) 02:13, 10 August 2009 (UTC))

Clarification of intro please

Could someone please rewrite this sentence in a manner that's less grammatically ambiguous?

"In other words, it is a function that takes a vector as its argument or input and returns a scalar, or say the function of function."

What is the phrase "or say the function of function" providing an alternative for? Gwideman (talk) 09:02, 2 December 2009 (UTC)

I agree: the sentence made no sense. I have rewritten it, and I hope clarified it. JamesBWatson (talk) 15:13, 14 December 2009 (UTC)

Broader?

In functional analysis, the functional is also used in a broader sense as a mapping from an arbitrary vector space into the underlying scalar field

How is this broader than the definition that comes before it?:

a functional is traditionally a map from a vector space to the field underlying the vector space

68.239.116.212 (talk) 18:31, 6 December 2009 (UTC)

My guess is that this is the result of the first paragraph being edited, and the second paragraph being left in a state which refers to the earlier version. I have removed the superfluous wording. JamesBWatson (talk) 15:18, 14 December 2009 (UTC)

OK, cool Gwideman (talk) 22:28, 14 December 2009 (UTC)

Linear Functionals

Hi all,

I've added a link to "Linear Functionals" in the See Also section. I'm aware that this could be problematic, but the Linear Functionals article has a sentence of sound clarification at the top of the article, so it seems to me like the link is a net positive.

If you feel otherwise, please edit for clarity, rather than just removing the link. (Or at least, if you do remove the link, please mention why, here on the talk page.) Thanks! —Preceding unsigned comment added by 128.163.130.137 (talk) 20:12, 26 January 2010 (UTC)

n-Dimensional?

Isn't this the arclength in two dimensional space?:

$f\mapsto \int _{x_{0}}^{x_{1}}{\sqrt {1+|f'(x)|^{2}}}dx$

The article says it is in n-dimensional space. Noodle snacks (talk) 20:56, 17 March 2010 (UTC)

There's nothing in that equation specific to *two* dimensions. The integral is along one dimension. The square (the root and the exponent) comes from the metric of Pythagoras theorem, which you can verify in 1, 2, and 3 dimensions. If f' was simply a single number (as would be the case in 2d) then the absolute sign brackets would be redundant. That should get you part way to answering your question ;) Cesiumfrog (talk) 06:29, 25 May 2010 (UTC)

Article might as well be hieroglyphics

Someone asid "As an applied mathematician I do not understand what is wrong with this entry."

What's wrong is that you have to be a mathematcian to understand it, perfessor.

You need to instantiate the concept with an example. For example,the article says "the mapping of a function to the value of the function at a point is a functional"

An example would be: The brightness of the sky directly above the eiffel tower in 1990 is a function (of time). Its value depends on the weather, the season, the time of day, etc.

Graphed, this function is a wavy line on a (very) long paper strip that passed through a brightness-meter in the eiffel tower. It recorded the sky-brightness for all of 1990. The long paper strip is rolled up for easy storage in boxes.

I also have nine similar rolls of paper, since I was measuring brightness for the entire decade.

Now, if I want to create a functional, I'll associate the paper roll created in 1990 to the brightness on some particular value of the 1990 graph. Let's say we associate (map) the paper roll from 1990 to the sky brightness at 3:42 PM on June 12, 1990. At that moment in 1990, the sky brightness was 247 lumens, so I write "247" on the box containing the 1990 paper roll.

Then I decide to associate the 1991 data with the sky brightness at 1:32 AM on November 3, 1991, which was 46 lumens, so I write "46" on the box containing the 1991 data.

I do the same thing for the other eight boxes, and I've created a functional.

That functional is useless and ridiculous.

Give a different example that's not useless, but is instead very helpful in some way. Helvitica Bold (talk) 08:40, 24 March 2011 (UTC)

Someone asid "As an applied mathematician I do not understand what is wrong with this entry."

"What's wrong is that you have to be a mathematcian to understand it, perfessor"

Quite right, or as has often been said, a professor or researcher may have intimate knowledge of his field but also may be completely inept to teach it intelligibly to students or the lay person. MxBuck (talk) 20:54, 24 February 2022 (UTC)

Edit/ Correction Request

The section on Vector Scalar Product states (Oct 12,2013):"The set of vectors such that this product is zero is a vector subspace of X, called the null space or kernel of X" I have looked at the definition of a kernel and see that it is defined ON a MAP of A → B not on A, B or A & B. If I am correct then the above statement, that a set of vectors is a kernel of X, is clearly wrong. X is NOT a map so the set of vectors can NOT be X's kernel.173.189.79.137 (talk) 18:03, 12 October 2013 (UTC)

• (four years later) Thanks, you are correct, the section doesn't make sense. And since it doesn't cite any sources, and doesn't seem to have anything to do with the article, I will just remove it, rather that try to fix it. Eleuther (talk) 17:37, 9 August 2017 (UTC)

Basic definition seems to be wrong

The term functional is used in math in at least two incompatible ways. In linear algebra, it refers to a linear function from a vector space into its field of scalars. This definition is useful for studying algebraic notions such as duality, etc. In functional analysis, however, a functional must always map into an ordered field, because its purpose is to study convexity, rather than duality.

