Talk:Discontinuous linear map

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[edit] Name issues

This article was originally called a linear functional which is not continuous. The article was proposed for renaming, here are some replies below, followed by discussion at the math wikiproject. Oleg Alexandrov (talk) 00:30, 4 January 2006 (UTC)

OK, OK, not that I care much, but the existing title, A linear functional which is not continuous looks nicer to me (but I am biased, I wrote this). Oleg Alexandrov (talk) 17:35, 26 December 2005 (UTC)

See discussion at Wikipedia talk:WikiProject Mathematics#A linear functional which is not continuous. Oleg Alexandrov (talk) 03:39, 27 December 2005 (UTC)

Yes, see the above wikiproject talk page for the result WhiteNight ^{T | @ | C} 20:37, 31 December 2005 (UTC)

[edit] Discussion of the name copied here from Wikipedia talk:WikiProject Mathematics

I wrote the article A linear functional which is not continuous only to immediately discover on its talk page a suggestion to move it to Non-continuous linear functional (darn, everybody should be drunk and sleeping this post-Christmas morning, not checking the recent changes). What do people think (if it matters at all)? Oleg Alexandrov (talk) 17:36, 26 December 2005 (UTC)

Isn't there a mechanism for providing both names for the article? Both seem fine... Randall Holmes 17:40, 26 December 2005 (UTC)

By the way, I've been steadily editing through this period, and it has been awfully quiet :-) Happy Hanukkah! Randall Holmes 17:41, 26 December 2005 (UTC)

Yeah, still quiet. :) Yes, there is a mechanism for providing both names, it is called a redirect, see Wikipedia:Redirect. So I guess the argument is about which is the primary meaning, for all that's worth. Oleg Alexandrov (talk) 02:31, 27 December 2005 (UTC)

I like the first name. Compare with stuff like An infinitely differentiable function that is not analytic. Also, is "non-continuous" a word? Shouldn't that be "discontinuous"? -lethe ^talk 03:02, 27 December 2005 (UTC)

Personally, I don't like the idea of a name for an encyclopedic article beginning with "A". Also, I think the original proposed name is too long. I'd go for Non-continuous linear functional. But maybe it's just me. --Meni Rosenfeld 12:49, 29 December 2005 (UTC)

I agree with Meni; initial articles are okay if they are title of a book or work of art, but I don't think they belong in a general article. Even if it has to do with showing existence. Isn't there some other way to word it? Gene Nygaard 21:37, 30 December 2005 (UTC)

Hi there :) - the article was moved to A linear map which is not continuous - is that what people have decided on? Right now it looks like no consensus, but I thought I'd just give everyone a buzz... WhiteNight ^{T | @ | C} 23:28, 30 December 2005 (UTC)

How about linear map which is not continous, dropping the leading "A" which makes people uneasy. The other option seems to be discontinous linear map, based on above. Oleg Alexandrov (talk) 00:27, 31 December 2005 (UTC)

Discontinuous linear map/Discontinuous linear functional and Infinitely differentiable non-analytic function seem reasonable. We anyway don't expect people to search for and bump into these articles directly, and we are probably going to use these as examples to show that being a "linear map" doesn't imply being continuous, and smoothness doesn't imply analyticity, in main articles on linear maps and analytic functions, so I guess the title is not all that crucial. Bottomline: Unless we are missing on something important, the shorter the better. deeptrivia (talk) 00:44, 31 December 2005 (UTC)

I moved the article A linear functional which is not continuous to discontinuous linear map which seems to address all concerns on this page. I made a bunch of other alternative titles redirect to it. Oleg Alexandrov (talk) 19:57, 31 December 2005 (UTC)

However, I find Infinitely differentiable non-analytic function a very clumsy name for An infinitely differentiable function that is not analytic. Oleg Alexandrov (talk) 19:57, 31 December 2005 (UTC)

[edit] functionals versus functions

I think the preference for functionals over functions in this article is a bit odd, considering that the latter includes the former as a special case, the proof is essentially the same, and functions are at least as common and important (if not more so). -lethe ^talk 07:26, 27 December 2005 (UTC)

Well, how do you call a linear function from a normed space to the base field? I thought it was a linear functional. Would "scalar-valued linear function" sound better? I am inclining to think so.

By the way, Lethe, you added the paragraph:

More generally, one may consider linear maps between any topological vector spaces. For example, the derivative is linear but not continuous on a space of functions with the uniform norm. Linear transformations of the vector space of real numbers over the rationals are generally not continuous real functions, although the existence of a basis for this vector space and the construction of these functions requires the axiom of choice.

I find this a bit confusing, because while the first sentence talks about linear topological spaces, all the text after that is still about normed spaces. Wonder what you think. Oleg Alexandrov (talk) 15:58, 27 December 2005 (UTC)

The paragraph contains two examples: the derivative, and the Hamel basis for the reals. Both of those are normed vector spaces. Neither map is a functional, instead they are both linear maps (they map to the vector space, not to the underlying field of scalars). I agree that functionals are maps to the field (no need to refer to them as "scalar-vauled"). My point is that linear functionals are a special case of linear maps, and I think that this article should be written about linear maps. The whole point of this article is to talk about the observation that linear does not imply continuous in infinite dimensions. This is true for all linear maps, not just linear functionals, and I'd like it better if the more general (and in my opinion, more interesting) case were discussed, and then functionals can be mentioned as a special case. -lethe ^talk 16:20, 27 December 2005 (UTC)

OK, I got that. :) I will rename the article. But my question still remains: why do you start a paragraph saying "More generally, one may consider linear maps between any topological vector spaces. " if you don't plan to talk about topological vector spaces. Maybe you should say "other examples of maps between normed spaces...". What do you think? Oleg Alexandrov (talk) 18:18, 27 December 2005 (UTC)

Oh, is your question about the phrase "more general"? Like you fail to see how it's more general, since they're still normed spaces. So it's more general because they're maps between two TVSes instead of maps to the underlying field, not because the spaces are not normable. The spaces are normable, so that part isn't more general. Perhaps the language could be made clearer? -lethe ^talk 18:25, 27 December 2005 (UTC)

