Borel's lemma

In mathematics, Borel's lemma, named after Émile Borel, is an important result used in the theory of asymptotic expansions and partial differential equations. In mathematics, Borel's lemma, named after Émile Borel, is an important result used in the theory of asymptotic expansions and partial differential equations. Suppose U is an open set in the Euclidean space Rn, and suppose that f0, f1 ... is a sequence of smooth functions on U. If I is an any open interval in R containing 0 (possibly I = R), then there exists a smooth function F(t, x) defined on I×U, such that for k ≥ 0 and x in U. Proofs of Borel's lemma can be found in many text books on analysis, including Golubitsky & Guillemin (1974) and Hörmander (1990), from which the proof below is taken. Note that it suffices to prove the result for a small interval I = (−ε,ε), since if ψ(t) is a smooth bump function with compact support in (−ε,ε) equal identically to 1 near 0, then ψ(t) ⋅ F(t, x) gives a solution on R × U. Similarly using a smooth partition of unity on Rn subordinate to a covering by open balls with centres at δ⋅Zn, it can be assumed that all the fm have compact support in some fixed closed ball C. For each m, let where εm is chosen sufficiently small that for |α| < m. These estimates imply that each sum