Thus, in FA, a functional on a complex vector space must map into $\mathbb {R}$, not into the field of scalars $\mathbb {C}$, because $\mathbb {C}$ cannot be ordered. Furthermore, in FA, functionals are not assumed to always be linear, and are not assumed to be defined on the entire vector space (see Hahn–Banach theorem). In other words, the LA and FA concepts of functional are quite different things.

The first sentence of the article seems to be conflating these two incompatible concepts. However, I'm not sure how to fix the article in a properly cited way, so I'm soliciting help on the issue here. Eleuther (talk) 15:10, 9 August 2017 (UTC)

Incorrect correction to lead

Hi, OnRandom, thanks for trying to improve the article, but your "correction" seems to be based on grammatical confusion. My text was

• In modern linear algebra, it refers to a linear mapping from a vector space V into its field of scalars, i.e., to an element of the dual space V*.

The antecedent of "it" here is the term functional, i.e., the term refers to an element of the dual space. The text is quite clear. Your replacement is

• In modern linear algebra, it refers to a linear mapping from a vector space V into its field of scalars, i.e., it is a member of the dual space V*.

Here, the antecedent of the first "it" is still the term, but the second "it" doesn't have a proper antecedent, unless you're really trying to say that the term itself is a member of the dual space, which doesn't make sense. So I think it's necessary to revert this "correction." Yours truly, Eleuther (talk) 10:16, 8 July 2018 (UTC)

Oops. Thanks for clarifying! Onrandom (talk) 10:26, 8 July 2018 (UTC)

Grammar seems confusing regarding "function of function"

Hi, I was confused by the use of the word "when" in the following sentence. --- Commonly, the space {\displaystyle X} X is a space of functions; thus the functional takes a function for its input argument, when it is sometimes considered a function of a function (a higher-order function). --- I believe you are saying that when the space X is a space of functions, the functional is often referred to as a "higher order function" or as the "function of a function". That is, the word "when" was used to mean "and under these circumstances" - I just find it unclear and distracting from the intended illumination of the topic. I may also be completely misunderstanding the intention of this text. --BrinHB (talk) 23:58, 6 January 2019 (UTC)

Hi, BrinHB, thanks. I agree that the language is bad. This is partly my fault. I'm working on a fix, which I hope to submit momentarily. Eleuther (talk) 22:09, 7 January 2019 (UTC)

In Computer Science

In computer science, I've seen the word `functional` used to refer to higher-order functions. Indeed, the wikipedia article higher-order functions use this word. However, I see no mention of this meaning of `functional` on this page. Unless I've missed something which is obvious to someone with a more maths background. I think this page would be improved by adding a reference to this meaning of the word. Arvidj (talk) 12:55, 19 June 2019 (UTC)

I have now added this definition to the page. I had issues with inserting references, but here are three different sources:

- In the article "Can programming be liberated from the von Neumann style", higher-order functions are referred to as "functional forms", of which this usage probably is derived.
- In the book "Semantics with application" by Nielson & Nielson, this usage is common. 
- Finally, within the Bird-Meertens Formalism, this usage is common. See for instance the article "Algorithmics: Towards programming as a mathematical activity", page 301.  — Preceding unsigned comment added by Arvidj (talk • contribs) 10:52, 20 June 2019 (UTC) 

At the top of the Intro you will see that there is a disambiguation page for Functional that already includes Computer Science. I suggest you move your section off this page onto the Computer Science reference. This is to maintain the integrity of this topic in the overlapping domains of applied mathematics and physics. MxBuck (talk) 21:13, 24 February 2022 (UTC)

Disambiguation or article. Should be one or the other but not both.

This article's introduction currently reads like a disambiguation page. It lists several concepts that go by the name "functional". But the article's body ignores all but one of these. This article should either be a disambiguation page or an article but not both.

Proposal: I suggest that this article become a disambiguation page and that the content in this article's body be moved into its own article. Possible names for this new article: Functional (functional analysis) or Functional (calculus of variations).

Opinions? Suggestions? Mgkrupa 20:03, 7 January 2022 (UTC)