So you changed "topological vector spaces" to "normed spaces". I guess that's OK, but I don't want the reader to get the impression that the norm is somehow important to this argument. The examples can easily be made in non-normable spaces (differentiation is also not continuous on the space of smooth functions), it's just that the normed spaces are most handy. I'm going to generalize to a non-normed space, but I think you'd like the more pedestrian example better. -lethe ^talk 19:08, 27 December 2005 (UTC)

[edit] structural changes

So I've done a lot of structural changes to the article. Here's a list:

I've put the examples ahead of the proof of existence
I've made the article primarily about linear maps instead of linear functionals.
I didn't like Oleg's insistence on only normed vector spaces, so I chose a larger space of functions which is not normable for my derivative example. The norm example is still there as well, because I think Fréchet spaces and Banach topologies are not so familiar to people
I removed the norm in the proof, so that it's a map V → V instead of a map V → R.
I added the example from locally convex topological vector space of a space for which no linear maps are continuous.

lethe ^talk 19:40, 27 December 2005 (UTC)

[edit] Rewrite

I see you made a lot of changes. That's good, but I would argue that the article is more complicated now (Frechet spaces) and does not keep the focus I had in mind so much anymore. My main purpose was that given any infinite-dimensional normed space, you can find a noncontinous operator on that space. Instead, now you focus on giving concrete examples for some concrete spaces, but the important fact that one can find such an example in any space is marginalized. Wonder what you think. Oleg Alexandrov (talk) 19:44, 27 December 2005 (UTC)

Well, if you don't want the Frechet space example, we don't have to keep it. It doesn't matter which functional space you choose, the derivative is discontinuous on almost all of them. Furthermore, if you'd like the proof of the theorem to go before the examples, I won't object either (though I thought you'd like that. Haven't I seen you moving examples to the top of articles on my watchlist recently?) -lethe ^talk 19:59, 27 December 2005 (UTC)

I am very happy with the Frechet space example, but not right on top, as Frechet spaces are more compliated beasts than Banach spaces. So, I do appreciate the example!

And yes, I do love examples. The only problem I have is that I would like the point of this article to be about the general fact that in any space one may find a noncontinous linear functional. On the other hand, I agree with you that simpler concrete examples are more helpful than the general example. I will think more about it.

So basically you want to use this article to expound on the claim that for every Banach space, the continuous dual is a proper subset of the algebraic. I kinda want to toss into this article everything we could say about continuous versus discontinuous linear maps for any TVSes. We can surely accommodate both purposes. -lethe ^talk 20:39, 27 December 2005 (UTC)

By the way, I moved the article to a new name, as you see. Would you like some other name? I will bypass the redirects later. Oleg Alexandrov (talk) 20:18, 27 December 2005 (UTC)

Right now, I'm thinking about your proof, whether it can be extended and we can say that every TVS has to have a discontinuous functional. That will probably have some bearing on what I want to do. -lethe ^talk 20:39, 27 December 2005 (UTC)

[edit] Mistake in the article?

The introduction now states that

If the space is finite-dimensional, then all topologies compatible with the algebraic structure agree.

I think that this is false. If one considers on the real line the finest possible topology, when any set is open, then R becomes a linear topological space not identical to the usual topological structure on R, no?

By the way, I would suggest that since normed spaces are simpler and used more, maybe the article should talk only about normed spaces in the first part, and move discusson of more complicated spaces later. Comments? Oleg Alexandrov (talk) 04:57, 28 December 2005 (UTC)

[edit] Proposed changes

So, it seems that given a linear topological space, even of infinite dimension, all the linear maps defined on that space can be continous, if for example the topology has all sets as open (discrete topology). So things are rather complicated.

I suggest doing some changes to this article which will not alter the order much, and putting linear topological spaces towards the bottom. So, I would view the article as having the following structure:

Introduction (basically written by Lethe, I will just shorten it a bit)
Concrete discontinous linear maps on normed spaces (Lethe's contribution)
Abstrat (but general) discountinous maps on normed spaces (my contribution)
The case of Frechet spaces: concrete discontinuous maps (Lethe's example) and general abstract discontinous maps (I think the abstract example in the case of normed spaces is adaptable to Frechet spaces, with some changes)
The case of general linear topological changes (where things are messy, not much can be said)

In short, most of my changes will just move the derivative as example of discontinous operator in Frechet spaces down, and I would introduce some sectioning. Wonder if there are comments. Oleg Alexandrov (talk) 01:43, 30 December 2005 (UTC)

Hi Oleg. I'm going through some sort of dramatic upheaval in my "real world" life right now, so I haven't been able to participate in the re-write here that I wanted to. Assuming that things work out, I'll be back, but in the mean time, feel free to do whatever you need to do here. In the mean time, let me say, yes, I made a gross error: all norms are equivalent in finite dimensions, not all topologies. I think you've already corrected this, but that fact weakens my case for the full generality. At this point, I'm on board for the article being about normed spaces. The original reason for my recitence thereon was that the theorem "all inf-dim vector spaces (resp. normed spaced) have discontinuous linear functionals seems to be nonconstructive; seems to rest on AC. If that is the case, then it's actually not nearly as interesting a fact as this article seems to warrant. But in most cases, we know it to be true without AC (i.e., using the derivative). Anyway, do what you like, I'll come back later and rewrite if I feel the need, but as of right now, I might be stuck offline for a while.

PS, read this with care, it was written drunk. -lethe ^talk 07:17, 30 December 2005 (UTC)

[edit] Issues

I did some rewriting in this article. I would like to say that I removed the part about discontinuous linear maps over the field of rationals. While they may be interesting, they are not relevant to functional analysis, where any serious work is done over the real or complex numbers for good enough reasons.

Now, about the following text:

The derivative operator is a linear map on a space of differentiable functions which in most cases will not be continuous. For example, it is not continuous in the Fréchet space D^∞(R) of smooth functions on the real line with its inductive limit of Banach topologies.

