An example about the normality of \(D(K,M)\). (Q1432383)

Let \(A\) be a Banach algebra of continuous functions on a compact Hausdorff space \(X\). \(A\) is said to be normal on \(X\) if for each closed subset \(F\) of \(X\) and \(x_0\in X\setminus F\), there exists an \(f\in A\) such that \(f(x_0)=1\) and \(f(x)=0\) for all \(x\in F\). Let \(K\) be a perfect, compact subset of the complex plane and \(M=\{M_n\}_{n=0}^{\infty}\) a sequence of positive numbers. Let \(D^{\infty}(K)\) denote the space of infinitely complex differentiable functions on \(K\), \(f^{(n)}\) the \(n\)-th complex derivative of \(f\), and \(| f |_K=\sup\{| f(x)|: x\in K\}\). \(D(K, M)\) is defined as follows: \[ D(K, M)= \left\{f\in D^{\infty}(K): \sum_{n=0}^{\infty} M_n^{-1} | f^{(n)}|_K< \infty \right\}. \] It is easy to see that if the interior of \(K\) is nonempty, then \(D(K, M)\) is not normal on \(K\). In the paper under review, the author gives an example of \(D(K, M)\) that is normal on \(K\).