Exercise 15, § 4

Let $\mathbf{U}$ be the set of complex numbers of modulus 1. Let $K_1,K_2$ be the compact subsets of $\mathbf{C}^2$ given by

a) Now prove that the set of germs in a neighbourhood of $K_1$ of polynomial functions on $\mathbf{C}$ with complex coefficients is not dense in $\mathscr{O}(K_1)$.

b) Now prove that the set of germs in a neighbourhood of $K_2$ of polynomial functions on $\mathbf{C}$ with complex coefficients is dense in $\mathscr{O}(K_2)$.