I could not prove that the derivative will be discontinuous. Actually, if by those fancy Banach topologies is meant the seminorms given by

$p_{K, n}(f)=\sup_{0\le m\le n, x\in K}|f^{(m)}(x)|$

where K are compact sets and n is integer (that is, uniform convergence over compact sets for sequences of functions and their derivatives), then it looks to me that the derivative operator is actually continuous. But I don't have much experience in this spaces, I may be wrong. Comments? Oleg Alexandrov (talk) 19:01, 1 January 2006 (UTC)

Hmm, yes. I stuck that example in there because I wanted an example of a TVS which was not normed. I just assumed that since the derivative is not continuous in the operator topology, it wouldn't be continuous in this coarser topology, but that assumption is not right (for example, every map is continuous on an indiscrete space). So um, yeah, if you think it is continuous, we should 86 it. But yeah, the Banach topology is the thing you mention. Not too familiar a space, but still my most familiar non-normable TVS, from distributions. Why don't we have an article on that? -lethe ^talk 08:20, 3 January 2006 (UTC)

[edit] One more issue

I have a question about the text:

The space of measurable functions on the unit interval with quasinorm given by

$||f|| = \int_I \frac{|f(x)|}{1+|f(x)|}dx$

is not a locally convex space, and the only continuous functional on this space is 0.

I wonder, which is the range of those functionals? I mean, you can always find a nonzero functional from the space to itself, that being the identity. So, do you mean scalar-valued functionals? Oleg Alexandrov (talk) 16:07, 3 January 2006 (UTC)

I suppose you meant to say codomain, not range, since the range of the zero function or functional is always the singleton {0}. Anyway, we're talking about functionals here, not functions, so the codomain should be the field of scalars, which I believe in this case is meant to be R. -lethe ^talk 18:57, 3 January 2006 (UTC)

Or maybe you meant to ask what is the codomain of f, the elements of the space? Those are measurable functions on the unit interval, so it's their domain and codomain. -lethe ^talk 19:00, 3 January 2006 (UTC)

OK, so now you see why I named this article originally a linear 'functional'... Because I did mean scalar-valued transformations. But I realized I don't like the word functional anymore, so I am going to reword the article. (And yes, I did mean codomain of the functional, thanks!) Oleg Alexandrov (talk) 22:18, 3 January 2006 (UTC)

Well this is what originally prompted me to edit in the first place. If the article is about functionals, then it's awkard to mention functions, whereas if the article is about functions, it's no problem to talk about functionals, because a functional is is just a function V → W with W=K. My concern was that the scalar-valuedness of the function is in no way essential to the failure of continuity, and writing only about the restricted set of functions which happen to be functionals is therefore misleading. On the other hand, I feel like an example of functionals like the above in an article on functions should be perfectly at home. -lethe ^talk 23:42, 3 January 2006 (UTC)

See the new sentence, which is my attempt to address this issue. -lethe ^talk 23:50, 3 January 2006 (UTC)

OK, so the concrete examples is a functional (although I don't think it is worth mentioning it), and the abstract example is fully general f:X->Y. How's that? Oleg Alexandrov (talk) 00:26, 4 January 2006 (UTC)

[edit] maps again

So you say that "Lethe's text [...] is not right. The derivative is not mapping the C^1 functions to C^1 functions." It's true that the derivative is not onto. It maps continuously differentiable functions to continuous functions. ~~So? That doesn't stop it from being a map, or somehow change it into a continuous map.~~ But actually, I thought we were dealing with smooth functions, because that's where the wikilink pointed. Either way, smooth functions, or continuously differentiable functions, the thing still isn't continuous in the operator norm. I was pretty surprised when I first learned that the derivative is not continuous, and I think this fact deserves special mention (i.e. beyond the nonconstructive general example). If you think it doesn't deserve mention, then remove it, I won't object. But don't remove it for being wrong, since I don't think it is wrong. (Unless I'm mistaken again. In which case, obviously it has to go.) -lethe ^talk 02:02, 4 January 2006 (UTC)

I wasn't thinking clearly there. For some reason I was having this brain-fart where I was assuming that C⁰(I)⊆C¹(I), so that the derivative could just be a non-surjective map. Duh. Now I see why it was wrong before I changed it to smooth. But it's OK now, I think. -lethe ^talk 04:35, 4 January 2006 (UTC)

Or else if you didn't want to restrict to smooth functions, it could be a map C¹ → C⁰ —the preceding unsigned comment is by Lethe (talk • contribs)

Smooth functions are fine. I think that is better, as it is simpler when the domain and codomain of the map are the same, unlike C¹ → C⁰ when they are different. Oleg Alexandrov (talk) 18:51, 4 January 2006 (UTC)

Your example is helpful, thank you! I removed it because I did not know how to fix it, but your fix was very simple.

Note that the derivative has a problem. If you want it to be a discontinuous map, its domain must not be a Banach space. That's why I like so much the ugly abstract example using the axiom of choice, it actually works for Banach spaces too, so one may not blame lack of of completeness on the existence of discontinuous operators. Oleg Alexandrov (talk) 03:08, 4 January 2006 (UTC)

I don't understand this last comment. We've decided that, for example, the derivative is discontinuous on the space of functions on a compact space with uniform norm. This is a Banach space, so why are you saying that its domain must not be Banach for discontinuity? -lethe ^talk 19:45, 4 January 2006 (UTC)

The space of C^1 functions with the uniform norm is not Banach. Its completion is C^0.

