In mathematics, the (linear) Peetre theorem, named after Jaak Peetre, is a result of functional analysis that gives a characterisation of differential operators in terms of their effect on generalized function spaces, and without mentioning differentiation in explicit terms. The Peetre theorem is an example of a finite order theorem in which a function or a functor, defined in a very general way, can in fact be shown to be a polynomial because of some extraneous condition or symmetry imposed upon it. This article treats two forms of the Peetre theorem. The first is the original version which, although quite useful in its own right, is actually too general for most applications. The original Peetre theorem Let M be a smooth manifold and let E and F be two vector bundles on M. Let be the spaces of smooth sections of E and F. An operator is a morphism of sheaves which is linear on sections such that the support of D is non-increasing: supp Ds ⊆ supp s for every smooth section s of E. The original Peetre theorem asserts that, for every point p in M, there is a neighborhood U of p and an integer k (depending on U) such that D is a differential operator of order k over U. This means that D factors through a linear mapping iD from the k-jet of sections of E into the space of smooth sections of F: where is the k-jet operator and is a linear mapping of vector bundles. Proof The problem is invariant under local diffeomorphism, so it is sufficient to prove it when M is an open set in Rn and E and F are trivial bundles. At this point, it relies primarily on two lemmas: Lemma 1. If the hypotheses of the theorem are satisfied, then for every x∈M and C > 0, there exists a neighborhood V of x and a positive integer k such that for any y∈V\{x} and for any section s of E whose k-jet vanishes at y (jks(y)=0), we have |Ds(y)|<C. Lemma 2. The first lemma is sufficient to prove the theorem. We begin with the proof of Lemma 1. Suppose the lemma is false. Then there is a sequence xk tending to x, and a sequence of very disjoint balls Bk around the xk (meaning that the geodesic distance between any two such balls is non-zero), and sections