A Müntz space having no complement (Q1060364)

A classical theorem of Müntz says that given a set A of positive numbers for which \(\sum_{\lambda \in A}1/\lambda <\infty\), then the closed span S(A) of the monomials 1, \(x^{\lambda}\); \(\lambda\in A\), is not all of C[0,1]. An example is displayed here of a set A such that S(A) is not complemented in C[0,1]. The proof relies on the following Lemma: there exists a \(c>0\) such that any projection of C[0,1] onto the span of 1, \(x^ N\), \(x^{2N},...,x^{kN}\) has norm \(>c \log k\).