Summary: We prove two general theorems, which appear to be very useful in the investigation of the Hyers-Ulam stability of a higher-order linear functional equation in single variable, with constant coefficients. We give several examples of their applications. In particular, we show that we obtain in this way several fixed point results for a particular operator. The main tool in the proofs is a complexification of a real normed (or Banach) space \(X\), which can be described as the tensor product \(X \otimes \mathbb R^2\) endowed with the Taylor norm.