The aim of this note is to give a type of characterization of Banach spaces in terms of the stability of functional equations. More precisely, we prove that a normed space $X$ is complete if there exists a functional equation of the type $$\sum_{i=1}^{n}a_if(\varphi_i(x_1,\ldots,x_k))=0 \qquad(x_1,\ldots,x_k\in D)$$ with given real numbers $a_1,\ldots,a_n$, given mappings $\varphi_1\ldots,\varphi_n\colon D^k\to D$ and unknown function $f\colon D\to X$, which has a Hyers--Ulam stability property on an infinite subset $D$ of the integers.