  The main goal of this note is to show that (not necessarily holomorphic)
multipliers of a wide class of normed spaces of continuous functions over a
connected Hausdorff topological space cannot attain their multiplier norms,
unless they are constants. As an application, a contractive multiplication
operator is either a multiplication with a constant, or is completely
non-unitary. Additionally we explore possibilities for a multiplication
operator to be (weakly) compact and (uniformly) mean ergodic.