Actually, the derivative is a closed operator, so, if it is defined on a Banach space it is automatically continuous, by the closed graph theorem. Hope I got it right. :) Oleg Alexandrov (talk) 21:51, 4 January 2006 (UTC)

Oh, right, C¹ isn't complete. I'm all over the place in here. But now what about the derivative on the Banach space L²? That space is complete, and the derivative isn't continuous there, I don't think. -lethe ^talk 22:11, 4 January 2006 (UTC)

Oh, well I guess the derivative isn't defined on all of L². -lethe ^talk 22:23, 4 January 2006 (UTC)

So the upshot is: the derivative fails to be continuous if either it's defined only on a dense subset of the domain, or if the domain isn't complete. How did you know it was a closed map? -lethe ^talk 09:24, 5 January 2006 (UTC)

Well, the closed operator definition applied to derivative would state that if $f_n\to f\,$ and $f'_n\to g\,$ then one must have

f

be differentiable and $f'=g.\,$ That is indeed true and rather easy to check if the convergence is in the uniform norm. I think. Oleg Alexandrov (talk) 17:06, 5 January 2006 (UTC)

OK, I see. The theorem of uniform convergence means that the map is closed, then the closed operator theorem says that if the domain is complete and the map defined everywhere, then the map is continuous. So you like the abstract example better, since it works in complete spaces, but I don't like the abstract example so well, since it is nonconstructible. But I wonder if discontinuous linear functions exist without AC. Actually, I vaguely recall hearing once that with a strong enough notion of constructiveness (like maybe Kronecker's, who rejects the axiom of infinity? I forget) then all maps are continuous. Weird. -lethe ^talk 19:38, 5 January 2006 (UTC)

[edit] derivative in Frechet space

Oleg, you say "it looks to me that the derivative operator is actually continuous" in the space of smooth functions of compact support. I want to investigate this matter. I poked around in my Reed and Simon last night, and it seems that we have a boundedness criterion for continuity in TVSes in terms of their seminorms very like the familiar one we have for Banach spaces.

Firstly, there's an easier space than the space of smooth functions of compact support with its inductive limit; namely the functions of rapid decrease, smooth functions all of whose derivatives fall off faster than any polynomial. This is the guy whose dual space is the tempered distributions. I think it might also be called the Schwarz space, but I'm not sure; R&S don't mention it and Wikipedia doesn't seem to know either. Anyway, this is a Fréchet space with a family of seminorms given by

$\| f \|_{a,b}=\sup | x^a D^bf|.$

So the derivative map will be continuous in this space If for all (c,d) there is some finite collection (a_i, b_i) such that

$\|Df\|_{c,d} \le M(\|f\|_{a_1,b_1}+\dotsb + \|f\|_{a_n,b_n})$

which will be the case if we can find an M such that

$x^cD^{d+1}f \le x^aD^bf.$

But simply take a=c, and b to be d+1, and the inequality is saturated with M=1. Basically, since the derivative is already in the definition of the seminorm, all it does is increment the index to the next seminorm in the family. So I conclude that yes, Oleg, you're right, the derivative is continuous in this top. Unless the quantifiers go the other way round, it wasn't crystal clear in the text. -lethe ^talk 09:21, 5 January 2006 (UTC)

Right. So, the derivative is not such a bad operator. It is not continuous, but close. :) And both on Banach spaces and Frechet spaces. Oleg Alexandrov (talk) 21:48, 5 January 2006 (UTC)

[edit] it's harder to find discontinuous everywhere-defined maps than you think

So I've been thinking about your point a lot today, Oleg. There are only two ways to have a discontinuous linear operator: "holes" in the domain (either the domain is not complete, or the operator is not defined everywhere. I've decided today that these two cases are essentially the same.) The other is by invoking the axiom of choice, which will allow us to prove the existence of (though not construct) examples even in complete spaces.

You are of the opinion that reliance on holes in the space for discontinuity makes the example less interesting. And well, I agree with that. But I also think that examples that rely on choice are somewhat pathological. I mean, they're nonconstructible, and it's well known that choice gives you lots of conconstructive counterintuitive things. Anyway, we can debate about that later. What I want to draw to your attention now is the fact that I'm thinking that the example you have in the article now isn't so good: it suffers from both defects.

You define T(e_n) = n ||e_n||. Consider the element

$f=\sum^\infty_n \frac{e_n}{n^2\|e_n\|}.$

This element is in the space by completeness, but T(f) is the harmonic series: it diverges. Your map is also not everywhere defined. In fact, the reason this thing fails to be continuous is not because of the arbitrariness introduced by AC, but rather for exactly the same reason that the derivative is discontinuous: it runs off to infinity, so completeness dictates that it cannot be everywhere defined (or else we have to remove points from our domain). -lethe ^talk 00:48, 7 January 2006 (UTC)

I think my example is fine. (e_n) is not a Hibert space basis, there is no Hilbert space there. It is a plain sequence of linearly independent elements (to be completed later by a basis), and each vector is a finite linear combination of elements of the basis. And it is everywhere defined, it takes the value zero at the elements of the basis which are distinct from (e_n). So, it is a linear operator in the best tradition of linear operator over vector spaces (this one can be defined even over vector spaces with the field being $\mathbb Z_2$ if you wish. :) Oleg Alexandrov (talk) 01:02, 7 January 2006 (UTC)

But it doesn't take the value 0 on sums of e_n. I've given you an example of such a sum where your operator diverges. (and I don't understand why you mention Hilbert spaces. I didn't assume a Hilbert space in my construction of an element which is not in the domain of T. And as far as Z2 goes, that's not an ordered field, so I don't know how we could define a norm over it) -lethe ^talk 01:08, 7 January 2006 (UTC)

Edit conflict:

The upshot is, at the least, we need to mention in the article that the existence of these things is related to two things. First, as already stated, that the space be infinite dimensional (without this, they don't exist). But second, if the space is incomplete, then we can construct discontinuous linear functions, but in complete spaces, their existence depends on AC (and so are nonconstructible). I think I could actually give a general construction in an incomplete space, though I'm not sure it belongs in the article. By the way, doesn't completeness require a uniform structure? Can we even talk about completeness in a general topological space? -lethe ^talk 01:06, 7 January 2006 (UTC)

You seem to be biased against the axiom of choice. :) I think it is used to prove the Hahn-Banach theorem, the existance of a basis in any vector space, Zorn's lemma, etc. You cannot do serious functional analysis without it.

