Stability of the Quadratic Functional Equation

Abstract

The quadratic functional equation

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

(3.1)

clearly has 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

$$B\left( {x,y} \right) = \left( {\begin{array}{*{20}{c}} 1 \\ - \\ 4 \end{array}} \right)\left( {f\left( {x + y} \right) - f\left( {x - y} \right)} \right)$$