Validating Originality in Mathematical Proofs: A Guide for Researchers

[Jimmy Snyder]

I believe I have a new method of proof for an already existing theorem. The theorem itself is quite elementary and the standard proof for it, that I have seen in several books, is not difficult. Nonetheless, I believe my proof is simpler yet. My question is: How do I find out if my proof has ever been published before?

In addition, I have a proof for what is a new theorem to me in the sense that I have never seen it before. How do I find out if it is really new? Moreover, I don't know if it really rises to the level of new "theorem". It's also quite elementary and the kind of thing that could be found in a textbook in the exercises. How do mathematicians tell the world when they have a new but elementary theorem?

 

[matt grime]

Well, firstly it would help if you stated the theorem. Ideally offering the proof would be best too. No one will steal it. We don't tell the world of new elementary theorems. If the style of proof is novel then it may be of interest, but you should apply it to more complicated areas. Contrary to popular opinion, you do not publish everything you discover. That would be almost certainly unhelpful.

There are various journals of recreational mathematics that may be interested if it is indeed a novel approach. Or the proof may be useful for pedagogical reasons.

 

[Jimmy Snyder]

Well, firstly it would help if you stated the theorem.

Thank you for responding. And please accept my apology for not satisfying your natural curiosity in this regard. If I did as you suggest, then you would certainly be able to do the research for me and give me the verdict. If you mean that you would be willing to do that, then please be assured that I do appreciate the offer. However, at this time I do not think I will avail myself of your services. I asked because I want to know how to do the research myself. Is there a repository of proven stuff? Is there some way to systematically search what is known?

Contrary to popular opinion, you do not publish everything you discover. That would be almost certainly unhelpful..

This rings true and the thought had occurred to me. My "theorem" looks more like something you would find in the exercises section of a textbook than stated as a theorem. However, it is not hard to imagine that there could be practical applications (meaning easier proofs of other theorems) for it. I just can't think of any right now. And even as a exercise it still might be new. So what should I do, call up the author of the next textbook in the field and suggest that they include my result as an exercise? How would I know if someone else had already done the same thing and scooped me? Is there a repository of exercises that textbook authors can go to for ideas?

 

[Icebreaker]

If you are writing a math book, and the theorem in question is called up, you can certainly use your own proof. Proving one single elementary theorem (that has already been proven) certainly isn't enough to publish a paper. But as recreational math, it's certainly fun. Why don't you post it here?

 

[Jimmy Snyder]

If you are writing a math book, and the theorem in question is called up, you can certainly use your own proof. Proving one single elementary theorem (that has already been proven) certainly isn't enough to publish a paper. But as recreational math, it's certainly fun. Why don't you post it here?

These are good insights.

I still wait for an answer to my questions. How do you do this kind of research? Has anyone here published a paper. How did you determine that your result had not already been published? Did you just submit it and let the publisher do the search? If so how does the publisher do it?

 

[DeadWolfe]

Matt Grime is a professor, and so I assume he has published some papers.

BTW Matt, what do or have you done research in?

 

[shmoe]

The easiest way is to ask an expert in the area. This is essentially what happens when you submit an article to a journal and it gets sent off to the referees.

You can use MathSciNet to search for all relevant articles on your topic. That's probably the easiest way to find material on your own. I think you may need a subscription to access it though, I've never tried from a non university network so I'm not positive.

 

[mathwonk]

well since you do not want to submit your result for our judgment, you might submit it to a journal, and get a referee's report. but they will only send it to someone like us for an opinion.

Experts know so much about a given area that they are very likely to know whether the result is new or not. In fact even some of us would probably be able to tell you immediately whether your result is new or not, if it is anywhere near our area.

Recently when I discovered a proof of something that had been around 15 or 20 years, I wondered if the experts had known it before, so I asked one of them. It turned out it was new. Sometimes what I have done is not new, but I try not to be discouraged even so, because usually I get enough momentum from the discovery to continue further and do something soon that is really new.

The same thing happened to someone else I became acquainted with recently. He asked me about something he had done in my area and I was not impressed because I had also done it earlier, and by a much easier method, and never bothered to publish it. On the other hand he pushed it a little further and published it, in a nice little paper, but still nothing to impress me.