Well, no, I'm not really. I like the axiom of choice. But I do think it should be used with care, and only used when necessary, and its use should be noted everytime. Sure, we need it for Hahn-Banach and lots of analysis. But I think this article could be of interest to more than just analysts, therefore I think we ought to straighten this out. -lethe ^talk 01:19, 7 January 2006 (UTC)

About completness, I think you can talk about it in any linear topological space. What you need though, is sequences which are indexed over more than just the integers, things called filters I think (see section on topology in that article). Oleg Alexandrov (talk) 01:11, 7 January 2006 (UTC)

whoa, I don't think I agree with you there. You need those things only in spaces that are not second countable. Anyway, I think now that any topological vector space automatically has a unique uniform structure, due to the algebraic structure. Of course, the topological structure is still required, so one cannot speak of Cauchy sequences (or nets) in a general vector space (as far as I know). -lethe ^talk 01:16, 7 January 2006 (UTC)

Take a closer look at the example of the vector not in the domain of T. If the example holds, perhaps we should replace it with an example that is everywhere-defined (I did not understand your previous objections). I think a proper example will be much more nonconstructive than this one. -lethe ^talk 01:24, 7 January 2006 (UTC)

OK, I dont' really know what uniform structure means.

Now, you say:

$f=\sum^\infty_n \frac{e_n}{n^2\|e_n\|}.$

This element is in the space by completeness, but T(f) is the harmonic series: it diverges.

I disagree. Your statement that it diverges is based on the equality

$T(f)=T\left(\sum^\infty_n \frac{e_n}{n^2\|e_n\|}\right)=\sum^\infty_n T\left(\frac{e_n}{n^2\|e_n\|}\right)$

which implicitely assumes continuity, and that is an assumption we don't make, obvioulsy. So I don't think that's a counterexample. Oleg Alexandrov (talk) 01:27, 7 January 2006 (UTC)

Sorry about the Hilbert business. Now, my example is correct. Just don't think about topology for a while, all my example uses is the fact that given a basis over a vector space, then, by defining an arbitrary function at those basis elements, it can be uniquely extended to a linear function over the vector space. It is that simple. :) Oleg Alexandrov (talk) 01:27, 7 January 2006 (UTC)

No, sure. I get the part about defining it on some set of vectors, and extending it outside by Hahn-Banach. It's the defined everywhere part that I was arguing. You seem to have found the flaw in my reasoning. I will think on it some more. -lethe ^talk 01:32, 7 January 2006 (UTC)

There is no Hahn-Banach, OK? :) It is a purely algebraic construction. :) Oleg Alexandrov (talk) 01:34, 7 January 2006 (UTC)

Wait, no. I'm still with you. Sorry, I thought Hahn-Banach was the one where you have every linearly independent set inside a basis. Which you do have to use, right? Also, now I get why you keep bringing up that your operator is zero outside of the sequence. My sum isn't an element of the sequence, even though it is an infinite sum of elements, it is not in the algebraic span. Right? Maybe I'm getting up to speed now. -lethe ^talk 01:37, 7 January 2006 (UTC)

Right. You cannot say

$f=\sum^\infty_n \frac{e_n}{n^2\|e_n\|}$

because you don't even know that the space is complete, so f might not exist. But if it exists, it can be written uniquely as

$f=c_{k_1}e_{k_1}+\cdots +c_{k_p}e_{k_p}+d_{l_1}f_{l_1}+\cdots +d_{l_q}f_{l_q}$

where the e's are from the sequence (e_n), and the f's are from the completion of that sequence to a basis. Oleg Alexandrov (talk) 01:49, 7 January 2006 (UTC)

OK. I'm totally on board the example now. Of course, the example is in a Banach space, which is complete, so f does exist, right? -lethe ^talk 02:03, 7 January 2006 (UTC)

It does not have to be a Banach space, it can be a normed space. So f is not guaranteed to exist. Oleg Alexandrov (talk) 03:32, 7 January 2006 (UTC)

Oh right. I had got it into my head that we were only concerned with that example in the complete case, but of course it still applies in incomplete spaces. Cool. -lethe ^talk 04:18, 7 January 2006 (UTC)

[edit] moving on

Well. That was the most rapid-fire talk page exchange I've ever had. Thanks for straightening me out, Oleg. I still want to belabor the difference between the constructible case and the nonconstructible case. Tell you what, I'll make some edits, and then we'll see if you don't like them. -lethe ^talk 05:02, 7 January 2006 (UTC)

Well, if I don't like it, I change it. :) If you want to discuss first, that works too. :) Oleg Alexandrov (talk) 05:05, 7 January 2006 (UTC)

OK, well, I know you didn't respond too enthusiastically when I brought up this distinction before, but let me just run this proposed intro by you:

In mathematics, linear maps form an important class of "simple" functions which preserve the algebraic structure of linear spaces and are often used as approximations to more general functions (see linear approximation). If the spaces involved are also topological spaces (see topological vector space), then it makes sense to ask whether all linear maps are continuous. It turns out that for normed spaces (which are the simplest and most used topologocial vector spaces), a discontinuous linear function can be proven to exist, and in the case that the space is not complete, explicit examples can be given, while for Banach spaces their existence relies on the axiom of choice.

Also, I'd like to restore my old example about the Hamel basis for the reals over the rationals. I think it wasn't worded very nicely when I first put it in, I tried to fit it into a single sentence. But it's an important example, commonly encountered. People will expect to see it, and it provides a model for the more general case.

And, just tonight I found a rather different example from the derivative of a constructible map which is kind of fun, that might be nice to include. It's a linear map whose graph is dense in XxY, therefore in some sense maximally discontinuous. The example is a map which is the identity on polynomials. It maps a polynomial, viewed as a map on [0,1], and therefore an element of C[0,1], to the same polynomial in C[2,3].

what do you think? -lethe ^talk 06:09, 7 January 2006 (UTC)

I don't understand why you would like the example of the Hamel basis for the reals over the rationals. The rationals are not a complete field, and I don't think those maps are interesting in the functional analysis context.

About the distinction between complete spaces and normed spaces which are not complete, that would just complicate the introduction I would argue. I don't mind you adding such an example, but talking about it in the introduction could be too much I think. One may say in the introduction that various special cases will be discussed, etc.

The example with polynomial, well, that seems to be a map which is not defined over the whole domain, rather a dense set. Of course bad things will happen! :)

OK, let me put it that way. The main purpose I had in mind when writing this article was the following: infinite-dimensional spaces are weird beasts in many respects. No matter if the space in question is Banach (or even Hilbert, with countable base!), no matter that it is over the complex numbers, which is algebraically closed, assume all the nice things you may feel like it, the fact is that if the dimension is infinite, it means that there exist linear maps which are not continuous, and one has to live with that.

