On Linear Functional Equations and Completeness of Normed Spaces

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 n ∑ i=1 aif(φi(x1, . . . , xk)) = 0 (x1, . . . , xk ∈ D) with given real numbers a1, . . . , an, given mappings φ1 . . . , φn : D k → D and unknown function f : D → X, which has a Hyers–Ulam stability property on an infinite subset D of the integers.