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Added functional. A bit sketchy.
Is that wise to restrict the meaning of functional to linear maps?
@David:
There's a reason why they're called continuous linear functionals (because not all functionals are continuous or linear by definition). Granted, I don't know what a nonlinear functional actually is, but for sure they exist.
The continuous/not-necessarily-continuous distinction is important, and I’m curious as to the phrasing “In the case that V is a TVS, then this map is necessarily continuous.”. The way I read that is that there is a theorem that for all topological vector spaces then any functional is continuous. This is certainly not true.
We could avoid the linear/non-linear issue simply by adding the header “Functionals in functional analysis” in front of what is currently there. That would allow us to later add “Functionals in X” if anyone ever finds out that someone has used them in non-linear theory. Certainly, if I have my functional analysis head on then when I hear “functional” then I understand “linear functional and almost certainly continuous” (i.e. unless I’ve been told to assume otherwise then I assume continuous).
We could avoid the linear/non-linear issue simply by adding the header “Functionals in functional analysis” in front of what is currently there.
I agree. I don’t know what a non-linear functional is, possibly some non-linear operator $V \to k$, for some space of functions $V$? Can someone give me an example of a non-linear functional. Since the page was being linked to from the page on locally convex TVS, I assumed the intended meaning was as given i.e. linear functionals.
I wrote
“In the case that V is a TVS, then this map is necessarily continuous.”
to indicate that if you are working with TVSs, then you want the functionals to be continuous. Perhaps a line drawing the distinction between working in $Vect_k$ and $TVS_k$, if the latter denotes the category of TVSs over $k$ and continuous maps, would be better. I should have said ’we demand this map is continuous’, instead of being ’necessarily continuous’.
OTOH, perhaps one wants measurable functionals or the like. A chap at uni was working on stochastic DEs on separable Hilbert spaces, and I’m sure that got pretty funky.
Can someone give me an example of a non-linear functional.
Most everything considered in variational calculus is nonlinear. Most action functionals are non-linear.
The German Wikipedia site on its page on functionals has a dedicated section on nonlinear functionals.
ah, thanks, Urs. Been doing too much topology - I forget about actual numbers sometimes :) Would it be too much of a stretch to say that nonlinear functionals come to the fore outside of the realm of ’nice’ vector spaces (e.g. Hilbert)? Will fix up the entry a bit later today. I did say it was sketchy!
Well, it’s really a very different sense of the word ’functional’, right? Those action functionals are frequently defined on domains that are not vector spaces in any reasonable sense; rather, the domains can be things like spaces of paths in Riemannian manifolds. functional might in fact be a candidate for disambiguation. (?)
might in fact be a candidate for disambiguation.
the idea did cross my mind, but it seems we need more disambiguation than I thought. Apart from action functionals, is there something inbetween? Can we say that functionals (in the sense other than linear maps to $\mathbb{R}$ or the base field) are generally maps from a space of functions (paths/loops/surfaces in a manifold) to $\mathbb{R}$? Or are more general configuration spaces allowable as domain?
In the case of linear functional, doesn’t this provide a nice opportunity to express it from the nPOV? Isn’t the ground field $k$ a special object (terminal object?) in $Vect_k$ so linear functionals are a comma category (or something) over this terminal object? Or something…
There should be some very nice statement along those lines.
linear functionals are a comma category (or something) over this terminal object?
exactly! I meant to say this, but forgot. Well, at least, linear functionals are objects of the slice/comma category…
Well, it’s really a very different sense of the word ’functional’, right?
It’s a different specific sense, but isn’t it still the same general sense: function of a “continuous number arguments” that is continuous or smooth in some sense?
Maybe I am wrong, but my impression has been that this is what happened historically: people started thinking about functions whose domains are “as big as a typical function space is” and called those “functional”s.
Then later it was discovered that a particular strong theory of these beasts is available in the simple special case (as usual) where the functional is linear on a topological vector space.
I don’t know, if that’s not the story and the term “functional” was really invented twice, independently, then I’d be surprised, but then we should still have a remark about that in the entry.