Then however he took up an unsolved conjecture of mine, I read his work and made some suggestions, and before long he had actually solved it, thus going beyond what I had been able to do, by his methods. I.e. he had some different methods, and eventually they yielded something new. Then I was impressed.

So i suggest you try to publish your result and see what happens. Even if some expert has done it and not published it, that does not do the rest of us any good. We who have not done it would enjoy seeing it published.

Some of my small remarks, improvements on existing results, have appeared as credited remarks in books by others. Some of my contributions have been made spontaneously from the audience during talks at meetings, and have appeared without credit in other people's works. Some of my proofs of known results I have told people over the phone and they have wound up in others papers.

If this sort of thing would annoy or frustrate you, then you should try to put your own name on your ideas in some way before giving them away. But if you are like me, you may find that communicating your ideas actually stimulates more creativity, even if it costs you some credit. In this situation, it seems better to try to remain as creative as possible, even if someone else gets some of the credit.

 

[Jimmy Snyder]

you might submit it to a journal, and get a referee's report.

While the thought of someone 'stealing' my idea certainly doesn't appeal to me, that's not my real issue. I don't want my idea to die on the vine.

Yours is the kind of answer I was looking for. But how do I package my idea. I could write it up like a full-blown article with a title, an abstract, references, etc. But that seems a little pretentious for such a squeeky little mouse like I've got. Can I just send a letter to the editor? If they read it and find merit in it what will they do? Publish the letter as is (in that case I better make the letter nearly as formal as an article). Will they dress it up and publish it in their own format? And the $64,000 question. Will they try to determine if it is new?

You said they might send it to this forum for that purpose. If they do, what will you do?

In hindsight, it seems that I could have gotten this far much faster if I hadn't mentioned that I had an idea, but simply asked how this kind of research is done.

 

[matt grime]

First, to clarify, I am certainly not a Professor, not even with the American meaning of the word.

The simplest thing to do is to ask someone knowledgeable in the area if it is new. (I've no idea why discussing it would make it "die on the vine".) If they do not know then they may know someone who does. Soon it will become apparent what is known in the area, and if it is publishable, but if it's only a proof of one small result then almost certainly it will not be of interest to a research journal: they stake their reputations on publishing important new pieces of mathematics.

A recreational journal may be best if you really want to write to someone about it, and a letter may be a good way of doing it, a note to the editor. It will not be rewritten, though it will be proof read and reviewed - any changes you will be asked to make (eg, putting it in a format they can publish).

Google can be your friend too, and someone has mentioned math.sci.net the AMS search engine. There is also Arxiv.

After submission it will not be submitted to a forum, it will be sent to a professional mathematician such as Mathwonk to review and decide if it meets certain criteria (that will often depend on the journal). Some are more lenient than others. As the result is not new, then almost certainly you will have to demonstrate the novelty of the proof, which is why I think a recreational journal may be more relevant.

I would perhaps suggest sending it privately to me, or if he we were to agree, mathwonk. One of us will at least be able to tell you where to look to see if it is new. Even if it is just the statement of the theorem.

So, to sum up the adivice:

Simply ask someone who might know.

 

[Jimmy Snyder]

While the thought of someone 'stealing' my idea certainly doesn't appeal to me

I wish I hadn't written that as it is entirely misleading. If by 'steal' you mean quote without attribution, I find that results like mine (if indeed it is 'my' result) are routinely quoted without attribution. My object is not to patent my idea but to get it 'stolen' in the most expeditious manner.

It is conceivable that when the next author gets ready to write the next elementary textbook in functional analysis, they will have come across my little piece. If so, it is conceivable that they would replace the 'standard' proof with mine because my proof is simpler. If they were to mention my name, of course it would be a feather in my cap, but it is pointless daydreaming to think that they actually would. They don't attribute the current standard proof to anyone as it is now.

What is the best strategy to get my idea to the attention of that mathematician who is just about to start writing an elementary textbook? Shall I dash off a letter to the editor of "The Journal of Functional Analysis"? Will they put it "out there" so that people will see it? Will they take the time to find out if this idea has already been seen? If so, how?

