Validating Originality in Mathematical Proofs for Researchers

[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.

 

[matt grime]

Finally got my brain in gear (which is odd given it's not even 10:30), and see what I was misreading yesterday.

 

[Jimmy Snyder]

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

Thanks. I can't speak for anyone else but I had no trouble getting in. You just need to register, and it's free. The article gives a fascinating insight into how theorems come to be.

 

[honestrosewater]

What a coincidence- I just happened upon a http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.ndjfl/1039293064?abstract= in a logic journal that says:

There is nothing original in it [this paper], beyond the easy observation that Lyndon's theorem gives the distribution laws. I wrote it up for publication because I became aware that the facts have not reached print. In fact they haven't even reached the grapevine. Recently I came across a team whose research project revolved around answering the question which Lyndon's theorem has already answered.

The author wasn't unknown (I'm actually reading one of his books now), but I don't imagine that's the reason they published the paper.

 

[mathwonk]

jimmy, that interview was with Raoul Bott. I have a personal story concerning him. Over 30 years ago I was a temporary instructor at a small college, struggling to learn as much math as possible by teaching as many new courses every year as they would allow, and volunteering to teach others as overloads.

Desiring to make my explanations clear, I tried to understand the material, and began constructing new proofs as you are doing. In an attempt to discern what in the world the green/stokes theorems were good for, I noticed with glee that they could be used to prove that the circle does not deform away from the origin, and hence one deduces the fundamental theorem of algebra, and brouwer's (smooth) fixed point theorem.

In summer, I persuaded my college to fund my attendance at a differential topology course for graduate students in Mexico, where Bott was lecturer. One day walking outside I told him my proof. Putting his arm kindly around my shoulders, he asked "Isn't that the usual proof?" I don't know, I said with subdued excitement, I made it up myself. What a privilege to be in his company for a few minutes.

I felt inspired by the whole meeting because the students there were not much stronger I thought than me. Moreover the hard work I had done at understanding differential methods made it clear to me what the points were that some of the lecturers were making. I could see that I had achieved the same view of some elementary things as they seemed to have. I stretched myself to take in the new ideas they spoke of, like characteristic classes for foliations, and as obstructions to embeddings of manifolds.


Within 5 more years I had a PhD (in a different area), and in a few more years I was an academic visitor at Harvard where I saw Bott again, feeling almost like a colleague. He did not remember me but I did not mind, being in heaven.

Now I am a tenured professor of mathematics at a university.

If you keep pushing your curiosity, it will lead you further, in some way.

best wishes

 

[Jimmy Snyder]

I published myself.

Thanks to everyone who helped me with this question. Here is a link to my web page where I have written up the theorem for all the world to see. If anyone has comments or suggestions for improvement, I would be most gratified to hear of them.

http://www.jimmyrules.com/operator.pdf

I just got finished reading Professor Kubrusly's "Elements of Operator Theory". I found a lot of typos and other errata in the book and sent them to him. Like so many authors of math and physics texts, he is very approachable and expressed his gratitude for the information I sent him. I also sent him the link above but he has not gotten back to me about it. His book, like most others in the field, has a proof of the second theorem in my paper that relies on the polarization identity.

 

[Jimmy Snyder]

Follow up:

The essential features of my proof were published by Martin Schechter in the book "Operator Methods in Quantum Mechanics", in 1981. His statement of the theorem is in Lemma 1.5.1 on page 8. His proof on page 10 scoops me by 24 years. I don't know if this reference is the original or the repetition of a proof that is even older.

I came across this book by accident. I chose it practically at random to better understand QM.

 

[mathwonk]

one of my first attempts at a thesis turned out to be due to,hurwitz in the 1800's.