Apart from action functionals, is there something inbetween? Can we say that functionals (in the sense other than linear maps to $\mathbb{R}$ or the base field) are generally maps from a space of functions (paths/loops/surfaces in a manifold) to $\mathbb{R}$?
Good question. The best formalizaton that I am aware of of the notion of functional as it appears in variational calculus is in terms of “functional forms on spaces of sections of jet bundles”. See for instance the text
in particular the beginning of chapter 3.
I suppose pretty much everything that has ever and will ever be called an “action functional” fits into this. So this would be my best bet for what the mathematical formalism underlying “action functionals” is.
see what I did to functional
Looks good. I fixed up the sentence that Andrew objected (mildly) to: functionals on TVSs are no longer ’necessarily’ continuous, only ’(usually) required to be’ continuous.
It is the same with operators. One can consider nonlinear operators on Hilbert spaces for example, and in nonlinear functional analysis people sometimes do. Unadorned word functional almost always assumes contininuous, but linear not so often, and more often than for operators in general. For me a functional is an operator from V to k.
But I disagree with Urs that functional means that the space is big like mapping space. Functional is extremely frequently used word in finite dimensional linear algebra (including over number fields, finite fields etc.), and the fact that physicist did not care about linear algebra much until they got into mapping spaces is a subculture in physics only.
But I disagree with Urs that functional means that the space is big like mapping space.
Well, typically it does. Why else would one say “functional” instead of just “function”? I can’t stop anyone from using it for functions, but my impression is that historically it was introduced for “functions of functions”.
For what it’s worth, the MathWorld article on Functional
starts with talking about general continuous maps and variational calculus;
then mentions that the term “functional” refers to “function on a space of functions”;
and only then mentions that there are also linear functionals.
I agree with Urs at #13 and #18, and was about to write something about the history myself. I just don’t think there’s any very compelling point there for modern mathematics (plus, ’functional’ is also used for finite-dimensional situations, so I knew someone would disagree had I written this myself).
In the stuff on snowglobe models for the lambda theory of freshman calculus, James Dolan had used the terms ’function’, ’functional’, ’functionalal’, ’functionalalal’, etc. for morphisms whose domain was of higher and higher order type! :-)
Zoran,
I see you edited the entry, now the first sentence is
[A] Functional is just a $k$-valued function on a $k$-vector space.
But we keep discussing here that this is not true, in general: the energy-functional,the standard example of a functional considered in variational calculus, is one on the space of path in some Riemannian manifold. This is not a vector space.
Very few of the functionals studied in variational calculus and physics are functionals on vector spaces. And if their domains happen to carry the structure of a vector space, then it does not actually matter for the theory.
So I suggest we revert the entry to the previous style: we start by saying that functionals are essentially functions, and then say that there are special cases, linear functionals, functional forms on jet bundles, etc.
James Dolan had used the terms ’function’, ’functional’, ’functionalal’, ’functionalalal’, etc. for morphisms whose domain was of higher and higher order type! :
So this is an argument for the point that “functional” refers to “function on a space of functions” and is not just a synonym for “function”.
Zoran,
another thing: why do we have to say
Physicists and engineers usually talk about functions in terms of variables;
in the second sentence? I would rather not, this has nothing particularly to do with the concept of functional. (And do you really mean “coordinates”? )
@Urs #22: yes, that’s what Dolan was referring to, albeit somewhat whimsically. Anyway, I agree with having mention of the historical usages (as in calculus of variations), and I like your earlier revision; I would second a motion to roll back.
I don’t know the history, I read and hear lots of number theory talks everything is finite or over other fields and word functional is used all the time; I work most of my life in algebra and in physics; and the quotes about dimensionality and “field content” are from that context. Mainly in physics-motivated setups the things grew further to nonlinear extentsions like, and only there I see that fetishization of word functional, like something of higher order (hence fetishization like Dolan’s alalalals).
The basic situation is linear. Like operators. Did you hear operator in programmiong languages and alike ? Though it was earlier in functional analysis and on vector spaces.
