Talk:Bounded operator

From Wikipedia, the free encyclopedia

WikiProject Mathematics
Mathematics Portal
This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating: Start Class Mid Priority  Field: Analysis

Does the Laplacian map onto L^2? ie is L^2 this the range or is it simply the codomain? Lupin 14:19, 2 Mar 2005 (UTC)

Good point. I think it could be proved that under certain conditions the Laplacian is actually surjective, but this could be hard and is not the point of the example. So I reworded the example to not imply the Laplacian is surjective. Oleg Alexandrov 16:04, 2 Mar 2005 (UTC)

Contents

[edit] Defined versus bounded

I think you are making a mistake here. An operator which is defined on its domain is not necessarily bounded on that domain. It is essential that the M in the estimate must be independent of the v.

Do you refer to the first paragraph? In the inequality there, I think it was clear that M is independent of v. I also made a small change for clarification. I wonder what you think. Thanks. Oleg Alexandrov 15:49, 15 Mar 2005 (UTC)

Sorry for having bothered you. I didn't look carefully enough, your text appears correct to me now. You can delete this comment if you want. Bas Michielsen.

[edit] Continued

There exists M such that BLAH BLAH BLAH for all x.
There exists M such that (BLAH BLAH BLAH for all x).
(There exists M such that BLAH BLAH BLAH) for all x.
There exists M such that for all x BLAH BLAH BLAH.
For all x there exists M such that BLAH BLAH BLAH.

The first version above is ambiguous! It can mean either the second version or the third. The last two are unambiguous. That is why my recent edit is important. That may explain the misunderstanding above. Michael Hardy 01:49, 16 Mar 2005 (UTC)

Thanks, looks better the new way. Oleg Alexandrov 01:59, 16 Mar 2005 (UTC)

[edit] Rename page to continuous linear operator

At the moment this page deals with continuous linear operators between normed vector spaces which are exactly the bounder linear operators. This is not true in general and continuous linear operators between topological vector spaces need not be bounded. Continuous linear map redirects here so either we create a new article continuous linear operator or we rename this article to continuous linear operator and rewrite the content. I would prefer the second option and treat bounded linear operator as a special case of continuous linear operator but this might increase the abstractness of the article. Any opinions (especially from a physicist) ? MathMartin 11:07, 25 Apr 2005 (UTC)

Well, I am not a physicist (an applied mathematician if you wish). My own opinion, is that this article should not become more general and more abstract. I have some grasp on linear topological spaces and all those exotic definitions of bornologic things and stuff, but I always thought those were very abstract beasts hard to get an intuition on.
I would rather suggest a new article be created, called say continuous operators in topological vector spaces or so. This because, as you yourself mentioned, things can become quite subtle in such a space, and, I think many exotic things can happen which do not happen in a normed space. In that new article all the fine details can be tackled, while bounded operator would stay the way it is, discussing the much simpler and the much more widespread case of normed spaces.
That's just an opinion. Even though I wrote a good chunk of this article, I will not mind if it is decided to have it rewritten. Oleg Alexandrov 15:26, 25 Apr 2005 (UTC)
It's hard to form an opinion without knowing what MathMartin wants to write about, but given the two options presented I'd favour the creation of continuous linear operator, which starts with a note that continuous is equivalent to bounded for normed spaces (with a link to this article) and then goes on to talk about the general case. -- Jitse Niesen 22:04, 25 Apr 2005 (UTC)
What a short and concise way to say what I was trying in three paragraphs. :) Oleg Alexandrov 22:15, 25 Apr 2005 (UTC)
Ok, perhaps it is better to create continuous linear operator discussing the general case and link to this article. I will create just a small stub and flesh out the article in the next few weeks. I am currently working on topological vector space articles and I need some place to talk about the morphisms (linear continuous operators) between those spaces. Can you given me some hints on the terminology ? Is operator still the correct word to use in this context or would it be better to use mapping or morphism ? MathMartin 11:13, 26 Apr 2005 (UTC)
New continuous linear operator article is fine with me. Don't know what to say about the terminology though. I thought MFH knew something about these things. You could let us know when you start the article, so that I can put it on my watchlist. :) Oleg Alexandrov 17:25, 26 Apr 2005 (UTC)

Whatever is decided about the page and whether the focus is on topological vector spaces, the statement in the article that "A linear operator is bounded if and only if it is continuous" is wrong, unless you qualify it correctly (e.g. it's true in a normed space, I think). Lavaka 06:11, 17 September 2006 (UTC)

[edit] Error in article?

