Polynomial approximation of nonlinear operators in space C (Q1076922)

Let F be a continuous nonlinear operator from a compact subset K of the space of continuous functions C on [0,1] into C. A polynomial operator on C is defined as \[ P(u)=\sum a_{j_ 1...j_ m}[g_ 1(u)]^{j_ 1}...[g_ m(u)]^{j_ m} \] where \(g_ k\) are bounded linear operators on C. It is shown that for any \(\epsilon >0\) there exists \(\delta >0\) and a polynomial operator P such that for all \(x\in K\) and \(\Delta\) \(x\in C\) such that \(\| \Delta x\| \leq \delta\) we have \(\| F(x)-P(x+\Delta x)\| <\epsilon.\) The proof suggests the actual construction of the operator P.