# Stability of the Quadratic Functional Equation

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## Abstract

The quadratic functional equation
clearly has

$$
f\left( {x + y} \right) + f\left( {x - y} \right) - 2f\left( x \right) - 2f\left( y \right) = 0$$

(3.1)

*f*(*x*) =*cx*^{2}as a solution with*c*an arbitrary constant when*f*is a real function of a real variable. We define any solution of (3.1) to be a*quadratic function*, even in more general contexts. We shall be interested in functions*f: E*_{1}*→ E*_{2}where both*E*_{1}and*E*_{2}are real vector spaces, and we need a few facts concerning the relation between a quadratic function and a biadditive function sometimes called its*polar*. This relation is explained in Proposition 1, p. 166, of the book by J. Aczél and J. Dhombres (1989) for the case where*E*_{2}=*R*, but the same proof holds for functions*f: E*_{1}*→ E*_{2}. It follows then that*f: E*_{1}*→ E*_{2}is quadratic if and only if there exists a unique symmetric function*B: E*_{1}×*E*_{1}*→**E*_{2}, additive in*x*for fixed y, such that*f*(*x*) =*B*(*x*,*x*). The biadditive function*B*, the polar of*f*, is given by$$B\left( {x,y} \right) = \left( {\begin{array}{*{20}{c}}
1 \\
- \\
4
\end{array}} \right)\left( {f\left( {x + y} \right) - f\left( {x - y} \right)} \right)$$

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## Copyright information

© Springer Science+Business Media New York 1998