International Journal of Pure and Applied Mathematics Volume 107 No. 2 2016, 381-391 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: 10.12732/ijpam.v107i2.8 P A ijpam.eu QUADRATIC FUNCTIONAL EQUATION ON ORTHOGONALITY VECTOR SPACES Sayed Khalil Ekrami 1 § , Madjid Mirzavaziri 2 1 Department of Mathematics Payame Noor University P.O. Box 19395-3697, Tehran, IRAN Department of Pure Mathematics Ferdowsi University of Mashhad P.O. Box 1159, Mashhad 91775, IRAN Abstract: Let (X, ⊥) be a real vector space of dimension at least 3, with the orthogonality defined on it by: (i) for all x ∈ X, x ⊥ 0 and 0 ⊥ x, (ii) for all x, y ∈ X \{0}, x ⊥ y if and only if x, y are linearly independent. We show that any orthogonally quadratic mapping on X is a quadratic mapping. Also we prove the Hyers-Ulam stability of orthogonally quadratic functional equation and the Hyers- Ulam stability of orthogonally pexiderized quadratic functional equation. AMS Subject Classification: 39B52, 39B55, 39B82 Key Words: quadratic functional equation, orthogonality space, stability 1. Introduction and Preliminaries As long as we are working in inner product spaces, usually there is no doubt what kind of orthogonality relation we have in mind. Namely, it is the one derived from an inner product and then vectors x and y are orthogonal (x ⊥ y) if and only if 〈x, y〉 = 0. Received: December 3, 2015 Published: April 9, 2016 c  2016 Academic Publications, Ltd. url: www.acadpubl.eu § Correspondence author