Orthogonally additive and orthogonally increasing functions on vector spaces

A real-valued function /: X-> R on an inner product space X is orthogonally additive if f(x + y) = /(JC) + f(y) whenever xly. We extend this concept to more general spaces called orthogonality vector spaces. If X is an orthogonality vector space and if there exists an orthogonally additive function on X which satisfies certain natural conditions then there is an inner product on X which is equivalent to the original orthogonality and /(JC) = \ f for all JC E X. We next consider a normed space X with James' orthogonality. A function f: X-^R is orthogonally increasing if f(x + y)^f(x) whenever JC 1 y. Orthogonally increasing functions on normed spaces are characterized.