Exercise 12, § 4

Let $U$ be a nonempty open set in $\mathbf{C}$. Now prove that there exists no norm on $\mathscr{O}(U)$ for which $\mathscr{O}(U)$ is a Banach algebra. (Using Theorem 1 of I, p. 29, prove that the functions of $\mathscr{O}(U)$ would be bounded.)