On the nonexistence of a projection from functions of 𝑥 to functions of 𝑥ⁿ

Abstract

The subspace ϕ ∗ C ∞ ( R 1 ) ⊂ C ∞ ( R 1 ) {\phi ^ \ast }{C^\infty }({{\mathbf {R}}^1}) \subset {C^\infty }({{\mathbf {R}}^1}) of all C ∞ {C^\infty } functions of ϕ ( x ) = x n , n = 1 , 2 , 3 , … \phi (x) = {x^n},n = 1,2,3, \ldots , is a closed subspace of C ∞ ( R 1 ) {C^\infty }({{\mathbf {R}}^1}) by Glaeser’s Composition Theorem. We prove that for n > 2 n > 2 there does not exist a linear continuous projection π \pi from C ∞ ( R 1 ) {C^\infty }({{\mathbf {R}}^1}) onto ϕ ∗ C ∞ ( R 1 ) {\phi ^ \ast }{C^\infty }({{\mathbf {R}}^1}) .