 

[Icebreaker]

James Stewart revises his Calculus books once in a while. Try contacting him if you think he might be interested in your idea.

http://www.stewartcalculus.com/index.php

edit: that website doesn't seem to offer a way of contacting him, but perhaps you could try contacting McMaster university, where he resides.

 

[Jimmy Snyder]

As the result is not new, then almost certainly you will have to demonstrate the novelty of the proof

How would I go about doing that?

 

[matt grime]

Writers will try and give the credit to the author of a proof if it is known who came up with the proof originally, however, many of the results in the textbooks you're thinking of were written at least 100 years ago and have been constantly revised and improved by many different people at many different times, so it no longer makes sense to attribute it to anyone.

If you do indeed have a novel and simple proof of a standard result, then many people will be keen to use it and will give credit where it's due. And the best way to do that is to simply tell people about it. Also remember that ideas like this can be reproduced for academic purposes using the "fair use" clause, so you cannot expect it to be only ever used with your name attached. That we try and go out of our way to give credit where its due says something about mathematicians.

 

[Icebreaker]

The suspense is killing me! Now I'm really curious about this proof. Get it published!

 

[Jimmy Snyder]

James Stewart revises his Calculus books once in a while. Try contacting him if you think he might be interested in your idea.

That is a really good idea. I am going to pursue this angle. As the field is functional analysis, I should think John Conway might be a good choice. He would know if the idea was new, and he might even know the name of the author of the next textbook. Unless someone here suggests another better name, that is exactly what I am going to do.

 

[mathwonk]

I suggest sending it to michael spivak, as he is to me a more authoritative calculus author than stewart. I.e. Stewart is a good author, but when you really want an expert opinion, I recommend sending it to the best.

Spivak is also a published and acknowledged researcher in differential topology, having studied with both Raoul Bott and John Milnor. To this day his name remains attached to the construction of the "Spivak normal fibre bundle", which is an abstract way of associating a "normal sphere bundle" to a Poincare manifold with boundary which is not actually embedded anywhere, as I recall. [see Topology (6), 1967., apparently his PhD thesis]

best wishes

or just private message it to Matt. Matt has a laudable startup project of assembling original approaches to known results, and is likely to know many different proofs of them.

Another expert, and friend of MIke Spivak, is Theodore Shifrin, University of Georgia, also a top instructor, published researcher in differential geometry, and distinguished textbook author.

I am a mathematician, but I feel kind of like a narrow specialist, and mostly know about my little area, and not even the latest stuff in that.

OOps, sorry, I had not read your last post. Conway sounds like a great choice. Of course now that I know it is Functional Analysis, you might look first at Riesz Nagy, and Lang's Analysis II, and Loomis' Abstract Harmonic Analysis, maybe Yoshida, and even Courant-Hilbert, and some newer books, just to see if it really is different from the usual standard presentations.

Sorry, I did not mean to say they would send your idea to this forum, I meant "us" in the sense of us professional mathematicians, just assuming there are others here.

I can tell you now though that if you have say a new proof of the open mapping theorem or the closed graph theorem or something minor like that, it is not publishable research. That is the kind of thing someone might mention in class, as "I just made this up for you guys". But new facts about Fredholm operators e.g. are certainly still an area of research.


By the way your title was a misnomer, you asked not how to do research but how to publish putative research. As to how to do it, there is a lesson in your experience to date. Namely your creative mind tends to exp[lore possibilities suggested by what you read and hear, and try to improve upon it. hence your research is going to be a slight extension oiften of whatever you are reading and hearing. Thus if you mainly read textbooks, you are likely at best to produce small improvements on very well known results.


So to maximize the chance of doing new research, you need to read new papers which contain the latest ideas, and go to meetings where the latest results are presented. To take possibly an absurd example, you had no chance of doing what Wiles did on Fermat's problem if you did not know about Frey's idea and Ribet's result on stable elliptic curves associated to candidate solutions of Fermat.

