Peetre theorem
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 "i" 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 R and "E" and "F" are trivial bundles. At this point, it relies primarily on two lemmas: We begin with the proof of  Lemma 1. It is easy to check that formula_10 is linear: Finally, we check that formula_13 is local in the sense that formula_14. If formula_15, then formula_16 such that formula_17 in the ball of radius formula_18 centered at formula_19. Thus, for formula_20, for formula_22, and hence formula_23. Therefore, formula_24. So by Peetre's theorem, formula_13 is a differential operator.