whole of Rn. Since Borel's lemma is local in nature, the same argument shows that if is a (bounded or unbounded) domain in Rn with smooth boundary, then any smooth function on the closure of can be extended to a smooth function on Rn. Seeley's result for a half line gives a uniform extension map which is linear, continuous (for the topology of uniform convergence of functions and their derivatives on compacta) and takes functions supported in [0,R] into functions supported in [−R,R] To define set where φ is a smooth function of compact support on R equal to 1 near 0 and the sequences (am), (bm) satisfy: tends to ; for with the sum absolutely convergent. A solution to this system of equations can be obtained by taking and seeking an entire function such that That such a function can be constructed follows from the Weierstrass theorem and Mittag-Leffler theorem. It can be seen directly by setting an entire function with simple zeros at The derivatives W '(2j) are bounded above and below. Similarly the function meromorphic with simple poles and prescribed residues at By construction is an entire function with the required properties. The definition for a half space in Rn by applying the operator R to the last variable xn. Similarly, using a smooth partition of unity and a local change of variables, the result for a half space implies the existence of an analogous extending map for any domain in Rn with smooth boundary. See also The Kirszbraun theorem gives extensions of Lipschitz functions. Notes References Theorems in analysis