For example one of my own better results occurred as follows: a famous person asked me to preview a preprint he had received from someone else famous, proving an outstanding and long standing problem in a special case. I read it carefully and learned the ideas, and gave a talk on it at a major university. Then some time later, maybe a year or more, I was able to remember the main idea and use it to help prove something else similar, but technically more difficult. So I benefited both from early acquaintance with the idea and from studying it carefully enough for my talk to remember it and be able to use it in a new situation.

It is very difficult to compete on current topics with people who are well located. E.g. at a top department, you never have to wonder whether what you are doinbg is new, as someone nearby can tell you immediately. Also you can acquire most known results quickly just by asking, so you save enormous amounts of time, and as a result you spend all your own energy doing new things, or trying to.

When back in the real world, you must develop discipline to continue to learn and read and try to keep up in some fashion, more on your own. You form learning groups with people at about your own level. You present things to each other and try to resist discouragement from experts who may bolster their own egos by implying you do not know anything and never will.

It takes work and stamina, and is very rewarding.

In my own case e.g. I need to break my addiction to this forum, stop downloading things I already know to other people, getting simutaneously frustrated with [a few] people getting their noses out of joint about matters they do not understand, and start trying to upload some new ideas and techniques and problems.

Ironically though, it is by giving the research advice above here, that I am hearing it myself.

This site is a nice place to spend some time, all in all.

Peace

 

[Jimmy Snyder]

Thank you mathwonk and all the others who have been so patient with me and given me such wonderful insights and perspectives.

just private message it to Matt.

Actually, I already sent it off to Professor Conway. I don't know if he will get back to me, but whether he does or not, my second step will be to post my stuff here. Please be patient and prepare to be disappointed when you see how minor a thing we are discussing.

I can tell you now though that if you have say a new proof of the open mapping theorem or the closed graph theorem or something minor like that, it is not publishable research.

It's not anything that major, I'm afraid. It's a very basic fact concerning self-adjoint operators.

By the way your title was a misnomer, you asked not how to do research but how to publish putative research.

Well, yes and no. I was not referring to research in the sense of discovering new things, but research in the sense of finding out if something is already known. Like library research. My question is still somewhat open. How do you find out if a result is known? I consider "Ask Conway." to be a partial answer. However, I am beginning to get the sense that there is no better answer.

As to how to do it, there is a lesson in your experience to date. Namely your creative mind tends to explore possibilities suggested by what you read and hear, and try to improve upon it. hence your research is going to be a slight extension often of whatever you are reading and hearing. Thus if you mainly read textbooks, you are likely at best to produce small improvements on very well known results.

As I said, the existing proof for this very well known result is not a difficult one. I was reading a book that stated the result and a hint on how to prove it. I couldn't prove it the way the author intended because, although I didn't realize it at the time, the hint was wrong. So I proved it my way and kept wondering what proof the author had in mind. Then later I found the proof in another book and realized what the author meant to say. I also realized that my proof was simpler, or at least different. Moreover, my proof can be used to prove another theorem that seems potentially useful. On the other hand it's possible that the only use for it is to be an exercise in a problem set for beginners. Or perhaps it is too trivial even for that modest purpose. I don't know, I'm no mathematician. Even at my best, I am unlikely to ever produce any more improvements of any kind.

 

[Jimmy Snyder]

try to resist discouragement from experts who may bolster their own egos by implying you do not know anything and never will.

Yes, well whatever psychological problems these egomaniacs may suffer from, the basic fact remains, doesn't it. In spite of the best efforts of generous mentors, I don't know anything and never will.

 

[mathwonk]

i seem to have given exactly the wrong impression.

anyone who reads actively as you have been doing, and tries to not just assimilate other people's mathematics, but do his own mathematics, is already a mathematician.

you should keep it up.

i am myself extremely proud of my proof in calculus, that the derivative of any function always satisfies the mean value property, whether that derivative is or is not continuous; although this fact has been known for ages and is proven quite easily in standard calculus books (although i cannot find any now).