Your philosophy seems to be different. You would like to explore cases when things are not so nice (the space is not Banach, or the field is the rationals (neither complete nor algebraically closed), or the map is partially defined). Of course then bad things will happen, no surprise. This is intelectually interesting, but not very useful I would think for somebody doing functional analysis.

That is to say, I don't mind if this article grows, as long as the main point is not obscured. Now, what the main point of this article should be, that's probably the main question, and I don't insist that my viewpoint is the best. Oleg Alexandrov (talk) 06:23, 7 January 2006 (UTC)

Well, that's it in a nutshell. I would like to see an article covering everything there is to say about linear maps which are not continuous, and not just one about Banach spaces. I'm sure we can find an arrangement for additional examples that you'll be happy with. I guess the more contentious issue is whether the matter of choice should be brought up in the intro. Just as it can't happen without infinite dimensions, it can't happen without choice (unless the space is not complete). So from one viewpoint, choice is just as integral to the counterintuitiveness that we're seeing as infinity. From the viewpoint that choice is always implicitly assumed (which I guess is the case among analysts), this probably doesn't need to be brought up as a primary concern. For comparison, I notice that Hahn–Banach theorem doesn't mention choice in the intro (or at all). Vitali set does. These are other areas of analysis where choice is required. So I dunno.

On another note, you've reminded me of something else. For a long time, I've wanted to see a page called something like infinite dimensional, where we collate all the weird things that can happen with infinite dimensional Lie groups, vector spaces, and maybe even sets. I started a version on my local computer a while back. I should get back to that. -lethe ^talk 07:15, 7 January 2006 (UTC)

OK, so we have to get used to the fact that the axiom of choice is necessary.

Your example about a linear discontinuous function on an incomplete space could go under my general example using the axiom of choice I would guess, as the latter is more important (works more generally, and for more useful spaces).

Then, one could see what happens in Frechet spaces. I believe my example can be generalized there. Instead of $\|e_n\|$ which shows up there, one could use the countable number of seminorms $q_1, q_2, \dots, q_n, q_{n+1}\dots$ and replace $\|e_n\|$ with

$\max(q_1(e_n), q_2(e_n), \dots, q_n(e_n)).$

I believe you're right, this could easily be generalized to Fréchet spaces (which have a countable family of seminorms) and even to lovally convex TVS (which have family of seminorms which need not be countable); I don't think the countablility of the seminorms is used here. And of course, it won't work for a TVS which is not locally convex, for that has no seminorms, and every functional is (rather, gives rise to) a seminorm. So we might mention the Fréchet LCTVSes, but I don't think we have to include the proofs. But first, let's be sure those proofs are there. Are they? Am I right that it works also for locally convex TVS? -lethe ^talk 19:06, 7 January 2006 (UTC)

With that in mind, if we are sure that the proof does work in the Fréchet case, and the LCTVS case, I propose that we replace the sentence "It turns out that for normed spaces (which are the simplest and most used topologocial vector spaces)" with either "…for most topological vector spaces" or else with "…for normed spaces", and then when we get to the proof for the normed space, we say something like "for simplicity, we assume the space is normed, though the proof can be generalized to LCTVS". What do you think? -lethe ^talk 19:10, 7 January 2006 (UTC)

Then, one could have your example with spaces of measurable functions which are not locally convex and there are no linear continuous functionals except zero.

Afterward we could put all those exotic examples with the linear maps which are partially defined and the maps over the rationals. How's that sound? Oleg Alexandrov (talk) 17:38, 7 January 2006 (UTC)

Yeah, we can say in the intro that axiom of choice kicks in in places, and that this is unavoidable in infinite-dimensional spaces. Oleg Alexandrov (talk) 17:40, 7 January 2006 (UTC)

I'm sure we can arrive somewhere that we'll both be happy with. I've heard that there are schools of thought that like to do analysis without choice. It's a fringe POV, but mentioning choice makes the article more useful to those people (and students aspiring to become those people), and I think it will do no harm, as long as we try to keep the main flavor for proper analysts. I went through an axiomatic set theory phase when I was in college, and that flavors a lot of my thought on maths, but I was never a constructivist. I'll try to be patient though, I guess I sometimes have a tendency to make drastic ill-advised edits. -lethe ^talk 19:06, 7 January 2006 (UTC)

Now that both of us have made our postions clear, good luck working on it. :) Oleg Alexandrov (talk) 20:20, 7 January 2006 (UTC)

[edit] more ZF friendly

I changed a bit the general example to be less non-constructive. :DD —The preceding unsigned comment was added by Scineram (talk • contribs) .

And I undid your change. I believe that it is an unnecessary complication to insist on constructivism. The axiom of choice is a well-established tool in functional analysis, and I see no need to avoid making use of it. As such, any infinite dimensional space has an infinite sequence of linearly independent elements, no need to restrict oneself only to vector spaces for which such a sequence can be explicitely constructed. Oleg Alexandrov (talk) 19:18, 27 February 2006 (UTC)

Except as I guess you already know, I disagree with you, Oleg, and agree with Scineram that this page may have interest not only for analysts (and also think that even analysis articles should note the use of AC). It's on my to-do list. -lethe ^{talk +} 19:53, 27 February 2006 (UTC)

Enjoy working on that. :) Oleg Alexandrov (talk) 19:59, 27 February 2006 (UTC)

[edit] well

I've done some changes. I restored the reals over rationals example, as I think it's important. I've added a section near the bottom about choice considerations, and I tried to refrain from mentioning the axiom by name as much as possible throughout the rest of the article. Was it enough? -lethe ^{talk +} 17:46, 8 March 2006 (UTC)

I wanted to incorporate scineram's recent comment into the section on AC that I've added to the article, but I can't convince myself that the comments are right. He claims that given an infinite sequence of independent vectors, define the operator to be 0 outside the sequence. But does this work? It contradicts my statement that there are no constructible examples over a complete domain. On the other hand, I don't see why it shouldn't work. scineram, you're not still around, are you? -lethe ^{talk +} 18:15, 8 March 2006 (UTC)

