Abstract
Any orthogonally additive injection of a real inner product space of dimension at least 2 onto an abelian group is additive.
Highlights
Let E be a real inner product space of dimension at least 2
Where a and b are additive functions uniquely determined by f
Remark 1 from [1] provides an example of an orthogonally additive function f : E → E which is injective and not additive and Remark 2 from [1] provides an example of an orthogonally additive function which maps E onto E and is not additive
Summary
Let E be a real inner product space of dimension at least 2. A function f mapping E into an abelian group is called orthogonally additive, if f (x + y) = f (x) + f (y) for all x, y ∈ E with x ⊥ y. See [3, Corollary 10] and [2, Theorem 1], that every orthogonally additive function f defined on E has the form f (x) = a( x 2) + b(x) for x ∈ E, (1)
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