Solution and stability of finite dimensional quadratic functional equation in Banach spaces.

The purpose of this current work, we introduced the finite dimensional quadratic functional equation of the form φ ( ∑ i = 1 m s i) + ∑ 1 ≤ i < j ≤ m φ (− s i − s j + ∑ l = 1 i ≠ j ≠ 1 m s 1) = ( − m 2 + 9 m − 18 2) ∑ 1 ≤ i < j ≤ m φ (s i − s j) + ( m 3 − 9 m 2 + 26 m − 16 2) ∑ i = 1 m φ (s i) when m is a positive integer with n is greater than three. And also we investigated its Hyers-Ulam stabilities in Banach spaces through two different techniques. [ABSTRACT FROM AUTHOR]