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According to Wikipedia, the Bohr-Mollerup Theorem (discussed previously on MO here) was first published in a textbook. It says the authors did that instead of writing a paper because they didn't think the theorem was new.

What other examples are there of significant theorems that first saw the light of day in a textbook? (I'm assuming Wikipedia is right about Bohr-Mollerup.)

I recognize that the word "significant" is imprecise; I have in mind theorems that mathematicians have picked up on and used in their own work, but I'm open to other interpretations.

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Would you consider the Disquisitiones Arithmeticae as a textbook? – Pierre-Yves Gaillard Sep 17 '10 at 6:23
How do you distinguish textbook from monograph? In some subjects this is obvious, but in some cases, I wouldn't be sure. – Andrés E. Caicedo Sep 17 '10 at 6:35
I should have asked: Would you consider Euclid's Elements as a textbook? Wikipedia writes: "Euclid's Elements is the most successful and influential textbook ever written": - And also "The Disquisitiones Arithmeticae is a textbook ...": – Pierre-Yves Gaillard Sep 17 '10 at 11:45
How about theorems first published in internet forums? I think there have been some theorems first published in Wikipedia articles, and under the norms of Wikipedia, that's considered a reason to delete them from the article. If the article is about only a new theorem that has not appeared elsewhere, that's grounds for deletion of the article. One mathematician whom I met some years ago, who didn't understand the rules, made a preprint of his into a Wikipedia article not long ago. It was of course deleted. – Michael Hardy Sep 17 '10 at 12:37
Bourbaki, Zariski-Samuel, Cartan-Eilenberg, EGA. – Pierre-Yves Gaillard Sep 17 '10 at 13:32
up vote 16 down vote accepted

I recall that, and Wikipedia independently confirms that L'Hôpital's rule first appeared in a textbook, apparently the first textbook on differential calculus: Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes published by Guillaume de l'Hôpital and made up of content mostly provided by Johann Bernoulli, who was on retainer to l'Hôpital, more or less, for this purpose.

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There have been many good answers. I've chosen to accept this one because the theorem in question is so well-known. – Gerry Myerson Nov 26 '10 at 1:32

Long ago, I proved that every derivation from $C^{k+1}$ functions to $C^k$ functions is given by a $C^k$ vector field. (The same fact with $\infty$ in place of $k$ and $k+1$ is, of course, classical.) The first (and only, as far as I know) publication is in the book "Manifolds, Tensor Analysis, and Applications" by Abraham, Marsden, and Ratiu (p.235).

The story behind this is that, at the time, I was sharing an office with Bill Floyd; Tudor Ratiu, whose office was just down the hall, was working on this book. Of course, he knew about the $C^\infty$ version of the result, but he stopped by to ask Bill about the $C^k$ version, and I happened to be there too. Neither Bill nor I knew anything about it, but later (the same evening, I think, but my memory may be playing tricks here) I worked out a proof. When I told Tudor about it, he asked if he could put it into the book, and I said sure. I think the proof in the book is more streamlined than my original argument.

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This post contain a real analytic… – Ali Taghavi Nov 23 '14 at 15:50

It happened to me once. While visiting the Institute for Advanced Studies at the Hebrew University of Jerusalem in 1976-77 I answered a question from the preliminary manuscript of volume 1 of Lindenstrauss-Tzafriri by constructing Banach spaces not isomorphic to Hilbert spaces all of whose subspaces have the approximation property. They replaced the question with one of my examples in the published book. I delayed writing the paper, which appeared several years later (1980). Actually I wrote the paper only because L-T had included the simplest rather than the most interesting example (which had the property that every subspace of every quotient has a Schauder basis).

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Out of curiosity, are these weak Hilbert spaces? – Yemon Choi Sep 17 '10 at 18:44
@Yemon: The ones in L-T are not, but the main one in my paper was the first weak Hilbert, the 2 convexification of modified Tsirelson that I had earlier constructed for a different purpose (and which later was proved by Casazza-Odell to be the 2 convexification of Tsirelson with an equivalent norm). – Bill Johnson Sep 18 '10 at 9:36

Jean-Pierre Serre had a lot of original theorems (some due to him and some due to John Tate) published originally in textbooks. To give two examples, his book "Cohomologie Galoisienne" ("Galois cohomology") contain, I believe, the first published proof of many theorems of Tate that form the socle of the theory. His book "abelian $l$-adic representations and elliptic curves" contains the proof of his theorem that two elliptic curves over a number field with non-integral j-invariants that have the same $L$-functions are isogenous.

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I have a personal example, with by Matrices; Theory and Applications (GTM 216, Springer-Verlag, 2000). A couple of years ago, I found a proof of almost sure convergence of the Jacobi method for computing the spectrum of a Hermitian matrix, when one uses the random strategy. I was not sure of the novelty of it (could anyone confirm it?), and I just included it in the second edition, which is going to appear in a month or two.

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The proof that Reed's conjecture holds fractionally (i.e. for the fractional rather than regular chromatic number) has only been published in this textbook.

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I'm not sure if this is the intended question, but an interesting question which has been occupying my mind recently is that concerning theorems which have only appeared in book form, be it in a textbook or a monograph. Most of these results are classifications taking the form of a long list.

A couple of cases come to mind. First of all, the classification of riemannian space forms in the book by Wolf Spaces of constant curvature has, to my knowledge, only appeared in that book.

A second example with which I am very familiar is the classification of four-dimensional homogeneous (pseudo-)riemannian manifolds by Boris Komrakov Jr, which is only available in a publication of the International Sophus Lie Institute. (It was not even available online last time I checked, but someone who was visiting the institute kindly brought me back a photocopy.)

Presumably such results have not been peer-reviewed or at least not to the standards of a journal. So short of reproducing the result oneself, should one trust it and use it in one's own research?

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Many combinatorial results of Ian Macdonald seem first published in his monographs, such as symmetric functions and hall polynomials. I think he might have first announced them in conferences. I guess monographs are not really textbooks.

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