So you need choice to extend a linear map on a subspace to a linear map on the whole space. You need choice even if you want it to be 0 outside the subspace. So I think scineram's comments for making the construction ZF friendly are not correct. -lethe ^{talk +} 18:33, 8 March 2006 (UTC)

[edit] completeness not relevant

I'm not sure why you say "completeness has nothing to do with it". It's not not an equivalent condition, that's true. Incompleteness does not imply constructibility. On the other hand, completeness does imply no constructibility. Therefore to say "it has nothing to do with the constructibility" cannot be right. Why do you think so? Would you rather an equivalent condition were mentioned in the intro? I don't know a short one, but I do think that completeness is an important criterion here. -lethe ^{talk +} 23:43, 8 March 2006 (UTC)

Sorry, I was terse. The thing is that the statement you put

If the space is complete, however, such maps can be proven to exist but not explicitly constructed; their existence relies on the axiom of choice.

is not correct. You can find such a map easily say in L^2(R), by considering the sequence of characterstic functions of the intervals [n, n+1] (that's the key step requiring the axiom of choice usually, but not here).

I belive this example is more constructive and more useful than the one where you can find a strange discontinuous map because your space fails to be complete, no? Oleg Alexandrov (talk) 01:26, 9 March 2006 (UTC)

You still require choice for that example. To extend your operator. And I don't consider the derivative to be a "strange" map. -lethe ^{talk +} 01:27, 9 March 2006 (UTC)

Just so we're clear on this, the work of Gamir constitutes a proof that choice is required for discontinuous maps on complete spaces. So please don't look for constructive examples; there aren't any. -lethe ^{talk +} 01:31, 9 March 2006 (UTC)

Oops, you are right, sorry. :) I guess I take the axiom of choice for granted (it's a free country, choice is a given :)

Allright, the article reads well so far, and I just hope you won't be tempted to preach the constructibility bible too loudly. :) Oleg Alexandrov (talk) 01:38, 9 March 2006 (UTC)

Thanks. I'm pretty pleased with how it's turned out, but we can tone down the constructivism. Do you think there's too much currently? I tried to confine all such discussion to a single section. -lethe ^{talk +} 01:41, 9 March 2006 (UTC)

[edit] inline tex

I really don't like inline tex, but I don't know how to make an overbar without it. So I used it in my new addition. I guess maybe there are alternative notoations for closures. -lethe ^{talk +} 23:17, 14 March 2006 (UTC)

Inline TeX is fine, as long as it does not become inline PNG. The math style manual about avoiding inline PNGs is just a recommendation, so if PNGs can't be avoided, then so be it. :) Oleg Alexandrov (talk) 00:38, 15 March 2006 (UTC)

Well I perversely have my settings set to "always render PNG", which perhaps explains why I dislike inline TeX so much. Also, did you see that in that WP:MSM, they say that the phrase "note that..." or "it should be noted that..." are to be avoided? I use those all the time. Yep, I'm definitely a bad person. -lethe ^{talk +} 02:08, 15 March 2006 (UTC)

I don't argue that you are not a bad person, you are indeed. :) But the reasons are not the ones you present. You don't like the axiom of choice, that's what makes you baddest person. :) Oleg Alexandrov (talk) 02:30, 15 March 2006 (UTC)

[edit] A few questions

The paragraph:

In functional analysis, one often considers operators defined only on a proper subset of the domain. Every discontinuous linear operator on an incomplete space can also be viewed as a densely defined operator on the completion of the space. It will still be discontinuous, but not everywhere defined. Discontinuous operators on complete spaces that are defined everywhere on the domain require a little more work.

from the text is in my view somewhat confusing. I don't see the need to view a disctontinuous operator on an incomplete space as an operator on a bigger complete space on which it is still discontinous. I would suggest all this paragraph be replaced by one sentence saying that now we will construct discontinuous operators on complete spaces; without going in the complications of not everywhere defined operators and completions. Oleg Alexandrov (talk) 23:12, 19 March 2006 (UTC)

Yes, I see your point, and I agree. I guess I introduced that text because of a point you made to me in our discussion: the derivative example isn't quite as interesting, since it's not defined on a complete space. I then learned about the theorem, which implies that the derivative will never be defined on a complete space, which surprised me a little. I guess the business about viewing it as an operator on its completion was my attempt to convey the dichotomy between the complete and incomplete cases. But the way the article is now, that idea is discussed later on. I don't remember if that was the case when I introduced that part, but anyway, you're right, it doesn't really belong there. Take a look at my attempt to redress your concern. By the way, how do you feel about the section on closed operators? -lethe ^{talk +} 09:50, 20 March 2006 (UTC)

Thanks for addressing the issue. I made some small changes in the section about closed operators. It looks good. Cheers, Oleg Alexandrov (talk) 03:00, 21 March 2006 (UTC)

[edit] twma's example

A few days ago, someone came to the math reference desk (eventually it'll be in the March 2006 archive) looking for help on finding a discontinuous linear map. I directed him or her to this article, but he was particular about having one for the space of smooth functions of compact support (the space whose dual is the space of distributions). This space is an LF-space, so would be an example of a discontinuous linear functional on a space more general than a Banach space. After a few days of discussion, he came up with this example. I don't feel particularly compelled to include it in the article, (and I don't expect you will either Oleg). It takes a bit longer to construct than the corresponding Banach space, and in its essence, is no different from the one we already have: make a convergent sequence of independent vectors, then extend to a basis. Nevertheless, I think it's a nice example, and once you unpack the construction, it's pretty simple. So I thought I'd include it on the talk page here, for the sake of posterity. Who knows, maybe at some point in the future, it will make sense to put it in the article. -lethe ^{talk +} 16:33, 20 March 2006 (UTC)

Here is an example of a discontinuous linear form on the test space D(R) on the real line R. For every compact subset K of R, D(K) is the vector space of all test functions with support contained in K. The topology of D(K) defined by the seminorms $\|\varphi\|_\alpha=\sup\{|(d/dx)^\alpha\varphi(x)|:x\in K\}$ for all integers α ≥ 0 coincides with the subspace topology induced by the test space D(R) equipped with the locally convex inductive topology induced by D(K) for all compact subsets K of R.

