On the completeness of normed spaces

Let ( G , + ) be an abelian group and let E be a normed space. A mapping f : G → E is called ε -quadratic if for a given ε > 0 it satisfies ‖ f ( x + y ) + f ( x − y ) − 2 f ( x ) − 2 f ( y ) ‖ ≤ ε for all x , y ∈ G . In this work, we show that E is complete if every ε -quadratic mapping f : X → E can be estimated by a quadratic mapping, where X is N 0 or a finitely generated free abelian group.