I just thought my proof was a little simpler, and more natural, and I bask in the accompanying feeling that perhaps I sometimes have a good insight.


here it is: look at rolle's theorem: it says that if a differentiable function takes the same value twice, then it has derivative zero somewhere in between.

well if you think about it that implies the intermediate value property for the derivative of the function.

I.e. if the derivative is positive at a, and negative at b, then f cannot be monotone on the interval [a,b] so f must take the same value twice in there, hence the derivative must equal zero in there.

but that is the intermediate value property, i.e. if the derivative is negatiove somewhere and positive somewhere else, then it must equal zero in between.

no big potatoes maybe, but I am very proud of it, because i thought of it myself.

 

[cepheid]

That was interesting and pretty easy to follow, mathwonk. Thanks.

Just a really obvious question...but thought I'd make sure I had it completely:

This seems to be key idea in going from Rolle's thm => Intermediate value thm:

then f cannot be monotone on the interval [a,b] so f must take the same value twice in there

I got to thinking...does this have to be true? Not monotonic means neither wholly increasing nor wholly decreasing...but both. So if the function is increasing and then later decreasing or vice versa...the only way it couldn't take on the same value twice, I realized is if it had some discontinuity in between...in which case none of the conditions we laid out for the theorem would be satisfied anyway...right? So it certainly has to be true if f is differentiable on that interval.

 

[mathwonk]

thats right. a differentiable function is also continuous, so if it is not monotone (and continuous) then it takes the same value twice by the intermediate value theorem for continuous functions. this is actually somewhat tedious to prove carefully.

 

[matt grime]

Given yo've explained how you came up with the proof, I would be interested in reading it. This web thing at Mathwonk alludes to is, if i get it going properly, intended to contain discussions about mathematics, original (in the 'novel' sense of the word) proofs, and lots of examples and questions for people to use in whatever way they see fit (as a teacher or a student - i keep writing things for teaching purposes, then losing them; this is simply an attempt to keep them all together). And the whole point about novel proofs is that they may often be the best way of teaching something, since they may more accurately reflect our thought processes for getting the result on our own. As I said in an earlier post, a lot of the proofs for results in textbooks have been continually improved and revised, but strictly in the sense of "made shorter and more slick"; they are not necessarily better to learn.

Here's a good example of using something slightly odd (transfinite induction for those in the know) to prove something well known (a continuous function on [0,1] is bounded) that is usually proven by invoking a larger and too powerful theorem (the continuous image of a compact space is compact):

http://www.dpmms.cam.ac.uk/~wtg10/bounded.html

 

[mathwonk]

I enjoyed that link, Matt. Here is the variation of the convergent series proof I came up with recently to present to people with minimal background in topology. it is not too different but I just eliminated all ideas one might learn in a later course, and realized that I am so brainwashed by those ideas (compactness, least upper bounds, convergent sequences) that I forget these arguments are actually very elementary.

The whole thing rests on how one defines a real number. So if we agree with the class that a real number is simply an infinite decimal, then it is easy to write the usual proof without using any advanced concepts.

Assuming a function is unbounded on a closed bounded interval, we want to find some point such that f is unbounded on every neighborhood of that point. We construct that point as a decimal, one entry at a time.


I.e. Let f be unbounded on [0,1], then there is some subinterval of length 1/10 on which it is unbounded, say f is unbounded on [.3,.4]. Then write down the number .3. This is our first approximation to the desired number.


Next subdivide the smaller interval [.3,.4] again into ten equal parts, and one deduces that f is unbounded on some subinterval of [.3,.4] of length 1/100. say f is unbounded on the interval [.36,.37]. then write down the number .36. This is our second approximation.

Continuing we produce an infinite sequence c = .36..., i.e. a real number, with the following properties:

1) the number c = .36... lies in the interval [0,1],
2) on every interval containing c, no matter how small, f is unbounded.


Hence f is not locally bounded at c, so f is not continuous at c. Thus if is continuous at every point of [0,1], then f is bounded on [0,1].

 

[mathwonk]

Diatribe and recommendation:

This proof is so easy looking to me that I fail to see any longer why this argument should be omitted from non honors calc courses, as it is at my university, so I tried to introduce it. What I found was that my class had great difficulty even understanding the meaning of the word "bounded". They thought in terms of the boundedness of the domain interval, and had trouble realizing that the boundedness of a function referred to the interval of values.