And do you really mean “coordinates”?
You can call them coordinates of course, but coordinate can be anything; by variable I mean real or complex variables. Like function of several complex variables.
I reedited the entry to accomodate largely your complaints.
I read and hear lots of number theory talks everything is finite or over other fields and word functional is used all the time
Yes, yes, of course, and we know all about that.
Mainly in physics-motivated setups the things grew further to nonlinear extentsions like, and only there I see that fetishization of word functional, like something of higher order (hence fetishization like Dolan’s alalalals).
I don’t know for sure, but I suspect you’ve got the history precisely backwards. For some historical information, see this page (from a website on earliest uses of mathematical terms). Apparently the earliest use of ’functional’ is due to Hadamard, from the very beginning of the twentieth century, where he is discussing precisely developments of the calculus of variations (a much, much older subject). (Or perhaps the noun form is due to Hadamard’s student Fréchet, but it was adapted from the adjectival form and applied to the same subject.) The appropriation of the noun to linear contexts was made later, around the 1920’s, and spread into mathematical English by the 1930’s. Or so that is my understanding – please correct me if I’m wrong.
And, you know, words like “fetishization” sound mean-spirited (especially when applied to good friends like Jim Dolan). Please refrain from using them, or if you feel you must, please consult a dictionary so that you know exactly what you are saying. Do you in fact know Dolan? Could you say with certainty when he is guilty of ’fetishizing’ something?
I don’t know for sure, but I suspect you’ve got the history precisely backwards.
You are proibably quite right historically. I am not ghiving any historical claims on early history, but only my feeling on social differences after the canonical functional analysis literature in mid 20th century. Fetishization points to giving hi long miraculous names to simple concepts, because they are rarely used or need special precaution. I read rather many elementary textbooks on physics in my hi school and university years and recall very right what kind of awe, warning and pre-pre-preparations were usually made when words like “functional” and especially “functional derivatives” are used. I do not know person Dolan personally, and did not know John Baez personally until a single recent conference (very strange why would that have anything to do with opinions about the conotations of words). On the other hand, I had a common acquaintance with Hadamard (believe or not) but he (the acquaintance) was killed by a street thief in 1993.
Finally, please do not expect me to use dictionary to be absolutely precise when using forum-level discussions. It is practical impossibility.
It occured to me, you may have heard of my acquaintance in fact (I knew him from a 1991 conference in Zagreb), as you are interested in foundations. There is something called…Kurepa axiom, if it may ring the bell. Unfortunately few months ago we had to say final farewell to his nephew who was also a good mathematician.
Fetishization points to giving hi long miraculous names to simple concepts… I read rather many elementary textbooks on physics in my hi school and university years and recall very right what kind of awe, warning and pre-pre-preparations were usually made when words like “functional” and especially “functional derivatives” are used. I do not know person Dolan personally… (very strange why would that have anything to do with opinions about the conotations of words).
Okay, thanks for explaining what you meant. But Jim Dolan of all people would be one of the last people I’d pick as manifesting the behavior implied by that connotation, and therefore it was unfortunate that you mentioned his “fetishization” particularly.
Anyway, Urs’s comment in #13 that apparently caused you to disagree was primarily a historical remark, so that’s what I thought we were talking about. All of it is worth remarking on in the Lab, IMO.
I think I am prepared to agree that in modern parlance, the word is likelier to carry the linear functional meaning than not – but I hardly think that people who nevertheless do recall the older calculus of variations meaning are falling prey to fetishistic impulses. On the contrary, people like Lawvere (and Dolan, for that matter) strive to understand such things in simple conceptual ways,without pretense. So who’s fetishizing? Nobody here, I hope! :-)
Edit: no, I hadn’t heard of Kurepa, unfortunately.
All of it is worth remarking on in the Lab, IMO.
Everything what one knows is worth mentioning in a structured text I think.
Đuro Kurepa got his PhD in Paris in 1933 I think. Hadamard is the person who helped him a lot though was not his official thesis mentor; even his first introduction to the library of Ecole Normale (I think) was by Hadamard.
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