A reader emailed the following comments to Wikipedia. Please review and correct the article, as appropriate:

I think there is a mistake in the article at Bounded operator. It is claimed there, in the fourth example, that "if a linear operator L has as domain and range Banach spaces, then it will be bounded". This is false, as shows the following (quasi-explicit) example: consider l^2, the vector space of square summable sequences (u_n)_{n\in\mathcal{N}} (of real numbers); it has a natural scalar product which makes it a Hilbert space, and a natural Hilbert basis (the canonical basis (e^k)_{k\in\mathcal{N}}, e^k_n = 0 if n is different from k, 1 if n=k). The axiom of choice allow to consider an algebraic basis of l2 containing (e^k)_{k\in\mathcal{N}}. Now, the linear map A from l2 to \mathcal{R} which sends ek to k + 1 and sends other elements of the basis to zero is well-defined, linear, one-to-one from l2 to \mathcal{R}, but not continuous.

I suppose that the mistake of the author comes from Banach's theorem, which asserts "any linear one-to-one continuous map from a Banach space to another has continuous inverse".

Let me add an erratum to my previous mail. In the end of the counter-example I gave, don't read "the linear map A from l2 to \mathcal{R} which sends ek to k + 1 and sends other elements of the basis to zero is well-defined, linear, one-to-one from l2 to \mathcal{R}, but not continuous", but replace "one-to-one" with "onto" --I think this is now a correct counter-example.

(Please excuse any errors in the markup; I don't understand a word of this.) Kelly Martin 14:31, 5 October 2005 (UTC)

It was many times when I thought of that example 4. It looked suspicious to me and I never managed to prove it. But for the record, it was not me who wrote it, I just moved it from operator norm. See this version of operator norm, before I ever contributed anything to that article. Oleg Alexandrov 16:34, 5 October 2005 (UTC)
Actually, I doubt that the linear map in the mail quoted by Kelly is well defined. Consider the sequence (un) with un = 1 / (n + 1)2. This sequence is in l2 (and even in l1), but Au = \sum_{n=0}^\infty (n+1)u_n = \sum_{n=0}^\infty \frac{1}{n+1} is not defined. -- Jitse Niesen (talk) 17:14, 5 October 2005 (UTC)

The map is rather well defined I think. The element u you mentioned above can be written as a finite linear combination of elements e_n of the orthonormal Hilbert space and of other elements which complete the orthonormal Hilbert space to an algebraic basis. Oleg Alexandrov 17:22, 5 October 2005 (UTC)

Hmm, you have a point. I ignored the expression "algebraic basis", which I did not understand, but I see now that this is what I'd call a Hamel basis. -- Jitse Niesen (talk) 18:16, 5 October 2005 (UTC)

[edit] Bounded operator

Hi Taku. I am really not sure that the proof of equivalence of bounded operators and continuous operators belongs in the very introduction to that article. Actually, I would argue that it does not belong anywhere in the article, except maybe at the bottom, as it obstructs the flow of the article. So I think it is good that you removed that. Oleg Alexandrov (talk) 05:06, 14 January 2006 (UTC)

I don't know why the proof is not important. It's not that trivial for those who haven't studied it, and that a bounded operator is continuous and vice versa is an important fact if it is not surprising. Besides giving some proof is the best way to show that the continuity at one point is the sufficient condition, which in my opinion should be mentioned. -- Taku 05:46, 14 January 2006 (UTC)
Yeah, I agree it looks good where you put it now. Some more wording may need to be changed, I will see if I get to it tomorrow. Oleg Alexandrov (talk) 06:32, 14 January 2006 (UTC)
Sorry, Oleg, for nicking this task away. By the way, I agree that the proof should be included because "bounded \iff continuous" is such an important equivalence. I removed the sentence
"In general, a linear operator L is continuous everywhere if and only if L is continuous at some point x0. For if L is continuous at x0, then L is continuous at 0 since L(x + h) − L(x) = L(h). It then follows for any vector x, \| L(x_0 + h) - Lx \| = \| Lh \| goes to 0 as \| h \| goes to 0."
since I am not sure why it is needed. -- Jitse Niesen (talk) 15:26, 14 January 2006 (UTC)
I thought that the exposition was not nicely put too. But again I think the article needs to cover more elementary facts related to continuity. I was going to check but I think we can show the continuity is actually uniform continuity. Those basic facts. I also think the article can use an exposition from the point of view of a neighborhood. That is, more topological approach. That can help clarify the concepts. Just some ideas. -- Taku 12:10, 15 January 2006 (UTC)
I think the fact that continuous somewhere implies continuous everywhere belongs at continuous linear operator, as the more topological approach (whatever that means). I saw that article was not linked, so I fixed that. -- Jitse Niesen (talk) 14:33, 15 January 2006 (UTC)
Agree with Jitse. Oleg Alexandrov (talk) 17:35, 15 January 2006 (UTC)
Obviously, I didn't know that article, and didn't read the previous discussion. I agree that technically continuous linear operator does not have to be defined between normed spaces (I mean true all you need to define continuity is some sort of topology), but I wonder when I see the term continuous linear operator I always sort of tacitly assume that is defined between normed spaces. I am not suggesting a merger, so this is just a thought :) -- Taku 22:45, 15 January 2006 (UTC)