Let ρ be a test function with ρ(0) = 1 and ρ(x) = 0 for all |x|>1. Choose 0<a_n+1<b_n+1<a_n<b_n ≤ 1 for all n ≥ 0, for example recursively by b₀=1, a_n=b_n/2 and b_n+1=a_n/2. Let c_n=(b_n+a_n)/2 and r_n=(b_n – a_n)/2. For g_n(x)=ρ[(x – c_n)/r_n], we have |g_n|₀ ≥ g_n(c_n) = 1. Then each $f_n=\frac{g_n}{n\sum_{\beta=0}^n\|g_n\|_\beta}$ is a test function with support contained in [a_n,b_n] ⊆ K=[-1,1]. For each α and all n > α, we have ||f_n||_α ≤ 1/n → 0. Now f_n → 0 in D(K) and hence also in D(R). Next because all intervals [a_n,b_n] are disjoint, the sequence {f_n} is linearly independent and hence it can be extended to an algebraic basis B of D(R). Define T(φ)=1 for all φ in B and extend T to a required linear form on D(R). Twma 08:24, 19 March 2006 (UTC)

I agree that it is not worth including, at least for now. I think it just provides a concrete way of constructing a sequence of linearly independent vectors in a space a bit more general than Banach, everything else seems to be the same; that if I read it right. Oleg Alexandrov (talk) 02:53, 21 March 2006 (UTC)

[edit] Beyond normed spaces

The article now has the sentence:

All of the comments for the existence of discontinuous linear maps on normed spaces apply to any topological vector space; given an appropriate basis, one can define an unbounded operator, and unbounded operators are discontinuous for any TVSes.

I am not sure that's true. Take an infinite-dimensional real vector space, and consider the discrete topology on it, such that every set is open. Then, any linear map from that space (actually, any map at all) is continuous. No? Oleg Alexandrov (talk) 15:43, 23 April 2006 (UTC)

Yeah, I guess all maps are continuous on a discrete space. I guess the problem is that a discrete space is not locally convex. -lethe ^{talk +} 23:43, 23 April 2006 (UTC)

It is locally convex, isn't it? The singletons form a basis for the topology and are convex. A Geek Tragedy 19:13, 5 March 2007 (UTC)

[edit] equivalence of norms

The equivalence of norms in the proof that all maps on finite-dimensional spaces are continuous is used for convenience only. it is not essential to the proof, which could be modified to work with any norm with more work. I just thought the proof was simpler, easier to understand, using the l1 norm. The real reason the proof fails for infinite dimensions is that the norm can't be bounded, it has no supremum. Given any norm, any linear map for which the supremum does exist will be bounded and therefore continuous. Only maps for which this supremum does not exist will be discontinuous. This is seen in the construction of discontinuous linear maps as well, where you have to construct a sequence for which sup ||f(e_i|| does not exist. In short, finite sets have suprema, infinite sets need not. -lethe ^{talk +} 05:37, 13 May 2006 (UTC)

I wonder how to modify the proof. It seems to me that the l1 norm is the largest norm, so I don't know how to make the inequality hold for smaller norms. Maybe this is another way to beat the proof? I stand by my comment that the failure of the supremum is the "important" reason, whatever that might mean. -lethe ^{talk +} 05:53, 13 May 2006 (UTC)

Good points. I tired to tweak the proof a bit. Wonder what you think. Oleg Alexandrov (talk) 06:03, 13 May 2006 (UTC)

I like the new edit. It does seem a little unnatural to assume a particular norm at the outset, and this makes it clearer exactly what that norm does for us. But I'm still curious, can we beat the proof this way? Is there a linear map which is unbounded in the l1 norm, but bounded in the l∞ norm, say? I'm gonna go to the bar and think about it. -lethe ^{talk +} 06:12, 13 May 2006 (UTC)

[edit] Finite rank

Per this edit, any linear map defined on some infinite-dimensional normed space with real values is continuous. But that's not true, I thought that this article was providing a counterexample. :) Oleg Alexandrov (talk) 23:17, 14 May 2006 (UTC)

Geez, I don't know what happened there. Gone for a six. -lethe ^{talk +} 10:52, 15 May 2006 (UTC)

[edit] strong vs weak

It says that BP is a weak negation of AC. But not-LM and not-BP are weak consequences of AC, thus BP and LM are STRONG negations of AC. They are at the bottom of the tree. http://www.math.vanderbilt.edu/~schectex/ccc/excerpts/acchart.gif

(I've copied this here from User talk:Lethe; this is probably the more appropriate forum.)

There are a couple of really odd things about the way that section is worded. For one thing, the Baire property is not any form of negation of the axiom of choice; it's a property of sets of reals (or more generally, subsets of a topological space). The "strong" negation of AC would be that every set of reals has the Baire property; the negation is not the Baire property itself.

Perhaps worse is the line about how

Solovay's result shows that it is not necessary to assume that all infinite-dimensional vector spaces admit discontinuous linear maps,

What's that supposed to mean, exactly? I'd say it shows almost the opposite—namely that you do have to assume something, beyond ZF, if you want to prove there are always such maps. (It certainly doesn't imply, by itself, that there don't have to be such maps, for if you believe AC is true—as Solovay, of course, does—then you'll believe there always are such maps.)

Finally, there's a problematic remarks about how such maps always have to be "nonconstructible", without specifying what "constructible" means. --Trovatore 04:16, 7 August 2006 (UTC)

I strongly have no clue what you guys are talking about (admittedly, that's a weak point). Oleg Alexandrov (talk) 15:45, 7 August 2006 (UTC)

[edit] Functional Analysis Article

"It turns out that the derivative of a function at a certain point is really a CONTINUOUS linear map." I'm quoting from the Functional Analysis article. (I added the caps to continuous.) Is the quoted sentence true?Rich 07:30, 19 September 2006 (UTC)

that short paragraph is somewhat outta place, IMHO. the Frechet derivative of a map between Banach spaces at a given point, if it exists, is by definition a bounded operator. on the other hand, the differential operator on, say, C[0,1] with the sup norm, is definitely not bounded. Mct mht 07:59, 19 September 2006 (UTC)

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