A real-valued function f : X → R
on an inner product space X is orthogonally additive if f(x + y) = f(x) + f(y)
whenever x ⊥ y. 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
f(x) = ±∥x∥2 for all x ∈ 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 x ⊥ y. Orthogonally increasing functions on normed spaces are
characterized.
