Corollary 2

Let E be a complex Hilbert space and let $\mathscr{L}(E)_n$ be the set of normal endomorphisms of E. Endow the space $\mathscr{C}(\mathbf{C})$ with the topology of compact convergence. The mapping of $\mathscr{C}(\mathbf{C})\times \mathscr{L}(E)_n$ into $\mathscr{L}(E)$ defined by $(f, u)\mapsto f(u)$ is continuous.