Theorems first published in textbooks? 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.
 A: 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Presumably such results have not been peer-reviewed or at least not to the standards of a journal.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?
A: 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.
A: 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.
A: Intersection theory has of course a long tradition, starting from the work of Schubert, Poincaré and many more, but the book Intersection theory by Fulton is by now considered the gold reference. Many results in the book were new at the time. Citing from the introduction

Several theorems are new or stronger than those which have appeared before, and some proofs are significantly simpler. Among the former are a new blow-up formula, a stronger residual intersection formula, and the removal of a projective hypothesis from intersection theory and Riemann-Roch theorems; the latter includes the proof of the Grothendieck-Riemann-Roch theorem. Some formulas from classical enumerative geometry receive a first modern or rigorous proof here.

A: 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.
A: 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.
A: 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).
A: 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.
A: 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.