The result was we spent the whole hour trying to clarify the meaning of the term "bounded", which I also believe to have been a good outcome, as we always expect our non honors classes to at least grasp the statement of the theorem I was proving.

I may be admitting my own unique weakness here, but it is tremendously difficult for me to realize that a large number of students cannot understand a statement, like:

"f is bounded on the interval I, iff the set of values {f(x) for all x in I} is contained in a bounded interval,

or equivalently, if there exist numbers K < L, such that K < f(x) < L for every x in I."

This was not grasped by some, even after an entire hour spent discussing its meaning, illustrating it, and rephrasing it various ways. I am coming from a background where the teacher simply wrote such things in the board once and that was it.

The complete loss of a tradition in which concepts are discussed in words, leaves our students with no ability to absorb any concepts at all. Literally we have long been teaching them in high schools here, only to make trivial calculations, like "the area of a circle is A = πr^2, and r = 3, what is A?". This kind of thing is a complete abdication of responsibility by the high school, and now also the college teaching fraternity.

 

[mathwonk]

One can also prove that a continuous function assumes a maximum on a closed bounded interval in a similar way, without mentioning least upper bounds as follows:

First prove the "intermediate value theorem", that if f is continuous on [0,1], and if f(0) = a < b = f(1), then for each number c between a and b, there is some d between 0 and 1 such that f(d) = c.

I.e. the continuous image of an interval is always an interval. This proof is exactly like the previous one. I.e. subdivide [0,1] into tenths, and find either a point of form .x where f equals c, or else find an interval like [.3,.4] such that c is between f(.3) and f(.4). Continuiing, either one finds a finite decimal d where f(d) = c, or an infinite decimal d = .3..., such that f(x) is both less than c and greater than c, on every neighborhood of d, hence by continuity, f(d) = c.


I advise skipping this proof since the idea is not too different from the previous one.


Then one proves that a continuous f achieves a maximum on [0,1] as follows:

If f is continuous on [0,1], then by the two previous results, the set of values of f there is a bounded interval, i.e. (a,b), or [a,b), or (a,b], or [a,b]. We claim it is the latter.


For example to prove b is a value, consider g(x) = 1/[f(x)-b]. Since the function f(x) comes arbitrarily near to b, if f(x) never equals b, the function g would be unbounded on [0,1] and yet continuous, a contradiction.

Of course this is the usual proof, but stripped of a reference to the least upper bound of the set of values of f. That reference is completely unnecessary, but because of my education, it is usually automatic.

Of course we are hiding here the fact that every interval has one of the forms above. I.e. why could there not be an interval of R that does not have an upper endpoint? This is of course the completeness of the reals, but it is obvious to the students, and if desired, the problem can be mentioned openly, and one can admit that one is taking for granted the fact that there are no missing points on the real line.


The student is less concerned with why a closed bounded interval has the form [a,b], than with why, after assuming that, the f above actually assumes the value b.

These arguments are so elementary, that to me it would be ridiculously condescending to assume a non honors calculus class should not see them. Experience of course may argue differently.


I am obviously very proud of even these tiny modifications of existing proofs. I conjecture that these elementary versions were in fact common in the very old days, before the introduction of modern terminology, designed to simplify and clarify the arguments. The result being that the arguments were made more abstract, and all but incomprehensible and inaccessible to all but the top students.

 

[mathwonk]

Since I have just presented rpoofs of the three main theorems that are normally omitted in non honors calculus. but in a way I consider accessible, with some effort, by any good calculus student, I would appreciate some feedback from anyone as toi whetehr they are or are not able to follow the arguments.

 

[Jimmy Snyder]

Since I have just presented rpoofs of the three main theorems that are normally omitted in non honors calculus. but in a way I consider accessible, with some effort, by any good calculus student, I would appreciate some feedback from anyone as toi whetehr they are or are not able to follow the arguments.