[edit] A to denote linear operator instead of L

A minor typographical proposal. Can we use the letter A to denote a linear operator in the article as par operator norm. I also think A is more common than L. (I noticed people use L to denote a set of linear operators or for the matter, bounded linear operators but not a linear operator in general). -- Taku 22:51, 15 January 2006 (UTC)

I don't care either way. As long as you change all, to keep consistent. Oleg Alexandrov (talk) 02:11, 16 January 2006 (UTC)

[edit] Lipschitz

I did this revert, as to me it appears that introducing one more notion, of Lipschitz continuity is not helpful. The introduction I think should be a simple as possible, and the point of the two original short sentences was to

  • Make it clear that bounded operator ≠ bounded function
  • linear bounded operator is the same as linear continuous operator

I don't see how the new concept of Lipschitz continuity helps with that; the reader may not even know what that is, and I think it distracts from the point the intro is trying to make. Comments? Oleg Alexandrov (talk) 03:02, 10 March 2006 (UTC)

The reason I made the change is this: the article currently states that the continuity is a consequence of the linearity of the bounded locally bounded function. This is not correct, or at least it's misleading. When one recognizes that the boundedness condition is simply a strong continuity condition, then continuity is automatic. Now, if you want to keep the intro as simple as possible, why is there this stuff about locally bounded functions? I consider locally boundedness more obscure than Lipschitz which seems somewhat common to me. Maybe you want to mention local boundedness so that the name bounded doesn't confuse people. I want to mention Lipschitz so that continuity seems natural. But I could be persuaded to move it down the article. You're not arguing for complete noninclusion, are you? -lethe talk + 04:06, 10 March 2006 (UTC)

Verbatim, the article used to say:

A consequence of the linearity property is that a linear operator is bounded if and only if it is continuous.

It says just that, a linear operator is bounded if and only if it is continuous. Now I removed the words "A consequence of the linearity property", so I think it is more clear.

Yeah, including something about Lipschitz further down may be good, although I don't see why you would do that. Maybe except if you wish to provide a complete proof of linearity+local boundedness= linearity + continuity. Oleg Alexandrov (talk) 04:24, 10 March 2006 (UTC)

I think it's a nice thing to include, but I agree with Oleg that it is not as important as local boundedness. Local boundedness needs to be mentioned early on because it answers the question "why is it called bounded if it is not bounded as a function?" I added a sentence on Lipschitz continuity further down. I'm not quite happy with the location though. -- Jitse Niesen (talk) 05:46, 10 March 2006 (UTC)

[edit] Old error

I was reading the discussion above about the apparently erroneous statement that all linear operators between Banach spaces are bounded. After all my recent work with discontinuous linear map, I feel I understand these sorts of questions better; clearly the statement is wrong if we assume AC, and it appears to do so, since the Baire category theorem uses AC. So this isn't a constructivism issue. The material was originally added to operator norm by Charles Matthews. I'd be pleased if we could fix his work and find a home for it. In particular, this notion of "condensing singularities" sounds fun to me. Now, as for what he was going for with this, the closest I could find is the uniform boundedness theorem, which relies on the Baire category theorem, and says that a family of maps is pointwise bounded if and only if the family is uniformly bounded. For a single map, I guess this says that a map is bounded if and only if it's bounded, so this isn't what he was going for. -lethe talk + 11:34, 10 March 2006 (UTC)

At that point I was probably trying to reconstruct something that appears in one of the Halmos books, for which you need to find another Halmos book to get the proof. This point has been discussed before though, and I thought was corrected. I wonder where that was ... Charles Matthews 11:42, 10 March 2006 (UTC)
Would that probably be his Hilbert space book then? -lethe talk + 11:46, 10 March 2006 (UTC)