I was able to follow all three arguments. The combination of the fact that your style of presentation is non-standard and my frozen brain is used to standard textbook style slowed me down, but I still got it. I am just as curious about your proofs as I am about my own. How do you find out if they are original?

 

[matt grime]

I think I have one problem with the proof.

I agree patrially with the first argument, that the image of f *contains* an interval, however, I don't see how you can exclude the possibility that the image of f is not R.

ie we are assuming that f(0) and f(1) are the end points of the image, which is not necessarily true.

 

[matt grime]

You cannot guarantee that no one else has thought of them. Though it is almost certain that no one else has published them.

 

[Jimmy Snyder]

Professor Conway's response (slightly edited)

Mr Snyder,

This is nice. I cannot tell you whether it
is totally new, but I suspect it is not. Proofs of basic facts like this
abound. You could try to send it in as a classroom note or something to the
Monthly, but perhaps I will pass on to you a bit of advice my thesis adviser
gave me over something similar (but different). Save it and hold on to it,
because if you stick with this business you are likely to have an occasion
where you can use it.

Peace,
John

Well, if Professor Conway says it's nice, who am I to argue. I intend to accept this informed opinion that my result is probably not new.

 

[Jimmy Snyder]

Snyder's nice theorem.

Theorem: Let X be an inner product space with inner product <., .>, and let S be a linear operator on X. Then the following three statements are equivalent.

(i) I am <Sx, y> = I am <x, Sy> for all x, y in X
(ii) Re <Sx, y> = Re <x, Sy> for all x, y in X
(iii) <Sx, y> = <x, Sy> for all x, y in X

Note that (iii) is the condition for S to be a self-adjoint operator.

Proof:

(iii) implies (i) is trivial. (iii) implies (ii) is trivial. (i) and (ii) implies (iii) is trivial. I will show that (i) implies (ii). Suppose (i) is true. Then for all x, y in X

Re <Sx, y> = I am i <Sx, y> = I am <Sx, -iy> = I am <x, -iSy>
= I am i <x, Sy> = Re <x, Sy>

(ii) implies (i) by a similar string of equations. Q.E.D.

It is concievable that someone would need to prove that a certain operator was self-adjoint and this theorem would cut the work in half. Here is just such an example.

Theorem: With X and S as above, <Sx, x> real for all x in X implies S is self-adjoint.

This is a well know basic fact about self-adjoint operators and can be found in just about any elementary textbook that covers inner-product spaces or Hilbert spaces.

Proof:

For all x and y we have

<S(x + y), x + y> = <Sx, x> + <Sy, y> + <Sx, y> + <Sy, x>

Take the imaginary part of this equation and apply the hypothesis.

0 = I am <Sx, y> + I am <Sy, x> = I am <Sx, y> - I am <x, Sy>.

So I am <Sx, y> = I am <x, Sy> for all x, y in X. Apply the nice theorem. Q.E.D.

In several books, I have seen this theorem proved by invoking a theorem known as the polarization identity for operators. That proof is also an easy one. However the polarization identity for operators itself, while easy, is tedious to prove. In addition to the polarization identity for operators, there is another theorem called simply the polarization identity. The error in the book I was reading was that it invoked this similarly named theorem and did not contain the needed one.

 

[JasonRox]

I didn't read the whole post, but I read to the point where someone mentionned that mathematicians don't tell the world everything new they find.

This is true. I have developped shorter proofs myself, but I see no gains in publishing it. The fact that it is proven is good enough, unless the proof itself results in unlocking a mystery in mathematics.

 

[mathwonk]

Matt, if you were talking to me, I believe what I have proved is:
1) the continuous image of a closed bounded interval is a bounded set.
2) the continuous image of an interval is an interval.
(Hence the continuous image of a closed bonded interval is a bounded interval. It remains to show it contains its endpoints.)
3) The continuous image of a closed bounded set is a bounded interval and contains its endpoints.
QED.

 

[mathwonk]

in regard to doing research, there is a wonderful interview with a master researcher of the second half of the 20th century, Raoul Bott. (He is still continuing too.)

there is a link to it at:
www.ams.org/notices/200104/fea-bott.pdf

If you can't get on there i could send it to you.